Digfilt

Digital Filters with MATLAB® ∗

Ricardo A. Losada†
The MathWorks, Inc.

May 18, 2008

∗MATLAB® and Simulink® are registered trademarks of The MathWorks, Inc.
†Ricardo Losada is with the Signal Processing and Communications develop-
ment team at The MathWorks, Inc.

Contents

I Filter Design 8
1 Basic FIR Filter Design 9
1.1 Why FIR filters? . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2 Lowpass filters . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.2.1 FIR lowpass filters . . . . . . . . . . . . . . . . . . . . 11
1.2.2 FIR filter design specifications . . . . . . . . . . . . . 11
1.2.3 Working with Hertz rather than normalized frequency 14
1.3 Optimal FIR filter design . . . . . . . . . . . . . . . . . . . . . 15
1.3.1 Optimal FIR designs with fixed transition width and
filter order . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.3.2 Optimal equiripple designs with fixed transition width
and peak passband/stopband ripple . . . . . . . . . . 22
1.3.3 Optimal equiripple designs with fixed peak ripple
and filter order . . . . . . . . . . . . . . . . . . . . . . 26
1.3.4 Constrained-band equiripple designs . . . . . . . . . 27
1.3.5 Sloped equiripple filters . . . . . . . . . . . . . . . . . 28
1.4 Further notes on equiripple designs . . . . . . . . . . . . . . 32
1.4.1 Unusual end-points in the impulse response . . . . . 32
1.4.2 Transition region anomalies . . . . . . . . . . . . . . . 33
1.5 Maximally-flat FIR filters . . . . . . . . . . . . . . . . . . . . . 35
1.6 Summary and look ahead . . . . . . . . . . . . . . . . . . . . 37

2 Basic IIR Filter Design 39
2.1 Why IIR filters . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.2 Classical IIR design . . . . . . . . . . . . . . . . . . . . . . . . 40
2.2.1 Cutoff frequency and the 3-dB point . . . . . . . . . . 41
2.2.2 Butterworth filters . . . . . . . . . . . . . . . . . . . . 41
2.2.3 Chebyshev type I filters . . . . . . . . . . . . . . . . . 42

CONTENTS 3

2.2.4 Chebyshev type II designs . . . . . . . . . . . . . . . . 44
2.2.5 Elliptic filters . . . . . . . . . . . . . . . . . . . . . . . 47
2.2.6 Minimum-order designs . . . . . . . . . . . . . . . . . 48
2.2.7 Comparison to FIR filters . . . . . . . . . . . . . . . . 50
2.3 IIR designs directly in the digital domain . . . . . . . . . . . 52

3 Nyquist Filters 56
3.1 Design of Nyquist filters . . . . . . . . . . . . . . . . . . . . . 57
3.1.1 Equiripple Nyquist filters . . . . . . . . . . . . . . . . 58
3.1.2 Minimum-order Nyquist filters . . . . . . . . . . . . . 61
3.2 Halfband filters . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.2.1 IIR halfband filters . . . . . . . . . . . . . . . . . . . . 63

4 Multirate Filter Design 67
4.1 Reducing the sampling rate of a signal . . . . . . . . . . . . . 68
4.1.1 Decimating by an integer factor . . . . . . . . . . . . . 68
4.1.2 Decimating by a non-integer factor . . . . . . . . . . . 79
4.2 Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.2.1 Fractionally advancing/delaying a signal . . . . . . . 80
4.2.2 Increasing the sampling-rate of a signal . . . . . . . . 82
4.2.3 Design of FIR interpolation filters . . . . . . . . . . . 86
4.2.4 Design of IIR halfband interpolators . . . . . . . . . . 89
4.2.5 Design of interpolators when working with Hertz . . 90
4.3 Increasing the sampling rate by a fractional factor . . . . . . 92
4.4 Fractional decimation . . . . . . . . . . . . . . . . . . . . . . . 95

5 Multistage/Multirate Filter Design 99
5.1 Interpolated FIR (IFIR) designs . . . . . . . . . . . . . . . . . 100
5.1.1 Further IFIR optimizations . . . . . . . . . . . . . . . 102
5.1.2 Multirate implementation of IFIR design . . . . . . . 104
5.2 Multistage/Multirate Designs . . . . . . . . . . . . . . . . . . 108
5.2.1 Setting the number of stages . . . . . . . . . . . . . . 111
5.3 Multistage/Multirate Nyquist filters . . . . . . . . . . . . . . 112
5.3.1 Using IIR halfband filters . . . . . . . . . . . . . . . . 113
5.4 Multistage interpolation . . . . . . . . . . . . . . . . . . . . . 115

Digital Filters with MATLAB Ricardo A. Losada

4 CONTENTS


6 Special Multirate Filters 120
6.1 Hold interpolators . . . . . . . . . . . . . . . . . . . . . . . . 121
6.2 Linear interpolators . . . . . . . . . . . . . . . . . . . . . . . . 125
6.3 CIC interpolators . . . . . . . . . . . . . . . . . . . . . . . . . 130
6.3.1 Design of CIC interpolators . . . . . . . . . . . . . . . 132
6.3.2 Gain of CIC interpolators . . . . . . . . . . . . . . . . 134
6.3.3 Further details of CIC filters . . . . . . . . . . . . . . . 136
6.4 CIC decimators . . . . . . . . . . . . . . . . . . . . . . . . . . 137
6.4.1 Design parameters . . . . . . . . . . . . . . . . . . . . 138
6.5 CIC compensators . . . . . . . . . . . . . . . . . . . . . . . . . 139
6.6 Farrow Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
6.6.1 Higher-order polynomials . . . . . . . . . . . . . . . . 144
6.6.2 Design of Farrow fractional delays . . . . . . . . . . . 146
6.6.3 Multirate Farrow filters . . . . . . . . . . . . . . . . . 147
6.6.4 Polynomial interpolation and maximally flat filtering 149
6.6.5 Using Farrow sample-rate converters in multistage
designs . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

II Filter Implementation 154
7 Implementing FIR Filters 155
7.1 Some basics on implementing FIR filters . . . . . . . . . . . . 155
7.1.1 Direct-form filter structure . . . . . . . . . . . . . . . . 156
7.1.2 Symmetric direct-form filter structure . . . . . . . . . 157
7.1.3 Transposed direct-form filter structure . . . . . . . . . 158
7.2 Fixed-point implementation . . . . . . . . . . . . . . . . . . . 160
7.2.1 Quantizing the filter’s coefficients . . . . . . . . . . . 160
7.2.2 Fixed-point filtering: Direct-form structure . . . . . . 165
7.2.3 Fixed-point filtering: Transposed direct-form structure168
7.2.4 Quantization of the output signal . . . . . . . . . . . . 169
7.2.5 Evaluating the performance of the fixed-point filter . 171

8 Implementing IIR Filters 177
8.1 Some basics of IIR implementation . . . . . . . . . . . . . . . 177
8.1.1 The use of second-order sections . . . . . . . . . . . . 177


CONTENTS 5

8.1.2 Allpass-based implementations . . . . . . . . . . . . . 178
8.2 Fixed-point implementation . . . . . . . . . . . . . . . . . . . 181
8.2.1 Fixed-point filtering . . . . . . . . . . . . . . . . . . . 182
8.2.2 Autoscaling . . . . . . . . . . . . . . . . . . . . . . . . 187
8.2.3 Evaluating filter performance using the magnitude
response estimate . . . . . . . . . . . . . . . . . . . . . 188

III Appendices 190
A Summary of relevant filter design commands 191
A.1 Filter Design (fdesign) . . . . . . . . . . . . . . . . . . . . . . 191
A.1.1 Setup design specifications . . . . . . . . . . . . . . . 191
A.1.2 Design options . . . . . . . . . . . . . . . . . . . . . . 192
A.1.3 Design analysis/validation . . . . . . . . . . . . . . . 193
A.2 Selecting filter structure . . . . . . . . . . . . . . . . . . . . . 193
A.3 Scaling IIR SOS structures . . . . . . . . . . . . . . . . . . . . 194
A.4 Designing multirate filters . . . . . . . . . . . . . . . . . . . . 195
A.5 Converting to fixed point . . . . . . . . . . . . . . . . . . . . 195
A.6 Generating Simulink blocks . . . . . . . . . . . . . . . . . . . 197
A.7 Graphical User Interface . . . . . . . . . . . . . . . . . . . . . 198

B Sampling, Downsampling, Upsampling, and Analog Reconstruc-
tion 199
B.1 Sampling an analog signal . . . . . . . . . . . . . . . . . . . . 201
B.1.1 Bandpass sampling . . . . . . . . . . . . . . . . . . . . 205
B.2 Sampling a discrete-time signal: downsampling . . . . . . . 206
B.2.1 Filtering to avoid aliasing when downsampling . . . 208
B.2.2 Downsampling bandpass signals . . . . . . . . . . . . 209
B.3 Increasing the sampling rate of a signal . . . . . . . . . . . . 209
B.4 Reconstructing an analog signal . . . . . . . . . . . . . . . . . 211
B.5 Practical sampling and analog reconstruction . . . . . . . . . 212
B.5.1 Oversampling . . . . . . . . . . . . . . . . . . . . . . . 214

C Case Study: Comparison of Various Design Approaches 219

D Overview of Fixed-Point Arithmetic 222
D.1 Some fixed-point basics . . . . . . . . . . . . . . . . . . . . . 223


6 CONTENTS

D.2 Quantization of signals and SNR . . . . . . . . . . . . . . . . 223
D.2.1 Quantizing impulse responses for FIR ﬁlters . . . . . 227
D.3 Fixed-point arithmetic . . . . . . . . . . . . . . . . . . . . . . 228
D.3.1 Fixed-point addition . . . . . . . . . . . . . . . . . . . 228
D.3.2 Fixed-point multiplication . . . . . . . . . . . . . . . . 230
D.4 Quantization noise variance . . . . . . . . . . . . . . . . . . . 232
D.5 Quantization noise passed through a linear ﬁlter . . . . . . . 233
D.5.1 Computing the average power of the output noise . . 234
D.6 Oversampling noise-shaping quantizers . . . . . . . . . . . . 234


Preface

This document∗ constitutes a tutorial on design and implementation of
digital filters in MATLAB. The tutorial is based on functionality from the
Filter Design Toolbox™.
The document covers the design of FIR and IIR single-rate and mul-
tirate filters. It also discusses advanced design techniques such as multi-
rate/multistage decimation/interpolation and the use of special multirate
filters such as allpass-based polyphase IIR filters, CIC filters, and Farrow
filters.
The tutorial focuses on practical aspects of filter design and implemen-
tation, and on the advantages and disadvantages of the different design
algorithms. The theory behind the design algorithms is kept to a minimal.

∗This document may be updated from time to time. The latest version can be found at
www.mathworks.com/matlabcentral

Chapter 1

Basic FIR Filter Design

Overview
In this chapter we discuss the basic principles of FIR filter design. We con-
centrate mostly on lowpass filters, but most of the results apply to other
response types as well. We discuss the basic trade offs and the degrees
of freedom available for FIR filter design. We motivate the use of optimal
designs and introduce both optimal equiripple and optimal least-squares
designs. We then discuss optimal minimum-phase designs as a way of
surpassing in some sense comparable optimal linear-phase designs. We
introduce sloped equiripple designs as a compromise to obtain equiripple
passband yet non-equiripple stopband. We also mention a few caveats
with equiripple filter design. We end with an introductory discussion of
different filter structures that can be used to implement an FIR filter in
hardware.
The material presented here is based on [1] and [2]. However, it has
been expanded and includes newer syntax and features from the Filter
Design Toolbox.

1.1 Why FIR filters?
There are many reasons why FIR filters are very attractive for digital filter
design. Some of them are:

• Simple robust way of obtaining digital filters

10 Basic FIR Filter Design

• Inherently stable when implemented non recursively

• Free of limit cycles when implemented non recursively

• Easy to attain linear phase

• Simple extensions to multirate and adaptive filters

• Relatively straight-forward to obtain designs to match custom mag-
nitude responses

• Some vendors and specialized hardware only support FIR

• Low sensitivity to quantization effects compared to many IIR filters

FIR filters have some drawbacks however. The most important is that
they can be computationally expensive to implement. Another is that
they have a long transient response. It is commonly thought that IIR fil-
ters must be used when computational power is at a premium. This is
certainly true in some cases. However, in many cases, the use of multi-
stage/multirate techniques can yield FIR implementations that can com-
pete (and even surpass) IIR implementations while retaining the nice char-
acteristics of FIR filters such as linear-phase, stability, and robustness to
quantization effects.∗ However, these efficient multistage/multirate de-
signs tend to have very large transient responses, so depending on the
requirements of the filter, IIR designs may still be the way to go.
In terms of the long transient response, we will show in Chapter 2
that minimum-phase FIR filters can have a shorter transient response than
comparable IIR filters.

1.2 Lowpass filters
The ideal lowpass filter is one that allows through all frequency compo-
nents of a signal below a designated cutoff frequency ωc , and rejects all
frequency components of a signal above ωc .
∗ That being said, there are also modern IIR design techniques that lend themselves to
efficient multirate implementations and are extremely computationally efficient. We are
referring here to more traditional IIR designs implemented using direct-form I or II struc-
tures, possibly in cascaded second-order section form.


1.2 Lowpass filters 11

Its frequency response satisfies

1, 0 ≤ ω ≤ ωc
HLP (e jω ) = (1.1)
0, ωc < ω ≤ π
The impulse response of the ideal lowpass filter (1.1) can easily be
found to be [3]

sin(ωc n)
hLP [n] = , −∞ < n < ∞. (1.2)
πn

1.2.1 FIR lowpass filters
Because the impulse response required to implement the ideal lowpass
filter is infinitely long, it is impossible to design an ideal FIR lowpass filter.
Finite length approximations to the ideal impulse response lead to the
presence of ripples in both the passband (ω < ωc ) and the stopband (ω >
ωc ) of the filter, as well as to a nonzero transition width between the pass-
band and stopband of the filter (see Figure 1.1).

1.2.2 FIR filter design specifications
Both the passband/stopband ripples and the transition width are unde-
sirable but unavoidable deviations from the response of an ideal lowpass
filter when approximating with a finite impulse response. Practical FIR
designs typically consist of filters that meet certain design specifications,
i.e., that have a transition width and maximum passband/stopband rip-
ples that do not exceed allowable values.
In addition, one must select the filter order, or equivalently, the length
of the truncated impulse response.
A useful metaphor for the design specifications in FIR design is to think
of each specification as one of the angles in a triangle as in Figure 1.2 ∗ .
The metaphor is used to understand the degrees of freedom available
when designating design specifications. Because the sum of the angles is
fixed, one can at most select the values of two of the specifications. The
third specification will be determined by the design algorithm utilized.
∗ For the ripples we should more generally speak of some measure or norm of them.
The peak ripple corresponding to the L∞ -norm is the most commonly used measure, but
other norms are possible.



Zero−Phase Response
1.2

1
Passband ripple

0.8

Ideal
Amplitude

0.6 lowpass
filter
0.4 Transition
width
0.2

0
Stopband ripple

−0.2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Normalized Frequency (×π rad/sample)

Figure 1.1: Illustration of the typical deviations from the ideal lowpass filter when ap-
proximating with an FIR filter, ωc = 0.4π.

Moreover, as with the angles in a triangle, if we make one of the specifica-
tions larger/smaller, it will impact one or both of the other specifications.

Example 1 As an example, consider the design of an FIR filter that meets the
following specifications:

Specifications Set 1

1. Cutoff frequency: 0.4π rad/sample

2. Transition width: 0.06π rad/sample

3. Maximum passband/stopband ripple: 0.05

The filter can easily be designed with the truncated-and-windowed impulse re-
sponse algorithm (a.k.a. the “window method”) if we use a Kaiser window∗ :
∗Notice that when specifying frequency values in MATLAB, the factor of π should be
omitted.


1.2 Lowpass filters 13

Filter order

Transition peak passband/
width stopband ripple

Figure 1.2: FIR design specifications represented as a triangle.

Fp = 0.4 - 0.06/2; Fst = 0.4 + 0.06/2;
Hf = fdesign.lowpass('Fp,Fst,Ap,Ast',Fp,Fst,0.05,0.05,'linear');
design(Hf,'kaiserwin');

The zero-phase response of the filter is shown in Figure 1.3. Note that since we
have fixed the allowable transition width and peak ripples, the order is determined
for us.
Close examination at the passband-edge frequency∗ , ω p = 0.37π, and at the
stopband-edge frequency, ωs = 0.43π, shows that the peak passband/stopband
ripples are indeed within the allowable specifications. Usually the specifications
are exceeded because the order is rounded to the next integer greater than the
actual value required.

∗ The passband-edge frequency is the boundary between the passband and the transition
band. If the transition width is Tw , the passband-edge frequency ω p is given in terms of
the cutoff frequency ωc by ω p = ωc − Tw /2. Similarly, the stopband-edge frequency is
given by ωs = ωc + Tw /2.



Zero−phase Response

1

0.9

0.8

0.7
Amplitude

0.6

0.5

0.4

0.3

0.2

0.1

0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Figure 1.3: Kaiser window design meeting prescribed specifications.

1.2.3 Working with Hertz rather than normalized frequency
In many applications the specifications are given in terms of absolute fre-
quency in Hertz rather than in terms of normalized frequency. Conversion
between one and the other is straightforward. Recall that normalized fre-
quency is related to absolute frequency by

2π f
ω=
fs

where f is absolute frequency in cycles/second, f s is the sampling fre-
quency in samples/second, and ω is normalized frequency in radians/sample.
Suppose the specifications for a filter design problem include a pass-
band frequency of 250 Hz, a stopband frequency of 300 Hz, and a sam-
pling frequency of 1 kHz. The corresponding normalized passband and
stopband frequencies are 0.5π and 0.6π respectively.
However, it is not necessary to perform such conversion manually. It
is possible to specify design parameters directly in Hertz.
For example, the following two filters H1 and H2 are identical.

Hf = fdesign.lowpass('Fp,Fst,Ap,Ast',.5,.6,1,80);


1.3 Optimal FIR filter design 15

H1 = design(Hf);
Hf2 = fdesign.lowpass('Fp,Fst,Ap,Ast',250,300,1,80,1000);
H2 = design(Hf2);

Notice that we don’t add 'Fs' to the string 'Fp,Fst,Ap,Ast' (or any
other specification string) when we specify parameters in Hertz. Simply
appending the sampling frequency to the other design parameters indi-
cates that all frequencies specified are given in Hertz.

1.3 Optimal FIR filter design
While the truncated-and-windowed impulse response design algorithm
is very simple and reliable, it is not optimal in any sense. The designs
it produces are generally inferior to those produced by algorithms that
employ some optimization criteria in that it will have greater order, greater
transition width or greater passband/stopband ripples. Any of these is
typically undesirable in practice, therefore more sophisticated algorithms
come in handy.

1.3.1 Optimal FIR designs with fixed transition width and
filter order
Optimal designs are computed by minimizing some measure of the de-
viation between the filter to be designed and the ideal filter. The most
common optimal FIR design algorithms are based on fixing the transition
width and the order of the filter. The deviation from the ideal response is
measured only by the passband/stopband ripples. This deviation or error
can be expressed mathematically as [4]

E(ω ) = Ha (ω ) − HLP (e jω ), ω∈Ω

where Ha (ω ) is the zero-phase response of the designed filter and Ω =
[0, ω p ] ∪ [ωs , π ]. It is still necessary to define a measure to determine “the
size” of E(ω ) - the quantity we want to minimize as a result of the op-
timization. The most often used measures are the L∞ -norm ( E(ω ) ∞ -
minimax designs) and the L2 -norm ( E(ω ) 2 - least-squares designs).



In order to allow for different peak ripples in the passband and stop-
band, a weighting function, W (ω ) is usually introduced,

EW (ω ) = W (ω )[ Ha (ω ) − HLP (e jω )], ω∈Ω

Linear-phase designs

A filter with linear-phase response is desirable in many applications, no-
tably image processing and data transmission. One of the desirable char-
acteristics of FIR filters is that they can be designed very easily to have
linear phase. It is well known [5] that linear-phase FIR filters will have
impulse responses that are either symmetric or antisymmetric. For these
types of filters, the zero-phase response can be determined analytically
[5], and the filter design problem becomes a well behaved mathemati-
cal approximation problem [6]: Determine the best approximation to a
given function (the ideal lowpass filter’s frequency response) by means of
a polynomial (the FIR filter) of a given order. By “best” it is meant the one
which minimizes the difference between them - EW (ω ) - according to a
given measure.
The equiripple design implements an algorithm developed in [7] that
computes a solution to the design problem for linear-phase FIR filters in
the L∞ -norm case. The design problem is essentially to find a filter that
minimizes the maximum error between the ideal and actual filters. This
type of design leads to so-called equiripple filters, i.e. filters in which the
peak deviations from the ideal response are all equal.
The firls design implements an algorithm to compute solution for
linear-phase FIR filters in the L2 -norm case. The design problem is to find
a filter that minimizes the energy of the error between ideal and actual
filters.

Equiripple filters
Linear-phase equiripple filters are desirable because they have the small-
est maximum deviation from the ideal filter when compared to all other
linear-phase FIR filters of the same order. Equiripple filters are ideally
suited for applications in which a specific tolerance must be met. For ex-
ample, if it is necessary to design a filter with a given minimum stopband
attenuation or a given maximum passband ripple.



1.06

1.05 Kaiser−window design
Equiripple design
1.04

1.03

1.02
Amplitude

1.01

1

0.99

0.98

0.97

0.96
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

Figure 1.4: Passband ripple for of both the Kaiser-window-designed FIR filter and the
equiripple-designed FIR filter.

Example 2 The Kaiser-window design of Example 1 was of 42nd order. With
this same order, an equiripple filter (with fixed transition width) can be designed
that is superior to the Kaiser-window design:

Fp = 0.4 - 0.06/2; Fst = 0.4 + 0.06/2;
Hf = fdesign.lowpass('N,Fp,Fst',42,Fp,Fst);
Heq = design(Hf,'equiripple');

Figure 1.4 shows the superposition of the passband details for the filters de-
signed with the Kaiser window and with the equiripple design. Clearly the
maximum deviation is smaller for the equiripple design. In fact, since the filter
is designed to minimize the maximum ripple (minimax design), we are guaran-
teed that no other linear-phase FIR filter of 42nd order will have a smaller peak
ripple for the same transition width.
We can measure the passband ripple and stopband attenuation in dB units
using the measure command,

Meq = measure(Heq);



If we compare the measurements of the equiripple design to those of the Kaiser-
window design, we can verify for instance that the equiripple design provides a
minimum stopband attenuation of 29.0495 dB compared to 25.8084 dB for the
Kaiser-window design.

Least-squares filters
Equiripple designs may not be desirable if we want to minimize the en-
ergy of the error (between ideal and actual filter) in the passband/stopband.
Consequently, if we want to reduce the energy of a signal as much as pos-
sible in a certain frequency band, least-squares designs are preferable.

Example 3 For the same specifications, Hf, as the equiripple design of Example
2, a least-squares FIR design can be computed from
Hls = design(Hf,'firls');
The stopband energy for this case is given by
2 π 2
Esb = H (e jω ) dω
2π 0.43π

where H (e jω ) is the frequency response of the filter.
In this case, the stopband energy for the equiripple filter is approximately
3.5214-004 while the stopband energy for the least-squares filter is 6.6213e-005.
(As a reference, the stopband energy for the Kaiser-window design for this order
and transition width is 1.2329e-004).
The stopband details for both equiripple design and the least-squares design
are shown in Figure 1.5.

So while the equiripple design has less peak error, it has more “to-
tal” error, measured in terms of its energy. However, although the least-
squares design minimizes the energy in the ripples in both the passband
and stopband, the resulting peak passband ripple is always larger than
that of a comparable equiripple design. Therefore there is a larger distur-
bance on the signal to be filtered for a portion of the frequencies that the
filter should allow to pass (ideally undisturbed). This is a drawback of
least-squares designs. We will see in Section 1.3.5 that a possible compro-
mise is to design equiripple filters in such a way that the maximum ripple
in the passband is minimized, but with a sloped stopband that can reduce
the stopband energy in a manner comparable to a least-squares design.



0.15

Equiripple design
Least−squares design
Normalized Frequency: 0.4299316
0.1
Amplitude: 0.08960634
Amplitude

0.05
Amplitude: 0.0352379

0

−0.05
0.5 0.6 0.7 0.8 0.9 1

Figure 1.5: Comparison of an optimal equiripple FIR design and an optimal least-squares
FIR design. The equiripple filter has a smaller peak error, but larger overall error.

Using weights
Both equiripple and least-squares designs can be further controlled by
using weights to instruct the algorithm to provide a better approximation
to the ideal filter in certain bands. This is useful if it is desired to have less
ripple in one band than in another.
Example 4 In Example 2 above, the filter that was designed had the same ripples
in the passband and in the stopband. This is because we implicitly were using a
weight of one for each band. If it is desired to have a stopband ripple that is say ten
times smaller than the passband ripple, we must give a weight that is ten times
larger:
Heq2 = design(Hf,'equiripple','Wpass',1,'Wstop',10);
The result is plotted in Figure 1.6.
It would be desirable to have an analytic relation between the maxi-
mum ripples in a band and the weight in such band. Unfortunately no
such relation exists. If the design specifications require a specific maxi-
mum ripple amount, say δp in the passband and δs in the stopband (both
in linear units, not decibels), for a lowpass filter we can proceed as follows:



Passband and Stopband Ripples
0.1

0.08

0.06

0.04

0.02
Amplitude

0

−0.02

−0.04

−0.06

−0.08

−0.1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Normalized Frequency (× π rad/sample)

Figure 1.6: Passband and stopband ripples obtained from weighing the stopband 10 times
higher than the passband.

1. Set the passband weight to one.
δp
2. Set the stopband weight to δs .

since both the filter order and the transition width are assumed to be fixed,
this will not result in the desired ripples unless we are very lucky. How-
ever, the relative amplitude of the passband ripple relative to the stopband
ripple will be correct. In order to obtain a ripple of δp in the passband and
δs in the stopband we need to vary either the filter order or the transition
width.
The procedure we have just described requires trial-and-error since ei-
ther the filter order or the transition width may need to be adjusted many
times until the desired ripples are obtained. Instead of proceeding in such
manner, later we will describe ways of designing filters for given pass-
band/stopband ripples and either a fixed transition width or a fixed filter
order.
For least-squares designs, the relative weights control not the ampli-
tude of the ripple but its energy relative to the bandwidth it occupies. This



means that if we weigh the stopband ten times higher than the passband,
the energy in the stopband relative to the stopband bandwidth will be 10
times smaller than the energy of the ripples in the passband relative to the
passband bandwidth. For the case of lowpass filters these means that

Esb 2 π 2
= H (e jω ) dω
π − ωs 2π (π − ωs ) ωs

will be ten times smaller than
Epb 2 ωp 2
= H (e jω ) dω.
ωp 2π (ω p ) 0

Minimum-phase designs
One of the advantages of FIR filters, when compared to IIR filters, is the
ability to attain exact linear phase in a straightforward manner. As we
have already mentioned, the linear phase characteristic implies a symme-
try or antisymmetry property for the filter coefficients. Nevertheless, this
symmetry of the coefficients constraints the possible designs that are at-
tainable. This should be obvious since for a filter with N + 1 coefficients,
only N/2 + 1 of these coefficients are freely assignable (assuming N is
even). The remaining N/2 coefficients are immediately determined by the
linear phase constraint.
If one is able to relax the linear phase constraint (i.e. if the application at
hand does not require a linear phase characteristic), it is possible to design
minimum-phase equiripple filters that are superior to optimal equiripple
linear-phase designs based on a technique described in [8].

Example 5 For the same specification set of Example 2 the following minimum-
phase design has both smaller peak passband ripple and smaller peak stopband
ripple∗ than the linear-phase equiripple design of that example:

Hmin = design(Hf,'equiripple','Wpass',1,'Wstop',10,...
'minphase',true);

It is important to note that this is not a totally unconstrained design.
The minimum-phase requirement restricts the resulting filter to have all
∗ This can easily be verified using the measure command.



its zeros on or inside the unit circle.∗ However, the design is optimal in
the sense that it satisfies the minimum-phase alternation theorem [9].
Having smaller ripples for the same filter order and transition width
is not the only reason to use a minimum-phase design. The minimum-
phase characteristic means that the filter introduces the lowest possible
phase offset (that is, the smallest possible transient delay) to a signal being
filtered.

Example 6 Compare the delay introduced by the linear-phase filter of Example 2
to that introduced by the minimum-phase filter designed above. The signal to be
filtered is a sinusoid with frequency 0.1π rad/sample.

n = 0:500;
x = sin(0.1*pi*n');
yeq = filter(Heq,x);
ymin = filter(Hmin,x);

The output from both filters are plotted overlaid in Figure 1.7. The delay intro-
duced is equal to the group delay of the filter at that frequency. Since group-
delay is the negative of the derivative of phase with respect to frequency, the
group-delay of a linear-phase filter is a constant equal to half the filter order.
This means that all frequencies are delayed by the same amount. On the other
hand, minimum-phase filters do not have constant group-delay since their phase
response is not linear. The group-delays of both filters can be visualized using
fvtool(Heq,Hmin,'Analysis','Grpdelay');. The plot of the group-delays
is shown in Figure 1.8.

1.3.2 Optimal equiripple designs with fixed transition width
and peak passband/stopband ripple
We have seen that the optimal equiripple designs outperform Kaiser-window
designs for the same order and transition width. The differences are even
more dramatic when the passband ripple and stopband ripple specifica-
tions are different. The reason is that the truncated-and-windowed im-
pulse response methods always give a result with approximately the same
passband and stopband peak ripple. Therefore, always the more stringent
∗ Given any linear-phase FIR filter with non negative zero-phase characteristic, it is pos-
sible to extract the minimum-phase spectral factor using the firminphase function.



1
Linear−phase filtered sinusoid
Minimum−phase filtered sinusoid
0.8

0.6

0.4
Amplitude

0.2

0

−0.2

−0.4

−0.6

−0.8

−1

0 10 20 30 40 50 60 70 80 90 100
Sample index (n)

Figure 1.7: Sinusoid filtered with a linear-phase filter and a minimum-phase filter of the
same order.

Group Delay

20

Linear−phase filter
Group delay (in samples)

Minimum−phase filter
15

10

5

0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Figure 1.8: Group-delay of a linear-phase filter and a minimum-phase filter of the same
order.



peak ripple constraint is satisfied, resulting in exceeding (possibly signif-
icantly) all other ripple constraints at the expense of unnecessarily large
filter order.
To illustrate this, we turn to a different equiripple design in which both
the peak ripples and the transition width are fixed. Referring back to the
triangle in Figure 1.2, this means the resulting filter order will come from
the design algorithm.

Example 7 Consider the following specifications:



3. Maximum passband ripple: 0.13 dB

4. Minimum stopband attenuation: 60 dB

An equiripple design of this filter,
Fp = 0.375 - 0.15/2; Fst = 0.375 + 0.15/2;
Hf = fdesign.lowpass('Fp,Fst,Ap,Ast',Fp,Fst,0.13,60);
cost(Heq)
results in a filter of 37th order (38 taps) as indicated by the cost command. By
comparison, a Kaiser-window design requires a 49th order filter (50 taps) to meet
the same specifications. The passband details can be seen in Figure 1.9. It is evi-
dent that the Kaiser-window design over-satisfies the requirements significantly.

Minimum-phase designs with fixed transition width and peak passband/stopband
ripple
The same procedure to design minimum-phase filters with fixed filter or-
der and fixed transition width can be used to design minimum-phase fil-
ters with fixed transition width and peak passband/stopband ripple. In
this case, rather than obtaining smaller ripples, the benefit is meeting the
same transition width and peak passband/stopband ripples with a re-
duced filter order.



Magnitude Response
1.008

1.006
Kaiser window design
1.004
Equiripple design
1.002
Magnitude

1

0.998

0.996

0.994

0.992
0 0.05 0.1 0.15 0.2 0.25 0.3

Figure 1.9: Passband ripple details for both the Kaiser-window-designed FIR filter and
the equiripple-designed FIR filter. The Kaiser-window design over-satisfies the require-
ment at the expense of increase number of taps.

Example 8 Consider the following specifications:
The minimum order needed to meet such specifications with a linear-phase
FIR filter is 262. This filter must be the result of an optimal equiripple design.
If we relax the linear-phase constraint however, the equiripple design (based
on the algorithm proposed in [8]) results in a minimum-phase FIR filter of 216th
order that meets the specifications:
Hf = fdesign.lowpass('Fp,Fst,Ap,Ast',.12,.14,.175,60);
Hmin = design(Hf,'equiripple','minphase',true);
cost(Hmin)



Note that for these designs minimum-phase filters will have a much
smaller transient delay not only because of their minimum-phase property
but also because their filter order is lower than that of a comparable linear-
phase filter. In fact this is true in general we had previously seen that a
filter with the same transition width and filter order with minimum-phase
has a smaller delay than a corresponding linear-phase filter. Moreover,
the minimum-phase filter has smaller ripples so that the two filters are not
really comparable. In order to compare apples to apples, the order of the
linear-phase filter would have to increased until the ripples are the same
as those of the minimum-phase design. This increase in filter order would
of course further increase the delay of the filter.

1.3.3 Optimal equiripple designs with fixed peak ripple
and filter order
So far we have illustrated equiripple designs with fixed transition width
and fixed order and designs with fixed transition width and fixed peak
ripple values. The Filter Design Toolbox also provides algorithms for de-
signs with fixed peak ripple values and fixed filter order [10]. This gives
maximum flexibility in utilizing the degrees of freedom available to design
an FIR filter.
We have seen that, when compared to Kaiser-window designs, fixing
the transition width and filter order results in an optimal equiripple de-
sign with smaller peak ripple values, while fixing the transition width and
peak ripple values results in a filter with less number of taps. Naturally,
fixing the filter order and the peak ripple values should result in a smaller
transition width.

Example 9 Consider the following design of an equiripple with the same cutoff
frequency as in Example 7. The filter order is set to be the same as that needed for
a Kaiser-window design to meet the ripple specifications

Hf = fdesign.lowpass('N,Fc,Ap,Ast',49,0.375,0.13,60);

The comparison of this new design with the Kaiser-window design is shown in
Figure 1.10. The transition width has been reduced from 0.15π to approximately
0.11π.



1
Kaiser−window design
0.9 Equiripple design
0.8

0.7
Amplitude

0.6

0.5

0.4

0.3

0.2

0.1

0
0.32 0.34 0.36 0.38 0.4 0.42 0.44

Figure 1.10: Comparison of a Kaiser-window-designed FIR filter and an optimal equirip-
ple FIR filter of the same order and peak ripple values. The equiripple design results in a
reduced transition-width.

Minimum-phase designs with fixed peak ripple and filter order
Once again, if linear-phase is not a requirement, a minimum-phase filter
can be designed that is a superior in some sense to a comparable linear-
phase filter. In this case, for the same filter order and peak ripple value, a
minimum-phase design results in a smaller transition width than a linear-
phase design.

Example 10 Compared to the 50th order linear-phase design Heq, the following
design has a noticeably smaller transition width:

Hmin = design(Hf,'equiripple','minphase',true);

1.3.4 Constrained-band equiripple designs
Sometimes when designing lowpass filters (for instance for decimation
purposes) it is necessary to guarantee that the stopband of the filter begins
at a specific frequency value and that the filter provides a given minimum
stopband attenuation.



If the filter order is fixed - for instance when using specialized hard-
ware - there are two alternatives available in the Filter Design Toolbox for
optimal equiripple designs. One possibility is to fix the transition width,
the other is to fix the passband ripple.

Example 11 For example, the design specifications of Example 7 call for a stop-
band that extends from 0.45π to π and provide a minimum stopband attenuation
of 60 dB. Instead of a minimum-order design, suppose the filter order available is
40 (41 taps). One way to design this filter is to provide the same maximum pass-
band ripple of 0.13 dB but to give up control of the transition width. The result
will be a filter with the smallest possible transition width for any linear-phase FIR
filter of that order that meets the given specifications.

Hf = fdesign.lowpass('N,Fst,Ap,Ast',40,.45,0.13,60);

If instead we want to fix the transition width but not constrain the pass-
band ripple, an equiripple design will result in a filter with the smallest
possible passband ripple for any linear-phase FIR filter of that order that
meets the given specifications.

Hf = fdesign.lowpass('N,Fp,Fst,Ast',40,.3,.45,60);

The passband details of the two filters are shown in Figure 1.11. Note
that both filters meet the Specifications Set 2 because the order used (40) is
larger than the minimum order required (37) by an equiripple linear phase
filter to meet such specifications. The filters differ in how they “use” the
extra number of taps to better approximate the ideal lowpass filter.

1.3.5 Sloped equiripple filters
In many cases it is desirable to minimize the energy in the stopband of
a signal being filtered. One common case is in the design of decimation
filters. In this case, the energy in the stopband of a signal after being fil-
tered aliases into the passband region. To minimize the amount of alias-
ing, we want to minimize the stopband energy. Least-squares filters can be
used for this, however, the drawback is that the passband presents larger
fluctuations than may be desirable. An alternative is to design optimal



Magnitude Response (dB)

Fixed passband ripple
0.06 Fixed transition width

0.04
Magnitude (dB)

0.02

0

−0.02

−0.04

−0.06

0 0.05 0.1 0.15 0.2 0.25 0.3

Figure 1.11: Comparison of a two optimal equiripple FIR filters of 40th order. Both
filters have the same stopband-edge frequency and minimum stopband attenuation. One
is optimized to minimize the transition width while the other is optimized to minimize the
passband ripple.

equiripple filters but allowing for a slope in the stopband of the filter. The
passband remains equiripple thus minimizing the distortion of the input
signal in that region.
There are many ways of shaping the slope of the stopband. One way
[11] is to allow the stopband to decay as (1/ f )k , that is as a power of the
inverse of frequency. This corresponds to a decay of 6k dB per octave.
Another way of shaping the slope is to allow it to decay in logarithmic
fashion so that the decay appears linear in dB scale.
Of course there is a price to pay for the sloped stopband. Since the
design provides smaller stopband energy than a regular equiripple design,
the passband ripple, although equiripple, is larger than that of a regular
equiripple design. Also, the minimum stopband attenuation measured in
dB is smaller than that of a regular equiripple design.

Example 12 Consider an equiripple design similar to that of Example 2 but with
a stopband decaying as (1/ f )2 :
Fp = 0.4 - 0.06/2; Fst = 0.4 + 0.06/2;




0.5 Least squares
0.4
Sloped equiripple

0.3

0.2
Magnitude (dB)

0.1

0

−0.1

−0.2

−0.3

−0.4

−0.5

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

Figure 1.12: Passband details of a sloped optimal equiripple FIR design and an optimal
least-squares FIR design. The equiripple filter has a smaller peak error or smaller transi-
tion width depending on the interpretation.

Hf = fdesign.lowpass('N,Fp,Fst',42,Fp,Fst);
Hsloped = design(Hf,'equiripple','StopbandShape','1/f',...
'StopbandDecay',2);
results in a stopband energy of approximately 8.4095e-05, not much larger that
the least-squares design (6.6213e-005), while having a smaller transition width
(or peak passband ripple - depending on the interpretation). The passband details
of both the least-squares design and the sloped equiripple design are shown in
Figure 1.12 (in dB). The stopband details are shown in Figure 1.13 (also in dB).

If we constrain the filter order, the passband ripple, and the minimum
stopband attenuation, it is easy to see the trade-off between a steeper slope
and the minimum stopband attenuation that can be achieved. Something
has to give and since everything else is constrained, the transition width
increases as the slope increases as well.

Example 13 Design two filters with the same filter order, same passband ripple,
and same stopband attenuation. The slope of the stopband decay is zero for the
first filter and 40 for the second.




−20 Least squares
Sloped equiripple
−25

−30
Magnitude (dB)

−35

−40

−45

−50

−55
0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95

Figure 1.13: Stopband details of a sloped optimal equiripple FIR design and an optimal
least-squares FIR design. The overall error of the equiripple filter approaches that of the
least-squares design.

Hf = fdesign.lowpass('N,Fst,Ap,Ast',30,.3,0.4,40);
Heq = design(Hf,'equiripple','StopbandShape','linear',...
'StopbandDecay',0);
Heq2 = design(Hf,'equiripple','StopbandShape','linear',...
'StopbandDecay',40);

The second filter provides better total attenuation throughout the stopband. Since
everything else is constrained, the transition width is larger as a consequence.
This is easy to see with fvtool(Heq,Heq2).

It is also possible to design minimum-phase sloped equiripple filters. These
designs possess similar advantages over linear-phase designs as those de-
scribed for other equiripple designs when linearity of phase is not a design
requirement.



Impulse Response

0.1

0.08

0.06
Amplitude

0.04

0.02

0

−0.02

0 5 10 15 20 25 30 35 40
Samples

Figure 1.14: Impulse response of equiripple ﬁlter showing anomalous end-points. These
end points are the result of an equiripple response.

1.4 Further notes on equiripple designs
1.4.1 Unusual end-points in the impulse response
In some cases the impulse response of an equiripple design appears to be
anomalous. In particular, the end points seem to be incorrect. However,
these end points are in fact a consequence of an equiripple stopband. If
we remove the anomaly by say repeating the value in the sample closest
to the end point, the stopband is no longer equiripple.

Example 14 Consider the design of this lowpass ﬁlter with band edges that are
quite close to DC:

Hf = fdesign.lowpass('N,Fp,Fst',42,0.1,0.12);
fvtool(Heq,'Analysis','impulse')

The impulse response is show in Figure 1.14. Notice that the two end points seem
completely out of place.


1.4 Further notes on equiripple designs 33

If the anomalous end-points are a problem∗ , it is not always feasible to
modify them by replacing their value with that of their nearest neighbor
because sometimes it is more than just the two end points that are anoma-
lous. Moreover, simply using the nearest neighbor removes the equiripple
property of the stopband in an uncontrolled way. It is preferable to use a
sloped equiripple design to control the shape of the stopband and at the
same time remove the anomalies.
Example 15 Compare the magnitude response and the impulse response of these
two designs using fvtool.
Hf = fdesign.lowpass('N,Fp,Fst',800,.064,.066);
Heq = design(Hf,'equiripple','Wpass',1 ,'Wstop',10);
Heq2 = design(Hf,'equiripple','Wpass',1 ,'Wstop',10,...
'StopbandShape','linear','StopbandDecay',80);
fvtool(Heq,Heq2)
A look at the impulse response reveals that adding a sloped stopband has altered
several of the end points of the impulse response removing the discontinuity.

1.4.2 Transition region anomalies
In equiripple design, the transition band between passband and stopband
or vice versa is treated as a don’t care region for optimization purposes.
That is, the frequency response is not optimized between bands with the
hope that the response will smoothly transition between them. This is the
case for lowpass and highpass filters, but may not be the case for filters
with more than two bands when the width of the transition bands differ
[4].
A possible solution is to reduce the larger transition bands to make
them all the same width. This of course incurs in some degradation in
the performance of the filter that will depend on what constraints are im-
posed. If this is not feasible, another solution would be to include the
problematic transition band in the optimization by means of a arbitrary
magnitude design.
∗ For instance, when the impulse response is very long, sometimes only a subset of it is
stored in memory and the rest of the values are computed by curve fitting or some other
interpolation scheme. In such cases the anomalous points pose a problem because the
curve fitting is unlikely to reproduce them accurately.




0

−10

−20
Magnitude (dB)

−30

−40

−50

−60

−70
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Figure 1.15: Bandpass filter with transition-band anomaly due to having different
transition-band widths.

Example 16 Consider the design of a bandpass filter with a first transition band
0.1π rad/sample wide and a second transition band 0.2π rad/sample wide:

Hf = fdesign.bandpass('Fst1,Fp1,Fp2,Fst2,Ast1,Ap,Ast2',...
.2,.3,.6,.8,60,1,60);

The magnitude response is shown in Figure 1.15. The anomalies in the second
“don’t-care” band are obvious. The design can be fixed by making the second
transition band 0.1π rad/sample wide. Since this is a minimum-order design, the
price to pay for this is an increase in the filter order required to meet the modified
specifications:

Hf2 = fdesign.bandpass('Fst1,Fp1,Fp2,Fst2,Ast1,Ap,Ast2',...
.2,.3,.6,.7,60,1,60);
Heq2 = design(Hf,'equiripple');
cost(Heq)
cost(Heq2)


1.5 Maximally-flat FIR filters 35

Magnitude Response
1
10th−order maximally flat FIR filter
0.9 20th−order maximally flat FIR filter
0.8

0.7
Magnitude

0.6

0.5

0.4

0.3

0.2

0.1

0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Figure 1.16: Maximally-flat FIR filters. The smaller transition width in one of the filters
is achieved by increasing the filter order.

1.5 Maximally-flat FIR filters
Maximally-flat filters are filters in which both the passband and stopband
are as flat as possible given a specific filter order.
As we should come to expect, the flatness in the bands comes at the
expense of a large transition band (which will also be maximum). There is
one less degree of freedom with these filters than with those we have look
at so far. The only way to decrease the transition band is to increase the
filter order.

Example 17 Figure 1.16 shows two maximally-flat FIR filters. Both filters have
a cutoff frequency of 0.3π. The filter with smaller transition width has twice the
filter order as the other.

The Maximally-flat stopband of the filter means that its stopband at-
tenuation is very large. However, this comes at the price of a very large
transition width. These filters are seldom used in practice (in particular
with fixed-point implementations) because when the filter’s coefficients
are quantized to a finite number of bits, the large stopband cannot be



0
Double−precision floating−point
16−bit fixed−point

−50
Magnitude (dB)

−100

−150

−200

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Figure 1.17: A maximally-flat FIR filter implemented with double-precision floating-
point arithmetic and the same filter implemented with 16-bit fixed-point coefficients.

achieved (and often is not required anyway) but the large transition band
is still a liability.

Example 18 Compare the stopband attenuation of a maximally-flat FIR filter im-
plemented using double-precision floating-point arithmetic with that of the same
filter implemented with 16-bit fixed-point coefficients. The comparison of the two
implementations is shown in Figure 1.17. The fixed-point implementation starts
to differ significantly from the floating-point implementation at about 75-80 dB.
Nevertheless, both filters have the same large transition width.

The maximally-flat passband may be desirable because it causes min-
imal distortion of the signal to be filtered in the frequency band that we
wish to conserve after filtering. So it may seem that a maximally-flat pass-
band and an equiripple or perhaps sloped stopband could be a thought-
after combination. However, if a small amount of ripple is allowed in the
passband it is always possible to get a smaller transition band and most
applications can sustain a small amount of passband ripple.

Example 19 We can approach a maximally-flat passband by making the pass-
band ripple of an equiripple design progressively smaller. However, for the same


1.6 Summary and look ahead 37

−3
1 × 10
0
dB passband ripple
−10 1 × 10−8 dB passband ripple
−20
Magnitude (dB)

−30

−40

−50

−60

−70

−80

−90

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Figure 1.18: Equiripple filters with passband approximating maximal flatness. The better
the approximation, the larger the transition band.

filter order and stopband attenuation, the transition width increases as a result.
Consider the following two filter designs:

Hf = fdesign.lowpass('N,Fc,Ap,Ast',70,.3,1e-3,80);
Hf2 = fdesign.lowpass('N,Fc,Ap,Ast',70,.3,1e-8,80);
Heq2 = design(Hf2,'equiripple');

The two filters are shown in Figure 1.18. It is generally best to allow some pass-
band ripple as long as the application at hand supports it given that a smaller
transition band results. The passband details are shown in Figure 1.19.

1.6 Summary and look ahead
Understanding FIR filter design is a matter of understanding the trade offs
involved and the degrees of freedom available. A drawback of FIR filters
is that the tend to require a large filter order and therefore a high compu-
tational cost to achieve the specifications desired. There are many ways
of addressing this. One is to use IIR filters. Another is to use multistage



x 10
−4 Magnitude Response (dB)
6

1 × 10−3 dB passband ripple
4 1 × 10−8 dB passband ripple
Magnitude (dB)

2

0

−2

−4

−6
0 0.05 0.1 0.15 0.2 0.25

Figure 1.19: Passband details of equiripple filters with very small passband ripple. The
flatter passband is obtained at the expense of a larger transition band, i.e. a smaller usable
passband.

and/or multirate techniques that use various FIR filters connected in cas-
cade (in series) in such a way that each filter shares part of the filtering
duties while having reduced complexity when compared to a single-stage
design. The idea is that for certain specifications to combined complexity
of the filters used in multistage design is lower than the complexity of a
comparable single-stage design.
We will be looking at all these approaches in the following chapters.
We will then look into implementation of filters and discuss issues that
arise when implementing a filter using fixed-point arithmetic.


Chapter 2

Basic IIR Filter Design

Overview
One of the drawbacks of FIR filters is that they require a large filter order
to meet some design specifications. If the ripples are kept constant, the
filter order grows inversely proportional to the transition width. By using
feedback, it is possible to meet a set of design specifications with a far
smaller filter order than a comparable FIR filter∗ . This is the idea behind
IIR filter design. The feedback in the filter has the effect that an impulse
applied to the filter results in a response that never decays to zero, hence
the term infinite impulse response (IIR).
We will start this chapter by discussing classical IIR filter design. The
design steps consist of designing the IIR filter in the analog-time domain
where closed-form solutions are well known and then using the bilinear
transformation to convert the design to the digital domain. We will see
that the degrees of freedom available are directly liked to the design al-
gorithm chosen. Butterworth filters provide very little control over the
resulting design since it is basically a maximally-flat design. Chebyshev
designs increase the degrees of freedom by allowing ripples in the pass-
band (type I Chebyshev) or the stopband (type II Chebyshev). Elliptic
filters allow for maximum degrees of freedom by allowing for ripples in
both the passband and the stopband. In fact, Chebyshev and Butterworth
designs can be seen as a special case of elliptic designs, so usually one
should only concentrate on elliptic filter design, decreasing the passband
∗ However, see Chapter 5.

40 Basic IIR Filter Design

ripples and/or increasing the stopband attenuation enough to approach
Chebyshev or Butterworth designs when so desired.
After looking at classical designs, we move on to examine design tech-
niques that are based on optimization directly in the digital domain.

2.1 Why IIR filters
IIR filters are usually used when computational resources are at a pre-
mium. However, stable, causal IIR filters cannot have perfectly linear
phase, so that IIR filters tend to be avoided when linearity of phase is a
requirement. Also, computational savings that are comparable to using
IIR filters can be achieved for certain design parameters by using multi-
stage/multirate FIR filter designs (see Chapter 5). These designs have the
advantage of having linear-phase capability, robustness to quantization,
stability, and good pipeline-ability.
Another important reason to use IIR filters is their relatively small group
delay compared to FIR filters. This results in a shorter transient response
to input stimuli. Even if we are able to match or surpass the implemen-
tation cost of an IIR filter by using multirate/multistage FIR designs, the
transient response of such FIR designs tends to be much larger than that
of a comparable IIR design. However, minimum-phase FIR designs such
as those seen in Chapter 1, can have group-delays that are comparable or
even lower than IIR filters that meet the same specifications.

2.2 Classical IIR design
We begin by discussing filters that are designed in the analog-time domain
and then transformed to the digital domain by means of the bilinear trans-
formation. This methodology covers the following IIR design techniques:
Butterworth, Chebyshev type I, Chebyshev type II, and elliptic (Cauer).
We follow the methodology presented in [12]-[13] in which the 3-dB point
is always a design specification.


2.2 Classical IIR design 41

2.2.1 Cutoff frequency and the 3-dB point
For FIR filters, we have started by designating the cutoff frequency for an
ideal filter (i.e. one with zero transition-width). However, depending on
the design method, the cutoff frequency may lack any meaningful value.
For windowed-impulse-response designs, the magnitude response will
typically cross the cutoff frequency at the point where the magnitude is
equal to 0.5. In decibels, this point corresponds to 20 log10 (0.5) = −6.0206.
The reason for this is that with window-based designs, the passband and
stopband ripples are about the same. For other designs, such as equirip-
ple, the point at which the magnitude response crosses the middle of the
transition band will depend on the ratio of the passband and stopband
ripples.
By contrast, the convention for classical IIR designs is to define the
cutoff frequency as the point at which the power of an input signal is at-
tenuated by one-half. This means that the cutoff frequency in decibels cor-
responds to 10 log10 (0.5) = −3.0103 which we loosely call the 3-dB point∗ .
To avoid confusion, we will call the point at which the power is re-
duced by one-half the 3-dB point rather than the cutoff frequency. When
entering specifications, we use the string 'F3dB' rather than 'Fc'.

2.2.2 Butterworth filters
Butterworth filters are maximally-flat IIR filters. For this reason, the only
design parameters are the cutoff frequency and the filter order.
Example 20 Design a 7th order Butterworth filter with a 3 dB point of 0.3π.
Hf = fdesign.lowpass('N,F3db',7,0.3);
Hb = design(Hf,'butter');
The design is shown in Figure 2.1. The plot also includes an FIR filter designed
with a cutoff frequency of 0.3π. Notice that the attenuation at 0.3π is 3-dB for
the Butterworth filter and 6-dB for the FIR filter.
As with maximally-flat FIR filters, the flatness in the passband and
stopband cause the transition band to be very wide. The only way of re-
ducing the transition width is to increase the filter order.
∗ Just to be clear, the gain of the filter in decibels is -3.0103. The attenuation, i.e. the
distance between the passband gain and this point is approximately 3 dB since distance
is always positive.



0 Normalized Frequency: 0.2995605
Magnitude (dB): −2.958706 Butterworth filter
Normalized
−10 Frequency: 0.300415
FIR filter
Magnitude (dB): −6.104557
−20
Magnitude (dB)

−30

−40

−50

−60

−70

−80
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 2.1: 7th order Butterworth filter with a 3-dB frequency of 0.3π. Also shown is an
FIR filter with a cutoff frequency of 0.3π.

Example 21 Compare the 7th-order Butterworth filter of the previous example,
with a 12th-order Butterworth filter designed with the same 3-dB point. The
magnitude-squared responses are shown in Figure 2.2.
Hf2 = fdesign.lowpass('N,F3db',12,0.3);
Hb2 = design(Hf2,'butter');
The transition-width is decreased as the filter order increases. Notice that both
filters intersect at the point where the magnitude-squared of the filter is equal to
0.5, i.e., the point at which the power of the input signal is attenuated by one half.

2.2.3 Chebyshev type I filters
Chebyshev type I filters can attain a smaller transition width than a But-
terworth filter of the same order by allowing for ripples in the passband
of the filter. The stopband is, as with Butterworth filters, maximally flat.
For a given filter order, the trade-off is thus between passband ripple and
transition width.

Example 22 Compare the 7th-order Butterworth filter from previous examples
with a 7th-order Chebyshev type I filter with 1 dB of peak-to-peak passband ripple.



Magnitude Response (squared)
1
7th order Butterworth filter
0.9 12th order Butterworth filter

0.8

0.7
Magnitude squared

0.6

0.5

0.4

0.3

0.2

0.1

0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Figure 2.2: Comparison of a 7th-order and a 12th-order Butterworth filter. The filters
intersect at the 3-dB point.

Magnitude Response (squared)
1
Butterworth filter
0.9 Chebyshev type I filter
0.8
Magnitude squared

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Figure 2.3: Comparison of a 7th-order Butterworth and a 7th-order Chebyshev type I
filter. Once again, the filters intersect at the 3-dB point.

Hf = fdesign.lowpass('N,F3db,Ap',7,0.3,1);
Hc = design(Hf,'cheby1');

The magnitude-squared responses are shown in Figure 2.3. The passband ripple



of the Chebyshev filter can be made arbitrarily small, until the filter becomes the
same as the Butterworth filter. That exemplifies the trade-off between passband
ripple and transition width.
With IIR filters, we not only need to consider the ripple/transition-
width trade-offs, but also the degree of phase distortion. We know that
linear-phase throughout the entire Nyquist interval is not possible. There-
fore, we want to look at how far from linear the phase response is. A good
way to look at this is to look at the group-delay (which ideally would be
constant) and see how flat it is.
Of the classical IIR filters considered here∗ , Butterworth filters have the
least amount of phase distortion (phase non-linearity). Since Chebyshev
and elliptic filters can approximate Butterworth filters arbitrarily closely
by making the ripples smaller and smaller, it is clear that the amount of
ripple is related to the amount of phase distortion. Therefore, when we
refer to the trade-off between ripples and transition-width, we should re-
ally say that the trade-off is between the transition-width and the amount
of ripple/phase-distortion. As an example, we show the group-delay of
the Butterworth and Chebyshev type I filters of the previous example in
Figure 2.4. Notice that not only is the group-delay of the Chebyshev fil-
ter less flat, it is also larger for frequencies below the 3-dB point (which
is the region we care about). This illustrates that although the filters have
the same order, a Chebyshev type I filter will have a larger transient-delay
than a Butterworth filter.

2.2.4 Chebyshev type II designs
Unlike FIR maximally-flat filters, it is possible to attain very large stop-
band attenuations with Butterworth or Chebyshev type I filters even when
the filter is implemented using fixed-point arithmetic. Nevertheless, few
applications require attenuations of hundreds of decibels. With Butter-
worth filters, such large attenuations come at the price of a large transition
width. With Chebyshev type I filters, a trade-off can be made between
transition-width and passband ripple. However, attaining a small transi-
tion width may result in unacceptably high passband ripple.
∗ Bessel filters have maximally-flat group-delay responses. However, this property is
lost when the bilinear transformation is used to convert the filter from analog to digital.
It is possible to use other conversion techniques such as impulse-invariance with Bessel
filters [14].



Group Delay

30 Butterworth
Chebyshev type I

25

20

15

10

5

0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Figure 2.4: Group-delay responses for a 7th-order Butterworth and a 7th-order Cheby-
shev type I filter. Both filters have a 3-dB point of 0.3π.

In most cases a Chebyshev type II (inverse Chebyshev) filter is prefer-
able to both a Chebyshev type I filter and a Butterworth filter. These filters
have maximally-flat passband and equiripple stopband. Since extremely
large attenuations are typically not required, we may be able to attain the
transition-width required by allowing for some stopband ripple.

Example 23 Design a 6th order filter with a 3-dB point of 0.45π. The filter must
have an attenuation of at least 80 dB at frequencies above 0.75π and the passband
ripple must not exceed 0.8 dB.

Hf1 = fdesign.lowpass('N,F3db',6,0.45);
Hf2 = fdesign.lowpass('N,F3db,Ap',6,0.45,0.8);
Hf3 = fdesign.lowpass('N,F3db,Ast',6,0.45,80);
Hb = design(Hf1,'butter');
Hc1 = design(Hf2,'cheby1');
Hc2 = design(Hf3,'cheby2');

The three designs are shown in Figure 2.5. Only the Chebyshev type II filter
reaches the required attenuation of 80 dB by 0.75π.

The group-delay responses for the designs of the previous example are
shown in Figure 2.6. Although the Butterworth’s group-delay is slightly



0 Butterworth filter
Chebyshev type I filter
−10
Chebyshev type II filter
−20
Magnitude (dB)

−30

−40

−50

−60

−70

−80

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 2.5: 6th order Butterworth, Chebyshev type I, and Chebyshev type II filters with
a 3-dB point of 0.45π.

Group Delay
18
Butterworth
16 Chebyshev type I
Chebyshev type II

14

12

10

8

6

4

2

0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Figure 2.6: Group-delay responses for filters from Example 23.

flatter than that of the Chebyshev type II, the latter’s group-delay is smaller
than the former’s for most of the passband.
Even though all three designs in the previous example are of 6th order,
the Chebyshev type II implementation actually requires more multipliers
as we now explain.



All the designs result in three second-order sections. However, the
maximally-flat stopband of the Butterworth and Chebyshev type I filters is
achieved by placing multiple zeros at w = π. The result is that the numer-
ators for all three sections are given by 1 + 2z−1 + z−2 . The corresponding
coefficients, {1, 2, 1} require no actual multiplications (the multiplication
by two can be implemented with a simple shift). There are two multiplica-
tions for each denominator of each section plus one more multiplier that
provides the correct passband gain (0 dB)∗ . In other words, both the But-
terworth filter and the Chebyshev type I filter can be implemented with 7
multipliers. By contrast, the Chebyshev type II design requires, at least 9
multipliers to implement.

2.2.5 Elliptic filters
Elliptic (Cauer) filters generalize Chebyshev and Butterworth filters by al-
lowing for ripple in both the passband and the stopband of a filter. By
making the ripples smaller and smaller, we can approximate arbitrarily
close either Chebyshev or Butterworth filters using elliptic filters. The ap-
proximation is not just of the magnitude response, but the phase response
(and consequently the group-delay response) as well.
We have already said that most applications do not require arbitrar-
ily large stopband attenuations, hence Chebyshev type II filters are gen-
erally preferable to Chebyshev type I and Butterworth filters. However,
Chebyshev type II filters are maximally-flat in the passband and as a con-
sequence have rather larger transition bands. Since most applications can
sustain some amount of passband ripple, the transition band can be re-
duced by using an elliptic filter with the same amount of stopband attenu-
ation as a Chebyshev type II filter (and the same filter order). The reduced
transition band is attained by allowing for some amount of passband rip-
ple. As with Chebyshev type I filters, the passband ripples of elliptic fil-
ters cause the group-delay to become larger and more distorted than that
of Butterworth or Chebyshev type II filters.
Example 24 Design an elliptic filter that meets the specifications of Example 23.
Hf4 = fdesign.lowpass('N,F3db,Ap,Ast',6,0.45,.8,80);
He = design(Hf4,'ellip');
∗ For fixed-point implementations, it may be preferable to break-up this scaling gain into
several gains, one per section, because of dynamic range issues.



0
−10 Elliptic filter
−20
Magnitude (dB)

−30

−40

−50

−60

−70

−80

−90

−100
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Figure 2.7: Elliptic and Chebyshev type II filters with a 3-dB point of 0.45π.

The resulting filter is shown in Figure 2.7 along with the Chebyshev type II filter
we had already designed. The passband ripple allows for the 80 dB attenuation to
be reached by about 0.63π rad/sample, far before the required 0.75π. If the target
application does not benefit from this the passband ripple can be reduced in order
to have smaller passband distortion.

The group-delay of the elliptic filter is similar but even less flat than
that of the Chebyshev type I filter. Figure 2.8 shows the group-delay re-
sponses of the Chebyshev type I design of Example 23 and the elliptic
design of Example 24.

2.2.6 Minimum-order designs
We can use any of the design methods described above for design prob-
lems with fixed passband/stopband ripples and fixed transition-widths.
Naturally, elliptic filters will result in the lowest filter order due to the per-
missibility of ripples in both passband and stopband. In fact, elliptic filters
are the IIR version of optimal equiripple designs.

Example 25 Design a highpass IIR filter with a maximum passband ripple of 1
dB and a minimum stopband attenuation of 60 dB. The stopband must extend to
0.45π and the passband must begin at 0.55π.



Group Delay

20
Chebyshev type I filter
Elliptic filter
18

16

14

12

10

8

6

4

2

0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Figure 2.8: Group-delay response of elliptic and Chebyshev type I filters with a 3-dB
point of 0.45π.


0 Butterworth
Chebyshev type I
−10
Chebyshev type II
Elliptic
Magnitude (dB)

−20

−30

−40

−50

−60

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 2.9: Magnitude responses of highpass filters designed with all four classical design
methods.

Hf = fdesign.highpass('Fst,Fp,Ast,Ap',.45,.55,60,1);
Hd = design(Hf,'alliir');
The response of all four filters can be visualized using fvtool(Hd) (see Figure
2.9). Since we have designed all four filters, Hd is an array of filters. Hd(1)



is the Butterworth filter, Hd(2) is the Chebyshev type I, Hd(3) the Chebyshev
type II, and Hd(4) the elliptic filter. By inspecting the cost of each filter (e.g.
cost(Hd(2))) we can see that the Butterworth filter takes the most multipliers
(and adders) to implement while the elliptic filter takes the least. The Chebyshev
designs are comparable.

As with Chebyshev type II filters, elliptic filters do not have trivial nu-
merators of the form 1 − 2z−1 + z−2 ∗ . In contrast, the actual cost of Butter-
worth and Chebyshev type I filters is a little lower than what is indicated
by the cost function as long as we are able to implement the multiplica-
tion by two without using an actual multiplier (the cost function does not
assume that this is the case; it is implementation agnostic).

2.2.7 Comparison to FIR filters
We have said that one reason to use IIR filters is because of their small
implementation cost when compared to FIR counterparts. We’d like to
illustrate this with an example.

Example 26 Consider the following design specifications:



3. Maximum passband ripple: 1 dB


Fc = 0.0625;
TW = 0.008;
Fp = Fc-TW/2;
Fst= Fc+TW/2;
Ap = 1;
Ast= 80;
Hf = fdesign.lowpass('Fp,Fst,Ap,Ast',Fp,Fst,Ap,Ast);
∗ Notice the negative sign for the term 2z−1 . This is because we are designing highpass
filters rather than lowpass.



If we design an elliptic filter,
He = design(Hf,'ellip');
cost(He);
the cost method indicates that 20 non-trivial multipliers and 20 adders are needed
to implement the filter.
An equiripple design,
requires a whopping 642 multipliers and 641 adders. If we use a symmetric direct-
form structure to implement the filter,
Heqs = design(Hf,'equiripple','FilterStructure','dfsymfir');
the number of multipliers is halved to 321 (the number of adders remains 641).

Clearly there is not contest between the FIR and the IIR design. In the
following chapters we will see that with the use FIR multirate/multistage
techniques we will be able to achieve designs that are more efficient than
the elliptic design shown here in terms not of the number of multipliers,
but of the number of multiplications per input sample (MPIS). However,
we will then go on to show that by using efficient multirate IIR filters based
on allpass decompositions, we can achieve even lower MPIS. Note that for
the designs discussed so far, both the FIR and IIR cases (being single-rate),
the number of MPIS is the same as the number of multipliers.
In addition to computational cost, another common reason for using
IIR filters is their small group-delay when compared to FIR filters. How-
ever, let us compare with minimum-phase FIR filters via an example.

Example 27 Consider the following bandpass filter specifications:

1. Lower cutoff frequency: 0.35π rad/sample

2. Upper cutoff frequency: 0.55π rad/sample

3. Lower/upper transition width: 0.1π rad/sample




Group Delay

Chebyshev type II
Linear−phase FIR
25
Group delay (in samples) Minimum−phase FIR

20

15

10

5

0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Figure 2.10: Group-delays of FIR and IIR bandpass ﬁlters.

Fc1 = 0.35;
Fc2 = 0.55;
TW = 0.1;
Fst1 = Fc1-TW/2;
Fp1 = Fc1+TW/2;
Fp2 = Fc2-TW/2;
Fst2 = Fc2+TW/2;
Ap = 1e-2;
Ast = 40;
Hf = fdesign.bandpass(Fst1,Fp1,Fp2,Fst2,Ast,Ap,Ast);
In the passband, [0.4π, 0.5π ], of all classical IIR designs, the Chebyshev type II
will have the lowest group delay. However, a minimum-phase equiripple FIR ﬁlter
has an even lower group-delay in the passband as can be seen in Figure 2.10.

2.3 IIR designs directly in the digital domain
In addition to the classical methods based on bilinear transformation on
analog designs, the Filter Design Toolbox includes design algorithms based


2.3 IIR designs directly in the digital domain 53

on optimization directly in the digital domain. The advantage of these
algorithms is that they allow for independent control of the numerator
and denominator orders, providing more flexibility than classical designs.
Moreover, these digital-domain designs allow for optimization of different
norms, from least-squares (2-norm) all the way to infinity-norms (equirip-
ple).
We have already stated that elliptic filters are optimal equiripple IIR fil-
ters. Essentially the same filters can be designed use the iirlpnorm design
algorithm if the L∞ norm is used as the pth norm.
Example 28 The following design yields virtually the same filter as the elliptic
design of Example 24:
Hf5 = fdesign.lowpass('Nb,Na,Fp,Fst',6,6,.4425,.63);
Hd = design(Hf5,'iirlpnorm','Wpass',1,'Wstop',450);
Unlike classical IIR designs, the passband gain of the least pth norm design is not
bounded by zero dB. However, if we normalize the passband gain to 0 dB, we can
verify that this filter is virtually identical to the elliptic filter within a scale factor.
To illustrate the extra flexibility afforded by independent control of the
numerator and denominator orders, consider the following example in
which have a larger order numerator than denominator.
Example 29 We want to use the iirlpnorm function to design a filter with less
than 0.8 dB passband ripple and at least 80 dB attenuation at 0.75π and be-
yond. All designs we showed so far used a 6th order. The two designs that met
the specifications, elliptic and Chebyshev type II, would have failed to meet the
specifications had a 4th-order been used instead. Here, we will use a 6th-order
numerator and a 4th-order denominator and still meet the specs.
Hf5 = fdesign.lowpass('Nb,Na,Fp,Fst',6,4,.4425,.75);
Hd = design(Hf5,'iirlpnorm','Wpass',1,'Wstop',25);
The design is shown along with the Chebyshev type II filter designed previously
in Figure 2.11. The passband ripple is only about 0.043 dB. Despite using less
multipliers than the Chebyshev type II filter, the least pth norm filter has a steeper
transition width thanks to the fact that there is some passband ripple.
We should note that unlike the classical IIR designs, it is not possible
to directly control the location of the 3-dB point with the least pth norm
design algorithm.



0
−10 IIRLPNORM filter

−20
Magnitude (dB)

−30

−40

−50

−60

−70

−80

−90

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Figure 2.11: Least pth norm IIR filter with a 6th order numerator and a 4th order de-
nominator. Also shown is the Chebyshev type II filter design in Example 23.

Trade-offs and degrees of freedom in IIR filter design are essentially the
same as those in FIR filter design. This is particularly true when ripples
are allowed in the passband and/or stopband of the filter. Elliptic filters
are IIR optimal equiripple filters that use the smallest filter order possi-
ble to meet a set of specifications involving ripples and transition band.
Butterworth and Chebyshev filters are special cases of elliptic filters with
zero passband and/or stopband ripples. However, most applications can
sustain some amount of ripple, making elliptic filters usually the way to
go. Further flexibility can be obtained by controlling individually the nu-
merator and denominators orders. Least pth norm designs allow for such
control.
IIR filters (elliptic in particular) can meet a set of specifications with
far fewer multipliers than what an FIR filter would require. However,
there is a price to pay in terms of phase distortion, implementation com-
plexity, stability issues when using finite-precision arithmetic, pipeline-
ability and so forth. If the design specifications are well-suited for a mul-
tirate/multistage FIR filter design, we may be able to obtain more effi-
cient designs without the drawbacks of IIR filters. We will study such



approaches in the coming chapters. However, we point out once again
that these multistage implementations tend to have large transient delays,
so that if this is a critical factor, IIR ﬁlters may be preferable (although
minimum-phase FIR ﬁlters can be an interesting option as we have seen).


Chapter 3

Nyquist Filters

Overview
Nyquist filters are a special class of filters which are useful for multirate
implementations. Nyquist filters also find applications in digital commu-
nications systems where they are used for pulse shaping (often simultane-
ously performing multirate duties). The widely used raised-cosine filter is
a special case of Nyquist filter that is often used in communications stan-
dards.
Nyquist filters are also called Lth-band filters because the passband of
their magnitude response occupies roughly 1/L of the Nyquist interval.
The special case, L = 2 is widely used and is referred to as Halfband filters.
Halfband filters can be very efficient for interpolation/decimation by a
factor of 2.
Nyquist filters are typically designed via FIR approximations. How-
ever, IIR designs are also possible. We will show FIR and IIR designs
of halfband filters. The IIR case results in extremely efficient filters and
through special structures such as parallel (coupled) allpass-based struc-
tures and wave digital filters can be efficiently implemented for multirate
applications.
Later, in Chapter 5, we will see a special property of Nyquist filters
when used in multirate applications. The cascade of Nyquist interpolators
(or decimators) results in an overall Nyquist interpolator (or decimator).
This makes Nyquist filters highly desirable for multistage designs.
Moreover, as we will see, Nyquist filters have very small passband rip-

3.1 Design of Nyquist filters 57

ple even when the filter is quantized for fixed-point implementations. This
makes Nyquist filters ideally suitable for embedded applications.

3.1 Design of Nyquist filters
Although not strictly necessary, the most common design of a Nyquist fil-
ter is as a special case of a lowpass filter. Nonetheless, highpass and band-
pass designs are also possible. When designing a Nyquist filter, the value
of the band L must be specified. Nyquist filters will reduce the bandwidth
of a signal by a factor of L.
With Nyquist filters, the passband ripple and the stopband ripple can-
not be independently specified. Therefore, in all designs, only the stop-
band ripple (and/or the transition width) will be given. The passband
ripple will be a result of the design. In most cases, the resulting passband
ripple is very small. This is particularly true for IIR Nyquist filters.
Example 30 Design a Kaiser window 5th-band Nyquist FIR filter of 84th order
and 80 dB of stopband attenuation∗ ,
L = 5; % 5th-band Nyquist filter
f = fdesign.nyquist(L,'N,Ast',84,80);
h = design(f,'kaiserwin');
Both the transition-width and the passband ripple of the filter are a result of the
design. The filter has a cutoff frequency given by π/L (0.2π in this case). The
passband details of the magnitude response are shown in Figure 3.1. Note that the
peak to peak ripples are indeed quite small (on the order of 2 × 10−3 ).
A characteristic of Nyquist filters is that every Lth sample of their im-
pulse response is equal to zero (save for the middle sample). As we will see
when we discuss multirate filter design, this characteristic is desirable for
interpolation because it means that the input samples will pass through
the filter unchanged. Moreover, for both interpolation and decimation
purposes, it means that one of the L polyphase branches will result in a
simple delay (no multipliers/adders) reducing the implementation cost
∗ Close inspection of the stopband will show that the 80 dB attenuation is not quite
achieved. This happens sometimes with Kaiser-window designs because the relation
between the adjustable parameter in the Kaiser window and the stopband attenuation is
derived in somewhat empirical fashion.


58 Nyquist Filters

x 10
8

6

4
Magnitude (dB)

2

0

−2

−4

−6

−8
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14

Figure 3.1: Passband ripple details for 5th-band Nyquist FIR filter designed with a Kaiser
window.

for such branch. The impulse response for the filter designed in Exam-
pled 30 can be seen in Figure 3.2. Note that every 5th sample is equal to
zero except for the middle sample which is the peak value of the impulse
response.

3.1.1 Equiripple Nyquist filters
If instead of specifying the order and stopband attenuation of an FIR Nyquist
filter design we specify the order and transition width or the transition
width and the stopband attenuation, equiripple designs are possible. It
must be pointed out however that it is not trivial to design equiripple FIR
Nyquist filters given the time-domain constraint (the fact that the impulse
response must be exactly zero every L samples). Equiripple designs may
have convergence issues for large order/small transition width combina-
tions and may result in bogus filters. Filters with equiripple passband and
sloped stopband are also possible∗
Although equiripple designs are possible, their advantages relative to
∗ Except in the halfband (L=2) case. In that case, symmetry characteristics of the magni-
tude response about the 0.5π point result in same passband/stopband ripples. Therefore
a sloped stopband will also mean a sloped passband.


3.1 Design of Nyquist filters 59

Impulse Response
0.2

0.15
Amplitude

0.1

0.05

0

−0.05
0 10 20 30 40 50 60 70 80
Samples

Figure 3.2: Impulse response of 5th-band Nyquist FIR filter. Every 5th sample is equal
to zero (except at the middle).

Kaiser-window designs are not as clear for Nyquist filters as they are for
regular filter designs. In particular, the passband ripples of Kaiser-window
designs may be smaller than those of equiripple designs. Moreover, as we
will see in Chapter 4, the increasing attenuation in the stopband may be
desirable for interpolation and especially for decimation purposes.

Example 31 We will compare a Kaiser window design to both an equiripple de-
sign with and without a sloped stopband.

f = fdesign.nyquist(4,'N,TW',72,.1); % 4th-Band Nyquist filter
h2 = design(f,'equiripple');
h3 = design(f,'equiripple','StopbandShape','linear','StopbandDecay',20);

While it is clear that the Kaiser-window design has the smallest minimum stop-
band attenuation, it does provide increased stopband attenuation as frequency
increases. The magnitude responses are shown in Figure 3.3.
Moreover, the Kaiser window design results in better passband ripple perfor-
mance than either equiripple design (this can be verified with the measure com-
mand). The passband ripple details are shown in Figure 3.4.


60 Nyquist Filters


0 Kaiser window
Equiripple
−10
Sloped equiripple
−20
Magnitude (dB)

−30

−40

−50

−60

−70

−80

−90

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Figure 3.3: Comparison of Kaiser window, equiripple, and sloped equiripple Nyquist
ﬁlter designs of 72nd order and a transition width of 0.1π.

0.015
Kaiser window
Equiripple
0.01
Sloped equiripple
Magnitude (dB)

0.005

0

−0.005

−0.01

−0.015
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

Figure 3.4: Passband ripple details for ﬁlters designed in Example 31.


3.2 Halfband filters 61

3.1.2 Minimum-order Nyquist filters
It is possible to design either equiripple or Kaiser-window FIR filters that
meet a transition-band requirement as well as a minimum-stopband re-
quirement with minimum order. It is important to note that while the
minimum stopband attenuation is the same for both designs. the max-
imum peak-to-peak passband ripple of each design will be different (of
course so to will be the resulting filter order).
What is interesting is that while equiripple designs tend to be of smaller
order, the peak-to-peak passband ripple of Kaiser window designs tends
to be smaller. This coupled with the increased overall attenuation in the
stopband due to its non-equiripple nature, may make Kaiser window de-
signs appealing despite the slightly larger implementation cost.

Example 32 Consider the following design examples, compare the results of the
measure command and the cost command for both Kaiser window designs and
equiripple designs.

f = fdesign.nyquist(8,'TW,Ast',0.1,60);
f2 = fdesign.nyquist(4,'TW,Ast',0.2,50);
hk = design(f,'kaiserwin');
he = design(f,'equiripple');
hk2 = design(f2,'kaiserwin');
he2 = design(f2,'equiripple');

Both cases illustrate the trade-off between filter order and passband ripple for
Nyquist filter design choices.

3.2 Halfband filters
Halfband filter are a special case of Nyquist filters when L = 2. This fil-
ters reduce the bandwidth of a signal roughly by half and are suitable for
decimation/interpolation by a factor of 2 (see Chapter 4).
As previously stated, Nyquist filters are characterized by the fact that
every Lth sample of its impulse response (i.e. every Lth filter coefficient)
is equal to zero. In the case of halfband filters this fact is particularly ap-
pealing since it means that roughly half of its coefficients are equal to zero.
This of course makes them very efficient to implement. As such, halfband


62 Nyquist Filters

filters are widely used in particular for multirate applications including
multistage applications (see Chapter 5).
The cutoff frequency for a halfband filter is always 0.5π. Moreover,
the passband and stopband ripples are identical, limiting the degrees of
freedom in the design. The specifications set follow the usual triangle
metaphor shown in Figure 1.2, taking into account the limitations just de-
scribed. As such, three different specifications are available for halfband
designs. In each case, two of the three angles of the triangle are specified
(as just stated, the stopband ripple automatically determines the passband
ripple).

f = fdesign.halfband('N,TW',N,TW);
f2 = fdesign.halfband('N,Ast',N,Ast);
f3 = fdesign.halfband('TW,Ast',TW,Ast);

In all three cases, either Kaiser window or equiripple FIR designs are pos-
sible. Moreover, in the first case, it is also possible to design least-squares
FIR filters.
Unlike the general case, there are usually no convergence issues with
equiripple halfband filter designs. In addition to that, the fact the pass-
band ripple is the same as the stopband ripple means that regardless of
the design algorithm, the resulting peak-to-peak passband ripples will be
about the same.
So for halfband filters, the advantages of optimal equiripple designs
over Kaiser window designs resurface in a more clear manner than for
other Nyquist filters. For a given set of specifications, the equiripple de-
signs will have either a larger minimum stopband attenuation, a smaller
transition width, or a smaller number of coefficients.

Example 33 Consider Kaiser window and equiripple designs for the following to
cases:

f = fdesign.halfband('N,TW',50,0.1);
f2 = fdesign.halfband('TW,Ast',0.1,75);

Using the measure command in the first case shows that the resulting stopband
attenuation (and consequently the passband ripple) is better in the equiripple case.
Similarly, using the cost command in the second case shows that the same per-
formance is achieved with fewer coefficients in the equiripple case.


3.2 Halfband filters 63

Sloped equiripple designs are also possible, but in the case of halfband
filters, they will result in similarly sloped passbands, given the symmetry
constraints on halfband filters.

3.2.1 IIR halfband filters
Nyquist filters can also be designed as IIR filters. The most interesting case
is that of halfband IIR filters [20]-[21]. These filters are extremely efficient
especially when implemented as two allpass branches in parallel. For IIR
halfband filters, the parallel allpass decomposition (8.1) becomes
1 ˆ 2
H (z) = A 0 ( z ) + z −1 A 1 ( z 2 )
ˆ (3.1)
2
ˆ ˆ
with A0 (z) = A0 (z2 ) and A1 (z) = z−1 A1 (z2 ).
Such implementations are ideally suited for polyphase multirate ar-
chitectures (see Chapter 4). In Chapter 5 we will show that using IIR
halfband-based multirate/multistage designs result in extremely efficient
filters that outperform FIR counterparts.
IIR halfband filters have similar symmetry constrains as FIR halfbands.
However, in the case of IIR filters, the symmetry is about the half-power
point (3 dB point) rather than the half-magnitude point (6 dB point) of
FIR halfbands. Because of this symmetry, it is not possible to design half-
band Chebyshev filters. Only Butterworth and elliptic halfband filters are
possible from the classical designs. A design algorithm synthesized di-
rectly in the digital domain [22] results in approximately linear phase in
the passband of the filter (at the expense of either higher order or reduced
stopband attenuation compared to elliptic halfbands).
Unlike the FIR case, it is important to point out that IIR halfbands are
not necessarily Nyquist filters. In particular, Butterworth and elliptic de-
signs result in non-Nyquist halfbands. As a result of this, although they
can be used for polyphase decimation/interpolation, in the interpolation
case, it will not be true that the input samples are preserved unchanged at
the output.
ˆ
The reason some IIR halfbands are not Nyquist, is that A0 (z2 ) and
ˆ
A1 ( z 2 ) may not be equal to a delay. However, in the case of approximately
ˆ
linear phase designs, A0 (z2 ) = z− D for some integer D. Therefore, the
filter is Nyquist and when used for interpolation, the input samples are
preserved unchaged (just delayed).


64 Nyquist Filters

x 10
0

−0.5

−1
Magnitude (dB)

−1.5

−2

−2.5

−3

−3.5

−4

−4.5
0 0.05 0.1 0.15 0.2 0.25

Figure 3.5: Passband details of quasi-linear phase IIR halfband filter with 70 dB of stop-
band attenuation.

In all cases, given only moderate stopband attenuations, the resulting
passband ripples of IIR halfband filters is very small. This fact coupled
with the fact that (as previously stated) many applications do not require
hundreds of decibels of stopband attenuation, limits the appeal of Butter-
worth halfband filters.
Example 34 Consider the following quasi-linear phase IIR halfband design. The
fact the we specify a 70 dB minimum stopband attenuation results in a peak-to-
peak passband ripple of only 4 × 10−7 dB!
f = fdesign.halfband('N,Ast',12,70);
h = design(f,'iirlinphase');
The passband details are shown in Figure 3.5.

Minimum-order designs
Minimum-order designs provide a good framework to compare imple-
mentation cost of FIR halfband filters vs. IIR halfbands. Elliptic halfbands
are the most efficient while quasi-linear phase halfbands give up some ef-
ficiency in the name of phase linearity. Either case is significantly more
efficient than FIR equiripple halfband.



Group Delay

iirlinphase
40
ellip
35 equiripple

30

25

20

15

10

5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Figure 3.6: Group-delay comparison for IIR and FIR halfband filters.

Example 35 Compare the following three minimum-order designs,
f = fdesign.halfband('TW,Ast',0.1,60);
hlp = design(f,'iirlinphase');
hel = design(f,'ellip');
heq = design(f,'equiripple');
Using the cost command, we can see that the FIR equiripple design requires 35
multipliers, the quasi-linear phase IIR design requires 15 multipliers, while the
elliptic design requires only 6 multipliers.
A plot of the group-delay of all three filters shows that while the elliptic de-
sign clearly has the lowest group delay, it is also the one with the most nonlinear
phase. The passband group-delay of the quasi-linear phase design is almost flat as
expected (see Figure 3.6).
The passband ripple of the equiripple design is quite small, yet it is orders of
magnitude larger than the passband ripple of either IIR design.

Nyquist filters have very useful properties that make them worth consid-
ering. They tend to result in efficient designs because of the fact that every


66 Nyquist Filters

lth sample of their impulse response is zero. They also tend to have very
small passband ripples.
Halfband filters are particularly interesting given their high efficiency.
IIR halfband filters are even more efficient and have extremely small pass-
band ripples.
Given all their advantages, Nyquist filters (FIR or IIR) should be the
first choice for multirate applications (see Chapter 4). The fact that their
cutoff frequency is given by π/L results in transition-band overlap in dec-
imation applications. This however is not a problem given that no aliasing
will occur in the band of interest ((see Chapter 4 and Appendix B.)
Moreover, we will see (Chapter 5) that cascade multirate Nyquist fil-
ters possess an interesting property: the overall equivalent multirate fil-
ter is also Nyquist. For example, a decimation/interpolation filter with a
rate change of say 8 can be implemented as a cascade of three halfband
decimation/interpolation filters (each with a rate change of 2). The three
halfband filters in cascade act a single Nyquist filter (the equivalent im-
pulse response of the cascade will have a value of zero every 8th sample)
but can be significantly more efficient than a single-stage design. This is
particularly true if IIR halfband designs are used.


Chapter 4

Multirate Filter Design

Overview
Multirate signal processing is a key tool to use in many signal processing
implementations. The main reason to use multirate signal processing is
efficiency. Digital filters are a key part of multirate signal processing as
they are an enabling component for changing the sampling rate of a signal.
If multirate concepts are not fresh, it may be helpful to read through
Appendix B prior to reading through this chapter.
In this chapter we will talk about designing filters for multirate sys-
tems. Whether we are increasing or decreasing the sampling rate of a sig-
nal, a filter is usually required. The most common filter type used for mul-
tirate applications is a lowpass filter.∗ Nyquist filters, described in Chapter
3, are the preferred design to use for multirate applications.
We start by presenting filters in the context of reducing the sampling
rate of a signal (decimation). We want to emphasize the following: if you
reduce the bandwidth of a signal through filtering, you should reduce its
sampling rate accordingly.
We will see that the above statement applies whether the bandwidth
is decreased by an integer or a fractional factor. Understanding fractional
sampling-rate reduction requires understanding interpolation. We present
first interpolation as simply finding samples lying between existing sam-
ples (not necessarily increasing the sampling rate). We then use this to
∗ However, as we will see, highpass, bandpass and in general any filter that reduces the
bandwidth of a signal may be suitable for multirate applications.

68 Multirate Filter Design

show how to increase the sampling-rate of a signal by an integer factor.
Finally we extend this paradigm in order to perform fractional decima-
tion/interpolation.

4.1 Reducing the sampling rate of a signal
In the previous chapters we have discussed the basics of FIR and IIR fil-
ter design. We have concentrated mostly on lowpass filters. A lowpass
filter reduces the bandwidth of the signal being filtered by some factor.
When the bandwidth of a signal is reduced, we usually want to reduce
the sampling rate in a commensurate manner. Otherwise, we are left with
redundant data, given that we are over-satisfying the Nyquist sampling
criterion. Any processing, storage, or transmission of such redundant data
will result in unnecessary use of resources.
Specifically, if the input signal x [n] has a bandwidth Bx and the filtered
signal y[n] has a bandwidth By related to the input bandwidth by

Bx
=M
By

we should reduce the sampling frequency of y[n] by a factor of M.

4.1.1 Decimating by an integer factor
If M is an integer the procedure is straightforward. We can downsample
y[n] by keeping one of every M samples

ydown [m] = y[nM ].

The idea is depicted in Figure 4.1 and is referred to as decimation. The
filter H (z) is a lowpass filter that roughly reduces the bandwidth by a
factor of M. Once the bandwidth is reduced, the sampling-rate should
be reduced accordingly. The downsampler that follows H (z) reduces the
sampling rate by keeping one out of every M of its input samples.
However, the procedure as described so far is inefficient in that after
the filter processes a set of M samples, only one sample kept. All the com-
putation involved in obtaining the other M − 1 samples is wasted.


4.1 Reducing the sampling rate of a signal 69

Figure 4.1: Reducing the sampling rate after the bandwidth of the signal is reduced.

Figure 4.2: FIR filter implemented in direct-form followed by a downsampler.

Efficient FIR filters stemming from decimation

For FIR filters, a better paradigm is easy to implement in which we never
actually compute any of the samples that we would throw away anyway.
To illustrate the idea consider an FIR filter implemented in direct-form
followed by a downsampler by M as shown in Figure 4.2. There is no point
computing all additions/multiplications indicated only to throw out the
result of most of them. Instead, we can downsample before the multipliers
so that only computations that are required are performed [3], [23], [24].
The idea is shown in Figure 4.3.
For any filter of length N + 1 (order N), even though the filter still
requires N + 1 multipliers, they are only used once every M samples.
In other words, the average number of multiplications per input sample
+
(MPIS) is NM 1 . That is of course a savings of a factor of M on average.
Unfortunately, for IIR filters, it is not straightforward to implement the



Figure 4.3: Efficient FIR decimation.

procedure shown due to the feedback. There is a way to implement deci-
mation efficiently with certain IIR designs using an allpass-based decom-
position as described in §8.1.2. We will discuss these efficient multirate IIR
filters later.

Determining the decimation factor So far we have described a filter that
reduces the bandwidth by a factor M as if it were an ideal filter. Of course,
with practical filters we have to worry about the finite amount of atten-
uation in the stopband, the distortion in the passband, and the non-zero
width transition band between passband and stopband. We’d like to ex-
plore how does using a non-ideal filter come into play when we decimate.
In order to answer this, consider what happens to the output signal
once it is downsampled by M (see also Appendix B). In the same way
that sampling an analog signal introduces spectral replicas of the analog
signal’s spectrum, downsampling a digital signal introduces replicas of its
spectrum. In the case of downsampling, the number of replicas introduced



Spectral replicas after decimation
0

−10
Magnitude spectrum of signal after decimation
−20 Spectral replica
Spectral replica
Magnitude (dB)

−30

−40

−50

−60

−70

−80
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

Figure 4.4: Spectrum of decimated signal with ωst = π/M along with spectral replicas.

is equal to M − 1.
We’d like to compare two different cases, one in which the cutoff fre-
quency is π/M, and another in which the stopband-edge frequency is
π/M. First let’s look at the latter case.
For illustration purposes, assume M = 3 and assume that the spectrum
of the signal to be filtered is flat and occupies the entire Nyquist interval.
This means that the spectrum of the output signal (prior to downsam-
pling) will take the shape of the filter. The conclusions we will reach are
valid for any value of M and any input signal (but the aliasing will be
lower than what we show if the input signal is already somewhat attenu-
ated in the stopband; we are showing the worst-case scenario). The spec-
trum of the decimated signal along with the replicas are shown in Figure
4.4. Aliasing invariably occurs because of the finite amount of stopband
attenuation. However, presumably we select the amount of stopband at-
tenuation so that the aliasing is tolerable. Notice the problem with using
filters with equiripple stopband when decimating. The energy in the stop-
band is large and it all aliases into the spectrum of the signal we are in-
terested in. Also notice that the usable bandwidth extends from zero to
the passband-edge frequency. The passband-edge frequency in this case is
given by π/M − Tw , where Tw is the transition-width.



Spectral replicas after decimation
0

−10

−20 Magnitude spectrum of signal after decimation
Spectral replica
Magnitude (dB)

−30 Spectral replica

−40

−50

−60

−70

−80
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

Figure 4.5: Spectrum of decimated signal with ωc = π/M along with spectral replicas.

Now let’s look at the case where the cutoff frequency is set to π/M.
Recall that for FIR filters the cutoff frequency is typically located at the
center of the transition band. The spectrum of the decimated signal along
with its corresponding replicas for this case are shown in Figure 4.5. Since
the cutoff frequency is in the middle of the transition band, the transition
bands alias into each other. This is not a problem since that region is dis-
torted by the filter anyway so that it should not contain any information
of interest. Notice that we have used a sloped stopband to reduce the to-
tal amount of aliasing that occurs. Of course this could have been done
in the previous case as well. Also, the usable passband extends now to
π/M − Tw /2 where once again Tw is the transition width. This means
that the usable passband is larger in this case than in the case ωst = π/M
by Tw /2. In order to have the same usable passband in both cases, we
would have to make transition width of the case ωst = π/M half as wide
as the transition width of the case ωc = π/M. Given that for FIR filters
the filter order grows inversely proportional to the transition width, this
would imply having to increase the filter order by a factor of about two in
order to obtain the same usable bandwidth with ωst = π/M.
While both selections, ωst = π/M, or ωc = π/M are valid. There are
clear advantages to using ωc = π/M. In practice, we usually have some



amount of excess bandwidth in a sampled signal so that the aliasing that
occurs in the transition band does not affect the application. For instance,
consider an audio signal. The most common sampling frequency is 44.1
kHz, but the maximum frequency we are interested in is 20 kHz. That
means there is 4.1/2 kHz available at each side of the spectrum for transi-
tion bands of filters. See [11] for more on this.
All this to say that usually the decimation factor M should still be se-
lected as the ratio ωc /π even when the non-ideal nature of practical FIR
filters is taken into consideration. This fits in well with the use of Nyquist
filters. Since an Mth-band Nyquist filter has a cutoff frequency of π/M,
it is well-suited for applications in which we want to reduce the sampling
rate by a factor of M.

Example 36 A 4th-band Nyquist filter reduces the bandwidth by a factor of 4.
We can simultaneously reduce the sampling rate by a factor of 4 by designing a
decimator Nyquist filter,

f = fdesign.decimator(4,'Nyquist',4,'TW,Ast',0.1,60);

Computational savings with FIR decimators We’d like to return to the
comparison between an FIR and an IIR filter that meet the same set of
specifications. In §2.2.6 we saw that for the specification set 4 an elliptic
IIR filter requires 20 multipliers and 20 MPIS to implement assuming a
direct-form II second-order sections implementation. An optimal equirip-
ple filter requires 642 multipliers if we do not take advantage of the sym-
metry in the filter coefficients. However, given that the cutoff frequency
is π/16, we can use the efficient decimation techniques described above
to implement the FIR filter with an average of 40.125 MPIS even without
taking advantage of the symmetric coefficients.∗

M = 16; % Decimation factor
Fc = 0.0625;
TW = 0.008;
Fp = Fc-TW/2;
Fst = Fc+TW/2;
∗ An efficient decimation structure that does take advantage of the symmetry in the co-
efficients is shown on page 77 of [24].



Ap = 1;
Ast = 80;
Hf = fdesign.decimator(M,'lowpass','Fp,Fst,Ap,Ast',...
Fp,Fst,Ap,Ast);
Hd = design(Hf,'equiripple');
cost(Hd)
Through the use of multirate techniques, we are within a factor of two
of the number of MPIS required to implement an IIR filter that meets the
specifications. However, we still have perfect linear phase and no stability
issues to worry about.
For the case just discussed, a Kaiser-window-designed∗ Nyquist fil-
ter would require more computations per input sample than a regular
lowpass filter. The main reason is the passband ripple. With the given
specifications, the Kaiser-window design will have a mere 0.0015986 dB of
passband ripple. The cost to pay for this vast over-satisfying of the speci-
fications is a computational cost of 73.6875 MPIS.
In Appendix C we study various approaches to filter design for these
same specifications. We will see that multistage/multirate designs and in
particular multistage Nyquist multirate designs will be the most efficient
of all approaches looked at for these specifications. This is true despite the
fact that such multistage Nyquist designs also will vastly over-satisfy the
passband ripple requirements.

Decimating by a factor of 2
When reducing the bandwidth/decimating by a factor of two, if the cutoff
frequency is set to π/2 (and transition-band aliasing is tolerated), FIR or
IIR halfband filters can be used to very efficiently implement the filter.
In the FIR case, the halfband decimator by 2 takes full advantage of the
fact that about half the filter coefficients are zero.

Example 37 Consider the following design:
f = fdesign.decimator(2,'halfband','TW,Ast',0.1,80);
h = design(f,'equiripple');
cost(h)
∗ Equiripple designs do not converge for these specifications given the Nyquist con-
straint on the impulse response.



Figure 4.6: Allpass-based IIR polyphase implementation of decimation by 2.

The halfband is already efficient in that for a filter of length 95, it only requires 49
multipliers. By reducing the sampling rate, a further gain factor of two is obtained
in computational cost. The number of MPIS is 24.5.

In the IIR case, efficient allpass-based polyphase implementations are
possible for halfband designs. The idea is two make use of the fact that
conceptually, decimation is lowpass filtering followed by downsampling.
By using (3.1) and the Noble identities, it is simple to obtain the implemen-
tation shown in Figure 4.6. Note that use of the Noble identities means that
the allpass filters are now given in terms of powers of z rather than of z2 .
Let’s look at the computational cost of using an IIR halfband filter for the
specifications from the previous example.

Example 38 For the same specifications contained in the variable f above, we
can design either an elliptic halfband decimator or a quasi linear phase halfband
decimator:

h1 = design(f,'ellip');
h2 = design(f,'iirlinphase');
cost(h1)
cost(h2)

As usual, the elliptic design is the most efficient, requiring only 6 multipliers and
3 MPIS. If approximate linear phase is desired, the iirlinphase design requires
19 multipliers and 9.5 MPIS.



Magnitude spectrum of highpass−filtered signal
0

−10

−20
Magnitude (dB)

−30

−40

−50

−60

−70

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8

Figure 4.7: Spectrum of highpass-filtered signal.

Decimating with highpass/bandpass filters
The motivation to reduce the sampling rate whenever the bandwidth of
a signal is reduced remains true regardless of the shape of the filter used
to reduce the bandwidth. If we use a highpass or a bandpass filter to re-
duce the bandwidth by an integer factor M, we would still want to reduce
the sampling frequency accordingly. The resulting signal after downsam-
pling is called the baseband equivalent of a bandpass signal. Any further
processing at baseband is less costly because we use fewer samples once
the sampling frequency is reduced. The bandpass signal and its baseband
equivalent contain the same information. We could for instance generate
an analog bandpass signal that corresponds to the digital bandpass signal
by starting from the baseband equivalent.
As long as the filters are FIR, we can continue to use the efficient imple-
mentation we showed since the direct-form structure can be used regard-
less of the response type of the filter. If a highpass halfband filter is used,
efficient polyphase IIR designs are a possibility as well.

Example 39 For example, consider using a highpass filter that reduces the band-
width by a factor M = 3. If we do not downsample, and the input to the filter
has a flat spectrum, the output signal will have the spectrum of the filter. Some-
thing like what is shown in Figure 4.7. If we downsample the filtered signal by a



Spectrum after decimation
0

−10

−20
Alias from spectral replicas
Magnitude (dB)

−30 Magnitude spectrum of signal after decimation

−40

−50

−60

−70

−80
−0.3 −0.2 −0.1 0 0.1 0.2 0.3

Figure 4.8: Spectrum of downsampled highpass-filtered signal.

factor of M, the resulting spectrum is moved to baseband as shown (along with
the aliasing from replicas) in Figure 4.8. The spectrum is shown relative to the
high sampling frequency, but we only show the portion of the Nyquist interval
that corresponds to the downsampled signal. For this reason, the interval plotted
is [−π/3, π/3]. Notice that in this case, some of the aliasing comes from the
replicas and some of it is “self-inflicted” so to speak. Nevertheless, as before, the
amount of aliasing is controlled by the stopband attenuation. In this example, no
single replica introduces more than -60 dB of aliasing in the usable passband.
Notice that the previous example amounts to removing the “white
space” that arises after highpass filtering. Since the energy in the signal
is negligible in this white space, we best not waste resources by process-
ing information corresponding to that band.
There are some restrictions as to where the bandpass that is retained
lies relative to the sampling rate [24]. Even if the filter were ideal, its re-
sponse needs to be such that it retains a bandwidth M times smaller than
the Nyquist interval and lying in one of the intervals
kπ/M < |ω | < (k + 1)π/M k = 0, 1, . . . , M − 1.
The cases k = 0 and k = M − 1 correspond to lowpass and highpass
filters respectively. These cases are thus never a problem. However, if the



Spectrum after decimation
0

−10

−20 Alias from spectral replicas
Magnitude spectrum of signal after decimation
Magnitude (dB)

−30

−40

−50

−60

−70

−80
−0.3 −0.2 −0.1 0 0.1 0.2 0.3

Figure 4.9: Spectrum of bandpass-filtered signal downsampled by a factor M = 3.

bandpass filter does not meet the restrictions just mentioned, we should
still try to downsample as much as possible.

Example 40 Suppose we are interested in retaining the band between 0.35π and
0.5167π. The bandwidth is reduced by a factor M = 6. However, the band-edges
are not between kπ/M and (k + 1)π/M for M = 4, 5, 6. The band-edges do lie
between π/3 and 2π/3, so if we design a bandpass filter we can at least decimate
by 3.

TW = 0.1/6; % Transition width
Fc1 = 0.35;
Fc2 = 0.5167;
Hf = fdesign.decimator(M,'bandpass','Fst1,Fp1,Fp2,Fst2,Ast1,Ap,Ast2',...
Fc1-TW/2,Fc1+TW/2,Fc2-TW/2,Fc2+TW/2,60,.2,60);

The spectrum of the bandpass-filtered signal after downsampling is shown in Fig-
ure 4.9. The frequency is relative to the high sampling rate prior to downsampling.
Since we downsampled by M = 3, the new Nyquist interval after downsampling



will be [−π/3, π/3]. Notice that we haven’t been able to remove all the white
space since we could only downsample by 3 even though the bandwidth was re-
duced by 6.

Decimation when working with Hertz
So far we have used normalized frequency in all our decimation designs. If
we’d like to specify the frequency specifications in Hertz, we should sim-
ply keep in mind that the sampling frequency that is used is the sampling
frequency of the input signal (prior to downsampling).

Example 41 Suppose we have a signal sampled at 30 kHz and we reduce its
bandwidth by a factor of 3. The band of interest extends from 0 Hz to 4250 Hz.
In order to remove redundancy, we reduce the sampling-rate by a factor of 3 as
well. We use a 3rd-band Nyquist filter, therefore the cutoff frequency is equal to
Fs /(2M ) = 5000 Hz. The transition band is set to 1500 Hz so that it extends
from 4250 Hz to 5750 Hz.

M = 3;
Band = 3;
Fs = 30e3;
Hf = fdesign.decimator(M,'Nyquist',Band,'TW,Ast',1500,65,Fs);
Hd = design(Hf,'kaiserwin');

4.1.2 Decimating by a non-integer factor
If the bandwidth reduction factor is not an integer, we would still like to re-
duce the sampling frequency accordingly. A simple approximation would
be to downsample the output by the largest integer that is not greater than
the bandwidth reduction factor. This is less than optimal however and if
the bandwidth reduction factor is less than two, there would be no effi-
ciency improvements even if we reduce the bandwidth to almost (but not
quite) half the original bandwidth.
A better solution would be to reduce the sampling rate by a non-integer
factor as well. This implies computing some samples between existing
samples as shown in Figure 4.10. Computing new samples in between
existing samples amounts to interpolation. We will return to discuss frac-
tional decimation in §4.4, but before we do so, we’ll look at interpolation.



1
Original samples
0.9
Underlying signal
0.8 Fractionally−decimated samples
0.7

0.6
Amplitude

0.5

0.4

0.3

0.2

0.1

0
30 35 40 45 50 55
Time (samples)

Figure 4.10: Illustration of fractional decimation.

4.2 Interpolation
Roughly speaking, interpolation consists of computing new samples be-
tween existing samples. In the context of signal processing, ideally we
interpolate by ﬁnding the desired points on the underlying continuous-
time analog signal that corresponds to the samples we have at hand. This
is done without actually converting the signal back to continuous time.
The process is referred to as sinc interpolation since an ideal lowpass ﬁlter
(with a sinc-shaped impulse response) is involved (see Appendix B).

4.2.1 Fractionally advancing/delaying a signal
Usually interpolation is presented in the context of increasing the sampling-
rate of a signal, but strictly speaking, this is not necessary. Consider the
situation shown in Figure 4.11. We’d like to compute the interpolated sam-
ples from the existing samples. For each existing sample we compute a
new sample, therefore the sampling rate remains the same.
The interpolated samples in Figure 4.11 are found by fractionally ad-
vancing the signal in time by a factor α. Equivalently, the samples can
be thought to be obtained by fractionally delaying the signal by a factor


4.2 Interpolation 81

0.5
Underlying analog signal
0.4
Sampled signal
0.3 Fractionally advanced signal

0.2

0.1
Amplitude

0

−0.1

−0.2

−0.3

−0.4

−0.5
30 32 34 36 38 40 42 44 46 48 50
Time

Figure 4.11: Fractional advance by a factor α = 0.4.

β = 1 − α. Both α and β are a fraction between zero and one. Thus, if x [n]
is our signal prior to interpolation, the signal consisting of interpolated
values is computed by passing the signal through a filter with transfer
function
Hfrac (z) = zα .
Of course advancing a signal in time is a non-causal operation. Since
the advance is less than one sample, we can make it causal by delaying
everything by one sample,

Hcausal = z−1 zα = zα−1 = z− β .
which shows the equivalence of a causal advance by a factor of α and a
delay (notice the negative sign) by a factor of β = 1 − α.
By taking the inverse Fourier transform, the impulse response of the
fractional-advance filter zα can be easily found to be
sin(π (n + α))
hfrac [n] = , −∞ < n < ∞ (4.1)
π (n + α)
Not surprisingly, the ideal fractional advance has an infinite impulse
response that is also infinitely non-causal. We will look at various FIR and



Figure 4.12: Conceptual diagram for increasing the sampling rate.

IIR approximations to this filter later, but for now we will look at how a
bank of fractional advance filters can be used to increase the sampling-rate
of a signal.

4.2.2 Increasing the sampling-rate of a signal
In Appendix B we have already looked at what increasing the sampling-
rate of a signal means in the frequency domain. In short, increasing the
sampling-rate is accomplished by lowpass filtering the signal with a digi-
tal filter operating at the same rate we wish to increase the sampling rate
to∗ . The key idea is to digitally produce a signal that would be exactly the
same as a signal we had obtained by sampling a band-limited continuous
time signal at the higher sampling rate.
Conceptually, increasing the sampling rate is accomplished as depicted
in Figure 4.12. The upsampler inserts L − 1 zeros between existing samples
in order to increase the sampling rate of the signal. This step in itself does
not modify the spectral content at all. All it does, is to change the Nyquist
interval to encompass a bandwidth that is now L times larger than before
upsampling. Because of this upsampling, the new Nyquist interval now
contains L − 1 spectral replicas of the baseband spectrum. In order to in-
terpolate between the original samples, replacing the zeros that have been
inserted with actual interpolated samples, it is necessary to lowpass filter
the upsampled signal in such a way that the L − 1 unwanted spectral repli-
cas are removed (see Appendix B). The filter H (z) performs such lowpass
filtering.
In practice, increasing the sampling rate is never performed as shown
in Figure 4.12. The reason is that inserting all those zeros means the low-
∗ Whenever we increase the sampling rate, we do not lowpass filter in order to reduce
bandwidth. On the contrary, we want to use all the information in the existing signal in
order to compute the new samples. The reason we lowpass filter is to remove spectral
replicas that are encompassed by the new Nyquist interval when the rate is increased.
See Appendix B for more on this.



0.5
Underlying analog signal
0.4
Original samples
0.3
Interpolated samples
0.2

0.1
Amplitude

0

−0.1

−0.2

−0.3

−0.4

−0.5
34 36 38 40 42 44 46
Time

Figure 4.13: Illustration of ideal band-limited interpolation in the time domain.

pass filter would be performing a lot of unnecessary computations when
its input is just a zero. Instead, a so-called polyphase approach is taken in
order to efficiently increase the sampling rate. The polyphase approach is
easiest to understand by looking at things in the time domain.
The situation in the time domain is depicted in Figure 4.13 assuming
we want to increase the rate by a factor L = 5. We have seen that in order
to produce each sample between existing samples, we need a fractional
advance filter. Looking at Figure 4.13, we can see that we need to take
every input sample x T [n] and produce 5 output samples { x T ′ [m]}, m =
5n + k, k = 0, . . . , 4 as follows (note that T = 0.5 and T ′ = T/5 = 0.1):

• x T ′ [5n] = x T [n]
1
• x T ′ [5n + 1] = x T [n + 5 ]
2
• x T ′ [5n + 2] = x T [n + 5 ]
3
• x T ′ [5n + 3] = x T [n + 5 ]
4
• x T ′ [5n + 4] = x T [n + 5 ]



Figure 4.14: Bank of filters used for interpolation. Each filter performs a fractional ad-
vance of the input signal and has a different phase. The overall implementation is re-
ferred to as a polyphase interpolation filter. For every input sample, we cycle through all
polyphase branches.

In general, the ideal interpolator consists of a bank of L filters, Hk (z), k =
0, . . . , L − 1 which will fractionally advance the input signal by a factor
k/L, k = 0, . . . , L − 1. The outputs of the filters are then interleaved (i.e.
only one filter needs to operate per high rate output sample) to produce
the interpolated signal. This is depicted in Figure 4.14
The L filters that comprise the filter bank are the fractional advance
filters,
Hk (z) = zk/L , k = 0, . . . , L − 1.
Evaluating on the unit circle, we have

Hk (e jω ) = e jωk/L , k = 0, . . . , L − 1

so that each filter Hk (e jω ) is allpass, i.e. Hk (e jω ) = 1 and has linear phase,
arg{ Hk (e jω )} = ωk/L. ∗
∗ The term polyphase stems from this derivation. Each filter in the filter bank has a
different phase. The interpolator filter consists of L polyphase parallel branches, each
branch tasked with computing one of the L interpolated outputs. The time-domain view
of the ideal interpolation filter thus has the polyphase structure built-in.



Herein lies the impossibility of designing these filters in an exact man-
ner. We cannot design them as FIR filters because no FIR filter can be
allpass (except for a pure delay). We cannot design them as IIR filters, be-
cause no stable IIR filter can have linear phase. However, it is clear how
we want to approximate the ideal interpolation filter bank.
FIR approximations will produce the exact linear phase, while approx-
imating an allpass response as best possible. On the other hand, IIR ap-
proximations will be exactly allpass, while trying to produce the required
phase.
It is enlightening∗ to realize that the filters comprising the filter bank
are the polyphase components of the ideal interpolation filter derived in
(1.2)!
Indeed, the impulse response of each fractional advance filter in the
filter bank is given by the inverse DTFT,
+
1 π sin π LnL k
hk [n] = e jωk/L e jωn dω = +
2π −π π LnL k
this corresponds to taking the transfer function corresponding to the ideal
lowpass filter (1.1) with the cutoff frequency set to π/L and rewriting it in
the form
HLP (z) = . . . + h[0] + z− L h[ L] + . . .
+ z −1 ( . . . + h [ 1 ] + z − L h [ L + 1 ] + . . . )
.
.
.
+ z−( L−1) (. . . + h[ L − 1] + z− L h[2L − 1] + . . .)
which can be written as
HLP (z) = H0 (z L ) + z−1 H1 (z L ) + . . . + z−( L−1) HL−1 (z L ).
The sub-filters Hk (z) are the polyphase components (each one is pre-
ceded by a different phase, z−k ) of the original transfer function. The im-
pulse responses of each sub-filter correspond to the L decimated sequences
of the ideal impulse response by again writing uniquely m = Ln + k, k =
0, . . . , L − 1 in (1.2).
∗ The reality is that this should be expected given that we had already determined that
the ideal interpolator was an ideal lowpass filter. Yet it is nice to see that things fit in
even though we have approach the derivation of the ideal interpolator from two different
perspectives.



4.2.3 Design of FIR interpolation filters
While interpolation filters are simply lowpass filters that can be designed
with the various techniques outlined previously, the polyphase filters that
compose the ideal interpolation filter give some insight on things to be
looking for when designing interpolation filters.
Consider an interpolation by a factor of L. The ideal L polyphase filters
will have a group-delay given by
k
− L, k = 0, . . . , L − 1

For simplicity, consider an FIR approximation to the ideal interpolation
filter where the order is of the form N = 2LM. Then each polyphase filter
will have order N/L = 2M.
Note that the ideal interpolation filter is infinitely non-causal. After fi-
nite length truncation, it is possible to make the approximation causal by
delaying by half the filter order, N/2. However, because we will imple-
ment in efficient polyphase form, we can make each polyphase component
causal by delaying it by M samples.
The delay will mean the introduction of a phase component in the re-
sponse of each polyphase component. So that instead of approximating
the ideal fractional advance e jωk/L the polyphase components will approx-
imate e jω (k/L− M) . The group-delay will consequently be of the form
dφ(ω ) dω (k/L − M )
− =− = M − k/L.
dω dω
A problem that arises is that even though the FIR approximation to
the ideal interpolation filter is symmetric and thus has linear phase, the
polyphase components are not necessarily symmetric and thus will not
necessarily have exact linear phase. However, for each non symmetric
polyphase filter, there is a mirror image polyphase filter which will have
the exact same magnitude response with a mirror image group-delay that
will compensate any phase distortion.

Example 42 We design a Nyquist filter that interpolates by a factor L = 4. The
Nyquist filter operates at the high rate (four times the input rate). It is a 4th-band
filter with a cutoff frequency given by π/4. This filter will remove 3 spectral repli-
cas encompassed by the augmented Nyquist interval once the signal is upsampled.
The filter is implement in efficient polyphase manner.




0

−20
Magnitude (dB)

−40

−60

−80

−100

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Figure 4.15: FIR Nyquist interpolate by 4 filter designed with a Kaiser window. The
filter removes 3 spectral replicas and operates at four times the input rate.

L = 4; % Interpolation factor
Band = 4; % Band for Nyquist design
Hf = fdesign.interpolator(L,'Nyquist',Band,'TW,Ast',0.1,80);
Hint = design(Hf,'kaiserwin');
The resulting Nyquist filter is shown in Figure 4.15.

For the design from the previous example, it is worth taking a look at
the magnitude response and group-delay of the polyphase components of
the resulting FIR filter. The magnitude response is shown in Figure 4.16
while the group-delay is shown in Figure 4.17. The magnitude response
reveals that only one of the polyphase filters is a perfect allpass, while
the others are approximations that fade-off at high frequency. The magni-
tude responses of the 2nd and 4th polyphase sub-filters are identical. The
group-delays show a fractional advance of 0, 0.25, 0.5, and 0.75 for the
four polyphase sub-filters. This advance is relative to a nominal delay of
M = 13 samples. The filter length is 2LM + 1 = 105. Note that two of the
polyphase components do not have perfectly flat group delays. However,
the nonlinear shape of one compensates for the other so that overall the
interpolation filter has linear phase.




1

0

−1
Magnitude (dB)

−2

−3

−4 Filter #1: Polyphase(1)
Filter #1: Polyphase(2)
−5
−6 Filter #1: Polyphase(4)

−7
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Figure 4.16: Magnitude response of polyphase components for a Nyquist FIR interpolate-
by-four ﬁlter.

Group Delay

16 Filter #1: Polyphase(1)
15


13

12

11

10

9

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Figure 4.17: Group-delay of polyphase components for a Nyquist FIR interpolate-by-four
ﬁlter.



Notice that the polyphase components can be obtained by taking every
Lth sample of the lowpass impulse response as stated above. For example,
it is easy to verify this for the second polyphase:

p = polyphase(Hint);
isequal(p(2,1:end-1),Hint.Numerator(2:4:end)) % Returns true

In general, for all Nyquist designs, one of the polyphase components
will have perfectly flat magnitude response and group-delay. This polyphase
component is a simple delay which provides a fractional advance of zero.
The presence of this polyphase components ensures that the input sam-
ples are ”passed undisturbed” to the output (they are simply delayed).
In general, a non-ideal non-Nyquist lowpass filter will not have a perfect
delay as one of its polyphase branches and therefore will alter somewhat
the input samples. However, this is usually not a major issue in terms of
distortion. Nevertheless, Nyquist filters are often used because of their
reduced implementation cost given that one of the polyphase branches is
simply a delay.

4.2.4 Design of IIR halfband interpolators
As with decimation, the use of halfband filters for interpolation is enticing
given their efficiency. In the case of interpolation by a factor L = 2, FIR
halfband filters (being Nyquist filters) will have one of their polyphase
branches given simply by a delay.
IIR filters will not necessarily have one of their branches given by a
simple delay because we do not use their impulse response directly to im-
plement them. However, quasi-linear phase IIR filters are implemented in
such a way that one of their polyphase branches is a pure delay. Neverthe-
less, elliptic halfband interpolation filters that meet the same specifications
are still more efficient even though they do not have a simple delay as one
of its polyphase branches (and therefore, as stated in Chapter 4, elliptic
halfband filters are not Nyquist and as a result, they modify the input
samples when interpolating).
The allpass-based IIR polyphase structure used to implement interpo-
lation by two is shown in Figure 4.18. Note that because we have used
the Noble identities, the transfer functions of the allpass filters in (3.1) are
given as powers of z rather than powers of z2 .



Figure 4.18: Allpass-based IIR polyphase implementation of interpolation by 2.

Example 43 Design elliptic and quasi linear-phase IIR halfband interpolate-by-
two filters for the given specifications.

Hf = fdesign.interpolator(2,'halfband','TW,Ast',0.08,55);
Hlin = design(Hf,'iirlinphase');
Hellip = design(Hf,'ellip');
cost(Hlin)
cost(Hellip)

The two polyphase branches for either design are perfect allpass filters. The filters
deviate from the ideal interpolation filter in their phase response. The group-delay
of the polyphase branches for the quasi linear-phase IIR design is shown in Figure
4.19. Note that one of the branches is a pure delay and therefore has perfectly flat
group-delay. The group-delay of the polyphase sub-filters for the elliptic design is
shown in Figure 4.20. Note how neither of the polyphase components has a flat
group-delay in this case.

4.2.5 Design of interpolators when working with Hertz
So far we have presented the design of interpolation filters in the context
of normalized frequency. If working with Hertz, the key point to keep in
mind is that when increasing the sampling rate we should not reduce the
bandwidth of the signal. Once again, the interpolation filter is used not to
reduce bandwidth, but to remove spectral replicas.

Example 44 Suppose we have an audio signal sampled at 48 kHz and we want
to increase the sampling frequency to 96 kHz. The band of interest extends to 20



Group Delay
30
28

26

24

22

20

18

16

14

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Figure 4.19: Group-delay of polyphase components for a quasi linear-phase IIR halfband
interpolate-by-two filter.

Group Delay
30

20

15

10

5

0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Figure 4.20: Group-delay of polyphase components for an elliptic IIR halfband
interpolate-by-two filter.

kHz. Since we want to increase the sampling frequency by a factor of 2, we can
use a (FIR or IIR) halfband filter,




0

−20
Magnitude (dB)

−40

−60

−80

−100

0 5 10 15 20 25 30 35 40 45
Frequency (kHz)

Figure 4.21: Magnitude response of halfband filter used to increase the sampling rate by
2.

f = fdesign.interpolator(2,'halfband','TW,Ast',8000,96,96000);
H = design(f,'equiripple');

The magnitude response of the filter can be seen in Figure 4.21. Note that the
passband extends to 20 kHz as desired. The cutoff frequency for the filter is 24
kHz. The spectral replica of the original audio signal centered around 48 kHz will
be removed by the halfband filter.

4.3 Increasing the sampling rate by a fractional
factor
So far we have discussed increasing the sampling rate by an integer factor.
In some cases, we want to increase the sampling rate by a fractional factor.
This can be done trivially for rational factors.
Suppose we want to increase the rate by a factor of L/M with L > M.
A trivial way of doing this is to perform a full interpolation by a factor of
L and then discarding the samples we don’t want, that is we keep one out
of every M samples after interpolating.


4.3 Increasing the sampling rate by a fractional factor 93

Figure 4.22: Conceptual configuration for increasing the sampling rate of a signal by a
fractional factor. L > M.

Figure 4.22 illustrates the idea. The role of the upsampler+filter combi-
nation is the same as for the case when we increase the rate by an integer
factor. Once we have increased the rate, we discard samples we don’t
need, keeping only the ones required for the new rate.
While the procedure we have just described is trivial, it is inefficient
since many of the samples that have been computed via interpolation are
subsequently discarded. Instead, to implement fractional interpolation ef-
ficiently we compute only the samples we intend to keep.
Specifically, if the polyphase filters are H0 (z), H1 (z), . . . , HL−1 (z), in-
stead of using H0 (z) to compute the first output, H1 (z) to compute the
second output and so forth, we use H0 (z) for the first output, we skip
M − 1 polyphase filters and use H M (z) for the second output and so forth.
As an example, if L = 3 and M = 2, the sequence of polyphase filters
used are H0 (z), then skip H1 (z), then use H2 (z), then, for the next input,
skip H0 (z), then use H1 (z), then skip H2 (z), and then start again by using
H0 (z).
The idea in general is shown in Figure 4.23. The structure resembles
that of interpolation by an integer factor except that only the branches
that are going to produce an output we will keep are used for every input
sample.

Example 45 We will work with Hertz to illustrate the fact that when we inter-
polate (by either an integer or a fractional factor) we never reduce the bandwidth
of the input signal.
Suppose we have a signal sampled at 500 Hz. The band of interest for the
signal is from 0 to 200 Hz, i.e. a transition width of 100 Hz has been allocated.
Say we want to increase the sampling rate to 1600 Hz. We choose to use a 16th-
band Nyquist filter. The transition width is set to 100 Hz in order to not disturb
information contained in the band of interest.

L = 16;
M = 5;



Figure 4.23: Bank of filters used for fractional interpolation. Unlike interpolation by an
integer factor, not all polyphase branches are used for each input sample.

Band = L;
Fs = 500*L; % High sampling rate
TW = 100;
Ast = 80;
Hf = fdesign.rsrc(L,M,'Nyquist',Band,'TW,Ast',TW,Ast,Fs);
Hrsrc = design(Hf,'kaiserwin');

Note that the sampling frequency we had to set for the design was that corre-
sponding to full interpolation by 16, i.e. 16 × 500. This is because, as we have
explained, the filter is designed as a regular interpolate-by-16 filter, but is im-
plemented efficiently so that no sample to be discarded is actually computed. The
magnitude response of the filter is shown in Figure 4.24. The passband of the filter
extends to 200 Hz so that the band of interest is left undisturbed. The resulting
sampling frequency is 1600. However, this filter removes 15 spectral replicas as
if the sampling frequency was being increased to 8000. The downsampling by 5
produces 4 spectral replicas centered around multiples of the resulting sampling
frequency, i.e. 1600.


4.4 Fractional decimation 95


20

0
Magnitude (dB)

−20

−40

−60

−80

−100

0 0.5 1 1.5 2 2.5 3 3.5
Frequency (kHz)

Figure 4.24: Magnitude response of filter used for fractional interpolation. The passband
extends to 200 Hz so that the band of interest is left untouched.

4.4 Fractional decimation
We now return to the problem of decimating by a fractional factor. For
now, we will restrict the discussion to rational values. As we mentioned
before, in order to perform fractional decimation, it is necessary to com-
pute samples lying between existing samples. We now know how to do
so by using interpolation.
It turns out that the implementation we have just described for frac-
tional interpolation can be used for fractional decimation as well. The
only difference is that L < M in this case. (Conceptually, Figure 4.22 still
applies with the caveat that now L < M.)
In terms of designing the filter however, we need to be careful with
where we set the cutoff frequency since it is no longer true that we want
to keep the full spectrum of the input signal.
Indeed, fractional decimation is appropriate when the bandwidth of a
signal is being reduced by a fractional factor. We use fractional decimation
to reduce the rate by a corresponding factor.
Suppose for instance that we have a signal sampled at 1000 Hz. We
are interested in keeping the information contained from 0 to 280 Hz. We
decide to both lowpass filter the signal and reduce the bandwidth by a



factor of 7/10.

L = 7;
M = 10;
Band = M;
Fs = 1000*L; % High sampling rate
TW = 140;
Ast = 80;
Hf = fdesign.rsrc(L,M,'Nyquist',Band,'TW,Ast',TW,Ast,Fs);
Hrsrc = design(Hf,'kaiserwin');

Note that once again, the sampling frequency set for the filter designed
is set as if full interpolation were to be implemented. The choice of the
band however, is made as M rather than L (compare to the previous exam-
ple) so that the resulting filter reduces the bandwidth by a factor of L/M.
The transition width is chosen so that the passband of the filter extends to
the desired 280 Hz. The computation is simple, f c − TW/2, where f c is the
cutoff frequency which is given by f s /(2Band) = 350 Hz. Subtracting half
the transition width, i.e. 70, means the passband of the filter will extend
to 280 Hz as intended. The passband details for the filter can be seen in
Figure 4.25.
If the input signal had a flat spectrum occupying the full Nyquist in-
terval (worst case), after filtering and fractional decimation, the spectrum
of the signal would take the shape of the filter. Its spectral replicas would
be centered around multiples of the new sampling frequency, i.e. 700 Hz.
The baseband spectrum along with the first positive two spectral replicas
is shown in Figure 4.26. Note that as before, we have allowed for transition
band overlap, but the baseband of interest presents no significant aliasing.

In this chapter we have presented design considerations for multirate fil-
ters. In particular, it is important to understand how the cutoff frequency
for filters should be set commensurate to bandwidth reduction in the case
of decreasing the sampling rate. Also, we presented the rather obvious
fact that when increasing the sampling rate of a signal, we do not wish to
reduce its bandwidth.



16.9028

16.9026

16.9024
Magnitude (dB)

16.9022

16.902

16.9018

16.9016

16.9014

16.9012

0 0.05 0.1 0.15 0.2 0.25 0.3
Frequency (kHz)

Figure 4.25: Passband details of filter designed for fractional decimation. The passband
extends to 280 Hz so that the band of interest is left untouched.

Lowpass filters are the most common filters for multirate applications.
However, highpass and bandpass filters can also be used for both decreas-
ing or increasing the sampling rate of a signal. Although we did not touch
upon the case when increasing the sampling rate, using a bandpass rather
than a lowpass for such case is simply a matter of choosing which spectral
replicas should be removed.
We have presented Nyquist filters as the preferred type of lowpass fil-
ters for multirate applications, however any lowpass filter in principle can
be used.
When using Nyquist filters, equiripple FIR filters are not always the
best choice. Kaiser window designs may have a smaller passband ripple
along with increasing stopband attenuation that may be desirable for re-
moval of spectral replicas/aliasing attenuation.
IIR halfband filters have presented as an extremely efficient way of im-
plementing decimate/interpolate by two filters.
In the next chapter we will see that multirate Nyquist filters have the
interesting property that cascades of such filters remain Nyquist overall.
As such, we will see that to perform efficient multirate filtering, multistage
designs can be a very attractive solution. In particular, cascading efficient



20

0
Baseband spectrum
First replica
−20 Second replica
Magnitude (dB)

−40

−60

−80

−100
0 200 400 600 800 1000 1200 1400 1600 1800 2000
Frequency (Hz)

Figure 4.26: Baseband spectrum and two positives spectral replicas of fractionally dec-
imated signal. The input signal is assumed to have a ﬂat spectrum which occupies the
entire Nyquist interval.

halfband ﬁlters may be a very desirable way of implementing changes in
sampling rate by a factor given by a power of two.


Chapter 5

Multistage/Multirate Filter
Design

Overview
Generally, given a fixed passband ripple/stopband attenuation, the nar-
rower the transition band of a filter, the more expensive it is to implement.
A very effective way of improving the efficiency of filter designs is to use
several stages connected in cascade (series). The idea is that one stage
addresses the narrow transition band but manages to do so without re-
quiring a high implementation cost, while subsequent stages make-up for
compromises that have to be taken in order for the first stage to be effi-
cient (usually this means that subsequent stages remove remaining spec-
tral replicas).
In this chapter we start by discussing the so-called interpolated FIR
(IFIR) filter design method. This methods breaks down the design into
two stages. It uses upsampling of an impulse response in order to achieve
a narrow transition band while only adding zeros to the impulse response
(therefore not increasing the complexity). The upsampling introduces spec-
tral replicas that are removed by the second stage filter which is a lowpass
filter with less stringent transition bandwidth requirements.
We will show that multirate implementations of multistage designs are
the most effective way of implementing these filters. The idea follows
upon the notion we have already discussed in Chapter 4 of reducing the
sampling rate whenever the bandwidth of a signal is reduced.

100 Multistage/Multirate Filter Design

We then extend the multistage implementation to more than two stages.
Determining the optimal number of stages (and the decimation factor of
each stage) in order to most efficiently implement a filter for a given set
of specifications is a hard manual task. Fortunately, we show tools which
perform these designs automatically for us.
As mentioned in Chapter 4, Nyquist filters should be the preferred de-
signs used for multirate applications. The use of Nyquist filters in mul-
tistage implementations will be explored. The interesting property that
Nyquist filters retain the Nyquist property when implemented in a multi-
rate/multistage fashion is discussed. We will show that these designs are
very efficient.
A particular case of Nyquist filters which are very efficient for multi-
rate/multistage applications are halfband filters. We will see how we can
obtain very efficient designs using halfband filters to implement some (or
all) of the stages in a multistage design. Moreover, we can use IIR half-
band multirate filters implemented in polyphase form in order to increase
the efficiency of designs even further.
Finally, we show that efficient multistage implementations apply to in-
terpolation as well as decimation. Multistage interpolation implementa-
tions will typically be mirror images of multistage decimation. Whereas
with multistage decimation the simplest filter always should be the first
stage of the cascade (operating at the highest sampling frequency), with
multistage interpolation it is the last stage that contains the simplest filter
(since in this case it is the last stage that operates at the highest sampling
frequency).

5.1 Interpolated FIR (IFIR) designs
For any given FIR design algorithm, if the peak ripple specifications re-
main the same, the filter order required to meet a given specifications set
is inversely proportional to the transition width allowed.
When the transition width is small, such as in the Specifications Set 3,
the filter order required may be quite large. This is one of the primary
disadvantages of FIR filters. We have already seen that relaxing the linear
phase requirement results in a significant savings in the number of filter
coefficients.
The so-called interpolated FIR (IFIR) approach [25],[26],[27] yields lin-


5.1 Interpolated FIR (IFIR) designs 101

Figure 5.1: The IFIR implementation. An upsampled filter is cascaded with an image
suppressor filter to attain an overall design with a reduced computational cost.

ear phase FIR filters that can meet the given specifications with a reduced
number of multipliers.
The idea is rather simple. Since the length of the filter grows as the
transition width shrinks, we don’t design a filter for a given (small) tran-
sition width. Rather, we design a filter for a multiple L of the transition
width. This filter will have a significantly smaller length than a direct
design for the original (small) transition width. Then, we upsample the im-
pulse response by a factor equal to the multiple of the transition width, L.
Upsampling will cause the designed filter to compress, meeting the origi-
nal specifications without introducing extra multipliers (it only introduces
zeros, resulting in a larger delay). The price to pay is the appearance of
spectral replicas of the desired filter response within the Nyquist interval.
These replicas must be removed by a second filter (called in this context
the interpolation filter or image suppressor filter) that is cascaded with the
original to obtain the desired overall response. Although this extra filter
introduces additional multipliers, it is possible in many cases to still have
overall computational savings relative to conventional designs. The im-
plementation is shown in Figure 5.1.
The idea is depicted by example in Figure 5.2 for the case of an upsam-
pling factor of 3. The “relaxed” design is approximately of one third the
length of the desired design, if the latter were to be designed directly. The
upsampled design has the same transition width as the desired design.
All that is left is to remove the spectral replica introduced by upsampling.
This is the job of the image suppressor filter.
As an example of the computational cost savings, consider once again
the design Specifications Set 3. The number of multipliers required for a
single linear phase design was 263. An IFIR design can attain the same
specs with 127 multipliers when using an upsampling factor of 6:



1.5
1 Desired design

0.5
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
1.5
1 Relaxed design

0.5
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
1.5
image supressor Upsampled design
1
0.5
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
1.5
1 Resulting design

0.5
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 5.2: Illustration of the IFIR design paradigm. Two filters are used to attain strin-
gent transition width specifications with reduced total multiplier count when compared
to a single filter design.

Hifir = design(Hf,'ifir','UpsamplingFactor',6);
cost(Hifir)

The response of the upsampled filter and the image suppressor filter is
shown in Figure 5.3. The overall response, compared to a single linear
phase equiripple design is shown in Figure 5.4.

5.1.1 Further IFIR optimizations
A drawback in the IFIR design is that the passband ripples of the two fil-
ters are combined in a disorderly fashion. In the worst case scenario, they




0 Image supressor filter
Upsampled filter
−10

−20
Magnitude (dB)

−30

−40

−50

−60

−70

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Figure 5.3: Magnitude response of the upsampled filter and the image suppressor filter
in an IFIR design.

can add up, requiring the design to ensure that the sum of the two peak
passband ripples does not exceed the original set of specifications. Close
inspection of the passband of the overall design in the previous example,
shown in Figure 5.5, reveals a rather chaotic behavior (but certainly within
spec.) of the ripple.
Further optimized designs, [4], [28], attain a much cleaner passband
behavior by jointly optimizing the design of the two filters to work better
together. This results in a filter that can meet the specifications set with an
even further reduction in the number of multipliers. The savings are espe-
cially significant for the image suppressor filter, which is greatly simplified
by this joint optimization.
Utilizing this joint optimization, the Specifications Set 3 can be met
with only 74 multipliers, once again for an upsampling factor of 6.
Hifir = design(Hf,'ifir','UpsamplingFactor',6,...
'JointOptimization',true);
cost(Hifir)
The manner in which the two filters work together is best described




0 IFIR design
Conventional equiripple design
−10

−20
Magnitude (dB)

−30

−40

−50

−60

−70

−80
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Figure 5.4: Overall magnitude response of an IFIR design and a conventional equiripple
design. The IFIR implementation requires 127 multipliers vs. 263 for the conventional
implementation.

by looking at their magnitude responses, shown in Figure 5.6. By pre-
compensating for a severe “droop” in the image suppressor filter, a flat
passband can be achieved with dramatic savings in the number of multi-
pliers required for the image suppressor filter. Out of the 74 multipliers
required, 29 are for the image suppressor filter and 45 for the upsampled
filter. By contrast, in the previous IFIR design, 78 of the 127 multipliers
correspond to the image suppressor filter, while 49 correspond to the up-
sampled filter.
The passband details of the overall design show a nice equiripple be-
havior, hinting at a much better optimized design. The passband details
are shown in Figure 5.7.
The reader interested in further modifications/improvements to IFIR
filters is referred to [29].

5.1.2 Multirate implementation of IFIR design
When designing an IFIR filter, the upsampling factor L used must be such
that the (normalized) stopband-edge frequency ωs satisfies Lωs < π. This



0.08

0.06

0.04
Magnitude (dB)

0.02

0

−0.02

−0.04

−0.06

−0.08
0 0.02 0.04 0.06 0.08 0.1 0.12

Figure 5.5: Passband details of an IFIR design revealing a rather chaotic behavior of the
ripple.

implies that the bandwidth of the output signal would be reduced by a
factor of L.
It is convenient from a computational cost perspective to reduce the
sampling frequency of the filtered signal, since at that point the Nyquist
criterion is being unnecessarily over-satisfied. Subsequent processing of
the filtered signal without reducing its sampling rate would incur in un-
necessary (and expensive) redundant processing of information.
The idea is to downsample the filtered signal by a factor of L to match
the reduction in bandwidth due to filtering. If we denote by I (z) the image
suppressor filter and by U (z L ) the upsampled filter, we would have a cas-
cade of these two filters and a downsampler as shown in Figure 5.8. Using
the Noble identities, we can “commute” the downsampler and U (z L ) to
obtain the implementation shown in Figure 5.9. The combination of I (z)
and the downsampler form a decimator which can be implemented effi-
ciently in polyphase form.
Example 46 Consider the same specifications as before, but now casted as a dec-
imation filter design:
Hf = fdesign.decimator(6,'Lowpass',...




10 Image supressor filter
Upsampled filter
0

−10
Magnitude (dB)

−20

−30

−40

−50

−60

−70

−80

−90
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Figure 5.6: Magnitude response of the upsampled filter and the image suppressor filter
in an optimized IFIR design. The two filters are jointly optimized in the design to achieve
a specifications set with a reduced number of multipliers.

'Fp,Fst,Ap,Ast',.12,.14,.175,60);
Hifir = design(Hf,'ifir','UpsamplingFactor',6,...
'JointOptimization',true);
cost(Hifir)

Looking at the results from the cost function, we can see that while the num-
ber of multipliers remains 74 (the same as for a single-rate design), the number
of multiplications per input sample has been reduced substantially from 74 to
12.333. The number 12.333 is computed as follows. Of the 29 multipliers used
for the image suppressor filter, because of its efficient decimating implementa-
tion, only 29/6 multiplications are performed on average per input sample. Be-
cause of decimation, only one out of 6 input samples (to the entire cascade) ever
reaches U (z), so per input sample, the total number of multiplications on average
is 29/6 + 45/6 = 12.333.



0.1

0.08

0.06

0.04
Magnitude (dB)

0.02

0

−0.02

−0.04

−0.06

−0.08

−0.1
0 0.02 0.04 0.06 0.08 0.1 0.12

Figure 5.7: Passband details of an optimized IFIR design. The optimized design exhibits
nice equiripple behavior.

Figure 5.8: Cascading an IFIR implementation with a downsampler.

Figure 5.9: Interchange of the downsampler and the upsampled ﬁlter using the Noble
identities.



5.2 Multistage/Multirate Designs
The procedure outlined in the previous section can be generalized to mul-
tiple stages. There are a couple of ways of thinking about the philosophy
behind this. Keeping in mind that the image suppressor filter I (z) is noth-
ing but a lowpass filter, as we increase the upsampling factor L, so that
the upsampled filter U (z) becomes simpler, the order required by I (z) to
remove the spectral replicas as they come closer to each other increases as
well. In other words, decreasing the complexity and computational cost
of U (z) results in an increased filter order for I (z). If the specifications
for I (z) become too stringent, the cost of implementing this filter may out-
weigh the benefits of the savings afforded to U (z). In such a situation, one
may consider using an IFIR approach to implement I (z) itself. After all,
the original idea behind the IFIR approach is to implement a lowpass fil-
ter as a cascade of two filters, U (z) and I (z). Since I (z) is a lowpass filter
itself, why not apply the IFIR approach recursively and break up I (z) into
a cascade of two filters. Of course, of these two new filters, one of them is
also a lowpass image suppressor filter which can in turn be implemented
as a cascade of two filters and so on.
In general this line of thinking leads us to implementing a filter in mul-
tiple stages. The multirate implementation yields a multirate-multistage
design which lowpass filters data while decreasing the sampling rate at
each stage for maximum computational efficiency. Determining the op-
timal number of stages, in the sense that the computational cost is mini-
mized can be a difficult task. Fortunately, the Filter Design Toolbox pro-
vides tools to automate such designs. We will look at such tools in a mo-
ment, but before that, let’s look at a different line of thought that also re-
sults in multistage/multirate filter designs.
We know that the filter order is increased as the transition width of a
filter is decreased. When we talk about decreasing the transition width,
we are usually referring to the width in terms of normalized frequency.
In terms of absolute frequency, one could ask what is a narrow transition
width: 1 kHz? 100 Hz? maybe 10 Hz? The answer is: it depends. A tran-
sition width is narrow relative to the sampling frequency that is involved.
For example 10 Hz is probably quite narrow relative to a sampling fre-
quency in the gigahertz range, but it is not narrow relative to a sampling
frequency of say 50 Hz. One should assess whether a transition region is
narrow by considering the normalized transition width ∆ f / f s rather than


5.2 Multistage/Multirate Designs 109

the absolute transition width ∆ f .
In many applications, we cannot change the transition width required
for a filter design just because it makes the filter order too large. The ap-
plication sets the requirement for the transition width and this is typically
something that cannot be changed. What we can do however, is change
the sampling frequency so that the transition width is not narrow relative
to such sampling frequency. This is another way of looking at the idea
behind multistage/multirate designs. If a very narrow transition width is
required relative to the sampling frequency involved, the multistage ap-
proach is to not try to achieve the narrow transition in one shot. Instead,
we lower the sampling rate in stages by using simple lowpass filters un-
til we achieve a rate at which the required transition width is not large
relative to such rate. Then we design a filter that provides the required
transition.
The computational savings come from two factors. On the one hand,
the normalized transition width ∆ f / f s will be much larger since f s is
smaller, resulting in a filter of lesser order. On the other hand, because the
sampling rate is lower at this point, the number of input samples that have
to be processed by the filter is much smaller than if we were handling the
original large sampling rate. Of course there are other lowpass filters we
need to implement that handle the full high sampling rate and the large
number of input samples associated with it. However, these other filter
are implemented with large transition widths, so that their order is low
thus requiring a small number of operations per input sample.
In terms of the frequency response, the simple filters operating at high
sampling rates are used to remove spectral replicas from the filters oper-
ating at lower sampling rates downstream. The concepts are easily illus-
trated through an example.

Example 47 Consider the design of a lowpass filter with the following specifica-
tions:


1. Cutoff frequency: 550 Hz

2. Transition width: 100 Hz





5. Sampling frequency: 10 kHz

The specifications imply that the band of interest extends from 0 to 500 Hz.
Since the bandwidth is being reduced by about a factor of 8, we decide to design a
decimation filter with a decimation factor of 8 to meet the specifications and reduce
the sampling-rate after filtering accordingly.
If we were to design a single-stage equiripple decimator, the design would
take 343 multipliers and about 48 multiplications per input sample. In order to
design a general multistage/multirate filter and allow for the tools to determine
the optimal number of stages we use the following commands:

'Fp,Fst,Ap,Ast',500,600,.1,80,1e4);
Hmulti = design(Hf,'multistage');
cost(Hmulti)

The number of multipliers required is 114. The number of MPIS is 18.875.
By inspecting the resulting design, we can see that a 3-stage filter has resulted.
In this example, each stage has a decimation factor of 2, which accounts for the
overall decimation factor of 8 we required.∗
A plot of the magnitude response of each stage helps understand the behavior
of the design and the source for the computational savings. This is shown in Fig-
ure 5.10. The simplest filter of all operates at the highest rate as should usually
be the case. This first stage has a very wide transition band starting at 500 Hz
(therefore not affecting the band of interest) and ending at about 4450 Hz. The
second stage operates at 5 kHz given the decimation by two of the first stage. Its
normalized transition width is also quite wide, given its sampling frequency. The
job of this filter is to eliminate spectral replicas of the third stage centered around
odd multiples of the third stage’s sampling frequency. Because the second stage
has replicas centered about multiples of 5 kHz, it cannot eliminate the spectral
replicas of the third stage centered around even multiples of the third stage’s sam-
pling frequency. The first stage takes care of those. Notice that only the last stage
has the narrow 100 Hz transition width. By that time, the sampling frequency has
∗ For this example, a joint-optimized IFIR design is more efficient than a general mul-
tistage design. This can happen in some cases, due to the highly optimized interaction
between the two stages as previously discussed. This joint-optimization is not carried
out for general multistage designs. This is an important thing to keep in mind.


5.2 Multistage/Multirate Designs 111

0

−10
Stage one
−20 Stage two
−30 Stage three
Magnitude (dB)

−40

−50

−60

−70

−80

−90

−100
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
Frequency (kHz)

Figure 5.10: Multistage design of decimation filter for Specifications Set 6.

been reduced by a factor of 4, so the normalized transition width is 4 times larger
(100/2500 vs. 100/10000) than it would have been if we had done a single-stage
design. This accounts for most of the computational savings.
The overall magnitude dB response of the three-stage filter is obtained by
adding the responses of each stage. This is shown in Figure 5.11.

5.2.1 Setting the number of stages
If so desired, the number of stages can be manually controlled up to a cer-
tain point. However, note that certain restrictions apply. The decimation
factor must be a number that can be factored into as many factors as the
number of stages. In some cases, it is advisable to manually try a different
number of stages than the one automatically provided by the tools since
similar computational costs may be found for different number of stages
and other things (such as implementation complexity) may factor in to the
decision of which design to use.

Example 48 We can use the Nstages option to control the number of stages in
a design. Consider the following two designs for the same design specifications:



0

−20

−40
Magnitude (dB)

−60

−80

−100

−120

−140

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
Frequency (kHz)

Figure 5.11: Overall magnitude response of multistage design for Specifications Set 6.

'Fp,Fst,Ap,Ast',.05,.06,.5,70);
Hmulti = design(Hf,'multistage');
Hmulti2 = design(Hf,'multistage','Nstages',3);

The number of stages that is determined automatically for Hmulti is 4. How-
ever, we force Hmulti2 to have 3 stages. The computational cost of both designs
are quite comparable. However, the complexity of implementing 3 stages in hard-
ware may be less than that of implementing 4 stages. This could be the deciding
factor between one design and the other.

5.3 Multistage/Multirate Nyquist filters
Multirate Nyquist filters have the interesting property that if you cascade
two or more of them together (all decimators or all interpolators) the re-
sulting multistage/multirate filter is still a Nyquist filter [30]. This fact
allows us to extend the reasoning behind the use of multistage/multirate
designs to Nyquist designs.


5.3 Multistage/Multirate Nyquist filters 113

We already know that Nyquist filters are well-suited for (either deci-
mation or interpolation) multirate applications. The ability to break down
a multirate Nyquist design into several multirate Nyquist stages provides
very efficient designs. In many cases, halfband filters are used for the in-
dividual stages that make up the multistage design. Whenever a halfband
filter is used for a stage, there is the possibility to use an extremely efficient
IIR multirate halfband design as long as we allow for IIR filters as part of
the design.
Example 49 As an example, consider the design of a Nyquist filter that deci-
mates by 8. We will compare single-stage a multistage design to illustrate the
computational savings that can be afforded by using multistage Nyquist filters.
Hf = fdesign.decimator(8,'nyquist',8,.016,80);
H1 = design(Hf,'kaiserwin');
H2 = design(Hf,'multistage');
H3 = design(Hf,'multistage','Nstages',2);
The computational costs of the three designs are summarized in the following
table:
NMult NAdd Nstates MPIS APIS NStages
H1 551 550 624 68.875 68.75 1
H2 93 90 174 15.625 14.75 3
H3 106 104 182 17.125 16.75 2

The equivalent overall filters for all three designs are 8-band Nyquist filters.
The default multistage design, H2, is a 3-stage filter with each individual stage
being a halfband filter (each halfband filter is different though). H3 is 4-band
Nyquist decimator followed by a halfband Nyquist decimator.

5.3.1 Using IIR halfband filters
Whenever we have a halfband decimator (or interpolator) we may want
to consider using either an elliptic halfband filter (if phase linearity is not
an issue) or a quasi linear-phase IIR halfband filter (if phase linearity is
important) to increase computational savings.
As an added bonus, all Nyquist designs tend to have a very small pass-
band ripple. If we use IIR halfband filters, the passband ripple can be in
the micro-dB range or even smaller!



0 3−stage FIR Nyquist
3−stage Nyquist linphase IIR
3−stage Nyquist ellip IIR
Magnitude (dB)

−50

−100

−150
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Figure 5.12: Passband phase response for multistage Nyquist FIR and IIR filters.

In the previous example, if we substitute the halfband filters of each
stage in H2 with quasi linear-phase IIR filters,
H4 = design(Hf,'multistage','HalfbandDesignMethod','iirlinphase');
we can get saving close to a factor of 3 in both total number of multipliers
as well as MPIS.
If we can use elliptic halfbands,
H5 = design(Hf,'multistage','HalfbandDesignMethod','ellip');
the savings are about twice as good. The computational cost is shown
below
NMult NAdd Nstates MPIS APIS NStages
H4 35 70 75 5.625 11.25 3
H5 12 24 18 2.625 5.25 3

The magnitude response of the 3-stage Nyquist FIR design along with
the two 3-stage Nyquist IIR designs is shown in Figure 5.12. All three
designs behave similarly overall as expected since the specifications for
each halfband (each stage) are the same. However, although the 3-stage


5.4 Multistage interpolation 115

x 10

3−stage FIR Nyquist
1.5 3−stage Nyquist linphase IIR
1
Magnitude (dB)

0.5

0

−0.5

−1

−1.5

0 0.02 0.04 0.06 0.08 0.1 0.12

Figure 5.13: Passband phase response for multistage Nyquist FIR and IIR filters.

FIR Nyquist design has a small passband ripple, it is nowhere near the
tiny passband ripple of the IIR designs. The passband details are shown
in Figure 5.13.
As stated before, the computational savings of the elliptic design come
at the expense of phase non-linearity. On the other hand, the group-delay
of the elliptic design is much lower than the other two designs. The pass-
band group-delay is shown in Figure 5.14.
We can use realizemdl(H4) or realizemdl(H5) to generate Simulink
blocks that implement these multistage decimators using halfband IIR fil-
ters in efficient polyphase form.

5.4 Multistage interpolation
Just as with decimation, using multiple stages to perform interpolation
can result in significant implementation savings. The reason once again
has to do with the ratio of the transition width relative to the sampling
frequency.
To see this, consider implementing a filter that interpolates by a factor
of 8, increasing the sampling-rate from 1250 Hz to 10000 Hz with a cutoff



Group Delay
350

300

250

200 3−stage FIR Nyquist
3−stage Nyquist linphase IIR
150

100

50

0.02 0.04 0.06 0.08 0.1 0.12

Figure 5.14: Passband group-delay for multistage Nyquist FIR and IIR filters.

frequency of 625 Hz and a transition width of 312.5 Hz. Let us compare
the transition width, relative to the sampling frequency of a single-stage
design vs. the first stage of a multistage design. Suppose the first stage
of the multistage design is a halfband filter that interpolates by a factor of
two (from 1250 Hz to 2500 Hz).
Figure 5.15 shows a comparison of the two filters. They both have the
same transition width, yet the sampling frequency ratio is 4-to-1. There-
fore the ratio ∆ f / f s is four times smaller for the single-stage filter than for
the first stage of a multistage implementation. Accordingly, the number
of coefficients for the single-stage design is 142 (with 135 adders), whereas
the first stage of a multistage implementation has only 22 multipliers (and
21 adders). The fact that we have a savings of more than a factor of 4
in terms of number of multipliers is owed to the fact that we use a half-
band filter for the latter design which yields further savings. Note that in
general for polyphase interpolators, the number of coefficients equals the
number of multiplications per input sample (MPIS).
Of course, as is apparent from Figure 5.15, the two filters are not per-
forming the same job, so the comparison is unfair. While the single-stage
design removes all spectral replicas in the new Nyquist interval (with
f s = 10000), the halfband filter only removes some spectral replicas. There


5.4 Multistage interpolation 117

0

Single−stage design
First stage of a multistage design
Magnitude (dB) (normalized to 0 dB)

−20

−40

−60

−80

−100

−120
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
Frequency (kHz)


is clean-up left to be done by subsequent interpolation filters operating
at higher rates (after all, we have only interpolated to 2500 Hz with the
halfband filter).
Figure 5.16 shows a full 3-stage design (each stage being a halfband
interpolator) compared to a single-stage design. Both designs satisfy the
same transition width and stopband attenuation requirements. Overall
the multistage design requires 40 coefficients and 72 MPIS ∗ . The code that
was used for these two designs follows:

f = fdesign.interpolator(8,'Nyquist',8,'TW,Ast',312.5,80,1e4);
g = design(f,'multistage','Nstages',3);

As with multistage decimators, the use of IIR halfbands implemented
in efficient allpass-based polyphase structures can improve the efficiency
of designs beyond what can be achieved with FIR filters. For example, for
the same specifications, this design:
∗ Since each stage produces more samples at its output than the number of samples at its
input, for multistage interpolators the number of multiplications-per-input-sample to the
overall implementation is larger then the number of coefficients required for all stages.



20
Single−stage design
Three−stage design
0

−20
Magnitude (dB)

−40

−60

−80

−100

−120

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
Frequency (kHz)


hm = design(f,'multistage','Nstages',3,...
'HalfbandDesignMethod','iirlinphase');

results in a 3-stage filter with 16 coefficients and 30 MPIS.
Figure 5.17 shows how subsequent stages remove spectral replicas that
remain from the first stage. Notice how as the sampling-rate increases, the
transition-width for each corresponding stage also increases so that the
ratio ∆ f / f s remains large resulting in simple filters. For interpolation, the
first stage is usually the most involved, but this is OK since the sampling
frequency is the lowest at that point. The complexity of each subsequent
stage is reduced gradually since each filter has to operate at higher and
higher sampling frequencies.

We have shown how multistage implementation of filters can result in sig-
nificant computational savings. The savings are increased when multi-
stage implementation are combined with multirate filters. There is an op-
timal number of stages which provide maximum computational savings.




0

First stage
Second stage
−20
Third stage
Magnitude (dB)

−40

−60

−80

−100

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
Frequency (kHz)


This optimal number can be determined automatically by the design tools
discussed.
Multistage Nyquist filters provide very efficient implementations. This
is particularly true for halfband filters, especially IIR halfbands. These
multistage filters are equally adequate for efficient decimation or interpo-
lation purposes.
In the next chapter we will discuss special multirate filters that are well-
suited for multistage implementations. These include hold and linear in-
terpolator filters as well as CIC interpolation/decimation filters. We will
also discuss the use of Farrow filters for fractional decimation/interpolation.
These filters can be used to provide fractional sampling-rate changes by
an arbitrary factor (not necessarily a ratio of two integers). Moreover, they
can do so while using a small number of multipliers.


Chapter 6

Special Multirate Filters

Overview
In this chapter we look at some special filters that come in handy for mul-
tirate applications.
We first look at hold interpolators which perform basic interpolation by
simply repeating input samples L times (L being the interpolation factor).
Next, we look at linear interpolators and show how they can be thought
of as two hold interpolators “put together”.
Both linear and hold interpolators are attractive because of their sim-
plicity. We see that they are very crude approximations to an ideal lowpass
filter used for interpolation. Usually, because of their simplicity, they are
used at the last stage of a multistage implementation, operating at high
sampling rates.
We then move on to CIC interpolators and show how these are just
generalizations of hold/linear interpolators. By using multiple (more than
two) sections, CIC interpolators can obtain better attenuation than hold or
linear interpolators. The nice thing about CIC interpolators is that they can
be implemented without using multiplication. This is an attractive feature
for certain hardware such as FPGAs and ASICs because multipliers take
up quite a bit of area and are difficult to make to operate at very high clock
rates.
CIC filters can also be used for decimation. Unlike the interpolation
case, in decimation, CIC filters are usually used at the first stage of a multi-
stage design. This is because at that stage the sampling-rate is the highest,

6.1 Hold interpolators 121

1

0.9
Original samples
0.8
Held−interpolated samples

0.7

0.6
Amplitude

0.5

0.4

0.3

0.2

0.1

0
0 5 10 15 20 25
Time (n)

Figure 6.1: Interpolated samples using a hold interpolator with a factor L = 4.

making the multiplierless feature of CIC filters very attractive.
We then show how to design CIC compensators for either interpolation
or decimation. These filters are lowpass filters whose passband is shaped
to compensate for the droop in the passband of CIC filters. Of course,
since linear and hold interpolators are special cases of CIC interpolators,
CIC compensators can be used for those linear and hold cases as well.
Finally, Farrow filters are shown both for single-rate fractional-delay
applications as well as for changing the sampling-rate of signals (by inte-
ger or fractional factors). The main allure of Farrow implementations are
the low number of fixed multipliers that are required and the tunability of
the fractional delay while the filter is running without having to re-design.

6.1 Hold interpolators
The simplest interpolator that one can build is the one that performs a
(zero-order) hold of each sample, inserting as many repeated samples as
desired. For example, Figure 6.1 shows the result of a hold interpolator,
interpolating by a factor of 4.


122 Special Multirate Filters

Magnitude Response

4 Hold interpolator
Ideal interpolator
3.5

3

2.5
Magnitude

2

1.5

1

0.5

0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Figure 6.2: Magnitude response of a hold interpolator with a factor L = 4. The ideal
interpolator is shown for comparison.

Hold interpolators are attractive because they require no arithmetic in
order to be implemented. The hold interpolator is an FIR interpolator
whose impulse response is simply a series of L 1’s.

Hm = mfilt.holdinterp(4);
Hm.Numerator

As we know, the ideal interpolator is a lowpass filter that removes L − 1
adjacent spectral replicas of the signal being interpolated. The hold inter-
polator is a very crude approximation to the ideal interpolator as can be
seen with fvtool(Hm). Figure 6.2 shows the magnitude response of the
hold interpolator compared to the ideal interpolator for the case L = 4.
The hold interpolator has a very wide transition band, allows significant
high-frequency content to pass through, and distorts the original data in
the band of interest. Note that the fact that high frequency remains after
hold interpolation is obvious from the time-domain plot shown in Figure
6.1. The fast transitions between the last interpolated sample for a given
original sample and the first interpolated sample for the next original sam-
ple are indicative of high frequency content.


6.1 Hold interpolators 123

Magnitude Response
8
Hold interpolator
Multistage interpolator
7 Ideal interpolator

6

5
Magnitude

4

3

2

1

0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Figure 6.3: Hold interpolator compared with multistage interpolator for L = 8. For
reference, the ideal interpolator is shown.

For these reasons, hold interpolators are usually not used in isolation,
but rather are used as part of a cascade of interpolators (typically at the last
stage, which operates at the fastest rate and therefore benefits the most of
the fact that no multiplications are required). When used in such cascade
configurations, most of the high frequency content has been removed by
previous stages. Moreover, the distortion of the signal in the band of in-
terest is much less severe since the band of interest will occupy a much
smaller portion of the passband of the hold interpolator.
Example 50 Let us compare the effect of using a hold interpolator for a full in-
terpolation by a factor L = 8 vs. using a hold interpolator for a factor L3 = 2 in
conjunction with two halfband filters, each one interpolating by two.
Figure 6.3 shows a comparison of a possible multistage design including two
halfband filters followed by a hold interpolator vs. a single-stage interpolate-by-
8 hold interpolator design. For reference, the ideal interpolate-by-8 filters is also
shown. While the multistage interpolator does allow some high-frequency content
through, it is nowhere as bad as the hold interpolator acting alone. Also, assuming
the band of interest extends to 0.1π, the multistage interpolation introduces a
maximum distortion of about 0.15 dB while the hold interpolator introduces a
distortion of almost 2 dB.



10

0

−10

−20

First stage (halfband)
−30
Magnitude (dB)

Second stage (halfband)
−40 Third stage (hold interpolator)

−50

−60

−70

−80

−90

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Figure 6.4: Magnitude response of each of the three stages of the multistage design.

1.2
Held−interpolated samples
Multistage−interpolated samples

1

0.8
Amplitude

0.6

0.4

0.2

0
0 20 40 60 80 100 120
Time (n)

Figure 6.5: Comparison of interpolated data using a hold interpolator and a multistage
interpolator with L = 8.


6.2 Linear interpolators 125

One thing to keep in mind that since interpolation is usually used right be-
fore D/A conversion, the remnant high frequency content can be removed by an
analog (anti-image) post-filter. This will ensure a smooth analog waveform. As is
apparent from Figure 6.3, because of the multistage interpolation, the remaining
high frequency content will be far away from the band of interest (the passband of
the filter), making it easy for a low-order analog lowpass post-filter (with a wide
transition band) to remove said high frequencies.
Figure 6.4 shows the magnitude response of each of the three stages of the
multistage design. Note that because of the prior interpolation by 4 (provided by
the first two stages) the band of interest is only a small fraction of the passband of
the hold interpolator. For this reason, the passband distortion is minimal.
Figure 6.5 shows the result of filtering the same data with the hold interpola-
tor compared to using the multistage interpolator. While it is obvious that some
high frequency content remains in either case, it is also obvious that overall the
multistage-interpolated data is much smoother and therefore has much less high-
frequency content.

A hold interpolator can be thought of in polyphase terms. As usual, the
polyphase components are formed by taking every Lth value from the im-
pulse response. In this case, since the impulse response is a sequence of 1’s
of length L, each polyphase branch consists trivially of a single coefficient
equal to one. This of course is consistent with the notion that for every
input sample, each polyphase branch computes one of the interpolated
values. In this case, each interpolated value is compute by “multiplying”
the input sample by one.

6.2 Linear interpolators
After hold interpolators, the next simplest interpolation filter is the linear
interpolator. While the hold interpolator only uses one sample in order
to compute the interpolated values (and therefore has only one coefficient
per polyphase branch), in the case of linear interpolators, two adjacent
samples are used in order to compute the interpolated values. This of
course means that each polyphase branch will have two coefficients as we
will soon see.
Linear interpolation is very simple to understand. Consider the plot
in Figure 6.6 which depicts the case L = 5 (it is the same idea for other



2

Original samples
1.8
Linear−interpolated samples

1.6

1.4

1.2
Amplitude

1

0.8

0.6

0.4

0.2

0
0 1 2 3 4 5 6 7 8 9 10
Time (n)

Figure 6.6: Computing linearly-interpolated samples for L = 5.

values of L). For every two existing samples, the interpolated values are
computed by forming a weighted average of the existing samples. The
weights are determined by the proximity of the interpolated values to each
of the existing samples. In this case, the interpolated sample closest to an
existing sample is four times closer to that existing sample than it is to the
existing sample on its other side. Therefore, when we form the weighted
average in order to compute the interpolated sample, we weight one of the
existing samples four times more than the other. This results in coefficients
equal to 0.2 and 0.8 (they always add to 1) for that polyphase branch. The
coefficients for other polyphase branches are similarly computed based
on the ratio of the distance between the new interpolated sample and the
existing surrounding two samples.
This means that for the case L = 5, the 5 polyphase branches can triv-
ially be computed as:
L = 5;
Hl = mfilt.linearinterp(L);
p = polyphase(Hl)
Notice that the last polyphase branch is trivial (its coefficients are 1 and 0).



This allows for the existing samples to be contained unchanged within the
interpolated samples. That is, linear interpolators (and hold interpolators)
are special cases of Nyquist filters.
As usual, the overall interpolation filter is a lowpass filter whose coef-
ficients can be formed from the polyphase branches.

Hl.Numerator
ans =
Columns 1 through 6
0.2000 0.4000 0.6000 0.8000 1.0000 0.8000
Columns 7 through 9
0.6000 0.4000 0.2000

A linear interpolator can be thought of as the convolution of two hold
interpolators. Indeed, for the case L = 5, the coefficients of the FIR hold
interpolation filter are:

Hm = mfilt.holdinterp(5);
Hm.Numerator
ans =
1 1 1 1 1

And of course

1/5*conv(Hm.Numerator,Hm.Numerator)
ans =
Columns 1 through 6
0.2000 0.4000 0.6000 0.8000 1.0000 0.8000
Columns 7 through 9
0.6000 0.4000 0.2000

is exactly equal to Hl.Numerator.
As we know, convolution in time is equal to multiplication in frequency.
Therefore, the magnitude response of a linear interpolator is given by mul-
tiplying (or adding when working with dB) the magnitude response of
two hold interpolators. This can be easily verified with fvtool(Hm,Hl).
The plot comparing the magnitude response of both interpolators is shown
(in dB) in Figure 6.7 (we have normalized the passband gain to 0 dB to ease
the comparison).




Hold interpolator
0 Linear interpolator

−10 Magnitude (dB) (normalized to 0 dB): −12.09688

−20 Normalized Frequency: 0.59375
Magnitude (dB) (normalized to 0 dB): −24.17285

−30

−40

−50

−60
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Figure 6.7: Comparison of linear and hold interpolators for L = 5.

Because the magnitude response of the linear interpolator is twice that
of the hold interpolator (in dB), the linear interpolator provides more stop-
band attenuation and therefore better lowpass filtering than a hold inter-
polator. This should be obvious to see if you think of the situation in
the time domain. While linearly-interpolated samples still exhibit some
abrupt transitions which indicate some high frequency components re-
main, it is not nearly as abrupt as taking the same samples and interpo-
lating with a hold interpolator.
On the other hand, the passband distortion of the linear interpolator is
also twice that of the hold interpolator. However, as with hold interpola-
tors, linear interpolators are usually used as the last stage of a multistage
interpolation scheme. Therefore, the passband distortion due to linear in-
terpolation can be minimal in such configurations. Although the concept
of closeness is a relative notion, one can intuitively think that if samples
have been interpolated already so that they are very close to each other,
the error added by using linear interpolation rather than a higher-order
filter is much smaller than if we try to directly linearly interpolate samples
that are far apart in one shot.
The behavior is very similar to the comparison shown above between



20

0

−20
Magnitude (dB)

−40

−60

−80

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Figure 6.8: Magnitude response of a multistage interpolator consisting of a halfband
filter, followed by an interpolate-by-2 linear interpolator followed by an interpolate-by-2
hold interpolator.

using a single-stage interpolate-by-8 hold interpolator and using a mul-
tistage interpolator with a simplified (hold or linear) final stage. In fact,
one perfectly reasonable possibility is to have a linear interpolator in the
next-to-last stage of a multistage configuration and a hold interpolator in
the last stage.
Figure 6.8 shows just that configuration. In this case, we have obtained
it by replacing the halfband filter in the second stage of the multistage filter
used above, with a linear interpolate-by-two filter. Once again, if the inter-
polation is being done just prior to D/A conversion, it is assumed that the
analog anti-image post-filter will remove the remaining high-frequency
content∗
The idea of convolving hold interpolators in order to obtain better stop-
band attenuation can be generalized to more than two such interpolators.
CIC interpolation filters [31] can be seen as just that. The number of section
∗ We should note that part of the high-frequency content is attenuated by the staircase
reconstructor that is typically used in D/A conversion in order to make the signal analog.
A post-filter is used after the staircase reconstructor to smooth-out the analog waveform
by removing remaining high-frequency content. For more on this, see [3].



of the CIC interpolators will correspond to the number of hold interpola-
tors convolved. As with linear interpolators, multi-section CIC interpola-
tors attain better stopband attenuation at the expense of larger passband
distortion. In the next section we will take a look at CIC interpolators.
After that, we will look at the design of CIC compensators which pre-
equalize the passband∗ in order to compensate for the passband distortion
introduced by the CIC filters. Of course, if deemed necessary, this com-
pensators can also be used to compensate for the distortion introduced by
linear or hold interpolators.

6.3 CIC interpolators
Let’s perform some simple math to derive the CIC filters starting from a
hold interpolator. We have seen that a hold interpolator has an impulse
response of the form
h[n] = {1, 1, 1, . . . , 1}
where the number of 1’s is equal to the interpolation factor. The transfer
function of such a filter is given simply by

H (z) = 1 + z−1 + z−2 + . . . + z−( L−1) (6.1)

If we multiply both sides of (6.1) by z−1 we get

z −1 H ( z ) = z −1 + z −2 + . . . + z − L (6.2)

Subtracting (6.2) from (6.1) we get

H (z)(1 − z−1 ) = 1 − z− L

which means we can re-write H (z) as

1 − z− L
H (z) =
1 − z −1
If we think of this as two filters in cascade, H (z) = H1 (z) H2 (z), with
H1 (z) = 1 − z− L (the “comb” due to its magnitude response) and H2 (z) =
∗ Post-equalize in the case of decimation.


6.3 CIC interpolators 131

Figure 6.9: Conceptual implementation of a CIC interpolator. As with all interpolators,
conceptually it is just an upsampler followed by a lowpass filter (the combination of the
last two filters).

Figure 6.10: Actual implementation of a CIC interpolator obtained by use of Noble iden-
tities.

1/(1 − z−1 ) (the integrator), we have derived a cascaded integrator-comb
(CIC) filter.
When used for interpolation∗ , the CIC interpolator in conceptual form
operates as the block diagram shown in Figure 6.9. Using the Noble iden-
tities, we can interchange the upsampler and the comb filter to obtain the
configuration shown in Figure 6.10.
Of course so far we haven’t really done much other than show a rather
complicated way of implementing a hold interpolator. In reality, if we
wanted to implement a hold interpolator, we would skip the comb and
integrator filters and simply repeat the input sample as many times as we
wish to interpolate.
The real use of CIC filters comes from a generalization of the fact we al-
ready noted relating linear interpolators to hold interpolators. Recall that
we stated that a linear interpolator’s impulse response can be obtained by
convolving two impulse responses of hold interpolators. With CIC filters,
we convolve K such impulse responses. Since convolution in time equates
to multiplication in frequency, this means that the transfer function of a
CIC filter with K sections is given by elevating H (z) to the power of K,
K
K 1 − z− L
HCIC (z) = ( H (z)) =
1 − z −1
If we once again use the Noble identities, we can easily verify that an
∗ Keeping in mind that interpolation conceptually consists of upsampling followed by
lowpass filtering.



Figure 6.11: Implementation of a multi-section CIC interpolator.

Figure 6.12: Multiplierless linear interpolator.

implementation of an K-section CIC interpolator can be done as shown in
Figure 6.11.
Note that the following commands can be used to quickly build a Simulink
model of a CIC interpolator:
L = 4; % Interpolation factor
K = 3; % Number of sections
Hm = mfilt.cicinterp(L,1,K);
realizemdl(Hm)
Of course, a multi-section CIC interpolator multiplies the magnitude
response of the hold interpolator K times. Because CIC filters are mul-
tiplierless, the allure of these filters is that we can obtain good stopband
attenuation by using only adders and delays if we use several sections.
For this reason, CIC filters are often implemented at the last stage of a
multistage interpolation filter, operating at the fastest rate.
The cases K = 1 and K = 2 are special cases of CIC interpolators that re-
sult in hold and linear interpolators. In fact, the inner-most combination of
comb/upsampler/integrator, can be readily replaced with a hold interpo-
lator (saving a couple of adders and delays [32]. This technique allows us,
for instance, to implement a multiplierless linear interpolator using only 2
adders and 2 delays as shown in Figure 6.12.

6.3.1 Design of CIC interpolators
When designing CIC interpolators, we should think in the context of mul-
tistage designs. The design basically consists of determining how many



sections are needed in order to obtain a certain stopband attenuation. Un-
like conventional lowpass filters, this stopband attenuation is not attained
for the entire stopband. Instead, the attenuation will be attained only
where needed to suppress remnant high-frequency content left behind by
previous stages of interpolation.

Example 51 As if often the case, an example is the best way to understand this.
Suppose we want to interpolate by a large factor, say 64, in order to make our life
easy when designing an analog post-filter for D/A conversion. The following is a
possible set of specifications for our interpolator:

f = fdesign.interpolator(64,'Nyquist',64,'TW,Ast',0.0078125,80);

These specifications state the the cutoff frequency for the interpolator is π/64 and
that the passband-edge frequency is π/64 − TW/2 = 0.0117π. The minimum
stopband attenuation for the entire stopband region is 80 dB.
First, let’s design this filter using conventional Nyquist filters and let us con-
strain the design to be done in 3 stages:

Hc = design(f,'multistage','Nstages',3);
cost(Hc)

The result of this design is a interpolate-by-4 Nyquist filter, followed by a interpolate-
by-2 Nyquist (halfband) filter, followed by an interpolate-by-8 Nyquist filter. Over-
all, the interpolation factor is 64 as desired. The cost of implementing this filter is
110 coefficients and 408 MPIS.
Now, let’s try to replace the last stage with an interpolate-by-8 CIC filter. The
desired stopband attenuation is 80 dB. However, in order to set the passband-
edge frequency, we should use the overall desired passband-edge frequency, i.e.
0.0117π:

f2 = fdesign.interpolator(8,'CIC',1,'Fp,Ast',0.0117,80);
Hcic = design(f2); % Results in 4 sections

The following commands can be used to replace the 3rd stage with a CIC and
compare the two multistage designs:

Hc2 = copy(Hc);
Hc2.stage(3) = Hcic;
fvtool(Hc,Hc2)




0

Conventional 3−stage interpolator
−20
3−stage interpolator that includes a CIC filter

−40

−60

−80

−100

−120

−140

−160

−180

−200
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Figure 6.13: Comparison of a conventional 3-stage interpolation filter and one which
uses a CIC interpolator in its last stage.

The magnitude response of the two implementations is shown in Figure 6.13
(we have normalized the passband gain to 0 dB in order to ease the comparison).
Clearly, either filter meets the specs. However, the multistage filter that uses the
CIC in its last stage can be implemented with only 70 coefficients and 88 MPIS.
Figure 6.14 shows the magnitude of each of the 3 stages comprising the filter
Hc2. The last stage is the CIC filter. Note that the CIC filter does not provide
80 dB attenuation over the entire stopband. However, it does provide 80 dB or
more attenuation where needed, i.e. at the points 2kπ/L ± 0.0117π where k =
1, . . . , L − 1 and L = 8.

6.3.2 Gain of CIC interpolators
In the magnitude response plots we have shown so far, we have normal-
ized the passband gain of the filters to 0 dB in order to simplify visualiza-
tion of the designs.
However, CIC interpolation filters have a very large gain. This goes
beyond the usual gain of L that all interpolators have. To see why, recall
first the linear interpolator case.
We stated that a linear interpolator can be though of as two hold in-




0

−10

Stage 1

−20
Stage 2
−30 Stage 3 (CIC)

−40

−50

−60

−70

−80

−90

−100
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 6.14: Magnitude response of each stage of a 3-stage interpolation filter using a
CIC interpolator in its last stage.

terpolators convolved. However, in order to get the impulse response of
the linear interpolator correctly scaled, when we convolved the two hold
interpolators, we had to scale the result by a factor of L as well.
For example, take the case L = 3. A hold interpolator has impulse re-
sponse given by h[n] = 1, 1, 1. A plot of the magnitude response (not in
dB) shows that the gain of the filter at frequency 0, is 3. This can easily be
verified with the following commands:

Hh = mfilt.holdinterp(3);
fvtool(Hh,'MagnitudeDisplay','Magnitude')

If we convolve two such impulse responses, we get an impulse re-
sponse which is 3 times larger than that of the corresponding linear in-
terpolator.

conv([1 1 1],[1 1 1])
Hl = mfilt.linearinterp(3);
Hl.Numerator

However, the linear interpolator itself has a passband gain of 3:



Hl = mfilt.linearinterp(3);
fvtool(Hl,'MagnitudeDisplay','Magnitude')
Therefore the convolution of two hold interpolators has a gain of L2 =
9.
Since CIC filters are obtained by (unscaled) convolutions of hold in-
terpolators, their gain is given by LK . If we account for the fact that we
expect any interpolator to have a gain of L, the extra gain introduced by a
CIC interpolator is LK −1 .∗ For example, for L = 3 and K = 4, we can easily
see that the passband gain is 81:
Hcic = mfilt.cicinterp(3,1,4);
fvtool(Hcic,'MagnitudeDisplay','Magnitude')
The extra gain, that is LK −1 , can be computed simply by using the gain
command,
gain(Hcic)
ans =
27

6.3.3 Further details of CIC filters
Although CIC filters seem pretty nifty, it is worth noting that CIC filters
are unstable. The factor 1 − z−1 in the denominator means that the filter
has a pole on the unit circle (at z = 1). In fact, the only way for CIC filters
to work (meaning that their output will not grow without bound) is by
using fixed-point arithmetic (with overflows wrapping).
Also, note that CIC filters have been constructed by adding a pole and
a zero at z = 1. This pole/zero pair should cancel, yielding the traditional
FIR transfer function. However, for implementation, we do not cancel the
pole and instead implement the filter with recursion. The filter is still FIR
even though it has feedback†
∗ In our treatment of CIC interpolators so far, we have ignored the differential delay (we
have assumed it is one). The differential delay is the order of the delays in the differ-
ence (comb) part of the filter. Typical values for the differential delay are 1 and 2. We
will assume it is always 1 here. A value different to one will also affect the passband
gain. † The strict definition of FIR is a filter whose minimal realization has a nil-potent
state-transition matrix in its state-space representation (see [26] Ch. 13). The CIC imple-
mentation is not minimal since it allows for a pole/zero cancellation.


6.4 CIC decimators 137

6.4 CIC decimators
Consider a signal with two-sided bandwidth equal to f s and sampled at a
sampling rate f s . If we need to decimate by a factor M in order to reduce
the rate to a sampling frequency f d = f s /M, we need to first reduce the
two-sided bandwidth of the signal to f d as well.
CIC filters are attractive for decimation purposes due to their low com-
putational cost given that they do not require multipliers. However, com-
pared to an ideal brick-wall lowpass filter, the response of CIC filter pro-
vides a poor lowpass characteristic. Nevertheless, this can be compen-
sated for by the use of subsequent filters that operate at reduced sampling
rates and therefore do not incur a high computational cost.
Let us illustrate typical CIC behavior for a decimation problem by a
factor M = 4. Let us examine Figure 6.15. For this example, we have
assumed f s = 100 MHz.
The rectangular areas shown with a dashed line illustrate how the var-
ious shifted replicas would overlay had they been filtered with a perfect
brick-wall filter. Notice that in this case the various images would be just
adjacent to each other but there would be no overlap, i.e. no aliasing.
In contrast, the decimated version of the CIC filtered signal shows sig-
nificant overlap between spectral replicas. Note that the aliasing is maxi-
mum at the edges of the low-rate Nyquist interval, [− f d /2, f d /2]. On the
other hand, the aliasing is minimal at the center of the Nyquist interval (at
DC).
For this reason, coupled with the fact that the droop in the CIC fil-
ter is not as accentuated in the vicinity of DC, decimating CIC filters are
usually used when the ultimate band of interest is not the entire interval
[− f d /2, f d /2] but rather a (perhaps small) subset of it, say [− f p , f p ] where
f p ≪ f d /2.
It is assumed of course that the energy of the signal outside of the final
band of interest, that is, in the band [− f d /2, − f p ] ∪ [ f p , f d /2] will be re-
moved with either a single-rate lowpass filter, or, more likely, with a sub-
sequent lowpass decimation filter that will reduce the sampling rate to the
vicinity of 2 f p (this can be done of course with more than one subsequent
decimation filter resulting in an overall implementation with perhaps 3 or
4 stages). This subsequent filter will typically compensate for the notable
droop in the passband of the CIC filter as well.
It is clear from Figure 6.15, that the maximum amount of aliasing in



Filtered magnitude spectrum after decimation
50
Baseband filtered spectrum
First shifted spectrum
Final band of interest [0,Fp] Second shifted spectrum
Third shifted spectrum

0
dB

−50

−100

−150
0 10 20 30 40 50 60 70 80 90 100
Frequency (MHz)

Figure 6.15: Final band of interest of signal decimated with a CIC filter.

the final band of interest will occur at f p . The actual maximum amount of
aliasing at f p is due to the adjacent replica. For a CIC filter, the larger the
number of sections K, the smaller the gain for the replica at such frequency,
i.e. in order to limit the amount of aliasing, we would like to have a large
number of sections. However, the more sections we have, the larger the
droop of the filter in the band of interest that we need to compensate for.
In order to design a CIC filter, we reverse the previous statement. Namely,
given an amount of aliasing that can be tolerated, determine the minimum
number of sections K that are required. The full design parameters are
listed next.

6.4.1 Design parameters
At this point we can enumerate the design parameters for the filter. The
design parameters are:

• the edge frequency f p


6.5 CIC compensators 139

• the amount of aliasing tolerated at f p : Ast

• the decimation factor M

• optionally, the sampling frequency prior to decimation f s

As with the design of CIC interpolators, the use of stopband attenu-
ation is slightly different than with conventional lowpass filters. In this
case, it is not that the filter must provide a minimum attenuation of Ast
throughout the entire stopband. Instead, Ast is the minimum amount of
attenuation permitted for frequencies that will alias back into the band of
interest. Of course, at the end of the day, this last interpretation is the same
for any decimation filter.

Example 52 Given these parameters, a CIC decimation filter with the necessary
number of sections K such that the amount of aliasing tolerated at f = f p is not
exceeded can be designed as follows:

D = 1; % Differential delay
Fp = 2e6; % 2 MHz
Ast = 80; % 80 dB
Fs = 100e6; % 100 MHz
Hf = fdesign.decimator(M,'CIC',D,'Fp,Ast',Fp,Ast,Fs);
Hcic = design(Hf);

The resulting design that meets the required attenuation consists of 6 sections.

6.5 CIC compensators
CIC compensators are single-rate or multirate filters that are used to com-
pensate for the passband droop in CIC filters. In the case of CIC interpola-
tion, what is usually done is to pre-equalize for the droop in a prior stage
of a multistage design. In the case of decimation, we post-equalize in a
subsequent stage to the CIC decimator.
As we have already stated, the amount of droop will depend on the
number of sections K in the CIC filter. For the special case of linear and
hold interpolators, the number of sections is 2 and 1 respectively.



120

100

80

60
Magnitude (dB)

40

20

0

−20

−40

−60

−80
0 5 10 15 20 25 30 35 40 45
Frequency (MHz)

Figure 6.16: Overall two-stage design consisting of CIC decimator and compensator.

In the case of decimation, the larger the ratio of the ﬁnal passband of
interest f p to the original sampling rate f s , the smaller the amount of droop
since only a very small portion of the passband of the CIC ﬁlter is involved.
In some cases, we can do without the compensation altogether, since the
small droop may not warrant equalization.
Similarly, in the case of interpolation, the larger the overall interpola-
tion factor, the smaller the droop will be in the baseband spectrum. Again,
it may be so that we choose not to equalize if the overall multistage inter-
polation factor is large enough.

Example 53 As an example consider a possible compensator design for the CIC
decimator design of the previous section. Suppose we want to further decimate by
a factor of 4 for an overall decimation factor of 32 for the two stages.

K = Hcic.NumberOfSections; % Determine from previous design
M2 = 4; % Decimation factor for this stage
Fst = 4.25e6; % 4.25 MHz, TW = 2.25 MHz
Ap = 1; % 1 dB peak-topeak passband ripple
Hf2 = fdesign.decimator(M2,'ciccomp',D,K,'Fp,Fst,Ap,Ast',Fp,Fst,Ap,Ast,Fs/M);
Hd = design(Hf2);


6.6 Farrow Filters 141


0

−20

CIC Decimator
CIC compensator
−40

−60

−80

−100

−120

−140

−160

0 5 10 15 20 25 30 35 40 45
Frequency (MHz)

Figure 6.17: Magnitude response of each of the two stages in the design.

We can obtain the resulting overall decimator of cascading the two filters from:
Hcas = cascade(Hcic,Hd);
fvtool(Hcas) % Show overall response
Figure 6.16 shows the resulting filter after cascading the CIC decimator and
its compensator. Figure 6.17 shows how the CIC decimator attenuates the spectral
replicas from the compensator. Figure 6.18 shows the passband of the overall two-
stage decimator. Notice how the droop in the passband has been equalized. As
with CIC interpolators, CIC decimators have a large passband gain (for the same
reason). The gain can be found from gain(Hcic) and is once again given by MK .
In Appendix C, we will compare a multistage decimator design includ-
ing a CIC decimator and a CIC compensator with various other single- and
multistage designs.

6.6 Farrow Filters
We will now derive a filter that can be very effectively used for both frac-
tional advances/delays and changing the sampling rate of signals by arbi-
trary factors (not necessarily rational).



110

109.5
Magnitude (dB)

109

108.5

108

107.5

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
Frequency (MHz)

Figure 6.18: Equalized passband of two-stage design showing the effect of the CIC com-
pensator on the droop.

Farrow filters are basically obtained by (piece-wise) polynomial inter-
polation (curve fitting) on the input samples to the filter. Once a poly-
nomial has been fitted, it can be evaluated at any point. This allows for
interpolation at arbitrary locations between samples. Moreover, the im-
plementation of Farrow filters is essentially done using Horner’s method
of polynomial evaluation. The advantage of doing this is that it allows
for tuning of the location at which we wish to interpolate by changing
only one parameter in the filter. All filter coefficients remain constant.
Therefore, Farrow filters are suitable for instance to implement variable
fractional advance/delay filters.
Theoretically, polynomials of any order can be used to fit to the existing
samples. However, since large order polynomials tend to oscillate a lot,
typically polynomials of order 1, 2, or 3 are used in practice.
The easiest way to understand the derivation of Farrow filters is to
start with linear interpolators. For the moment, consider the case where
we want to interpolate solely for the purpose of introducing a fractional
advance/delay (i.e. without changing the sampling rate of the signal).
A linearly interpolated fractional advance/delay can be implemented
with two multipliers as we have already seen (this is just what each polyphase



Figure 6.19: Two-tap filter that can be used for fractional delay by linear interpolation.

Figure 6.20: Computing a fractionally advanced/delayed interpolated sample from its
two surrounding samples via linear interpolation.

branch of a linear interpolator does). Consider the two-tap filter shown in
Figure 6.19. In order to compute the interpolated values between a previ-
ous sample, x [n − 1], and a current sample, x [n], we weigh each sample
relative to the distance between it and the desired interpolated value. Fig-
ure 6.20 shows the resulting interpolated sample.
The filter as shown, performs a fractional delay of 1 − α (as mentioned
before, this is equivalent to a fractional advance of α if we allow for a full
sample delay for the sake of causality). The output is computed as

y[n] = (1 − α) x [n − 1] + αx [n]

Now, by simply re-writing the previous expression as

y[n] = x [n − 1] + α( x [n] − x [n − 1])



Figure 6.21: Fractional advance filter.

Figure 6.22: Fractional delay filter.

we see we can implement the filter as shown in Figure 6.21.
The advantage of this implementation is that there are no fixed mul-
tipliers. The desired fractional advance, α, can be thought of as a second
input to the filter. It can be tuned at any time, enabling a variable fractional
advance/delay.
It is of course trivial to re-wire the filter so that the input is the fractional
delay β = 1 − α rather than the fractional advance α:

y[n] = β( x [n − 1] − x [n]) + x [n]

This is shown in Figure 6.22.

6.6.1 Higher-order polynomials
As usual, better interpolation can be obtained at the expense of more com-
putation. Quadratic and cubic polynomial interpolation are very common
cases. Consider the cubic case. For this case, we use two samples to the



Figure 6.23: Fractional delay using cubic polynomials.

left and two samples to the right of where we wish to interpolate.∗ The
situation is depicted in Figure 6.23. Once we have four samples (such as
the first four input samples), we can interpolate, but we compute values
that lie between the two inner-most samples only (between the second
and third from the left). To interpolate, we fit a 3rd-order polynomial to
the four samples (the polynomial is unique) and we evaluate the polyno-
mial at the point we wish to interpolate. Then we advance one sample, the
leftmost sample is discarded and the right-most sample comes into play.
Once again, we fit the unique (new) 3rd-order polynomial to these four
samples. Once we have the polynomial, we interpolate by computing the
value of the polynomial between the two inner-most samples. After this,
we would advance another sample and so on.
As in the linear case, for higher-order polynomials, the Farrow struc-
ture makes use of Horner’s rule so that all coefficients in the filter are con-
stant while the fractional delay is an input that is tunable at run-time.

∗ As always, we must allow for sufficient delay in practice in order to make things causal.




Linear−Lagrange Farrow Filter
0 Cubic−Lagrange Farrow Filter

Magnitude (dB) −5

−10

−15

−20

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Figure 6.24: Magnitude response of linear and cubic fractional delays designed via the
Lagrange interpolation formula.

6.6.2 Design of Farrow fractional delays
The design of variable fractional delays using the Farrow structure and
Lagrange interpolation boils down to selecting the filter order (the poly-
nomial order). As we have already stated, the linear, quadratic, and cubic
cases are the most typical.
It is worth noting though, that for this structure, as the filter order N
increases, the number of coefficients increases as N 2 . The fractional delay
is specified in the design as a starting point. However, as we have already
said, the value can be changed at any time while the filter is running.

Example 54 Let us compare linear and cubic fractional delays using the La-
grange interpolation formula:

del = 0.5; % Desired fractional delay
Hf_lin = fdesign.fracdelay(del,1);
Hf_cub = fdesign.fracdelay(del,3);
Hlin = design(Hf_lin,'Lagrange');
Hcub = design(Hf_cub,'Lagrange');

As we know, ideal fractional delays should have an allpass magnitude re-
sponse, and a flat group-delay. Figure 6.24 compares the magnitude responses.



Group Delay

1.5

1.4
Linear−Lagrange Farrow Filter
1.3 Cubic−Lagrange Farrow Filter

1.2

1.1

1

0.9

0.8

0.7

0.6

0.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Figure 6.25: Group delay of linear and cubic fractional delays designed via the Lagrange
interpolation formula.

As can be seen, neither is a particularly good approximation to an allpass filter,
however, the cubic design fairs a little better. The group-delays are plotted in Fig-
ure 6.25. Both of them are ideal relative group-delays of 0.5. In the cubic case,
there is an extra one sample delay introduced by the higher order.

Simulink models for the Farrow filters resulting from these designs can
be obtained as usual by using the realizemdl() command (e.g. realizemdl(Hlin)
or realizemdl(Hcub) in the Example above).

6.6.3 Multirate Farrow filters
So far we have seen how to implement single-rate fractional delays with
Farrow filters. Farrow filters can be used for both increasing and decreas-
ing the sampling-rate of a signal (by either integer or fractional factors).
One of the key advantages of Farrow filters is that they can be used to
change the sampling-rate of a signal by an irrational factor without adding
complexity in terms of number of coefficients. In contrast, the previous
polyphase designs we had looked at could only change the rate by a fac-
tor given by the ratio of two integers. The best that could be done for
irrational factors, was to form an approximation to a rational number of
the form L/M. In order to approximate an irrational factor adequately, it



Figure 6.26: Multirate Farrow filter.

may be necessary for L and M to be very large integers. As a consequence
of this, the number of coefficients required may be quite large.
Nevertheless, it is worth keeping in mind that while the number of co-
efficients may be kept low by using Farrow structures, the actual number
of MPIS will depend on the sampling-rate conversion factor. To see this,
let’s describe how a multirate Farrow filter works.
Consider the modified diagram for the linear case of a Farrow frac-
tional delay shown in Figure 6.26. The dashed line separates the filter into
a section running at the input signal’s sampling-rate and a section run-
ning at the output sampling-rate. Note that we have re-labeled the output
to be y[m] rather than y[n]. This is due to different input and output rates.
Notably, the fractional delay, now denoted β m will now change at every
instant an output sample occurs.
Let’s walk through a simple case to get a feel for how this operates.
Let’s say that we are increasing the sampling-rate by a factor of 2. Since
for every input there are two outputs, the value held in the delay register
will be used twice. The first time an input is used, β m will take on the
value 0.5 and the output will be computed as

y[m] = 0.5( x [n − 1] − x [n]) + x [n] = 0.5x [n − 1] + 0.5x [n]

Before the input sample changes, one more output sample will be com-
puted. β m will take the value 0 and the output will simply be

y [ m + 1] = x [ n ];



Subsequently, the input sample will change, β m will be once again set
to 0.5 and so forth.
In summary, when increasing the sampling-rate by a factor of two, β m
will cycle between the values {0.5, 0} twice as fast as the input, producing
an output each time it changes.
In the general case, it is simply a matter of determining which values β
must take. The formula is simply

m fs
βm = mod 1
f s′

where f s is the input sampling rate and f s′ is the output sampling rate.

3f
Example 55 Let’s say we want f s′ = 5 s . Then β m will cycle through the values
{0, 2/3, 1/3}. The following code designs such a multirate filter (for the first-
order case) and filters some random data:

% Design first-order multirate Farrow filter
f = fdesign.polysrc(3,5);
f.PolynomialOrder = 1;
H = design(f,'lagrange');
% Filter some random data
x = randn(100,1);
y = filter(H,x);

Figure 6.27 shows partial plots of the input and output signals assuming f s =
1. Notice that the delay follows the values of β m that we have indicated.
Figure 6.28 shows partial plots for the cubic case rather than the linear case.
The improved smoothness of using a cubic polynomial is apparent from the figure.

6.6.4 Polynomial interpolation and maximally flat filtering
In this section we show that using polynomials as we have for piece-wise
curve fitting by interpolating N + 1 samples using an Nth-order polyno-
mial is equivalent to “traditional” interpolation from the signal processing
point of view if we use a maximaly-flat FIR filter as our lowpass filter.
To see this, consider interpolation by L = 4. Let’s first design a cubic-
polynomial Farrow filter to perform such interpolation.



Input signal
2
Output signal
Linear interpolant
1.5

1
Amplitude

0.5

0

−0.5

−1

−1.5

10 12 14 16 18 20 22 24
Time (sec)

Figure 6.27: Multirate Farrow ﬁltering using a linear polynomial.

2
Input signal
Output signal
1.5
Cubic interpolant

1
Amplitude

0.5

0

−0.5

−1

−1.5
8 10 12 14 16 18 20 22
Time (sec)

Figure 6.28: Multirate Farrow ﬁltering using a cubic polynomial.



% Design first-order multirate Farrow filter
L = 4;
M = 1;
N = 3; % Polynomial order
f = fdesign.polysrc(L,M);
f.PolynomialOrder = N;

To interpolate by 4, the fractional delay will take on values of {3/4, 1/2, 1/4, 0}
in a cyclic manner. If we set the fractional delay of the filter to each of those
four values and in each case compute the equivalent transfer function:

Hf = fdesign.fracdelay(0.75)
H = design(Hf,'lagrange');
[b0,a0] = tf(H)
H.FracDelay=0.5;
[b1,a1] = tf(H)
H.FracDelay=0.25;
[b2,a2] = tf(H)
H.FracDelay=0;
[b3,a3] = tf(H)

It is trivial to verify that b0,b1,b2,b3 correspond to the polyphase com-
ponents of the FIR filter returned by the intfilt() command:

bint = intfilt(L,N,'lagrange');

The magnitude response of the filter bint is shown in Figure 6.29. We
also show the ideal interpolation filter for reference. As previously noted,
the maximally flat characteristic has the undesirable effect of a wide transi-
tion band as a result. This tells us that polynomial interpolation produces
only acceptable filters for sampling-rate conversion applications (and frac-
tional delays). We had already seen in this in a different form in Figure
6.24, where we noted how far from an ideal allpass the fractional delays
produced by polynomial interpolation were.
In numerical analysis, better piece-wise curve-fitting is obtained by us-
ing techniques such as hermite polynomial interpolation and cubic splines.
In signal processing, typically we improve the interpolation by using fil-
ters with better transition-width than maximally-flat filters.



Magnitude Response
4
Maximally flat filter
Ideal interpolation filter
3.5

3

2.5
Magnitude

2

1.5

1

0.5

0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Figure 6.29: Magnitude response of a maximally-flat interpolate-by-4 filter correspond-
ing to cubic polynomial interpolation.

6.6.5 Using Farrow sample-rate converters in multistage de-
signs
We have seen how multistage/multirate designs can result in significantly
reduced implementation cost when compared to single-stage cases. How-
ever, multistage techniques are applicable only when the rate change fac-
tor is not a prime number. However, if we need to change the rate by a
prime number, we need not be “stuck” with a single-stage design. One
possibility, is to change the rate by a non-prime number near the prime
number we are looking for, and use a Farrow filter to change the final rate
so as to obtain the desired overall rate-change.

Example 56 Suppose we have a signal sampled at 680 kHz, but the band of inter-
est extends to 17 kHz only. We may have oversampled as signal as part of analog
to digital conversion for the purpose of simplifying the anti-aliasing analog filter.
We wish to reduce the bandwidth and the rate by decimating. The ideal decima-
tion factor would be M = 17. However, since 17 is a prime number, we can only
design a single-stage decimator such as:

Fs = 680e3; % 680 kHz
Fp = 17e3; % 17 kHz



TW = 6e3; % 6 kHz
Ast = 80; % 80 dB
Hf = fdesign.decimator(M,'Nyquist',M,'TW,Ast',TW,Ast,Fs);
Hd = design(Hf,'kaiserwin');
cost(Hd)

The cost of this design is of about 543 coefficients and 32 MPIS.
An alternative strategy may be to design a multistage filter that decimates by
16. In fact, due to the wide transition band of the Farrow filter that follows, we
may choose to have a smaller transition band here, say half the transition width:

Hf = fdesign.decimator(16,'Nyquist',16,'TW,Ast',TW/2,Ast,Fs);
Hmult = design(Hf,'multistage')
cost(Hmult)

The result is a multistage design with 4 halfbands in cascade. The cost is of
about 92 coefficients and 10 MPIS.
In order to reach the overall decimate-by-17 factor, we can design a cubic-
polynomial Farrow filter to perform a rate-change by a factor of 16/17:

f = fdesign.polysrc(16,17);

We can cascade this filter after the multistage decimator to achieve the goal we
want:

Hcas = cascade(Hmult,H);


Part II

Filter Implementation

Chapter 7

Implementing FIR Filters

7.1 Some basics on implementing FIR filters

There are several factors that influence the selection of the filter structure
used to implement an FIR filter. If the filter length is very large, it may be
preferable to use a frequency-domain implementation based on the FFT.
If it is to be implemented in the time domain, the target hardware is an
important factor.
From a design perspective, there is an important distinction between
minimum-phase and linear phase filters. Minimum-phase FIR filters do
not have any symmetry in their coefficients while linear phase FIR filters
have either symmetric or antisymmetric coefficients. Depending on the
target hardware, it may be possible to implement a linear-phase FIR filter
using less multipliers than the minimum-phase filter by taking advantage
of the symmetry even if the filter length of the linear-phase is larger. In
other words the implementation advantages of linear-phase filters may
offset the gains of a minimum-phase design making it preferable to use a
linear-phase filter even when linearity of phase is not critical for the appli-
cation at hand. Of course there are other reasons to use minimum-phase
filters that may give a compelling reason to do so. For instance, as we have
seen, minimum-phase filter introduce a minimal transient delay. This may
be important enough to stick with a minimum-phase design.

156 Implementing FIR Filters

7.1.1 Direct-form filter structure
Consider the design of an FIR filter,

Hf = fdesign.lowpass('N,Fp,Fst',3,.3,.8);
Heq = design(Hf,'equiripple','FilterStructure','dffir');

The filter order is very low for illustration purposes. The filter struc-
ture that is used to implement the filter is specified by the string 'dffir'
(which is the default) and corresponds to the so-called direct-form struc-
ture (also called a tapped delay line). To visualize the structure we can
create a Simulink block for the resulting filter using the realizemdl com-
mand,

realizemdl(Heq);

The resulting block is a subsystem. If we look inside the subsystem, we
will see the structure shown in Figure 7.1.
The number of delays is equal to the filter order and the number of co-
efficients (number of taps), which is one more than the number of delays,
determines the filter length.
The structure has some regularity in that a sample is read from memory
(a delay is simply a register), multiplied with a filter coefficient, and accu-
mulated to form the output. DSP processors have historically been built
with this multiply-accumulate (MAC) instruction in mind. The structure
requires a shift of the input data throughout all delays for each sample. To
minimize the amount of data copies, circular buffers are commonly used
[3].
A downside of the direct-form structure is that is does not take advan-
tage of the symmetry (or antisymmetry) in the coefficients of linear-phase
filters. So the cost of using this filter structure is maximum in terms of the
number of multipliers required as can be seen using cost(Heq). Never-
theless, many DSP processors have been architectured to implement this
structure efficiently, so that it should be used even for linear-phase filters
on that hardware. Of course the structure is well-suited for filters that
do not present symmetry in their coefficients such as minimum-phase FIR
filters.


7.1 Some basics on implementing FIR filters 157

Figure 7.1: A 4-tap FIR filter implemented in direct-form.

7.1.2 Symmetric direct-form filter structure
The design shown in the previous section was a linear-phase design. Since
the number of coefficients is four, there are two unique multipliers. The fil-
ter can be implemented using only two multipliers by using the symmetric
direct-form structure.

Hsym = design(Hf,'equiripple','FilterStructure','dfsymfir');
realizemdl(Hsym);
cost(Hsym)

The filter structure obtained in Simulink by using the realizemdl com-
mand is shown in Figure 7.2. In general, the number of multipliers re-



Figure 7.2: A 4-tap symmetric FIR filter implemented using the symmetric direct-form
filter structure.

quired is half∗ of the number required for the direct-form structure. Notice
that even though there are only two multipliers, there are still three delays
(same as for the direct-form structure) required since the number of delays
corresponds to the order of the filter.

7.1.3 Transposed direct-form filter structure
The direct-form structure has the disadvantage that each adder has to wait
for the previous adder to finish before it can compute its result. For high-
speed hardware such as FPGAs/ASICs, this introduces latency which lim-
∗ Rounded up if the number of coefficients is odd.


7.1 Some basics on implementing FIR filters 159

Figure 7.3: A 4-tap FIR filter implemented in transposed direct-form.

its how fast the filter can be clocked.
A solution to this is to use the transposed direct-form structure in-
stead. With this structure, the delays between the adders can be used
for pipelining purposes and therefore all additions/multiplications can be
performed in fully parallel fashion. This allows real-time handling of data
with with very high sampling frequencies.
The transposed structure for a 4-tap filter is shown in Figure 7.3. It
can be generated using the realizemdl command on a filter whose filter
structure has been selected to be transposed direct-form.

Htran = design(Hf,'equiripple','FilterStructure','dffirt');
realizemdl(Htran)



7.2 Fixed-point implementation
Implementing an FIR filter using fixed-point arithmetic is generally a rather
simple task. Some care must be taken when quantizing the coefficients so
that we use enough bits to achieve the desired stopband attenuation of
the filter and we scale the bits in the most beneficial manner (more on this
below).
Once the coefficients have been quantized, and given a fixed-point in-
put signal, it is easy to figure out what it would take to perform all mul-
tiplications/additions within the filter in a such a way that no roundoff
error is introduced (so-called full-precision mode).
With all arithmetic being performed with full precision, there will be bit
growth associated with filtering. In most cases, the output signal cannot
have the full wordlength resulting from this bit growth. We must throw
out bits at the output, usually to match the wordlength of the input signal.
To throw away bits, we simply remove LSBs. This will introduce some
quantization error at the output as we will see below. No overflow will
occur by throwing out LSBs.
Appendix D provides a brief review of fixed-point concepts that may
be useful prior to reading through the rest of this Chapter.

7.2.1 Quantizing the filter’s coefficients
The filter coefficients have to be quantized from double-precision floating
point in which they are designed into fixed-point representation with usu-
ally a smaller number of bits. We must make sure we make the most of the
limited number of bits we have.
The first thing to do is check if there are enough bits available to cover
the required dynamic range. When we quantize the impulse response of
an FIR filter, we should not expect to achieve the full 6 dB/bit∗ . This is not
surprising given the sinc-like shape of most FIR impulse responses. Most
of the values of the impulse response are small, so that the average signal
strength does not cover the available range.

Example 57 Let’s compute the SNR when quantizing the impulse response of an
FIR filter using 16 bits to represent the coefficients:
∗ See Appendix D.


7.2 Fixed-point implementation 161

Hf = fdesign.lowpass(0.4,0.5,0.5,80);
himp = Hd.Numerator; % Non-quantized impulse response
Hd.Arithmetic = 'fixed'; % Uses 16-bit coefficients by default
himpq = Hd.Numerator; % Quantized impulse response
SNR = 10*log10(var(himp)/var(himp-himpq))
SNR =
84.3067
When we quantize the impulse response of an FIR filter, we may use
the additive noise model to get an idea of how the frequency response is
affected. If the impulse response is given by h[n], the quantized impulse
response is
hq [n] = h[n] + e[n]
The additive noise, e[n], will affect the frequency response of the quan-
tized impulse response by adding a noise-floor of intensity ε2 /12, distort-
ing both the passband and the stopband of the filter. In most cases, the
effect on the stopband is most critical, reducing the attainable stopband
attenuation of the filter.
A good rule-of-thumb is that we can expect about 5 dB/bit when quan-
tizing an FIR filter, as long as we use the bits wisely. By this we mean, scale
them so that impulse response is representable with the bits we have with-
out wasting any bits.
The range of values that can be represented with a given wordlength/fraction
length combination falls between a negative and a positive power of two
(for instance the intervals [-0.125,0.125), [-0.5,0.5), [-1,1),[-8,8), etc)∗ . We
want to choose the smallest interval that contains the largest (absolute)
value of the impulse response. If we choose an interval larger than the
minimum, we are just wasting bits. If the largest value of the impulse re-
sponse is positive and a power of two, then it is somewhat of a judgement
call whether or not we want to allow for a small overflow error for that
one coefficient.
Example 58 Consider the following equiripple Nyquist design:
f = fdesign.nyquist(4,'TW,Ast',0.2,80);
∗ Note the open interval on the right. That’s because we cannot represent exactly that
value. How close we get will depend on the number of bits available.



The filter has a minimum attenuation of 80 dB. Its largest coefficient is 0.25.
Using the 5 dB/bit rule, we need at least 16 bits in order to provide the 80
dB of attenuation. Since the largest coefficient is 0.25, we can choose a fractional
length of 17 to scale the bits so that all values in the interval [-0.25,0.25) are
representable. In this example we choose to live with small overflow in the quan-
tization of the largest coefficient (we quantize the value to 0.25 − 2−17 ). Since the
value is a power of two, depending on the hardware, another option would be not
to implement this coefficient as a multiplier at all (instead simply perform a bit
shift). This is a nice property of all Nyquist filters, the largest coefficient is always
a power of two.
In order to set a 16-bit wordlength and a 17-bit fraction length, we perform
the following steps:

h.Arithmetic = 'fixed'; % Uses 16-bit wordlength by default
h.CoeffAutoScale = false;
h.NumFracLength = 17;

Note that there is automatic scaling of the coefficient bits by default. It is designed
to avoid overflow in the quantization while minimizing the interval so that the
bits are used as best possible. As we have said, strictly speaking the quantization
of the coefficient equal to 0,25 overflows with a fraction length of 17, so a fraction
length of 16 is used by default (which along with the wordlength, means any value
in the interval [-0.5,0.5) is representable without overflow).
The magnitude response of the quantized filter is shown in Figure 7.4. ∗ The
16 bits are adequate in order to preserve quite faithfully the intended magnitude
response of the filter.

To emphasize the point regarding the need to use both the right num-
ber of bits and use them wisely, consider what would have happened in
the previous example if instead of a fraction length of 17, we used a frac-
tion length of 15. The magnitude response for this case is shown in Figure
7.5. Notice that the quantized filter no longer meets the required 80 dB
stopband attenuation (the passband also has greater error than in the pre-
vious case).
∗ Note that this analysis along with most others (including the impulse response) only
takes into account the quantization of the coefficients. It does not take into account the
fact that there may be roundoff/overflow introduced by the multiplications/additions
when actually filtering data.



0

−10

−20

−30 Filter #1: Quantized
Magnitude (dB)

−40 Filter #1: Reference

−50

−60

−70

−80

−90

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Figure 7.4: Magnitude response of equiripple Nyquist filter quantized with 16-bit
wordlength and a fraction length of 17.

The reason we can no longer meet the stopband attenuation is that we
are essentially wasting two bits by using a fraction length of 15. The max-
imum roundoff error (assuming we round to the nearest quantized value)
can be as a large as 2−16 as opposed to 2−18 if we had used a fraction
length of 17. This increase in roundoff error results in raising the quanti-
zation noise floor in the frequency domain, reducing the attainable mini-
mum stopband attenuation.

Redundant bits in smaller coefficients
So far, we have assumed that the same number of bits are used to represent
each coefficient of an FIR filter. Because usually many of the coefficients
are quite smaller than the largest coefficient, many of the bits used for
these small coefficients are redundant and, depending on the hardware
used to implement the filter, could be removed without affecting the per-
formance.

Example 59 Let’s look again at what happens when we quantize the following
filter:



0

−10

−20

−30 Filter #1: Quantized
Magnitude (dB)

−40
Filter #1: Reference

−50

−60

−70

−80

−90

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Figure 7.5: Magnitude response of equiripple Nyquist filter quantized with 16-bit
wordlength and a fraction length of 15.

Hd.Arithmetic = 'fixed';
B = fi(Hd.Numerator,true,16,16);

The variable B contains the fixed-point numbers corresponding to the filter’s coef-
ficients. Because the largest value of the coefficients is about 0.434, they have been
quantized with a fractional length of 16 (the default wordlength is in turn 16 bits
which is in-line with the requirement for this filter following the 5dB/bit rule).
Now let’s look at the binary representation of say the first three coefficients:

B.bin
ans =
1111111111011100 1111111110100110 1111111111010100 ...

The repeated 1’s towards the left (the MSB) are all unnecessary in order to rep-
resent the value. Indeed, we could represent the first value simply as 1011100,
the second value as 10100110, and so forth. So in reality we need only 7 bits to
represent the first coefficient, 8 bits to represent the second coefficient, etc.



Indeed, note that 1111111111011100 with a fractional length of 16 corre-
sponds to the value -0.00054931640625. But this number can be represented
without loss of precision using only 7 bits:

C1 = fi(-0.00054931640625,true,7,16)
C1.bin

The full 16 bits are used for the largest coefficients. For instance for the middle
one (the largest one):

Hd.Numerator(30)
ans =
0.4339599609375
C30 = fi(0.4339599609375,true,16,16);
C30.bin
ans =
0110111100011000

which of course has no redundant bits.

The fact that we have redundant bits in many coefficients is another
way of understanding why we cannot achieve the full 6 dB-per-bit when
quantizing FIR filter coefficients. Effectively we don’t use all bits available
to represent most of the coefficients, thus the SNR decreases.
With certain hardware, it is possible to take advantage of the fact that
many coefficients have redundant bits by using smaller multipliers (in
terms of the number of bits) for these coefficients.

7.2.2 Fixed-point filtering: Direct-form structure
Consider an FIR filter implemented in direct form. Assume the input sig-
nal has been quantized already and the filter coefficients have been quan-
tized as well. It is possible to easily perform all additions and multipli-
cations within the filter structure in such a way that no roundoff error is
introduced.
If the number of bits used to represent the output is the same as the
number of bits used for the additions (the accumulations) then no round-
off error is introduced throughout the filter. This so-called full precision
mode represents the best possible result we can achieve when fixed-point



filtering with an FIR filter implemented in direct form. The only quanti-
zation error is due to the coefficient quantization (the quantization of the
input signal is considered separately as it is not affected by what happens
within the filter itself.
No overflows will occur in full precision mode because we assume we
will grow enough bits when adding (see below) to accommodate for signal
levels increasing.

Full precision products
In Appendix D, we saw that given two fixed-point numbers, one with
wordlength/fraclength given by { B1 , F1 } and the other with wordlength/fraclength
given by { B2 , F2 }, the product of the two numbers can be represented with
no roundoff by using a wordlength/fraclength of { B1 + B2 , F1 + F2 }.
For a direct-form FIR filter to be implemented with full precision, this
means that there will be bit growth once the input signal is multiplied with
the filter’s coefficients (see Figure 7.1).

Full precision additions
In Appendix D, we saw that in order to perform a series of N additions
without introducing round-off error, it is necessary to allow for a bit-growth
of ⌊log2 ( N )⌋ + 1 bits. In an FIR filter implemented in direct form, there are
a series of additions that occur after multiplying the input signal with the
coefficients (see Figure 7.1).

Example 60 Consider once again the following design:


The coefficients are represented with 16 bits. If the input is represented with 16
bits as well, full precision multiplications would mean that the values to be added
are represented with 32 bits. Since there are 58 additions to be performed, the
number of bits we need in order to perform the additions with full precision is

⌊log2 (58)⌋ + 1 = 6



As it turns out, it is possible to perform the additions with full-precision
by using even less bits than what it is specified by the formula ⌊log2 ( N )⌋ +
1. We will look at how to do so next.

Full precision multiplications/additions given the actual coefficient val-
ues
Let’s look again at the previous example to see if we actually need 32 bits
for the full-precision multiplications. The largest coefficient of the filter Hd
is 0.434. The coefficients are represented with { B1 , F1 } = {16, 16} and the
input is assumed to be represented with { B2 , F2 } = {16, 15}, i.e., the input
signal is assumed to fall within the interval [−1, 1).
The fraclength for the product is given by F1 + F2 = 31. Since the largest
possible value that any input value multiplied by any coefficient can pro-
duce is about 0.434, the interval used to represent the products must be
[−0.5, 0.5). This interval is covered by making the wordlength equal to
the fraclength. Since the fraclength is 31, the wordlength needed for full-
precision multiplications is also 31, not 32.
Now let’s move to full-precision additions. Let’s take a look at the
convolution equation for a length N + 1 FIR filter,
N
y[n] = ∑ h[m] x [n − m]
m =0

Given that we should know what are the maximum values that the
input can take, we want to determine what are the maximum numbers the
output can take. To do so, we use some simple math. We start by taking
the absolute value at both sides of the convolution equation,
N
|y[n]| = ∑ h[m] x [n − m]
m =0

Using the triangle inequality for the 1-norm, we have
N
|y[n]| ≤ ∑ |h[m] x[n − m]|
m =0
N
= ∑ |h[m]| |x[n − m]|
m =0



Let’s say that the input covers the range [− R/2, R/2). This means that
in the worst case,

R N R
|y[n]| = ∑ |h[m]| = h[n] 1
2 m =0 2

Thus the 1-norm of the impulse response provides a gain factor for the
output relative to the input. For example, if R/2 = 1, the maximum, value
the output can take is given by h[n] 1 . This tells us how many bits we
need to grow in order to perform full-precision arithmetic throughout the
filter.

Example 61 For the 59-tap filter Hd we have been using, we can compute the
1-norm of its impulse response as follows:
norm(Hd,'l1')
ans =
1.9904
This means that we need to ensure the additions can represent the range [−2, 2).
Since the multiplications fall in the interval [−0.5, 0.5), it is necessary to add
two bits in order to implement a full-precision filter. If we look at the values for
AccumWordLength and OutputWordLength when we write get(Hd), we see than
indeed these values have been set to 33 bits, i.e. two more bits than what is used
for the multiplications.

We have seen how we determine the number of bits required for full-
precision implementations. As a matter a fact, this value is still somewhat
conservative because it assumes the worst possible case for the input. If
we were to actually take into account typical values we think the input
can assume, we may even be able to reduce the required number of bits
further.

7.2.3 Fixed-point filtering: Transposed direct-form struc-
ture
In order to perform all operations with full precision, the settings for the
transposed direct-form structure shown in Figure 7.3 are almost identical
to the non-transposed direct-form case.



The only difference has to do with the wordlength necessary for the
state registers. In the direct-form case, the states needed to have the same
wordlength as the input signal in order to avoid roundoff. In the trans-
posed structure, the states must have the same wordlength as the wordlength
used for addition. This is typically more than twice the wordlength of the
input, so that the transposed structure clearly has higher memory require-
ments for a given filter.

7.2.4 Quantization of the output signal
In many cases, we need to reduce the number of bits at the output of the
filter for downstream processing. A common case is to use the same num-
ber of bits for the output as we did for the input. In the example above, the
input was represented with 16 bits, but the full-precision output required
33 bits. This means we may need to discard∗ 17 LSBs at the output of the
filter. The situation is depicted in Figure 7.6 for the direct-form structure.
The idea is the same for other structures. In that figure, we assumed that
the coefficients are represented with the same number of bits as the input
and all multiplications/additions are performed with full precision. G is
the number of bits that need to be added for the additions to be performed
without round-off as discussed above.
As discussed in Appendix D, the quantization can be modeled as addi-
tive white noise with variance ε2 /12 as long the number of bits discarded
is relative large (more than 3). For the direct-form structure, this means
we can replace the quantizer with an adder as shown in Figure 7.7.
In the example above, the full-precision version of Hd had 33 bits at the
output and a fractional length of 31. If we discard 17 LSBs, we will have
16 bits at the output and a fractional length of 14:
Hd.FilterInternals = 'specifyPrecision';
Hd.OutputWordLength = 16; % 33-16 = 17
Hd.OutputFracLength = Hd.OutputFracLength-17; % 14
The variance of the quantization noise that we introduced will be given
by:
∗ Note that by discarding, we do not necessarily mean simple truncation. It is possible
that when we discard, we perform some sort of rounding such as round to nearest. This
is controlled with the RoundMode setting once the filter internals have been set to specify
the precision.



Figure 7.6: Direct-form FIR filter with quantization at the output.

2ˆ(-14)ˆ2/12
ans =
3.1044e-10
We can also find this value by integrating the power-spectral density of
the output signal due to the noise signal e[n] by using (D.4). Note that the
transfer function between the noise input, e[n], and the output is simply
He (z) = 1. The integral of the PSD gives the average power or variance:
P = noisepsd(Hd);
avgpower(P)
ans =
3.0592e-10
This value is close to the theoretical value we expect from the additive
noise model as computed above.∗ The PSD is plotted in Figure 7.8. Note
that the PSD is approximately flat as it should be for white noise (the plot
can be obtained by calling the noisepsd command with no outputs).
∗ The PSD here includes a factor of 1/2π, therefore the intensity of the PSD is given by
the variance divided by 2π. When we integrate in order to compute the average power,
we should keep in mind that the variance has already been scaled by 2π.



Figure 7.7: Direct-form FIR filter with quantization at the output.

7.2.5 Evaluating the performance of the fixed-point filter
In order to evaluate how well the fixed-point filter compares to the ref-
erence floating-point filter, we can filter some test data and compare the
results. In order to isolate the behavior of the filter, it is important that the
same quantized input test data is used in all cases.
We would like to distinguish between three different cases:
• What we ideally would want to compute, yr [n]. This is the result of us-
ing the non-quantized original filter coefficients and performing all
multiplications/additions with infinite precision (or at least double-
precision floating-point arithmetic).

• The best we can hope to compute, yd [n]. This is the result of using the
quantized coefficients, but without allowing any further round-off
noise within the filter or at its output. All multiplications/additions
are performed with full-precision.

• What we actually compute, y[n]. This is the result we get with all final



Round−off Noise Power Spectrum
−100

−105

−110
Power/frequency (dB/rad/sample)

−115
Computed PSD
Theoretical PSD
−120

−125

−130

−135

−140

−145

−150
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 7.8: Power spectral density of the quantization noise at the output of an FIR filter.

fixed-point settings in our filter.

Clearly, what we actually compute can at best be the same as the best
we can hope to compute. In order to make the best we can hope to com-
pute closer to what we ideally would compute, we would have to use
more bits to represent the filter coefficients (and throughout the filter).

Example 62 Consider once again the following design:


As we have said, the quantized filter uses 16-bits to represent the coefficients by
default. A plot of the magnitude response of Hd shows that 16-bits are not quite
enough in this case to get the full 80 dB attenuation throughout the stopband.
Nevertheless, if we choose to use 16 bits for the coefficients, the best we can hope
to achieve can be computed for some test data as follows:

rand('state',0);
x = fi(2*(rand(1000,1)-.5),true,16,15);



Href = reffilter(Hd); % Compute reference filter
yr = filter(Href,x);
yd16 = filter(Hd,x); % Uses full-precision
var(yr-double(yd16))
ans =
4.8591e-10
The filter Hd performs all multiplications/additions using full precision because
the FilterInternals property is set by default to 'FullPrecision'.
If we decide to use say 18 bits for the coefficients instead of 16, we can get
closer to the result obtained with the reference filter,
Hd18 = copy(Hd);
Hd18.CoeffWordLength = 18;
yd18 = filter(Hd18,x); % Uses full-precision
var(yr-double(yd18))
ans =
2.4946e-11
Once we have settled the number of bits for the coefficients, we have
established a baseline for the best we can hope to achieve if we can’t use
full-precision. We then measure how well our actual results compare to
the best we can hope for.
Example 63 Continuing with our previous example. Suppose we can only keep
16 bits at the output and use 16 bit coefficients. In this example, this means we
need to remove 17 LSBs at the output as was shown in Figure 7.6. Once again,
this can be done as follows:
Hd.OutputFracLength = Hd.OutputFracLength-17;
If we now filter the same data and compute the energy in the error between the
best we can hope for and what we actually compute:
y = filter(Hd,x);
var(double(y)-double(yd16))
ans =
3.1004e-10
We of course obtain the same result we had previously computed for the variance
of the noise, i.e., 2ˆ(-14)ˆ2/12.



Magnitude Response Estimate

0 Full precision
16−bit output
−10 14−bit output
12−bit output
−20

−30
Magnitude (dB)

−40

−50

−60

−70

−80

−90

−100
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 7.9: Magnitude response estimates for various output wordlengths.

Using the magnitude response estimate to evaluate performance
Most of the analysis commands we perform on fixed-point filters only take
into account the coefficient quantization. This is true of the magnitude
response, impulse response, group-delay, and pole/zero computation.
The magnitude response estimate however, computes an estimate of
the magnitude response of the filter by actually filtering data through it.
The fixed-point filtering means that roundoff error may be introduced
because of discarding bits at the output or other roundoff if the addi-
tions/multiplications are not performed with full precision.
The closer the magnitude response estimate resembles the magnitude
response computed solely based on the quantization of the coefficients,
the more the actual filtering approaches the best we can hope to compute.
In this way, we can use the magnitude response estimate to evaluate the
performance of the fixed-point filter.

Example 64 Let us continue with the same design we have been using the last
several examples. If we start with full precision, the magnitude response estimate
is almost indistinguishable from the magnitude response:



−45
Full precision
14−bit output

−60

−65
Magnitude (dB)

−70

−75

−80

−85

−90

−95

−100
0.5 0.6 0.7 0.8 0.9 1

Figure 7.10: Stopband details of magnitude response estimates for various output
wordlengths.

Hd.Arithmetic = 'fixed'; % 16-bit coefficients
H = freqz(Hd);
He = freqrespest(Hd,1,'NFFT',8192);
norm(H-He)
ans =
0.0023
Now, we will compute the magnitude response estimate using always the same
16-bit coefﬁcients, but in each case discarding a different number of bits at the
output. Notice that we now use a relatively large number of trials, 50, in order
to get a reliable average from the estimate (the computation takes a while). We’ll
start by discarding 17 bits,
Hd.OutputFracLength = Hd.OutputFracLength-17;
He16 = freqrespest(Hd,50,'NFFT',8192);



We’ll repeat this for 14-bit and 12-bit outputs,

Hd.OutputWordLength = 14;
Hd.OutputFracLength = 12;
Hd.OutputWordLength = 12;
Hd.OutputFracLength = 10;

The plot of the various magnitude response estimates is shown in Figure 7.9. Note
the quantization at the output basically results in reduced stopband attenuation.
The stopband is shown in greater detail in Figure 7.10.


Chapter 8

Implementing IIR Filters

8.1 Some basics of IIR implementation
8.1.1 The use of second-order sections
All of the IIR designs we have discussed in Chapter 2 are performed us-
ing second-order sections by default. The full polynomials of the trans-
fer function are never formed. The main reason for this has to do with
hardware implementations of IIR filters. However, even from the design
perspective it is good practice to always keep the design in second-order
section and avoid computing the transfer function explicitly.
The reason is that forming the transfer function involves polynomial
multiplication, or equivalently convolution of the polynomial coefficients.
Convolution can introduce significant round-off error even with double-
precision floating-point arithmetic. This is mainly due to additions of large
numbers with small numbers (that become negligible when using finite
precision arithmetic).
Of course the effect gets worse as the filter order increases since we are
performing more and more convolutions. For low-order designs, it is not
absolutely necessary to use second-order sections, but there still may be
good reasons to do so from an implementation perspective.
Example 65 As a rather extreme example consider the following design:
Hc = design(Hf,'cheby1');
[b,a] = tf(Hc);

178 Implementing IIR Filters

0
Second−order sections
−100
Transfer function
−200

−300
Magnitude (dB)

−400

−500

−600

−700

−800

−900

−1000
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 8.1: A Chebyshev type I filter implemented in second-order sections and with the
transfer function.

The resulting filter order is very high (67). By forming the transfer function
using the tf command, we have completely distorted the response of the filter (see
Figure 8.1). Moreover, the filter has become unstable! The transfer function of the
polynomials is completely wrong as evidenced by the pole/zero plot (Figure 8.2).
Compare to the pole/zero plot of the second-order sections to see what it should
look like (Figure 8.3).

The most common second-order section structures are the direct-form I
and direct-form II. A two-section direct-form I filter is shown in Figure 8.4.
The structure uses more delays than the direct-form II structure because
the latter shares the delays between the numerator and denominator of
each section while the former does not. However, if we do not use the
scale value between sections, we can share the delays from one section to
the next, so that the direct-form I structure would only require two more
delays than the direct-form II rather than twice as many.

8.1.2 Allpass-based implementations
Second-order sections have been used for many years to implement IIR
filters. However, another possibility is to use allpass-based implementa-


8.1 Some basics of IIR implementation 179

Pole/Zero Plot
2 Filter #1: Zero
1.5
Filter #1: Pole

1
Imaginary Part

0.5

0

−0.5

−1

−1.5

−2

−4 −3 −2 −1 0 1 2
Real Part

Figure 8.2: Pole/zero plot of a Chebyshev type I filter implemented with the transfer
function.

Pole/Zero Plot
1
Filter #1: Zero
0.8
Filter #1: Pole
0.6

0.4
Imaginary Part

0.2
67
0

−0.2

−0.4

−0.6

−0.8

−1

−1.5 −1 −0.5 0 0.5 1 1.5
Real Part

Figure 8.3: Pole/zero plot of a Chebyshev type I filter implemented with second-order
sections.

tions. These have the advantage that they require less multipliers than
corresponding second-order sections implementations.
The idea is to implement the filter H (z) as the sum of two allpass filters



Figure 8.4: Direct-form I second-order sections structure.

A0 (z) and A1 (z),
1
H (z) = ( A0 (z) + A1 (z)) (8.1)
2
This decomposition, known as parallel allpass or coupled-allpass, can
be performed under certain conditions on H (z) [15], [16], [17]. It is know
that Butterworth, Chebyshev, and elliptic filters satisfy such conditions
and therefore can be implemented in such manner.
The allpass filters themselves can be implemented in various ways.
Lattice filters are one possibility. In the Filter Design Toolbox, the all-
pass filters are implemented using a minimal number of multipliers and
adders. In keeping with the good practices of not forming the full transfer
function, the allpass filters are implemented as cascades of low-order all-
pass sections. The delays are shared between sections whenever possible
in order to minimize the number of them required.

Example 66 To illustrate, consider the elliptic design of Example 26. When us-
ing the direct-form II second-order sections implementation we found that the
number of multipliers required was 20, the number of adders was also 20 and the
number of delays was 10. If we use cascaded low-order allpass sections,
He2 = design(Hf,'ellip','FilterStructure','cascadeallpass');
we can find through the cost function that the number of multipliers reduces to
12. The number of adders increases to 23, and the number of delays is 15.

To visualize the structure, we use the realizemdl command to gener-
ate a Simulink model of the filter. Even though the number of adders is
higher than with second-order sections, the saving in multipliers usually
more than compensates. Ultimately, this depends on the hardware used
to implement the filter. On a programmable DSP processor for instance,
an adder and a multiplier have the same cost. This is not the case with an
FPGA or an ASIC.



Example 67 For illustration purposes, we can easily get to the transfer function∗
of the allpass filters as follows:
Hf = fdesign.lowpass('N,F3dB,Ap,Ast',5,.3,1,60);
He = design(Hf,'ellip','FilterStructure','cascadeallpass');
[b1,a1] = tf(He.stage(1).stage(1)); % A0(z)
[b2,a2] = tf(He.stage(1).stage(2)); % A1(z)
Note that He.stage(1) contains the term A0 (z) + A1 (z) while He.stage(2)
is simply the 1/2 factor. If we look at the numerators b1 and b2, we can see that
they are reversed versions of a1 and a2 respectively. This is an indication of their
allpass characteristic.

Lattice wave digital filters
A variation of the cascaded allpass implementation is to use lattice wave
digital filters. The term lattice in this case refers to the fact that there are
two parallel allpass branches A0 (z) and A1 (z). It does not refer to the
lattice structure proposed by Gray and Markel [18].
To use these structures for the elliptic design of Example 26, it is simply
a matter of specifying the appropriate filter structure,
He3 = design(Hf,'ellip','FilterStructure','cascadewdfallpass');
This implementation requires the same 12 multipliers as the cascade
allpass does. However, the number of adders increases to 34. The number
of states is only 11, but the savings in number of states may not make up
for the additional adders required.
In terms of computational cost, the cascade allpass implementation
seems to be superior. However, the lattice wave digital filters are well-
scaled for fixed-point implementations, a fact that may make their use
compelling.
More about lattice wave digital filters can be found in [18] and [19] for
example.

8.2 Fixed-point implementation
Implementing IIR filters with fixed-point arithmetic is more challenging
than the FIR case. We have already seen reasons not to use the transfer
∗ This is not needed for implementation purposes.



function of IIR filters even before thinking of fixed-point implementation.
The case to avoid the transfer function becomes much stronger once fixed-
point is taken into account. The following example showcases this.

Example 68 Consider the following design:

He = design(Hf,'ellip'); % Already in SOS form

Let’s construct a direct-form II filter using the transfer function equivalent to the
SOS we have:

[b,a] = tf(He);
Htf = dfilt.df2(b,a);

Now let’s quantize the coefficients using 16 bits and compare the resulting mag-
nitude responses.

He.Arithmetic = 'fixed';
Htf.Arithmetic = 'fixed';

The magnitude responses are shown in Figure 8.5. Note that in this case, forming
the double-precision floating-point transfer function does not significantly alter
the magnitude response (try fvtool(b,a)). The deviation that we observe for the
transfer function in Figure 8.5 has to do with the quantization.

It is worth noting that the magnitude responses shown take into ac-
count only the quantization of coefficients. Any other round-off error in-
troduced by the fixed-point settings is not reflected in the magnitude re-
sponse analysis shown in fvtool.

8.2.1 Fixed-point filtering
In general, for fixed-point implementation, we want to stay away from
the transfer function because quantizing large polynomials introduces sig-
nificant errors when compared to quantizing second-order polynomials.
Nevertheless, with IIR filters, it is not enough to obtain a nice magnitude
response with the quantized coefficients.
Unlike FIR filters, it is not possible to implement IIR filters using full-
precision arithmetic since this would result in infinite bit growth due to




0

−10

−20
SOS filter
−30 TF filter
Magnitude (dB)

−40

−50

−60

−70

−80

−90

−100
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Figure 8.5: Comparison of magnitude response of the same filter quantized to 16-bit
coefficients. In one case, we use second-order sections (SOS) while in the other case we
use the transfer function (TF).

the feedback. Therefore, it is necessary to introduce quantization at the
feedback points within a filter structure. This quantization will of course
introduce roundoff error. In the worst case, the quantization produces an
overflow condition.
Nevertheless, as with FIR filters, we can consider the following com-
putations for the output:

• yr [n], what we ideally would want to compute. This is the result of us-
ing the non-quantized original filter coefficients and performing all
multiplications/additions with infinite precision (or at least double-
precision floating-point arithmetic).

• yd [n], the best we can hope to compute. Since full-precision is not an op-
tion with IIR filters, in order to obtain a baseline, we can use the
double() command on the filter to obtain a filter that has quan-
tized coefficients, but performs all multiplications/additions using
double-precision floating-point arithmetic. ∗
∗ Note that the commands reffilter() and double() differ in that the former uses the orig-
inal floating-point coefficients, while the later uses the quantized coefficients.



• y[n], what we actually compute,. This is the result we get with all final
fixed-point settings in our filter.

Let us consider the direct-form II second-order sections structure. For
each section, the feedback part comes before the shared states. Since we
need to quantize the result from the feedback loop prior to the values en-
tering the states, we want to ensure that round-off error at this point is
minimized.

Example 69 Let us repeat the previous design with a slight change. By default,
we scale and re-order the second-order sections in order to help obtaining good
fixed-point filtering results from the start. However, for now, let’s remove the
scaling/re-ordering to show how a good magnitude response is necessary but not
sufficient in order to obtain satisfactory results.

He = design(Hf,'ellip','SOSScaleNorm',''); % Don't scale

We’ll compute the ideal output, yr [n], and compare to the best we can hope
to compute, yd [n]. The difference between the two is due solely to coefficient
quantization.

rand('state',0);
x = fi(2*(rand(1000,1)-.5),true,16,15);
Hr = reffilter(He);
yr = filter(Hr,x); % Ideal output
Hd = double(He);
yd = filter(Hd,x); % This is the best we can hope for
var(yr-yd)
ans =
1.5734e-08

Now, let’s enable min/max/overflow logging and filter the same data with the
fixed-point filter:

fipref('LoggingMode','on')
ye = filter(He,x);
qreport(He)



If we look at the results from qreport, we can see that the states have overflown
37 out of 10000 times. Because of the overflow, it is not worth comparing ye to
yd.

In order to avoid overflow, we can use scaling to control the signal
growth at critical points inside the filter. The most stringent scaling, ℓ1 -
norm scaling, will ensure that no overflow can occur.

Example 70 Let us design the elliptic filter again, but using ℓ1 -norm scaling this
time. For now, we will disable re-ordering of the second-order sections:
s = fdopts.sosscaling;
s.sosReorder = 'none';
Hl1 = design(Hf,'ellip','SOSScaleNorm','l1',...
'SOSScaleOpts',s);
Hl1.Arithmetic = 'fixed';
Now let’s filter with this scaled structure,
yl1 = filter(Hl1,x);
qreport(Hl1)
Notice from the report that we have no overflows. Therefore we can compare the
output to our baseline in order to evaluate performance,
var(yd-double(yl1))
ans =
1.8339e-04

When doing fixed-point filtering, one needs to compromise between
maximizing the SNR and minimizing the possibility of overflows. The
ℓ1 -norm scaling we have used is good for avoiding overflows, but reduces
the signal levels too much, adversely affecting the SNR. A more commonly
used scaling (the default) is to use L∞ -norm scaling. However, this scaling
may result in overflow as the next example shows.

Example 71 Let’s design the filter and scale using L∞ -norm scaling. For now,
we still do not re-order the sections.
HLinf = design(Hf,'ellip','SOSScaleNorm','Linf',...
'SOSScaleOpts',s);
HLinf.Arithmetic = 'fixed';



Now let’s filter and look at the report.
yLinf = filter(HLinf,x);
qreport(HLinf)
The report shows that 8 overflows in the states have occurred.

So far, we have not re-ordered the second-order sections in the exam-
ples we have explored. Re-ordering can help improve the SNR, and, as the
next example shows, sometimes even help avoid overflow.

Example 72 Let’s once again use L∞ -norm scaling, but this time use automatic
re-ordering of the sections.
s = fdopts.sosscaling;
s.sosReorder = 'auto';
HLinf = design(Hf,'ellip','SOSScaleNorm','Linf',...
'SOSScaleOpts',s);
HLinf.Arithmetic = 'fixed';
Once again, let’s filter the same data, and look at the report.
yLinf = filter(HLinf,x);
qreport(HLinf)
Since we did not overflow, it is meaningful to compare the output to our baseline
to evaluate performance.
var(yd-double(yLinf))
ans =
1.4809e-05
As we can see, the result is more than an order of magnitude better than what we
got with ℓ1 -norm scaling.

Yet a less stringent scaling is to use L2 -norm scaling.∗ Yet the chances
of overflowing when using that type of scaling are even larger.
It is worth noting that in some applications, it is preferable to allow
for the occasional overflow in order to increase the SNR overall. The as-
sumption is that an overflow once in a while is not critical for the overall
performance of the system.
∗ This norm can be shown to be the same as ℓ2 -norm. This is Parseval’s theorem.



8.2.2 Autoscaling
The scaling we have seen so far is data agnostic. As a result sometimes
overflow occurs.
We can use the autoscale command in order to optimize the fixed-
point fractional length settings in a filter based on specific input data that
is given. Doing so, we can get even better results than if we use data-
agnostic scaling.

Example 73 Let’s try once again to design the filter without data-agnostic scal-
ing.

He = design(Hf,'ellip','SOSScaleNorm',''); % Don't scale

We know that this filter will overflow with the input data we have. However, if
we use this input data to autoscale,

Hs = autoscale(He,x);

and we filter the same data,

ys = filter(Hs,x);
qreport(Hs)

we can see from the report that now overflow has been avoided. We now evaluate
the output relative to our baseline,

var(yd-double(ys))
ans =
1.4144e-07

Which is of course the best result we have obtained so far.

Finally, if we combine L∞ -norm scaling, automatic re-ordering, and
autoscaling, we can easily verify that we can get even better results:

var(yd-double(ys2))
ans =
2.3167e-08



0 Best we can hope for
Autoscaled, re−ordered, L
∞
−10
Re−ordered L
∞
−20 Non−reordered l
1
−30
Magnitude (dB)

−40

−50

−60

−70

−80

−90

−100
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 8.6: Magnitude response estimates for various scaling cases.

8.2.3 Evaluating filter performance using the magnitude re-
sponse estimate
As we saw for FIR filters, one way to evaluate the performance of the
fixed-point IIR filter is to compute an estimate of the magnitude response
by filtering data.
Because the filtering is performed with fixed-point arithmetic, the mag-
nitude response estimate takes into account not only the quantization of
the coefficients, but also any roundoff error introduced within the filter.
The magnitude response estimate is particularly useful when the fixed-
point filter behaves in a weekly nonlinear manner.∗ This means that it
is most useful when no overflow occurs given that overflows introduce
significant nonlinear behavior.
As an example, Figure 8.6 shows the magnitude response estimate for
four cases corresponding to examples above. In all four cases shown, over-
flow is avoided. Once again, the roundoff error in the filtering has a net
effect of reducing the stopband attenuation. Moreover, unlike the FIR case,
∗ Quantization is a nonlinear operation. But a linear model that approximates the behav-
ior as additive white noise is adequate in many cases as we have seen.



2
Best we can hope for
Autoscaled, re−ordered, L
1.5 ∞
Re−ordered L
∞
1 Non−reordered l
1
Magnitude (dB)

0.5

0

−0.5

−1

−1.5

−2
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

Figure 8.7: Passband details of magnitude response estimates for various scaling cases.

the roundoff can also affect the passband response of the ﬁlter. The pass-
band details are shown in Figure 8.7.


Appendix A

Summary of relevant filter design
commands

A.1 Filter Design (fdesign)
A.1.1 Setup design specifications
Use fdesign objects to store design specifications. This will not design the
filter, only set the specs. In order to design you need to invoke the design
command (see below).

f = fdesign.lowpass('Fp,Fst,Ap,Ast',0.3,0.4,0.5,75);

To view all available frequency response designs (lowpass, highpass, etc)
type:

help fdesign/responses

To change specification set, it can be done a step at a time,

f.Specification = 'N,Fc,Ap,Ast';
f.FilterOrder = 60;
...

but it can also be done in one shot,

setspecs(f,'N,Fc,Ap,Ast',60,0.3,0.5,75);

192 Summary of relevant filter design commands

By default, frequency specifications are assumed to be normalized with an
implied scaling by π, so for example a value of 0.3 for Fc means that the
cutoff frequency is being set to 0.3π radians/sample. In order to specify
frequencies in Hertz, the sampling frequency must be appended,

setspecs(f,'N,Fc,Ap,Ast',60,30,0.5,75,200); % Fc = 30 Hz, Fs = 200 Hz

For a list of all available specification sets for a particular response,

set(f,'Specification')

A.1.2 Design options
To view available design methods for a particular response/specification
combination:

designmethods(f) % List all design methods
designmethods(f,'iir') % List only IIR designs
designmethods(f,'fir') % List only FIR designs

To design with default design options

h = design(f) % Use default design method
% For the following commands, h will be an array
h = design(f,'all') % Design for all design methods
h = design(f,'fir') % Design for all FIR methods
h = design(f,'iir') % Design for all IIR methods

To view design options and defaults for a particular design method:

designoptions(f,'equiripple')

For context-sensitive (response sensitive) help type:

help(f,'equiripple') % Specific help for lowpass/equiripple combo


A.2 Selecting filter structure 193

A.1.3 Design analysis/validation
To view various frequency-domain, time-domain, or pole-zero responses:

f = fdesign.lowpass('Fp,Fst,Ap,Ast',1.3e3,1.4e3,0.5,75);
h = design(f,'equiripple','StopbandShape','1/f','StopbandDecay',3);
fvtool(h)

In order to measure critical frequencies and/or ripple/attenuations:

measure(h)

To view a high-level implementation cost:

f = fdesign.halfband('TW,Ast',0.01,80);
cost(h)
ans =
Number of Multipliers : 463 % Number of non-zero and non-one coefficients
Number of Adders : 462 % Total number of additions
Number of States : 922 % Total number of states
MultPerInputSample : 463 % Non-zero/non-one multiplications per input sample
AddPerInputSample : 462 % Additions per input sample

For some general information about the design:

info(h)

A.2 Selecting filter structure
At design time, different filter structures can be specified. For FIR filters,
the default structure is direct-form, dffir. For symmetric filters (linear
phase), half the multipliers can be saved by using a symmetric FIR struc-
ture:

f = fdesign.lowpass('N,Fp,Fst',40,0.4,0.5);
h1 = design(f,'Wpass',1,'Wstop',10,'FilterStructure','dffir'); % default
h2 = design(f,'Wpass',1,'Wstop',10,'FilterStructure','dfsymfir');
cost(h1)
cost(h2)



For IIR filters, the default structure is direct-form II. IIR filters are de-
signed by default as cascaded second-order sections (SOS). Direct-form I
SOS structures use more states, but may be advantageous for fixed-point
implementations,

f = fdesign.lowpass('N,F3dB,Ap,Ast',5,0.45,1,60);
h1 = design(f,'FilterStructure','df1sos');
h2 = design(f,'FilterStructure','df2sos');
cost(h1)
cost(h2)

A.3 Scaling IIR SOS structures
By default, sos IIR designs using structures such as df1sos and df2sos are
scaled. The scaling uses L∞ -norm scaling, so that the cumulative magni-
tude response of the second-order sections (first section alone, first and
second sections, first-second-and-third, etc.) does not exceed one (0 dB):

f = fdesign.lowpass;
h1 = design(f,'ellip','FilterStructure','df1sos');
h2 = design(f,'ellip','FilterStructure','df2sos');
s = dspopts.sosview;
s.view='Cumulative'; % Set view to cumulative sections
fvtool(h1,'SOSViewSettings',s)
s.secondaryScaling = true; % Use secondary scaling point for df2sos

This scaling helps avoid overflows when the filter is implemented with
fixed-point arithmetic, but doesn’t totally eliminate the possibility of them.
To eliminate overflows, the more restrictive l1 -norm scaling can be used (at
the expense of reduced SNR due to sub-utilization of the dynamic range):

f = fdesign.lowpass;
h1 = design(f,'ellip','FilterStructure','df1sos','SOSScaleNorm','l1');
s = dspopts.sosview;
s.view = 'Cumulative';


A.4 Designing multirate filters 195

A.4 Designing multirate filters
To design multirate filters, select the multirate type, then the interpola-
tion/decimation factor(s), then any fdesign single-rate response,
f = fdesign.interpolator(4,'lowpass','N,Fc,Ap,Ast',60,1/4,0.5,75); % Interp by 4
f = fdesign.decimator(2,'halfband','TW,Ast',0.1,95); % Decim by 2
f = fdesign.rsrc(3,5,'nyquist',5,'TW,Ast',0.05,80); % Up=3 Down=5
After the specifications are set, the usual commands are available: design,
designmethods, designoptions, etc.
The cost command takes into account the polyphase savings of multi-
rate filters:
f1 = fdesign.interpolator(15,'lowpass','Fp,Fst,Ap,Ast',...
1/15-0.01,1/15+0.01,1,80);
h1 = design(f1,'multistage'); % Results in 2 stages
cost(h1) % Compare to a single-stage design: design(f1,'equiripple');
f2 = fdesign.decimator(8,'nyquist',8,'TW,Ast',0.03,80);
h2 = design(f2,'multistage'); % Result in 3 stages
cost(h2) % Compare to a single-stage design: design(f1,'kaiserwin');
% When applicable, the use of IIR halfbands results in huge implementation savings
% The implementation uses allpass-based polyphase branches
h3 = design(f2,'multistage','HalfbandDesignMethod','iirlinphase');
cost(h3) % If linear phase is not an issue, ellip halfbands would be even better

A.5 Converting to fixed point
In order to convert a design to fixed point, set the Arithmetic property,
h.Arithmetic = 'fixed';
or
set(h,'Arithmetic','fixed');
By default, the coefficients are quantized to 16 bits with automatic scal-
ing, and, for FIR filters, the filter internals (additions and multiplications)
are set in a such way that no loss of precision occurs (so-called “full pre-
cision”). The input signal to the filter is assumed to be a signal quantized



with 16 bits and covering the range [−1, 1). The LSB is scaled by 2−15 .
(Two’s-complement is assumed for all fixed-point quantities.)
Coefficient quantization affects the frequency (and time) response of
the filter. This is reflected in fvtool and measure,

f = fdesign.decimator(2,'Halfband','TW,Ast',0.01,80);
fvtool(h)
measure(h)
% Compare to
fvtool(h)
measure(h)

For FIR filters, in order to change fixed-point settings for the output and/or
additions/multiplications within the filter, change the filter internals first∗ ,

h.FilterInternals = 'SpecifyPrecision'

For example to view the effect solely due to the use of a smaller number
of bits for the output of a filter, one could write something like,

x = randn(1000,1); % Create a random input
xn = x/norm(x,inf); % Normalize to the range [-1,1]
xq = fi(xn,true,16,15); % Quantize to 16 bits. Cover the range [-1,1)
h = design(f,'equiripple','StopbandShape','linear','StopbandDecay',40);
yfull = filter(h,xq); % Full precision output (33 bits)
h.FilterInternals = 'SpecifyPrecision';
h.OutputWordLength = 16;
h.OutputFracLength = 13;
y16 = filter(h,xq); % 16-bit output
plot(double(yfull-y16)) % Plot the difference due to output quantization
∗ The LSB scaling for any fixed-point quantity can be changed by setting the appro-
priate “FracLength” parameter. Note that this quantity is not really a “length” since
it can be greater than the word length or even negative. For example a word length
of 4 bits and a “FracLength” of 8 means the bits (from MSB to LSB) are scaled by
−2−5 , 2−6 , 2−7 , and 2−8 respectively. (The negative scaling of the MSB is due to the
two’s-complement representation.)


A.6 Generating Simulink blocks 197

There are a couple of commands that can come in handy for analysis
when converting filters from floating point to fixed point: double, and
reffilter,
hd = double(h); % Cast filter arithmetic to double. Use quantized coeffs
hr = reffilter(h); % Get reference double-precision floating-point filter
The difference between hd and hr lies in the filter coefficients. The refer-
ence filter, hr, has the original “ideal” double-precision coefficients, while
the casted-to-double filter, hd, has a casted version of the quantized coeffi-
cients.
These functions can be used to compare what we ideally would like to
compute, with the best we can hope to compute∗ , and with what we can
actually compute. Continuing with the previous two examples:
yr = filter(hr,xq); % What we would ideally like to compute
yd = filter(hd,xq); % Best we can hope to compute. Same as yfull
plot(double([yr-yd,yr-y16]))

A.6 Generating Simulink blocks
There are two commands that can be used to generate blocks from floating-
point or fixed-point filters: block() and realizemdl():
f = fdesign.hilbert('N,TW',122,0.1);
h = design(f);
block(h)
f = fdesign.decimator(2,'halfband','TW,Ast',0.05,75);
h = design(f,'iirlinphase','FilterStructure','cascadeallpass');
realizemdl(h)
The difference between them is that block will use the Digital Filter block
(or one of the multirate blocks) from the Signal Processing Blockset while
∗ For FIR filters with multiplications/accumulations computed with more than 53 bits,
the best one can hope to compute should be determined with full-precision fixed-point
filtering rather than casting to double-precision floating point.



realizemdl will build the filter from scratch using basic elements like
adders/multipliers.
The use of the block command will generally result in faster simula-
tion speeds and better code generation. However, realizemdl provides
support for more filter structures and will permit customization of the re-
sulting block diagram (say to reduce the number of bits of some of the
multipliers of a filter or to rewire the filter in some way).

A.7 Graphical User Interface
Most of the commands discussed here are available through a GUI:

filterbuilder

Note that to design multirate filters, you should first select the response
(lowpass, halfband, etc) and then set the Filter Type to decimator, etc (if
available).
Once designed (and possibly quantized), the filter can be saved to the
workspace as an object named with the variable name specified at the top.


Appendix B

Sampling, Downsampling,
Upsampling, and Analog
Reconstruction

This appendix provides a short review of both theoretical and practical is-
sues related to sampling of a signal and reconstruction of an analog signal
from a sampled version of it. In general, understanding sampling and ana-
log reconstruction is essential in order to understand multirate systems. In
fact, as we will see, sampling an analog signal is just the extreme case of
downsampling a digital signal thereby reducing its sampling rate. Simi-
larly, analog reconstruction is just the edge case of increasing the sampling
rate of a signal.
The material presented here is standard in many textbooks∗ , but there
are a couple of practical issues that are often overlooked and that we wish
to highlight.
First, it is well-known that in order to avoid aliasing when sampling
it is necessary to sample at least twice as often as the highest frequency
component present. What is somewhat less well-known is that in lieu of
ideal brickwall anti-aliasing and reconstructing lowpass ﬁlters, we should
always sample a bit higher than twice the maximum frequency compo-
nent. As a rule of thumb, anywhere between 10 and 20 percent higher.
This is done to accommodate for the transition bands in the various ﬁl-
ters involved. Moreover, if we oversample by a factor of say two or four
∗ In particular the textbooks [3] and [11] are recommended in addition to the article [35].

200Sampling, Downsampling, Upsampling, and Analog Reconstruction

to ease the burden of the analog anti-aliasing filter, we still leave an extra
10-20 percent of room even though at this point we would be sampling
anywhere between 4 and 8 times higher than the highest frequency com-
ponent.
Moreover, since we know that the information in the excess frequency
bands is of no interest for our application, we allow for aliasing to occur
in this band. This enables us to sample at a rate that, while larger than
twice the frequency of interest, does not need to be as large as twice the
stopband-edge of the anti-aliasing filter. Equivalently, this allows for the
transition width of the anti-aliasing filter to be twice as large (resulting a
simpler, lower-order filter) than what it would be if did not permit aliasing
in the excess frequency bands.
As an example consider an audio signal. The frequency range of inter-
est is 0 to 20 kHz. Ideally we would bandlimit the analog signal exactly to
20 kHz prior to sampling with an anti-alias analog filter. We would then
sample at 40 kHz. Subsequently, in order to reconstruct the analog signal
from the samples, we would use a perfect lowpass analog filter with a cut-
off frequency of 20 kHz that removes all spectral replicas introduced by
sampling. In reality, we band-limit the signal with an analog anti-aliasing
filter whose passband extends to 20 kHz but whose stopband goes be-
yond 20 kHz by about 20 to 40 percent. Typical sampling rates for audio
are 44.1 kHz or 48 kHz, meaning that we have an excess bandwidth of
about 10 to 20 percent. The stopband-edge of the anti-aliasing filter will
be at about 24.1 kHz or 28 kHz respectively. These would also be the max-
imum frequency components of the audio signal that is bandlimited with
such anti-alias filter. Since we don’t quite sample at twice the maximum
frequency components, aliasing occurs. However, aliasing will only occur
in the excess frequency band, 20 kHz to 22.05 kHz or 20 kHz to 24 kHz
depending on the sampling frequency.
Other sampling rates used for audio are 88.2 kHz, 96 kHz, 176.4 kHz,
and 192 kHz. These sampling rates correspond to 2x or 4x oversampling,
but the important thing is that they still include an extra 10 to 20 percent
cushion in addition to the 2x or 4x oversampling.
The second issue we wish to highlight is that when choosing the stop-
band edge frequency for a decimation filter we can in many cases allow
for transition-band overlap (i.e. aliasing in the transition band). This issue
is related to the excess bandwidth resulting from the 10 to 20 percent over-
sampling that we have just mentioned and it is the reason the extra 10 to


B.1 Sampling an analog signal 201

20 percent is still present even if there is an oversampling of 2x or 4x.
Allowing for transition-band overlap (aliasing in the excess frequency
bands) when decimating means that the cutoff frequency (not the stopband-
edge frequency) for the decimation filter can be selected as π/M where M
is the decimation factor. This in turn means that efficient Nyquist filters
can be used for decimation. In particular, very efficient halfband filters
can be used when decimating by a factor of two.
We will elaborate on all this in §B.5.

B.1 Sampling an analog signal
We will provide only a brief overview of sampling. For a complete treat-
ment of the subject see [35]. Sampling is also covered in any signal pro-
cessing textbook (see e.g. [3]).
Consider an analog signal x a (t) with Fourier transform Xa ( f ) also called
its spectrum or frequency response,
∞
Xa ( f ) = x a (t)e−2πj f dt.
−∞

If we construct a sampled signal xs (t) from the original signal x a (t) in
the following way:
∞
xs (t) = ∑ x a (nT )δ(t − nT )
n=−∞

The spectrum of the sampled signal, Xs ( f ) is related to the spectrum of
the original signal by the Poisson summation formula,

1 ∞
Xs ( f ) = ∑ Xa ( f − k f s )
T k=−∞
(B.1)

where f s = 1/T is the sampling frequency.
Note that the sampled signal xs (t) is still an analog signal in the sense
that it is a function of the continuous-time variable t. Therefore, its spec-
trum is still given by the continuous-time Fourier transform,
∞
Xs ( f ) = xs (t)e−2πj f dt.
−∞



However, if we form the discrete-time sequence

x [n] = { x a (nT )} ∀n,

that is, we form a sequence by recording the value of the analog signal
x a (t) at each multiple of T, we can determine the continuous-time Fourier
transform of xs (t) by computing the discrete-time Fourier transform of
x [ n ],
∞
Xs ( f ) = X ( f ) = ∑ x [n]e−2πj f nT .
n=−∞

So even though xs (t) and x [n] are conceptually different, they have the
same frequency response.
Eq. (B.1) is valid for any signal x a (t) whether band-limited or not [3].
The terms Xa ( f − k f s ) are shifted versions of the spectrum of x a (t), and
are called spectral replicas. They are centered around multiples of f s . This
concept is key to understanding multirate signal processing so we’d like
to emphasize it. Any sampled signal has spectral replicas centered around
multiples of its sampling frequency. The higher the sampling frequency,
the further apart the spectral replicas will lie.
In general, when we form the sum of all the spectral replicas as in Eq.
(B.1), the replicas will interfere with each other due to frequency overlap-
ping. This interference is called aliasing. However, if the signal is ban-
dlimited in such a way that only one replica has a non-zero value for any
given frequency, then the replicas will not overlap and they will not inter-
fere with each other when we add them as in Eq. (B.1).
Obviously if Xa ( f ) = 0 ∀ f , the replicas will invariably overlap. How-
ever, if x a (t) is bandlimited, i.e. its spectrum is equal to zero for all fre-
quencies above a certain threshold and all frequencies below a certain
threshold, then by spreading the replicas enough apart from each other,
i.e. choosing a large enough value for f s , the replicas will not overlap
thereby avoiding aliasing.
It is easy to see that if the analog signal’s spectrum satisﬁes

fs
Xa ( f ) = 0, |f| > (B.2)
2
then the replicas will not overlap. Aliasing will thus be avoided if the
sampling frequency is at least equal to the two-sided bandwidth.



Figure B.1: Spectrum of band-limited analog signal with maximum frequency given by
f s /2.

Figure B.2: Spectrum of sampled signal with maximum frequency given by f s /2.

Figure B.1 shows the magnitude spectrum of an example of an analog
signal that satisfies Eq. (B.2). This case corresponds to the best known
case, i.e. f max = f s /2 and f min = − f s /2. The two-sided bandwidth, Bx , is
simply Bx = f max − f min = 2 f max = f s .
If the analog signal is not bandlimited, we must make it so by lowpass
filtering the signal with an analog filter prior to sampling. Such filter is
called an anti-alias filter for obvious reasons and its cutoff frequency must
be chosen such that it allows all frequencies we are interested in for a given
application to pass.
A critically sampled spectrum, i.e. one in which the sampling fre-
quency is exactly equal to the two-sided bandwidth is shown in Figure
B.2. The spectral replicas are adjacent to each other but do not overlap. If
we were to oversample the signal, that is if we choose f s′ > Bx , the sampled
spectrum would be something like what is depicted in Figure B.3. Notice
the “white-space” between replicas due to oversampling.



Figure B.3: Spectrum of an oversampled signal.

Figure B.4: Continuous-time signal with bandpass spectrum.

The interval [− f s /2, f s /2] is called the Nyquist interval. Since f s deter-
mines the number of samples per second, when comparing two Nyquist
intervals of different size, it is useful not only to realize that one encom-
passes a larger range of frequencies than the other, but also that it involves
a larger amount of samples. Therefore, in general, any digital processing
performed on a signal on a larger Nyquist interval requires more com-
putation than comparable signal processing performed on a signal that
occupies a smaller Nyquist interval. This means that we generally want
to avoid oversampling (i.e. avoid white-space in the spectrum). However,
oversampling can be useful in some particular cases.

The spectral replicas that appear in a sampled signal make the signal
periodic in the frequency domain. This periodicity means that any opera-
tion performed on a signal within the Nyquist interval will automatically
happen in all spectral replicas as well. We can take advantage of this fact
to efﬁciently process analog bandpass signal as is explained next.



Figure B.5: Spectrum of sampled bandpass signal with f s = 2 f c2 .

B.1.1 Bandpass sampling

Now consider the sampling of the bandpass signal depicted in Figure B.4.
If we were to sample at f s = f max − f min = 2 f c2 the resulting signal would
have the spectrum shown in Figure B.5. The white-space between replicas
tells us that the signal has been oversampled. In fact, for this signal we
can sample at a much lower rate as depicted in Figure B.6 and still avoid
aliasing.

Notice that by sampling at f s = 2 f c1 instead of at f s = 2 f c2 we have
effectively moved the bandpass spectrum to baseband. The Nyquist inter-
val being smaller, [− f c1 , f c1 ] rather than [− f c2 , f c2 ], means that if we want
to modify the signal in some way through signal processing algorithms it
will require less computation to do so than if we had sampled at f s = 2 f c2 .
Because of the periodicity of the spectrum, the processing we performed
at baseband is replicated in the bandpass replicas.

Notice that we have made some assumptions in order for bandpass
sampling to work. We have assumed that f c1 = f c2 − f c1 so that when we
sample at f s = 2 f c1 , all spectral replicas align nicely without overlapping.
This will happen for any f s such that f s = 2k ( f c2 − f c1), where k is some
positive integer. Nevertheless, aliasing may occur if we choose some other
sampling frequency even if it is larger than the minimal f s that can be
chosen to avoid aliasing.



Figure B.6: Spectrum of sampled bandpass signal with f s = 2 f c1 .

B.2 Sampling a discrete-time signal: downsam-
pling
We have seen that sampling an analog signal results in spectral replicas
and possible aliasing if the sampling frequency is not large enough.
We now move on to explore what happens when we sample a signal
that is already sampled, i.e. a discrete-time signal. More precisely, given a
sequence x [n], we downsample the sequence by a factor of M, where M is
a positive integer. In other words, we form the sequence xd [n] by keeping
one out of every M samples of x [n],

xd [n] = x [ Mn].

We will see that just as sampling an analog signal results in spectral
replication, sampling a discrete-time signal x [n] results in its spectrum,
X ( f ), being replicated as well. The replicas of X ( f ) are centered at multi-
ples of the sampling frequency of the downsampled signal.
Just as with sampling, the spectral replicas that form the spectrum
of a downsampled signal will overlap in general and generate aliasing
when added together. However, once again just as with sampling, if the
two-sided bandwidth is less or equal to the downsampled rate (within a
Nyquist interval), the spectral replicas do not overlap and no aliasing oc-
curs.
The fact that aliasing may occur when downsampling should be ob-
vious. After all, if x [n] was sampled at a frequency just high enough to
avoid aliasing, throwing out M − 1 out of every M samples is equivalent


B.2 Sampling a discrete-time signal: downsampling 207

Figure B.7: Critically sampled signal.

Figure B.8: Three times oversampled signal.

to having sampled the analog signal at a frequency M times lower than
what is required to avoid aliasing. On the other hand, if x [n] has been
oversampled by at least a factor of M, then no aliasing should occur when
downsampling.
In order to derive the relation between the spectra, consider two sam-
pled versions of x a (t). In one case, we sample with a sampling frequency
f s = 1/T, xd [n] = { x a (nT )}, while in the other case we sample with a sam-
pling frequency f s′ = M f s = 1/T ′ = M/T, x [n] = { x a (nT ′ )}.
For illustration purposes, assume (without loss of generality) that f s
corresponds to the critically sampled case so that all spectral replicas are
adjacent to each other. The signal x [n] is oversampled by a factor M.
As an example assume M = 3. The spectrum of xd [n], Xd ( f ), would be
something like that shown in Figure B.7. The spectrum of x [n], X ( f ), has
the same baseband replica, but the remaining replicas are centered around
multiples of f s′ as shown in Figure B.8.
In order to construct Xd ( f ) from X ( f ), we add X ( f ) with two shifted
versions of it, X ( f − f s ) and X ( f − 2 f s ). The resulting spectrum is shown
in Figure B.9.
Of course we can also show this formally for any value of M through



Figure B.9: Spectrum resulting from downsampling an oversampled signal.

the use of the Poisson summation formula. We know that
1 ∞
Xd ( f ) = ∑ Xa ( f − k f s )
T k=−∞

and
1 ∞
X( f ) = ∑ Xa ( f − k f s′ )
T ′ k=−∞
after some algebraic manipulation, we can arrive at the discrete version of
the Poisson summation formula,

1 M −1
Xd ( f ) = ∑ X ( f − k f s ).
M k =0
(B.3)

Note the scaling factor 1/M. When we downsample a signal, the base-
band replica retains the same shape (hence the same information) as long
as no aliasing occurs. However, all replicas are scaled down by the down-
sampling factor.

B.2.1 Filtering to avoid aliasing when downsampling
If the bandwidth of the signal x [n] is larger than 1/M times the Nyquist
interval, aliasing will occur if such signal is downsampled by a factor of M.
In order to avoid aliasing, a standard result in multirate signal processing
is to pass the signal x [n] through a lowpass filter prior to downsampling.
The role of this filter is akin to that of the analog anti-alias filter used to
bandlimit an analog signal prior to sampling. The cutoff frequency for the
filter should be π/M.


B.3 Increasing the sampling rate of a signal 209

While this result is correct, reversing the thinking on this is perhaps
more useful and sometimes overlooked. By filtering a signal in a way
that its bandwidth is reduced we are basically acknowledging that we no
longer are interested in part of its bandwidth (more precisely in the in-
formation contained therein). Therefore, if we reduce the bandwidth of a
signal by a factor of M, we should reduce its sampling rate accordingly
in order to reduce the size of the Nyquist interval. The point is we want
to eliminate white-space in the spectrum since we know that white-space
is akin to an oversampled signal and the larger Nyquist interval that goes
with an oversampled signal implies extra (unnecessary) computation.
The distinction is subtle but important. We are used to thinking: “I
need to downsample therefore I should filter first”. That’s fine, but we
should also think: “I have reduced the bandwidth of a signal therefore I
should downsample”. In fact, as we will see in Chapter 4, we should do
this even if the factor by which we reduce the bandwidth is not an integer.

B.2.2 Downsampling bandpass signals
Just as with sampling, under certain conditions it is possible to downsam-
ple a bandpass signal and avoid aliasing. If the signal is not bandpass
already, we can filter it with a bandpass filter that reduces the bandwidth
to 1/M times the Nyquist interval and then downsample it as long as the
passband of the resulting signal lies in one of the intervals

kπ/M < |ω | < (k + 1)π/M k = 0, 1, . . . , M − 1.

The downsampling will cause the bandpass spectrum to be moved to
baseband. This is a useful technique that allows a spectrum to be trans-
lated in frequency without the need for any multiplications. Without down-
sampling, it would be necessary to multiply the bandpass signal by a si-
nusoid if we wanted to translate its spectrum to baseband.

B.3 Increasing the sampling rate of a signal
We have seen that sampling the same analog signal at two different rates
produces the same baseband spectrum plus replicas centered around mul-
tiples of the sampling frequency. Figure B.7 shows the spectrum of a crit-



ically sampled signal, while Figure B.8 shows the spectrum of the same
signal oversampled.
In those figures, the oversampling factor is 3, but the general principle
applies to any integer oversampling factor L. The question is how to in-
crease the sampling rate of a signal without converting the signal back to
the analog domain and resampling at a higher rate.
Increasing the sampling rate is just a digital lowpass filtering problem.
The filter must keep the baseband spectrum and all replicas of Figure B.8,
but remove all remaining replicas, i.e. for the case of 3x oversampling,
remove all replicas in Figure B.7 that are not in Figure B.8.
In order to do so, the lowpass digital filter needs to operate at the high
sampling rate so that it has replicas of its own centered around multiples
of the high sampling rate that will preserve the sampled replicas at such
frequencies.
It is well-know that the cutoff frequency for the lowpass filter should
be π/L. However, rather than memorizing that formula it is helpful to
realize that the cutoff frequency must be selected in such a way that no in-
formation is lost. This should be obvious since we are trying to reconstruct
samples from the analog signal based on the samples we have. Therefore,
the baseband spectrum must be left untouched. The lowpass filter must
remove the L − 1 adjacent replicas.
The (ideal) lowpass filter specifications used to increase the sampling
rate by a factor L are thus,

L if | f | ≤ f s /2,
H( f ) = (B.4)
0 if f s /2 < | f | ≤ L f s /2.

where f s is the sampling rate before we increase the rate and L f s is the
sampling rate that results after increasing the rate. The gain of L in the
passband of the filter is necessary since as we saw the spectrum of the
low-rate signal has a gain that is L times smaller than that of the high-rate
signal.
The frequency response of the filter used to increase the sampling rate
by a factor L = 5 is shown in Figure B.10. The filter removes L − 1 spec-
tral replicas between the replicas centered around multiples of the high
sampling rate L f s .


B.4 Reconstructing an analog signal 211

Figure B.10: Frequency response of filter used to increase the sampling rate by a factor
L = 5.

B.4 Reconstructing an analog signal
Once signal processing algorithms have been performed on a sampled sig-
nal in order to modify it in some desired way, it may be necessary to re-
construct the analog signal corresponding to the samples at hand.
Just as with increasing the sampling rate of a signal, the mechanism to
reconstruct an analog signal is obvious if we look at the situation in the
frequency-domain. We’d like to take X ( f ), the spectrum of the sampled
signal x [n], and obtain Xa ( f ), the spectrum of the band-limited analog
signal x a (t). Since

1 ∞
X ( f ) = Xs ( f ) = ∑ X a ( f − k f s ).
T k=−∞

In the most common case, we need to remove all the replicas, Xa ( f −
k f s ), k = 0 and only retain the baseband spectrum Xa ( f ). However, if we
sampled a bandpass signal, and we wish to reconstruct it, we would want
to retain a replica for k = 0 instead.
In order to retain the baseband spectrum, we need a lowpass filter. The
lowpass filter must be analog in order for it to not have spectral replicas
of its own and thus be able to remove replicas centered at arbitrarily large
multiples of f s .
The role of the lowpass filter H ( f ) is depicted in Figure B.11. Its speci-
fications are
T if | f | ≤ f s /2,
H( f ) = (B.5)
0 otherwise.



Figure B.11: Spectral characteristics of lowpass filter used for analog reconstruction.

B.5 Practical sampling and analog reconstruction
The fact that we can reconstruct a band-limited analog signal from its sam-
ples provided we sample the signal frequently enough leads us to con-
clude that no information is lost when we sample the signal in such man-
ner. However, a truly band-limited signal requires a filter with infinite at-
tenuation in its stopband. Moreover, the signal must be of infinite length
in order to have a finite frequency spectrum, so even if could construct a
filter with infinite attenuation, it would not be enough to obtain a truly
band-limited signal. In reality there is always some degree of aliasing in-
troduced when sampling.
Nevertheless, let us assume that we had a perfectly band-limited ana-
log signal that we have sampled in such a way that no aliasing occurs. In
order to reconstruct the analog signal, we would require an ideal lowpass
filter with infinite attenuation and a perfect brickwall response with no
transition gap between passband and stopband.
In reality, both the anti-aliasing filter used to band-limit a signal prior
to sampling and the lowpass filter used for analog reconstruction have
both a finite amount of stopband attenuation and a non-zero transition
gap between passband and stopband.
The finite attenuation can be dealt with by achieving enough attenua-
tion so that the degrees of aliasing and distortion are tolerable. In many
cases, oversampling techniques are used so digital filters provide some of
the burden of attenuating the unwanted frequency components. Different
applications may require different attenuation levels.
In order to deal with the non-zero transition of the filters, excess band-
width is required. In other words, in practice, instead of sampling at twice
the largest frequency of interest, we give ourselves some extra room by


B.5 Practical sampling and analog reconstruction 213

Figure B.12: Practical sampling with excess bandwidth.

sampling at a slightly larger rate. The idea is illustrated in Figure B.12.
The upper graph in the Figure shows the analog spectrum | Xa ( f )| after
the analog signal has been filtered with an analog anti-aliasing prefilter.
The anti-alias filter has a passband that coincides with the band of interest.
Since the filter is non-ideal, it has a transition band that extends beyond
the band of interest. The further we can push out the transition band,
the easier it is to construct the analog prefilter (a lower order is required,
which in turn means fewer analog components). Notice that the sampling
frequency is not selected to be twice the stopband-edge of the anti-alias
prefilter, f max . It is selected to be twice the frequency that lies in the mid-
point between the maximum frequency of interest and f max . As a result,
sampling introduces aliasing in the excess frequency band. However, the
aliasing does not corrupt any of the frequencies of interest. The magnitude
spectrum of the signal sampled in the way just described, | X ( f )|, is shown
in the lower graph of Figure B.12.
The reason we select the sampling frequency in such a way that aliasing
occurs in the excess bandwidth region is that we would like to keep the
sampling frequency as low as possible. This is because we will have a
smaller number of samples to deal with. Any signal processing algorithms



performed on the sampled signal will be less computationally intensive as
a result of having fewer samples.
Once the signal needs to be converted back to the analog domain, a
lowpass analog reconstruction filter is needed. The design of this filter also
takes advantage of the excess bandwidth available. Figure B.13 shows the
requirements for a reconstruction filter. The filter takes the sampled signal
with magnitude spectrum | X ( f )|, and reconstructs the analog signal with
magnitude spectrum | Xa ( f )| by removing all spectral replicas. As with
the anti-alias prefilter, the passband of the reconstruction filter should ex-
tend up to the highest frequency of interest. The transition band require-
ments will depend on how harmful the frequencies present in the excess
bandwidth are for the application at hand. If necessary, the frequencies
between the band of interest and f s /2 can be attenuated digitally with a
digital lowpass filter prior to analog reconstruction. This would allow a
larger transition region for the reconstruction filter, making it simpler to
build.
Although we won’t get into details, it is worth noting that the recon-
struction filter is usually built in two stages. The first stage is usually a
simple staircase reconstruction that converts the signal to analog by hold-
ing each sample’s value until the next sample. This simple filter has a
poor lowpass response that allows for some high-frequency components
to remain. These high frequency components are removed by the second
stage filter. Increasing the sampling rate of the signal digitally prior to re-
construction further eases the job required by the reconstruction filter. We
will discuss this next.

B.5.1 Oversampling
As we pointed out in the precious section, the further we can push out the
stopband-edge frequency of the anti-alias analog filter, the easier it is for
such filter to be built. The idea can be extended to 2x or 4x oversampling
(or more). In such cases, the stopband-edge can be pushed out all the way
to two times or four times the maximum frequency of interest plus an extra
10 to 20 percent cushion.
There are two main consequences to doing this. First, since we over-
sample, we have more samples to deal with than we’d like. Usually we
avoid having more samples than necessary, but in this case we purposely



Figure B.13: Practical reconstruction with excess bandwidth.

generate more samples in order to simplify the anti-aliasing filter. Second,
since the anti-alias filter has a transition band that extends all the way to
twice or more the band of interest, there is poor out of band attenuation,
i.e. many undesired frequencies are present.
In order to deal with both of these issues, we use a combination of
digital lowpass filtering plus downsampling. The filtering takes care of
the poor out of band attenuation and allows for the downsampling to oc-
cur without aliasing corrupting the band of interest. Aliasing does occur
when downsampling, but it is limited to the transition band of the digital
filter, which overlaps with its spectral replicas. The whole idea is illus-
trated for the case 2x oversampling in Figure B.14. For this 2x case, the
digital filtering can be performed with a very efficient halfband filter. The
magnitude spectrum | X ( f )| shown in the Figure corresponds to the over-
sampled signal. The digital-lowpass-filtered signal’s spectrum is labeled
| X ′ ( f )|, while the downsampled signal’s spectrum is labeled | Xd ( f )|. The
implementation of the digital filtering and downsampling is always done
in an efficient manner (polyphase or otherwise) as explained in Chapter
4. Note that the spectrum | X ′ ( f )| is conceptual since in practice we do
not filter first and then downsample (that would be wasteful). It is shown
for illustration purposes. In reality we go directly from | X ( f )| to | Xd ( f )|



Figure B.14: Illustration of 2x oversampling plus decimation.

using an efficient decimation filter.
As we just saw, oversampling eases the job of the anti-aliasing ana-
log prefilter by placing some of the burden on the filter used for decima-
tion. Similarly, increasing the sampling rate can be used to ease the job of
the analog reconstruction filter when we are performing digital to analog
conversion. The idea is completely analogous to that of oversampling in
analog to digital conversion. The burden of removing spectral replicas is
shared between the digital filter used for increasing the sampling rate and
the analog reconstruction filter.
The digital filter removes the adjacent replicas. If the sampling rate
is increased by a factor of L, the filter removes L − 1 adjacent replicas.
The analog reconstruction filter has to removed the remaining replicas that
cannot be removed by the digital filter. Nevertheless, since the adjacent
replicas have been removed, this allows for a wide transition width for the



Figure B.15: Analog reconstruction by first increasing the sampling-rate by a factor of
L = 2.

analog reconstruction filter which, once again, enables us to use a simple
low-order filter that is easy (and cheap) to build.
The situation for the case L = 2 is illustrated in Figure B.15. | Xd ( f )|
represents the magnitude spectrum of the digital signal we wish to con-
vert to analog. This signal has been sampled (possibly downsampled as
well) in such a way that the excess bandwidth overlaps with the replicas
as previously discussed. |Y ( f )| represents an optional step. Depending
on the application, we may need to filter some of the components in the
band outside of the band of interest. This should be done at the lowest
rate possible so that the transition width relative to the sampling rate is
as large as possible (lowering the order required for the filter). Next, we
increase the sampling rate by two. The magnitude response of the filter
used to remove the adjacent replicas is shown by the dashed line. Note
that this filter also attenuates components out of the band of interest. It



can be designed so that it does all the work done to get |Y ( f )| in addi-
tion to removing spectral replicas in one step. The resulting magnitude
spectrum of the upsampled signal is represented by |Yu ( f )|. The remains
spectral replicas are now widely separated so that an analog reconstruc-
tion filter with a wide transition band can be used for the final digital to
analog conversion.
As mentioned before, the analog reconstruction filter is usually built
in two stages. The staircase reconstruction filter causes some distortion
in the band of interest. The more we increase the sampling rate prior to
reconstruction, the less the distortion. If necessary, the distortion can be
compensated for by pre-equalizing the signal digitally prior to sending it
through the staircase reconstructor. To do so, the filter used for increasing
the rate is designed in such a way that it boosts the signal over a part of
the band of interest that will be attenuated by the staircase reconstructor.
Finally, we mention that in practice noise-shaping sigma-delta quan-
tizers are usually used in conjunction with the techniques outlined here in
order to reduce the number of bits required to represent the signal at high
sampling rates. The noise-shaping allows for the use of a smaller number
of bits without decreasing the signal-to-noise ratio.


Appendix C

Case Study: Comparison of
Various Design Approaches

Consider the following lowpass specs:

1. Cutoff frequency: 3.125 MHz

2. Transition width: 0.4 MHz

3. Maximum passband ripple: 1 dB


5. Input sampling frequency: 100 MHz

The table below compares the implementation cost of various single-
rate designs. The following command is used for the speciﬁcations:

Fs = 100e6;
TW = 0.4e6;
Fp = 3.125e6 - TW/2;
Fst = 3.125e6 + TW/2;
Ap = 1;
Ast = 80;
Hf = fdesign.lowpass(Fp,Fst,Ap,Ast,Fs);

220 Case Study: Comparison of Various Design Approaches

NMult NAdd Nstates MPIS APIS NStages Decim. Fact
ellip 20 20 10 20 20 1 N/A
equiripple 642 641 641 642 641 1 N/A
IFIR 138 136 759 138 136 2 N/A
IFIR JO 115 113 680 115 113 2 N/A

IFIR JO denotes that the joint optimization option is used with the IFIR
design. See Chapter 5 for more on this.
Next, we compare 3 designs that include decimation by a factor of 15.
The following command sets the specs.:

Hf = fdesign.decimator(15,'lowpass',Fp,Fst,Ap,Ast,Fs);
equiripple 642 641 630 42.8 42.7333 1 15
IFIR JO 224 222 216 17.333 16.933 2 15
multistage 169 167 161 14.6 14.3333 2 15

Finally, we compare one CIC multistage design and 3 multistage Nyquist
designs. This line is used to set the specs for the Nyquist designs:

Hf = fdesign.decimator(16,'nyquist',16,TW,Ast,Fs);

For the CIC design, we break it down in 3 stages. The ﬁrst stage, which
is the CIC decimator, will provide a decimation of 4. The next stage, the
CIC compensator, will provide decimation of 2. The ﬁnal stage will be a
regular halfband, also providing decimation by 2.
The code is as follows:

% CIC decimator design
M1 = 4;
D = 1; % Differential delay
Hf1 = fdesign.decimator(M1,'cic',D,Fp,Ast,Fs);
Hcic = design(Hf1,'multisection');
% CIC compensator design
M2 = 2;
Nsecs = Hcic.NumberOfSections;
Hf2 = fdesign.decimator(M2,'ciccomp',D,...
Nsecs,Fp,Fs/(M1*M2)-Fp-TW,Ap,Ast,Fs/M1);
Hcomp = design(Hf2,'equiripple');


221

% Halfband design
M3 = 2;
Hf3 = fdesign.decimator(M3,'halfband',...
TW,Ast,Fs/(M1*M2));
Hhalf = design(Hf3,'equiripple');
Hcas = cascade(Hcic,Hcomp,Hhalf); % Overall design

Specifying the Nyquist band to be 16 implies that the cutoff frequency
π
is 16 which corresponds to the 0.0625π required. This value is almost al-
ways set to be the same as the decimation factor.
All Nyquist designs result in a 4-stage ﬁlter with halfband decimators
in each stage. Note that the Nyquist designs far exceed the passband rip-
ple speciﬁcation (1 dB) even though it is not set explicitly. In the case of
IIR halfbands, the passband ripple is of the order of 10−8 dB.

CIC multistage 86 94 166 6.0625 12.125 3 16
FIR equirip 98 94 180 10.325 9.375 4 16
IIR ellip 14 28 22 2.3125 4.625 4 16
IIR lin phase 37 74 80 3.8125 7.625 4 16


Appendix D

Overview of Fixed-Point
Arithmetic

Overview

In this appendix we provide a brief overview∗ of fixed-point arithmetic
and its implications on digital filters.
The basic trade-off that one should understand whenever dealing with
fixed-point arithmetic is maximizing SNR vs. minimizing the possibility
of overflows occurring. As is so often the case, these two requirements are
in opposition to each other and a compromise needs to be reached.
The SNR in this realm refers to signal to quantization noise ratio. The
SNR can be increased in two ways: increasing the signal power by max-
imizing the use of the available dynamic range or decreasing the noise
power by using more bits to represent data. Of course, using more bits
is not always an option. Hardware constraints may pre-determine how
many bits are available to represent a given quantity.
The issue with increasing the utilization of the available dynamic range
is that a signal can easily overflow when some arithmetic is performed on
it.

∗ For a more complete coverage, Chapter 2 of [3] and Chapter 8 of [36] are recommended.

D.1 Some fixed-point basics 223

D.1 Some fixed-point basics
Consider a register used to store a fixed-point number,

b B −1 b2 b1 b0
...
B−bits

The register has B bits (it has a wordlength of B), the value of the kth bit
(from right to left) is given by bk which can obviously be only 0 or 1. The
bits themselves are not enough to determine the value being held in the
register. For this, we need to define the scaling of the bits. Specifically, a
two’s complement fixed-point number stored in such a register has a value
given by
B −2
value = −bB−1 2B−1− F + ∑ bk 2 k − F (D.1)
k =0

where F is a (positive or negative) integer that defines the scaling of the
bits. F is referred to as a fraction length (or FracLength) but this is some-
what of a misnomer. The reason is that F is not necessarily a length (for
instance if F is negative or if F is greater than B). However, when F is
positive and F ≤ B, it will be equal to the number of bits to the right of the
binary point hence the term fraction length.
From (D.1), we can see that the value of a fixed-point number is deter-
mined by assigning weights of the form 2−m to each bit. The rightmost bit,
b0 has the smallest weight, 2− F . This bit is called the least-significant bit
(LSB). The leftmost bit, bB−1 , has the largest weight, 2B−1− F , which is why
it is called the most-significant bit (MSB).
Given the bit values, bk , the pair { B, F } completely characterizes a fixed-
point number, i.e. suffices in determining the value that the bits represent.

D.2 Quantization of signals and SNR
Fixed-point representation can be used to represent data in any range.
With two’s complement representation, the most negative value that can
be represented is given by the scaling of the MSB. The most positive value
that can be represented is given by the sum of all the weights of all bits
except for the MSB.


224 Overview of Fixed-Point Arithmetic

Figure D.1:

For example, with 4 bits available and a fraction length of 8 (so that
the scaling of the LSB is 2−8 and the scaling of the MSB is 2−5 ), the most
negative value that can be represented is −2−5 and the most positive value
that can be represented is 2−6 + 2−7 + 2−8 (which is equal to 2−5 − 2−8 ).
In total, we can represent 24 = 16 numbers between −2−5 and 2−5 − 2−8
with a resolution or increment of 2−8 between each number and the next
consecutive one.
In general, the number of values that can be represented within those
limits is given by 2B . The step size or increment between the values repre-
sented is given by ε = 2− F . So for a given dynamic range to be covered, the
number of bits determines the granularity (precision) with which we can
represent numbers within that range. The more bits we use, the smaller
the quantization error when we round a value to the nearest representable
value. As the number of bits increases, the SNR increases because the
quantization noise decreases.
Speciﬁcally, consider Figure D.1. The range that is covered is denoted
by R. The quantization step is denoted ε = 2− F . The number of values that
can be represented is given by 2B . There are only two degrees of freedom
in these quantities which are related by
R
=ε
2B
If we round all values to their nearest representable value, the maxi-
mum error introduced by such rounding is given by ε/2. If R is constant,


D.2 Quantization of signals and SNR 225

and B increases, then ε and consequently the largest quantization error
decreases.
A simple calculation of SNR when quantizing a signal can be per-
formed as follows [3]: If R is fully covered by the signal in question, the
signal strength is R. The quantization error covers the interval [−ε/2, ε/2],
i.e. it has a range of ε, so the SNR is given by
R
SNR = .
ε
In dB, we have

R
SNR = 20 log10 = 20 log10 (2B ) = 6.02B
ε
This result is known as the 6 dB/bit rule [3]. The result provides an up-
per bound on the achievable SNR when using B bits to cover a given range
R. To achieve this upper bound, it is necessary that the signal being quan-
tized can take any value within that range with equal probability. In other
words, it is necessary that the signal behaves like uniformly distributed
white noise over that range.

Example 74 Consider the following quantization of a signal which covers the
range [−1, 1) with equal probability,
x = 2*(rand(10000,1)-0.5);
xq = fi(x,true,16,15);
SNR = 10*log10(var(x)/var(x-double(xq)))
SNR =
96.3942
We use 16 bits to represent the signal and the step-size is 2−15 . The SNR we can
achieve is in-line with the 6 dB-per-bit rule.

Consider what would happen if instead of uniform white noise, we
take samples from a Gaussian white noise signal and ensure they fall within
the interval [−1, 1),

x = randn(10000,1);
x = x/norm(x,inf);
xq = fi(x,true,16,15);



Histogram of x
350

300

250

200

150

100

50

0
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
x

Figure D.2: Histogram of a white-noise signal normalized so that the maximum values
is ±1.

SNR = 10*log10(var(x)/var(x-double(xq)))
SNR =
89.9794

The SNR is no longer satisfying the 6 dB-per-bit rule. The reason is that
the signal very rarely takes values close to the largest available range (i.e.
close to -1 or 1). This can easily be veriﬁed by looking at the histogram:
[n,c]=hist(x,100);bar(c,n)
The histogram for the white-noise case is shown in Figure D.2. On
average the signal strength is not that large, while the quantization error
strength remains the same. In other words, the quantization error can be
equally large (on average) whether x takes a small or a large value. Since


D.2 Quantization of signals and SNR 227

x takes small values more often than it take large values, this reduces the
SNR.
Indeed, if we look at the var(x) for the uniform case, it is much larger
than var(x) in the Gaussian case, while var(x-double(xq)) is the same on
both cases.
While it may seem not so useful to consider white noise examples, it
is worth keeping in mind that many practical signals that are quantized,
such as speech and audio signals, can behave similar to white noise.
In summary, depending on the nature of the signal being quantized,
the achievable SNR will vary. In some cases, depending on whether the
application at hand permits it, it may be preferable to allow for the occa-
sional overflow with the aim of increasing the SNR on average for a given
signal.

D.2.1 Quantizing impulse responses for FIR filters
When implementing an FIR filter with fixed-point arithmetic, the first thing
we need to do is quantize the filter’s coefficients, i.e., its impulse response.
Since the impulse response of most FIR filters has a few large samples∗ and
many small samples, on average we can’t expect to get the full 6 dB/bit.
In practice we can expect somewhere between 4.5 and 5.5 dB/bit when
quantizing FIR filter coefficients.

Example 75 Consider the following highpass filter design:

Hf = fdesign.highpass('Fst,Fp,Ast,Ap',0.5,0.6,80,.5);
h = Hd.Numerator; % Get the impulse response
norm(h,inf)

The impulse response is shown in Figure D.3. The largest sample has a magnitude
of 0.434. Therefore, our fixed-point settings should be such to cover the range
[−0.5, 0.5) since this is the smallest power-of-two interval that encompasses the
values of the impulse response. This can be accomplished by setting F to be equal
to B.
Now let’s compute the SNR for various values of B and see what SNRs we
get. We will perform the following computation for different values of B:
∗ Typically, the middle sample dominates for linear-phase filters.



Impulse Response

0.5

0.4

0.3

0.2

0.1
Amplitude

0

−0.1

−0.2

−0.3

−0.4

−0.5

0 10 20 30 40 50
Samples

Figure D.3: Impulse response of highpass filter. The largest coefficient has a magnitude
of 0.434.

B = 8; % change to 10, 12, 16, 18, 24
F = B;
hq = fi(h,true,B,F);
SNR = 10*log10(var(h)/var(h-double(hq)))
If we divide the SNR by B, for the various values of B shown, we get the following
values of dB/bit: 4.7690, 4.9704, 5.1000, 5.3796, 5.3760, 5.5890.

D.3 Fixed-point arithmetic
Let’s look at a couple of things to keep in mind when performing addi-
tions/multiplications with fixed-point arithmetic.

D.3.1 Fixed-point addition
In general, when adding two fixed-point numbers, in order to avoid round-
off error it is necessary to increase the wordlength of the result by one bit.
This is to account for the possibility of a carry resulting from the addition.


D.3 Fixed-point arithmetic 229

For example suppose we want to add two 4-bit numbers that fall in the
range [−1, 1), say x1 = 0.875 and x2 = 0.75. x1 has the following binary
representation (with F = 3): 0111. x2 has 0110 as its binary representation.
The sum of the two is 1.6250. We need to represent it with 5 bits so that
no roundoff error is introduced. The binary representation of the result is
01101. Note that in general because of the possibility that the LSB is equal
to 1, it is not possible to discard the LSB without introducing roundoff
error, hence the bit growth.
Many filter structures contain a series of additions, one after the other.
However, it is not necessary to increase the wordlength by one bit for ev-
ery addition in the filter in order to avoid roundoff (i.e., in order to have
full precision). If N is the number of additions, it is only necessary to in-
crease the wordlength by ∗

G = ⌊log2 ( N )⌋ + 1. (D.2)

To see this, consider the addition of several numbers represented with
16-bit wordlength and a fraction length of 15. The worst case scenario is
when the actual numbers are the largest possible representable number
for a given format. Consider the largest positive value, 0.99† . If we add
two of these numbers, the result will be 1.999. To avoid overflow, it is
necessary to cover the range [−2, 2) in order to represent the result. In
order to have no loss of precision, i.e. avoid roundoff, we need to increase
the wordlength to 17 bits (and leave the fraction length fixed at 15). If we
now add 0.99 to the value 1.999 the result is 2.999. Since this value falls
outside the range [−2, 2), we need to further increase the wordlength by
one bit in order to represent the result with no loss of precision. With an
18-bit wordlength and a fraction length of 15 we can represent numbers
in the range [−4, 4). If we add 0.99 to the result we had so far, 2.999, we
get 3.999 as an answer. Since this number falls within the representable
range, it is not necessary to add an additional bit in order to represent this
number. The next addition would require another bit to be aggregated,
but the following three additions would not, and so forth.

∗ The operator ⌊·⌋ denotes the largest integer smaller or equal to the value operated on
(i.e. the floor command in MATLAB). † A similar argument can be made by using the
largest negative value.



Tree-adder view of multiple additions
An easy way of understanding this log2 bit growth when adding is to look
at what would happen if we used a tree-like structure to implement the
series of additions in a direct form FIR filter. Figure D.4 shows the case of
adding 4 B-bit numbers together. If we added one bit for every addition,
in order to obtain a full-precision result, we would need to add 3 bits in
total.
However, as we can see in Figure D.4, we can get a full-precision result
by adding only 2 rather than 3 bits overall. This takes into the account
the worst possible case, meaning that we need to add 1 bit every time two
numbers are added.
One could say the tree-adder view has the log2 bit growth built-in to
it. The idea still applies when there is not a power-of-2 number of values
to be added together. Some branches will be slightly incomplete, but the
required bit growth should still be correctly determined.
Note that the (D.2) holds for the worst case scenario in which the num-
bers being added are the of largest possible magnitude. In Chapter 7, we
will see that for FIR filters, we can reduce the bit growth required to avoid
roundoff by looking at the actual coefficients of the filter.

D.3.2 Fixed-point multiplication
Since multiplication is nothing but repeated addition, it should be ex-
pected that bit-growth is required if we want to multiply two numbers
without introducing loss of precision. Just as with addition, the bit growth
is independent of the fraclengths involved.
Let’s look at the worse case scenario if we multiply a number x1 , repre-
sented with B1 bits, with a number x2 , represented with B2 bits. Since the
bit growth is independent of the fraclength, consider the binary digits in-
volved. In the worst case, we are multiplying the bits in x1 times −2B2 −1 .
That means that we add (ignore the sign) x1 to itself 2B2 −1 times. Using
(D.2), the bit growth is B2 bits.
In terms of the fraclength, if x1 has a fraclength of F1 and x2 has a fr-
aclengh of F2 , the LSBs have weights of 2− F1 and 2− F2 respectively. The
product of the two will of course have a weight of 2−( F1 + F2 ) .
In summary, given two fixed-point numbers, one with wordlength/fraclength
given by { B1 , F1 } and the other with wordlength/fraclength given by { B2 , F2 },


D.3 Fixed-point arithmetic 231

Figure D.4: Tree-adder view of adding 4 B-bit numbers. Only B + 2 bits are needed for
a full precision result.

the product of the two numbers can be represented with no roundoff by
using a wordlength/fraclength of { B1 + B2 , F1 + F2 }.

For example, given a number represented with 16 bits and a fraction
length of 14 and another number represented with 18 bits and a fraction
length of 10, the product of the two numbers can be represented without
loss of precision by using a wordlength of 34 bits and a fraction length of
24.

As is the case with additions, in Chapter 7 we will see that for FIR fil-
ters, we can reduce the bit growth required to avoid roundoff when mul-
tiplying by looking at the actual coefficients of the filter.



Figure D.5: Quantization of a signal.

Figure D.6: Statistical model for the quantization of a signal.

D.4 Quantization noise variance
In many cases, a good model for the quantization error is as additive noise
to the signal being quantized [3].
Specifically, consider the quantization of the signal x [n] as shown in
Figure D.5. The operator Q{·} takes the input signal and produces a quan-
tized version of it, Q{ x [n]} = xq [n].
Statistically, we can model the effect of Q{·} as additive noise signal as
shown in Figure D.6.
Provided that the input signal changes fast enough, the noise signal
itself, e[n], can be modeled as uniformly distributed white-noise over the
range [−ε/2, ε/2].∗ The variance of the quantization noise signal can easily
be found to be ε2 /12 [3].

Reducing the number of bits used to represent a signal

The results we have presented so far apply to quantizing an analog signal
in order for it to be represented with fixed-point. This is of course impor-
tant when one is considering analog-to-digital conversion.
In digital filters, we don’t worry about the quantization of the input
signal since this occurs elsewhere. However, it is often the case that we
∗ Again, we assume here we are rounding to the nearest value when we quantize. If
instead we round towards −∞, i.e. we truncate, things change slightly.


D.5 Quantization noise passed through a linear filter 233

need to re-quantize a quantized signal, i.e. reduce the number of bits used
to represent it.
Examples where this is common include throwing out bits when mov-
ing data from an accumulator (all the additions in a direct-form FIR filter)
to the output of the filter because we need to preserve the input signal
wordlength at the output. A typical example is keeping only 16 bits for
the output from an accumulator that may have anywhere from 32 to 40
bits.
Another common case is in the feedback portion of an IIR filter since
otherwise the repeated additions would result in unbounded bit growth.
Of course the bits that we throw out when we re-quantize a signal are
the LSBs. This way the same range can be covered after re-quantization
and overflow is avoided. The precision is clearly reduced as we throw out
LSBs (the quantization step increases from 2− F to 2−( F−Q) ; where Q is the
number of bits that are thrown away.
The variance of the quantization noise that we have just derived doesn’t
quite apply for the case of re-quantizing a quantized signal [36]. However,
such variance becomes a very good approximation if at least 3 or 4 bits are
thrown out. Since we are typically removing anywhere from 16 bits on-
ward, the ε2 /12 value is usually used without giving it a second thought.
The value of ε in this case corresponds of course to the quantization step
after re-quantizing.

D.5 Quantization noise passed through a linear
filter
When analyzing how quantization affects the output signal of a filter, the
following result will come in handy: If a random signal, x [n], with power
spectral density (PSD) given by P xx (e jω ) is filtered through a filter with
frequency response H (e jω ), the output of the filter, y[n], will be a random
signal with PSD given by,
2
Pyy (e jω ) = H (e jω ) P xx (e jω )

In particular, if the input is a white-noise signal, such as the quantiza-
tion noise, its PSD will be constant, with intensity given by the variance



2
scaled by 2π, i.e. σx /2π. In this case, the PSD of the output will be
2
σx 2
Pyy (e jω ) = H (e jω ) . (D.3)
2π
Therefore, in this case the PSD will take the spectral shape of the squared
magnitude-response of the filter.

D.5.1 Computing the average power of the output noise
Once we have computed the PSD of the output due to the quantization
noise, we can compute the average power at the output due to such noise
by integrating the PSD,
π
PAVG = Pyy (e jω )dω
−π
In the case of FIR filters, when all arithmetic is performed with full
precision and therefore the only source of quantization noise is due to the
bits being discarded at the output, the transfer function between the noise
source and the filter output is trivially H (e jω ) = 1. If the PSD intensity is
σx /2π = ε2 /(2π · 12), the average power is simply
2

π ε2 2 ε2
PAVG = H (e jω ) dω = (D.4)
−π 2π · 12 12

D.6 Oversampling noise-shaping quantizers
A great deal of emphasis is placed throughout this document on multirate
filters. One of the primary uses of decimation filters is in easing the imple-
mentation of analog to digital conversion. This process of course involves
quantization of signals.
Similarly, one of the primary uses of interpolation filters is to help with
digital to analog conversion (it is one of the few reasons to oversample).
In either the decimation or interpolation case, the use of noise shap-
ing quantizers (or re-quantizers) in which multirate techniques are used
in conjunction with shaping the quantization noise (via highpass filtering)
can reduce the number of bits required to represent a signal without de-
creasing the SNR.
These schemes are widely used in practice. For more information see
[3].


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