Design of iir digital highpass butterworth filter using analog to digital mapping technique 0.81_

International Journal of Computer Applications (0975 – 8887)
Volume 52 – No. 7, August 2012

Design of IIR Digital Highpass Butterworth Filter
using Analog to Digital Mapping Technique
Subhadeep Chakraborty

Krishna Kumar Jha

Abhirup Patra

Department of ECE
Calcutta Institute of
Technology
Uluberia, Howrah, West
Bengal, India, PIN-711316

Department of MCA
Technology

Department of ECE
Technology

ABSTRACT
Infinite Impulse Response (IIR) filter is of recursive type filter.
The present output sample of an IIR filter depends on the
present input samples, past input samples and past output
samples. There are a number of techniques available to
determine the digital IIR Filters. This paper is based on the
computer based approach to design the digital IIR filter along
with the calculation of filter coefficients using the analog to
digital mapping technique. The program based on the proposed
algorithm is simulated in Matlab and found the result is
satisfying.

Keywords: IIR filter, Digital filters, Butterworth filter, High
pass filter, coefficient, analog to digital mapping

1. INTRODUCTION
Filters play a very important role in signal processing. In this
paper the IIR digital filter is discussed which is very essential in
Digital Signal Processing (DSP).In DSP, there are two type of
systems. The first type of system performs signal filtering in
time domain. They are known as Digital filters [1][9]. Another
type of system provide signal representation in frequency
domain. They are known to as Spectrum analyzer [1][3]. The
term IIR comes from infinite impulse response meaning that the
impulse response of filter is of infinite duration whereas the
impulse response of a FIR (Finite Impulse Response) filter is of
finite duration.
IIR filter processes certain properties such as width of the passband, width of the stop-band, maximum allowable ripple at
pass-band and maximum allowable ripple at stopband[1][2][11]. A preferred design of IIR filter can be done with
help of those properties [4]. There are various process to design
IIR Digital filter. Basically IIR digital filter is designed from an
analog filter. Then using analog to digital mapping technique or
frequency transformation an IIR Digital filter can be designed
suitably[2][4].
An analog filter generally constructed by resistors, capacitors
and op amps to produce the required filtering effect. Those filter
circuits can be widely used in reduction of noise, video signal
enhancement, graphic equalizers in hi-fi systems, and many
other areas. Those analog filters are actually designed as per
specific requirement for producing satisfying filtering output.
Digital filters perform many filtering works by replacing the
analog filters. The digital filters have many features for which
we can replace the analog filters and use the digital filters and
the features are high accuracy and reliability, small physical size
and reduced sensitivity to component tolerances or drift[5][13].
Analog filter can be designed by either active element or
passive element. When the design of an active filter is
completed, the cutoff frequency can be calculated. When the
cutoff frequency is obtained, the pass-band or stop-band

allowable frequency can easily be obtained. Now the IIR digital
filter can be implemented directly, such as for example if a
analog Low Pass Filter is designed and the designer wants to
design a High Pass Filter, the designer can apply mapping
technique and frequency transformation to transform from Low
Pass Filter to High Pass Filter or the designer can design from
analog High Pass Filter to High Pass Filter using frequency
transformation technique.[2]
There are two transformation techniques available that are
widely used are- Impulse invariant method and bilinear
transformation [2][4].

2. DIGITAL IIR FILTER DESIGN
An IIR filter, as discussed in the introductory part, can be
designed from active or passive element. When a voltage source
is applied across the input terminal, the filter becomes an active
filter. Now there are many type of IIR filters such as
Butterworth filter, Chebyshev filter, Elliptic filter etc. A
Butterworth filter designed by Opamp is shown in fig.1[5]

Fig.1 3rd Order IIR Highpass Butterworth Filter
In signal processing, the order of the FIR filter is always
higher than that of the IIR filter when we basically view the
same magnitude response. So in that case, the group delay of
FIR filter is large enough compared to IIR filter. So, in order to
process a signal processing with high-speed and with highprecision, it is very important to design the IIR filters [12].
Now from the Fig.1 [5], a digital IIR Filter can be
designed. The left hand section is a 1st Order section and the
right hand section is a 2nd order section. This is because the
general available designs for filters are the 1st Order filter and
the 2nd Order filter. Therefore by means of cascading those two
a 3rd Order filter can be designed [5].

6

A digital filter means its transfer function must be in the zplane, i.e. H(z). The impulse response h (n) for a realizable filter
is,

h(n) = 0

for n ≤ 0

Start

Specify filter parameters

… …(1.1)

A stable filter must satisfy the condition,

Calculation of transfer function in s-domain



 | h(n) | 
n 0

... …(1.2)

M

H ( z) 

 b( n) z

n

… …(1.3)

n 0
N

1   a ( n) z  n

Apply frequency
transformation

Yes
Apply analog to digital mapping

Realize filter structure

Analysis of digital filter

Re-realize

… …(1.3a)

Re-Calculate

B( z ) b(0)  b(1) z 1  b(2) z 2  .......  b(M ) z  M

A( z ) 1  a(1) z 1  a(2) z 2  ..............  a( N ) z  N

Set suitable cutoff
frequency

Calculation of filter coefficient

n 1



Re-Specify

Now the generalized transfer function [1][2][10] of an IIR
Digital filter is,

Direct
Realization
?

No

Where,
b(n) = Numerator coefficient of the filter
a(n) = Denominator coefficient of the filter

Implementation of digital filter

r

Realization

IIR filters have many advantages as follows[1]:It requires less number of arithmetic operations.
There are shorter time delays in these filters.
IIR Filters have similarities with analog filters.
These filters depend not only upon the input but
also upon previous output values.
They are more susceptible to noises.

A suitable digital filter can design by calculating the
numerator coefficient and denominator coefficient. There are
various techniques available for the design and calculation of
those coefficients. The algorithm proposed for this purpose is
smartly eligible to determine the filter coefficients and hence
helps to design the IIR Digital filter with correct specification
provided to it. This algorithm also provide with the transfer
function in the digital domain. The flowchart of the proposed
algorithm is shown in fig.2

Stop

Fig.2 Proposed algorithm
Now, with help of this algorithm, the designer can specify
the necessary parameters for a filter directly or the
transformation from Low pass filter to High pass filter or Low
pass filter to Low pass filter or High pass filter to Low pass
filter or High pass to High pass filter are also allowed.
In the proposed algorithm, the direct and indirect
realization of a digital filter can be performed. Let, the
parameters of an analog filter is specified. Then after calculation
the transfer function of the analog filter i.e. in
s-domain is
obtained. Then it’s the designer choice whether he is interested
in direct realization or in indirect realization. If the realization is
the indirect one, the suitable cutoff frequency must be specified
and on that the frequency transformation will be done. For
direct realization, the analog to digital mapping will be
performed on the calculated transfer function in s-domain so
that it produces the final transfer function in z-domain which
indicates the transfer function of a digital filter. Then by
calculation, the filter coefficient can be determined .The
stability of the digital filter is determined by the pole-zero plot.
So by applying the algorithm, the digital stable filter can be
determined.

7


3. ANALOG TO DIGITAL MAPPING
Analog to Digital mapping means, simply, transformation
of the transfer function of a specified circuit from the s-domain
to z-domain. If a filter i.e. its transfer function is designed in sdomain, it is called the analog filter. After mapping, when the
transfer function is finally designed in z-domain, this is called a
digital filter.
Now to perform mapping, let we consider the impulse
response of the filter in time domain is h(t). So the transfer
function corresponding to h(t) can be obtained by the Laplace
transform[6],i.e.


H ( s)  L{h(t )}   h(t ).e  st dt

… …(1.4)

0

Where,
s = complex variable
= σ + jω
Here h(t) is continuous. To obtain the discrete format of h(t) i.e.
h(n), substitute

monotonic amplitude frequency response at 0db. The
magnitude curve can be obtained from the software program
by providing the coefficients to it and the coefficient is
obtained by the mapping technique.
Basically the practical interest goes to the point that to
determine the filter coefficients properly that in case truly
help the designer to construct a filter of interest. By using the
proposed algorithm, the filter coefficient can be determined.

4.
SIMULATION
DISCUSSION

AND

The program for the design of IIR Butterworth High-pass
filter is simulated in MATLAB7 by choosing the proper
specifications such as Pass-band frequency, Stop-band
frequency, Maximum allowable Pass-band and Stop-band
ripples, so that the designed High-pass filter will be perfect.
The output graphs are shown from Fig.4 to Fig.11.
Table 1 gives the results of the coefficients of High-pass
Butterworth filter for order = 3 and order = 5.

… …(1.5)

t = nT

RESULT

Table.1 Result for coefficients

where,
T = sampling time

Filter name

Filter
order

So, by substituting t = nT, we get h(nT) from h(t). If the
sampling time T = 1 sec, then we obtain the discrete form of h(t)
as h(n). Now from h(n) we can easily design H(z), i.e.,
3

H ( z )  Z {h(n)} 



 h(n)z

… …(1.6)

n

n 

So, in this way, the transfer function of the digital filter can
be obtained and moreover the stable poles of s-plane are
mapped inside the unit circle of the z-plane[2]. Fig.3 shows the
concept of mapping from s-plane to z-plane and vice versa.
The Butterworth High pass filter response increases
logarithmically with increase in frequency and provides with

Numerator
coefficient
-0.1247, -0.4272,
-0.4776

-0.3616,
0.3602,
-0.2123

-0.5807,-0.424,
-1.548,-2.826,
-2.064

-1.471,1.912,
-1.855,1.355,
-0.8383

Butterworth
Highpass
Filter
5

Denominator
coefficient

jω

Im(z)
z-plane
x

x

s-plane

x
x

x x
Re(z)

σ

R=1
Fig.3 Mapping of poles

8

IIR Butterworth High-pass Filter(Order = 3)

Fig.4 Magnitude response(Order=3)

Fig.6 Phase response(Order=3)

Fig.8 Impulse response(Order=3)

IIR Butterworth High-pass Filter(Order = 5)

Fig.5 Magnitude response(Order=5)

Fig.7 Phase response(Order=5)

Fig.9 Impulse response(Order=5)

9


Fig.10 Pole-Zero plot(Order=3)

5. CONCLUSION
In this paper, the calculation and result for the coefficients of
IIR Butterworth High-pass filter using the analog to digital
mapping is shown. Those coefficients, that are determined by
the proposed algorithm, are truly necessary for designing the
filter. Now, if we look on the pole-zero plot of the filter, we
can see that the filters are stable. So, we can design an analog
filter to a stable digital filter by applying analog to digital
mapping technique and with help of suitable proposed
algorithm by which a stable filter as well as the optimum
values of its coefficients is obtained and the output figures are
presented in above figures, generated by Matlab 7.

6. REFERENCES
[1]. Ranjit Singh and Sandeep K. Arya, “Determining
Optimum coefficients of IIR Digital Filter using Analog
to Digital Mapping,” International
Journal of
Advancements in Computer Science and Information
Technology, Vol. 01,No. 01, September 2011 pp.19-23.
[2]. P. Ramesh Babu,”Digital Signal Processing”, Fourth
edition,
Scitech
Publication(India)
Pvt.
Ltd,
Chennai,2008.
[3]. R.S. Chauhan and Sandeep K. Arya,”Design of IIR digital
filter using analog to digital mapping”, Journal of Neural
Computing Systems, Vol. 03,No. 01,2010,pp. 51-55.
[4]. Yaduvir Singh, Sweta tripathi and Manoj Pandey, “
Analysis of Digital IIR filter with LabVIEW”,
International Journal of Computer Applications, Vol. 10,
No. 06, 2010, pp.23-30.
[5]. Ramakant A. Gayakwad, “Opamp and Linear Integrated
Circuit”, Fourth Edition, PHI Learning Private Limited,
New Delhi, 2010.
[6] A. Sudhakar and Shyammohan S. Palli, “Circuits and
Network”, Fourth Edition, TataMcGraw Hill Education
Private Limited, New Delhi, 2011.
[7]. Samarjeet Singh and Uma Sharma, “Matlab based Digital
IIR filter design”, International Journal of Electronics
and Computer Science Engineering, Vol. 01, No.
01,ISSN 2277-1956,pp.74-83

Fig.11 Pole-Zero plot(Order=5)

[8] Proakis, J. G. and Manolakis, D. G. 2007. Digital Signal
Processing: Principles, Algorithms, and Applications.
Pearson Education Ltd.
[9]

Ranjit Singh Chauhan and Sandeep Kumar Arya,
“Determine Optimal Coefficients of IIR Digital Filters
using Simulated Annealing”, International Journal of
Computer Applications (0975 – 8887) Volume 43–
No.10, April 2012 36

[10]

Amar
Palacherla, Microchip Technology Inc,
“Implementing IIR Digital Filters”, Microchip
Technology Inc.

[11] Ranjit Kaur, Manjeet Singh Patterh and J.S. Dhillon,
“Design of Optimal L1 Stable IIR Digital Filter using
Hybrid Optimization Algorithm”, International Journal
of Computer Applications (0975 – 8887) Volume 38–
No.2, January 2012
[12]

Yasunori Sugita and Toshinori Yoshikawa, “Design of
Stable IIR Digital Filters with Specified Group Delay
Errors”, International Journal of Information and
Communication Engineering 6:1 2010

[13] Gurleen Kaur and Ranjit Kaur, “Design of Recursive
digital filters using Multiobjective Genetic algorithm”,
Gurleen Kaur et al. / International Journal of Engineering
Science and Technology (IJEST), ISSN : 0975-5462 Vol.
3 No. 7 July 2011

AUTHOR’S PROFILE
Subhadeep Chakraborty, born in 1986, is Assistant
Professor in Calcutta Institute of Technology. He received the
B.Tech degree from Saroj Mohan Institute of Technology,
WBUT,India and M.Tech degree from Kalyani Govt.
Engineering College, WBUT, India in Electronics and
Communication Engineering in 2008 and 2010.The author has
been teaching in Calcutta Institute of Technology for 2 years.
His primary research interest includes Digital Signal
Processing, Embedded System and Microprocessor.

Krishna Kumar Jha received the B. Sc. degree from
University of Calcutta, India, 2001 and the Master in
Computer Application
[MCA] from Sikkim Manipal
University of Health, Medical & Technological Sciences,

10

Gangtok, India, 2008 and Master of Technology – Computer
Science and Application from University of Calcutta, India,
2011. Currently he is Asst Prof. in the Dept of MCA at
Calcutta Institute Of Technology, Uluberia, India. The author
has been teaching for the last 7 years in the field of computer
science. His primary research area includes ns2, WMN, cloud
computing, virtualization of Network.

Abhirup Patra is pursuing his B.Tech degree from Calcutta
Institute of Technology and this is his final year. His basic
interest includes Digital Signal Processing, Control System.

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