Digital filter structures Digital Signal Processing NIT DURGAPUR

2
2
Frequency Selective
LTI System
input
signal
output
signal
Filter
What are filters?
• Any Frequency Selective LTI System is a
Filter

3
3
Types of filters
• Low Pass Filter
• High Pass Filter
• Band Pass Filter
• Band Stop Filter
• All Pass Filter

4
4
Ideal Low pass filter
Pass Band
c  
|H()|
0
-c
-
c  
|H()|
0
-c
-
Practical Low pass filter
• passes all the frequencies
up to cut-off frequency (c)
with constant gain.
• rejects all frequencies above
c completely.
• abrupt transition from pass
band to stop band.
• passes all the frequencies
upto cut-off frequency (c).
• rejects all frequencies above
c.
• smooth transition from pass
band to stop band.
Smooth Transition
to 
to 

5
5
 c
-c
1
High pass filter

|H()|
0
Pass Band
 0
-  0
1
Band pass filter

|H()|
-  2 -  1  1  2
0
1
Band stop filter

|H()|
1
All pass filter

|H()|
0
to 
to 
to 
to -
to -
to -
 0
-  0
-  2 -  1  1  2
0

6
6
TYPES OF DIGITAL FILTER :
TYPES OF DIGITAL FILTER :
1.
1. FINITE IMPULSE RESPONSE
FINITE IMPULSE RESPONSE FILTER
FILTER
(FIR Filter)
(FIR Filter)
2.
2. INFINITE IMPULSE RESPONSE
INFINITE IMPULSE RESPONSE FILTER
FILTER
(IIR Filter)
(IIR Filter)

7
7
Digital Filter Structures
Digital Filter Structures
 The actual implementation of an LTI digital filter can be either in
software or hardware form, depending on applications
 In either case, the signal variables and the filter coefficients cannot
represented with finite precision
 However, a direct implementation of a digital filter based on either
the difference equation or the finite convolution sum may not provide
satisfactory performance due to the finite precision arithmetic
 Need to develop alternate realizations and choose the structure that
provides satisfactory performance under finite precision arithmetic
 A structural representation using interconnected basic building blocks is the
first step in the hardware or software implementation of an LTI digital filter

8
8
REALIZATION OF DIGITAL
FILTER
FILTER
y(n)= bk
x(n-k)
k = 
N-1 M-1
k = 
This is a difference equation for most practical cases, where
x(n) is input, y(n) is output and y(n-k) is previous output and
ak
and bk
are system coefficients
FIR Filter (All zero or moving average filter): Put ak
=0
y(n)=-  ak
y(n-k) + bk
x(n-k)
k = 
M-1

9
9
FILTER
FILTER
IIR Filter (All pole or autoregressive system): Put bk
=0 for K greater equal
to 1 and less equal to M
k = 
N-1
y(n)=- bk
y(n-k)+ b0
x(n)
k = 
N-1
IIR Filter (Pole-zero or autoregressive, moving average system):
y(n)=-  ak
y(n-k) + bk
x(n-k)
M-1
k = 
Factor influencing the choice of structures 1. Computational Complexity
2. Memory requirements and 3. Finite-word length effect

10
10
Realization
Realization
Structures of FIR systems:
1. Direct form
2. Cascade form
Structures of IIR systems:
1.Direct form I & direct form II
2. Cascade form
3. Parallel form

Basic Building Blocks
 The computational algorithm of an LTI digital
filter can be conveniently represented in block
diagram form using the basic building blocks
shown below
x[n] y[n]
w[n]
 A
x[n] y[n]
y[n]
1

z
x[n]
x[n] x[n]
x[n]
Adder
Unit delay
Multiplier
Pick-off node

 Advantages of block diagram representation
(1) Easy to write down the computational algorithm by
inspection
(2) Easy to analyze the block diagram to determine the
explicit relation between the output and input
(3) Easy to manipulate a block diagram to derive other
“equivalent” block diagrams yielding different
computational algorithms
(4) Easy to determine the hardware requirements
(5) Easier to develop block diagram representations from the
transfer function directly

Canonic and Noncanonic
Structures
Structures
 A digital filter structure is said to be canonic
if the number of delays in the block diagram
representation is equal to the order of the
transfer function
 Otherwise, it is a noncanonic structure

Structures
Structures
 The structure shown below is noncanonic
as it employs two delays to realize a first-
order difference equation
]
1
[
]
[
]
1
[
]
[ 1
0
1 




 n
x
p
n
x
p
n
y
d
n
y

Equivalent Structures
 Two digital filter structures are defined to be
equivalent if they have the same transfer
function
 We describe next a number of methods for
the generation of equivalent structures
 However, a fairly simple way to generate
an equivalent structure from a given
realization is via the transpose operation

Transpose Operation
(1) Reverse all paths
(2) Replace pick-off nodes by adders, and vice
versa
(3) Interchange the input and output nodes

 There are literally an infinite number of equivalent structures
realizing the same transfer function
 It is thus impossible to develop all equivalent realizations
 In this course we restrict our attention to a discussion of some
commonly used structures
 Under infinite precision arithmetic any given realization of a digital
filter behaves identically to any other equivalent structure
 However, in practice, due to the finite word length limitations, a
specific realization behaves totally differently from its other
equivalent realizations
 It is important to choose a structure that has the least quantization
effects when implemented using finite precision arithmetic
 One way to arrive at such a structure is to determine a large
number of equivalent structures, analyze the finite word length
effects in each case, and select the one showing the least effects

Basic FIR Digital Filter
Basic FIR Digital Filter
Structures
Structures
 A causal FIR filter of order N is
characterized by a transfer function H(z)
given by
which is a polynomial in
 In the time-domain the input-output relation
of the above FIR filter is given by
 

 N
n
n
z
n
h
z
H 0 ]
[
)
(
1

z
  
 N
k k
n
x
k
h
n
y 0 ]
[
]
[
]
[

20
20
Direct Form of FIR filters (Example)
Direct Form of FIR filters (Example)
H(z) = 1 + 3z -1
+ 2z -2
y(n) = x(n)
Z-1
Z-1
x(n) y(n)
+
+
3
x(n-1)
x(n-2)
1
2
+ 3x(n-1) + 2x(n-2)

Direct Form FIR Digital Filter
Structures
Structures
 An FIR filter of order N is characterized by
N+1 coefficients and, in general, require N+1
multipliers and N two-input adders
 Structures in which the multiplier coefficients
are precisely the coefficients of the transfer
function are called direct form structures

Structures
Structures
 A direct form realization of an FIR filter can
be readily developed from the convolution
sum description as indicated below for N =
4

Structures
Structures
 An analysis of this structure yields
which is precisely of the form of the convolution
sum description
 The direct form structure shown on the
previous slide is also known as a tapped
delay line or a transversal filter
]
[
]
[
]
[
]
[
]
[
]
[
]
[ 2
2
1
1
0 



 n
x
h
n
x
h
n
x
h
n
y
]
[
]
[
]
[
]
[ 4
4
3
3 


 n
x
h
n
x
h

Structures
Structures
 The transpose of the direct form structure
shown earlier is indicated below
 Both direct form structures are canonic with
respect to delays

25
25
H(z) =  h(k) . z-k
K=0
M-1
Cascade Form of FIR filters
= H1(z ) . H2(z) . … . Hp(z)
=  Hl(z)
l = 1
p
H2(z)
x(n) y(n)
H1(z) Hp(z)
Second Order System Using Direct Forms

26
26
H(z) = H1(z ) . H2(z)
H1(z) H2(z)
x(n) y(n)
+ y1(n)
Z-1
x(n)
y1(n) = x(n) + x(n-1)
+ y(n)
Z-1
x2(n)
y(n) = x2(n) + 2x2(n-1)
=y1(n)
2
1
H1(z)

Cascade Form FIR Digital Filter
Structures
Structures
 A higher-order FIR transfer function can
also be realized as a cascade of second-
order FIR sections and possibly a first-
order section
 To this end we express H(z) as
 where if N is even, and if
N is odd, with
 




 K
k k
k z
z
h
z
H 1
2
2
1
1
1
0 )
(
]
[
)
( 

2
N
K 
2
1

N
K
0
2 
K


Structures
Structures
 A cascade realization for N = 6 is shown below
 Each second-order section in the above
structure can also be realized in the
transposed direct form

Digital filter structures Digital Signal Processing NIT DURGAPUR

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