Digital Signal Processing - Frequency Response Digital Filter Designs

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DIGITAL SIGNAL PROCESSING
FREQUENCY RESPONSE
DIGITAL FILTER DESIGNS
Lectured by: Assoc. Prof. Dr. Thuong Le-Tien
National Distinguished Lecturer
September 2011
Lectured by Assoc.Prof.Dr. Thuong Le-Tien
Frames of the chapter:
1. Equivalent Descriptions of Digital Filters
2. Transfer functions
3. Sinusoidal Response
4. Poles and zeros designs
5. Deconvolution, inverse Filters, and Stability
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1. Equivalent descriptions of digital filters
With the aid of z-transform, several mathematically 
equivalent ways to describe and characterize FIR and 
IIR filters:
* Transfer function H(z)
* Frequency response H
* Block diagram realization and sample processing 
algorithms 
* I/O difference equation
* Pole/Zero pattern
* Impulse response h(n) 
* I/O convolution equation
The most important one is the transfer function 
because from it we can easily obtain all the others
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Equivalent description of digital filters
2. Transfer functions
Example  
1
1
8.01
25





z
z
zH
 
11
1
01
1
8.01
5.7
5.2
8.018.01
25









zz
A
A
z
z
zH
       nunnh n8.05.75.2  
        1111 258.0258.01   zzHzzHzzHz
       12518.0  nnnhnh 
     zXzHzY 
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       



j
j
e
e
H
z
z
zH










8.01
4.015
8.01
4.015
1
1
2cos211 aaae j   
 
64.0cos6.11
16.0cos8.015





H
Consider the transfer function:
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z 1
x(n) y(n)
v (n) = x(n 1) 1 w (n) = y(n 1) 1
2 0,8
5
- z 1-
The magnitude of the transfer function may be 
plotted with the help of the pole/zero geometric
Pole/zero pattern and magnitude response
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21
35
3
5
8,01
25
)()(
35
8,01
25
)()(
1
10










z
z
zHH
zHH
 
11
1
8.01
5.7
5.2
8.01
25







zz
z
zH
     zHzHzH 21 
  5.21 zH
   12 8.01/5.7  zzH
Example:
Parallel form realization of H(z)
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z 1
x n( ) y(n)
w0
0,8
7,5
w n-1( )
w1
-2,5
w n( )
-
Canonical form
Realization of H(z)
Transposed 
realization of H(z)
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z 1
x n( ) y(n)
w0
0,8
w n-1( ) w1
w n( )
2
5
-
z 1
x(n) y(n)
2 0,8
5
w (n)1
-
       zX
zazaza
zbzbzbb
zXzHzY
M
M
L
L





...1
...
2
2
1
1
2
2
1
10
       zXzbzbzbbzYzazaza LLMM   ......1 221102211
LnlnnMnMnn xbxbxbyayay   ...... 11011
LnlnnMnMnn xbxbxbyayay   ...... 11011
   
  MM
L
L
zazaza
zbzbzbb
zD
zN
zH





...1
...
2
2
1
1
2
2
1
10
* The transfer function of an IIR filter as the 
ratio of two polynomials of degrees, say L and M
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Example: Determine the transfer function of the third-
order FIR filter with impulse response: h = [1, 6, 11, 6]
* I/O equation:
y(n) = x(n) + 6x(n -1) + 11x(n -2) + 6x(n -3)
* Z-Transform: 
H(z) = 1 + 6z-1 + 11z-2 + 6z-3
* H(z) has one zero at z = -1, 
H(z) = (1 + z-1)(1 + 2z-1)(1 + 3z-1)
* substitute z = ej into the equation
    LLzbzbzbbzNzH
  ...22
1
10
Lnlnnn xb...xbxby   110
FIR filter:
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H() = (1 + e-j)(1 + 2e-j)(1 + 3e-j)
This is a low pass filter.
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Example: )4()()(  nxnxny
44 1
)(
)(
)()()()(   z
zX
zY
zHzXzzXzY
  jjjjj ejeeeeH 22224 )2sin(2)(1)(
j jz kezz jk   ,1,,13,2,1,0,1 4/24
Example: Determine the transfer function 
and causal impulse response of the filters:
(a) y(n) = 0.25y(n-2) + x(n)
(b) y(n) = - 0.25y(n-2) + x(n)
Solve:
(a) Z-transform 
Y(z) = 0.25Y(z)z-2 + X(z)
H(z) = Y(z)/X(z)
A1 = A2 = 0.5. The causal impulse response:
h(n) = A1(0.5)
nu(n) + A2(-0.5)
nu(n) 
 
1
2
1
1
2 5.015.0125.01
1
 





z
A
z
A
z
zH
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Pole/zero pattern and frequency
response of the filter
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(b) Y(z) = -0.25Y(z)z-2 + X(z)
 
1
*
1
1
1
2 5.015.0125.01
1
 





jz
A
jz
A
z
zH
           
     nun
nuenujAnh
n
jnnn
eùA
2/cos5.0
5.05.0Re25.0Re2 2/2




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4. Pole/zero designs
* First order filters
Pole/zero placement can be used to design filters such
as first order smoothers, notch filters and resonator
The general transfer function, where a, b are positive 
and less than one; gain factor G is arbitrary
   
1
1
1
1
8.01
4.015
8.01
25










z
z
z
z
zH  
 
1
1
1
1





az
bzG
zH
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Frequency response at lowest and highest 
frequencies:  = 0,  by setting z =  1
The attenuation of the highest frequency relative to 
the lowest one is:
  8.001.0 20/1/1  effna
       
a
bG
H
a
bG
H






1
1
 ,
1
1
0 
 
 
  
  ba
ab
H
H



11
11
0

  
  
4.0
21
1
18.01
8.011



b
b
b
   
1
1
8.01
4.01





z
zG
zH
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i
i
pmax
 eff
n
)/1ln(
)/1ln(
ln
ln





effn
Effective time constant
Define:
Where  is the desired 
level of smallest
* Resonators and equalizers
Example for a resonator
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0Re jp  0Re* jp Pole conjugate pair
     221111 1Re1Re1 00  



zaza
G
zz
G
zH
jj 
2
201 ,cos2 RaRa  
      jjjjjj eaea
G
ee
G
H
2
211Re1Re1
00  



 
  
1
Re1Re1 00
0 


 

jjjj ee
G
H
 
     2020
2
2
cos21cos21 RRRR
G
H




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Normalizing |H(0)|= 1 
Calculating for 
the width 3-dB 
dB
H
H
o
3
2
1
log10
)(
)(
log20 1010 








1
|)Re1()Re1(|
)(
00
0 


 

jjjj ee
G
H
2
1
|)(|
2
1
|)(| 20
2  HH
2
0 )2(cos21)1( RRRG 
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Solving of this equation:  = 2 – 1
When p closed to the unit circle:
  2(1 – R) 
|PQ| = 1 – R.
||||
)(
 ;
||||
)(
*
*
pzpz
G
zH
pzpz
G
zH
QQ
Q
AA
A




||
||
|)(|
|)(|
PA
PQ
pz
pz
zH
zH
A
Q
Q
A 



2
1
)(
)(

Q
A
zH
zH
2
1
||
||

PA
PQ |PA|= |PQ|
|AB| = 2|QA| = 2|PQ| = 2(1 – R)
 = |AB| = 2(1 – R)
For 3-dB 
condition or
)(sin
sin
)( 00
0


 nR
G
nh n
For a given , R can be calculated
12/12/2011 22
Example: Design a resonator with 2 poles, f0 = 500Hz , 
bandwidth  = 32Hz, fS­ = 10kHz.
e)(rad/sampl 1.0
2
0 

 
S
o
f
f
02,0
2



Sf
f
2(1 – R) = 0.02  R = 0,99 
then G = 0.0062; a1 = -1.8831; a2 = 0.9801
11 9801.08831.11
0062.0
)(
 

zz
zH
12/12/2011 23
00 *
11 ,
  jj rea rea 0  r  1
2
2
1
1
2
2
1
1
11
11
1
1
)Re1()Re1(
)1()1(
)(
00
00










zaza
zbzb
zz
zrezre
zH
jj
jj
2
201
2
201
,cos2
,cos2
rb rb
Ra Ra


))(cos21())(cos21(
))(cos21())(cos21(
|)(|
2
0
2
0
2
0
2
02
RRRR
rrrr
H


