Department of Computer Eng.
Sharif University of Technology
Discrete-time signal processing
Chapter 7:
FILTER DESIGN TECHNIQUES
Content and Figures are from Discrete-Time Signal Processing, 2e by Oppenheim, Shafer and Buck, ©1999-2000 Prentice Hall Inc.
Filter Design Techniques
DSP
• Filter:
– Any discrete-time system that modifies certain frequencies
• Frequency-selective filters pass only certain frequency
components and totally reject all others
• Filter Design Steps:
– Specification
• Problem or application specific
– Approximation of specification with a discrete-time system
• Our focus is to go from spec to discrete-time system
– Implementation
• Realization of discrete-time systems depends on target technology
• Desired filter is generally implemented with digital computation
– It is used to filter a sampled and digitized continuous-time signal
Chapter 7: Filter Design Techniques
1
DSP
Filter Design Techniques
• We have already studied the use of discrete-time systems to
implement a continuous-time system
– If our specifications are given in continuous time we can use:
xc(t)
C/D
xn
H(ej)
yn
D/C
yr(t)
 
H e j  H c  j / T 
 
– It can be noted that the specifications for filters are typically given in
frequency domain
Chapter 7: Filter Design Techniques
2
Filter Specifications
DSP
• Specifications
– Passband
0.99  H eff  j   1.01
0    2  2000 
– Stopband
H eff  j   0.001
• Parameters
2  3000   
1  0.01
 2  0.001
 p  2  2000 
• Specs in dB
 s  2  3000 
– Ideal passband gain
=20log(1) = 0 dB
– Max passband gain =
20log(1.01) = 0.086dB
– Max stopband gain =
20log(0.001) = -60 dB
Chapter 7: Filter Design Techniques
3
Discrete Signal Processing
7.1 Discrete-Time IIR Filter Design from
Continuous-Time Filters
Chapter 7: Filter Design Techniques
4
7.1.1
Filter Design by Impulse Invariance
DSP
• Remember impulse invariance
– Mapping a continuous-time impulse response to discrete-time
– Mapping a continuous-time frequency response to discrete-time
 
h  n  Td hc  nTd  H e
j
 
2
  Hc  j  j
Td
k 
 Td


k

• If the continuous-time filter is bandlimited to
H c  j   0
 
H e
j
   / Td
 
 Hc  j 
 Td 
 
• In this design procedure:
– Start with discrete-time spec in terms of  (Td has no role in the process)
– Go to continuous-time =  /T and design continuous-time filter
– Use impulse invariance to map it back to discrete-time = T
• Works best for practical filters due to possible aliasing
Chapter 7: Filter Design Techniques
5
DSP
Impulse Invariance of System Functions
• Develop impulse invariance relation between system functions
• Partial fraction expansion of transfer function of continuous-time
N
filter:
A
Hc  s   
k 1
• Corresponding impulse response:
k
s  sk
N
s t
 Ak e k
hc  t    k 1
 0

t0
t0
• Impulse response of discrete-time filter:
N
h  n   Td hc  nTd    Td Ak e
k 1

N
sk nTd
u  n   Td Ak e sk Td
k 1
 u n
n
• System function:
N
Td Ak
sk Td 1
z
k 1 1  e
H  z  
• Pole s=sk in s-plane transforms into a pole at e s T in the z-plane
k d
Chapter 7: Filter Design Techniques
6
DSP
Example
• Impulse invariance applied to Butterworth
0.89125  H  e j   1
H  e j   0.17783
0    0.2
0.3    
• Since sampling rate Td cancels out we can assume Td=1
• Map spec to continuous time
0.89125  H  j   1
H  j   0.17783
0    0.2
0.3    
• Butterworth filter is monotonic so spec will be satisfied if
Hc j0.2  0.89125 and Hc j0.3  0.17783
Hc j  
2
1
1  j / jc 
2N
• Determine N and c to satisfy these conditions
Chapter 7: Filter Design Techniques
7
DSP
Example Cont’d
• Satisfy both constrains
 0.2 
1 


 c 
2N
1




 0.89125 
2
and
 0.3 
1 


 c 
2N
1




 0.17783 
2
• Solve these equations to get N  5.8858  6 and   0.70474
c
• N must be an integer so we round it up to meet the spec
• Poles of transfer function
sk   1
1/12
 jc   c e j /12 2k 11
• The transfer function
H s 
for k  0,1,...,11
0.12093
s 2  0.364s  0.4945 s 2  0.9945s  0.4945 s 2  1.3585s  0.4945
• Mapping to z-domain



0.2871  0.4466 z 1
2.1428  1.1455 z 1
H z 

1  1.2971z 1  0.6949 z 2 1  1.0691z 1  0.3699 z 2
1.8557  0.6303z 1

1  0.9972 z 1  0.257 z 2
Chapter 7: Filter Design Techniques
8
Example Cont’d
Chapter 7: Filter Design Techniques
DSP
9
Filter Design by Bilinear Transformation
DSP
• Avoids the aliasing problem of impulse invariance
• Maps the entire s-plane onto the unit-circle in the z-plane
– Nonlinear transformation
– Frequency response subject to warping
• Bilinear transformation
2
s
Td
• Transformed system function
 1  z 1 

1 
1

z


2
H  z   Hc 
 Td
 1  z 1  

1  
1

z


• Again Td cancels out so we can ignore it
• We can solve the transformation for z as
1  Td / 2  s 1   Td / 2  jTd / 2
s    j
z

1  Td / 2  s 1   Td / 2  jTd / 2
• Maps the left-half s-plane into the inside of the unit-circle in z
– Stable in one domain would stay in the other
Chapter 7: Filter Design Techniques
10
DSP
Bilinear Transformation
• On the unit circle the transform becomes
1  jTd / 2
z
 e j
1  jTd / 2
• To derive the relation between  and 
2
s
Td
 1  e  j

 j
1

e


2



j



Td

 2e j / 2 j sin  / 2   2 j
 

tan
  j / 2

 
2
e
cos

/
2
T


2


d
• Which yields

2
 
tan  
Td
2
or
 Td 

 2 
  2arctan 
Chapter 7: Filter Design Techniques
11
Bilinear Transformation
Chapter 7: Filter Design Techniques
DSP
12
DSP
Example
• Bilinear transform applied to Butterworth
0.89125  H  e j   1
H  e j   0.17783
0    0.2
0.3    
• Apply bilinear transformation to specifications
2
 0.2 
0.89125  H  j   1
0    tan 

Td
2
 0.3
tan 
Td
 2
H  j   0.17783
 2 

  

• We can assume Td=1 and apply the specifications to
H c  j  
2
• To get
 2 tan 0.1 
1 


c


2N
1




 0.89125 
2
1
1    / c 
2N
 2 tan 0.15 
and 1  


c


Chapter 7: Filter Design Techniques
2N
1




 0.17783 
2
13
DSP
Example Cont’d
• Solve N and c
2
2
 
 

1
1


log  
  1  
  1 

  0.17783 
   0.89125 
 
N
 5.305  6
2log  tan  0.15  tan  0.1  
c  0.766
• The resulting transfer function has the following poles
sk   1
1/12
 jc   c e j /12 2k 11
for k  0,1,...,11
• Resulting in
Hc  s  

0.20238
s 2  0.3996s  0.5871 s 2  1.0836s  0.5871 s 2  1.4802 s  0.5871


• Applying the bilinear transform yields
H  z 

0.0007378 1  z
1  1.2686 z

1

1

6
 0.7051z 2 1  1.0106 z 1  0.3583 z 2
1  0.9044 z
1
1

 0.2155 z 2
Chapter 7: Filter Design Techniques


14
DSP
Example Cont’d
Chapter 7: Filter Design Techniques
15
DSP
7.2 Design of FIR Filters By
Windowing
Chapter 7: Filter Design Techniques
16
DSP
Filter Design by Windowing
• Simplest way of designing FIR filters
• Method is all discrete-time no continuous-time involved
• Start with ideal frequency response
   h ne
Hd e j 

d
 jn
n  

 
1
j
jn
hd n 
H
e
e
d
d

2  
• Choose ideal frequency response as desired response
• Most ideal impulse responses are of infinite length
• The easiest way to obtain a causal FIR filter from ideal is
hd n 0  n  M
hn  
else
 0
• More generally
hn  hd nwn
where
1 0  n  M
wn  
else
0
Chapter 7: Filter Design Techniques
17
DSP
Windowing in Frequency Domain
• Windowed frequency response H  e j 
1

2



  

H d e j W e j    d
• The windowed version is smeared version of desired
response
• If w[n]=1 for all n, then W(ej) is pulse train with 2 period
Chapter 7: Filter Design Techniques
18
Properties of Windows
DSP
• Prefer windows that concentrate around DC in frequency
– Less smearing, closer approximation
• Prefer window that has minimal span in time
– Less coefficient in designed filter, computationally efficient
• So we want concentration in time and in frequency
– Contradictory requirements
• Example: Rectangular window
   e
W e
j
M
n 0
e
 j M / 2
 j n
1  e  j  M 1

1  e  j
sin   M  1 / 2 
sin  / 2
Chapter 7: Filter Design Techniques
19
Rectangular Window
DSP
• Narrowest main lobe
– 4/(M+1)
– Sharpest transitions at
discontinuities in frequency
• Large side lobes
– -13 dB
– Large oscillation around
discontinuities
• Simplest window possible
1 0  n  M
w  n  
else
0
Chapter 7: Filter Design Techniques
20
Bartlett (Triangular) Window
DSP
• Medium main lobe
– 8/M
• Side lobes
– -25 dB
• Hamming window
performs better
• Simple equation
 2n / M

w  n    2  2n / M

0

0 n M /2
M /2 n M
else
Chapter 7: Filter Design Techniques
21
Hanning Window
DSP
• Medium main lobe
– 8/M
• Side lobes
– -31 dB
• Hamming window
performs better
• Same complexity as
Hamming
1 
 2 n  
 1  cos 
 0  n  M
w  n   2 
M



0
else

Chapter 7: Filter Design Techniques
22
Hamming Window
DSP
• Medium main lobe
– 8/M
• Good side lobes
– -41 dB
• Simpler than Blackman

 2 n 
0.54

0.46
cos
0nM



w  n  
 M 

0
else

Chapter 7: Filter Design Techniques
23
Blackman Window
DSP
• Large main lobe
– 12/M
• Very good side lobes
– -57 dB
• Complex equation

 2 n 
0.42  0.5cos  M  



0nM
w  n  
 4 n 
0.08cos



M 


0
else

Chapter 7: Filter Design Techniques
24
Incorporation of Generalized Linear Phase
DSP
• Windows are designed with linear phase in mind
– Symmetric around M/2
w  M  n 0  n  M
w  n  
0
else

• So their Fourier transform are of the form
 
 
 
W e j  We e j e  j M / 2
where We e j is a real and even
• Will keep symmetry properties of the desired impulse
response
• Assume symmetric desired response
 
 
H d e j  H e e j e  j M / 2
• With symmetric window

 
Ae e j  
1
2
  

j   
j
H
e
W
e
d
e
e


– Periodic convolution of real functions
Chapter 7: Filter Design Techniques
26
Linear-Phase Lowpass filter
DSP
• Desired frequency response
 
H lp e j
 j M / 2

  c
e

c    

 0
• Corresponding impulse
response
hlp  n 
sin c  n  M / 2  
  n  M / 2
• Desired response is even
symmetric, use symmetric
window
h  n 
sin c  n  M / 2  
  n  M / 2
w  n
Chapter 7: Filter Design Techniques
27
Kaiser Window Filter Design Method
DSP
• Parameterized equation
forming a set of windows
– Parameter to change mainlobe width and side-lobe
area trade-off
2
 
 I0  1   n  M / 2  

 M/2  
wn   


I0  

0

0nM
else
– I0(.) represents zerothorder modified Bessel
function of 1st kind
Chapter 7: Filter Design Techniques
28
Determining Kaiser Window Parameters
DSP
• Given filter specifications Kaiser developed empirical
equations
– Given the peak approximation error  or in dB as A=-20log10 
– and transition band width   s   p
• The shape parameter  should be

0.1102  A  8.7 
A  50

0.4
  0.5842  A  21  0.07886  A  21 21  A  50

0
A  21

• The filter order M is determined approximately by
M
A8
2.285
Chapter 7: Filter Design Techniques
29
Example: Kaiser Window Design of a Lowpass Filter
DSP
 p  0.4 , s  0.6 , 1  0.01,  2  0.001
• Specifications
• Window design methods assume   min(1 ,  2 )  0.001
• Determine cut-off frequency
– Due to the symmetry we can choose it to be
c  0.5
• Compute
  s   p  0.2
A  20log10   60
• And Kaiser window parameters
  5.653
M  37
• Then the impulse response is given as
2


n

18.5

 

I 0 5.653 1  

 sin 0.5  n  18.5   
 18.5  
 

h  n   
  n  18.5
I 0  5.653


0

Chapter 7: Filter Design Techniques
0nM
else
30
DSP
Example Cont’d
Approximation Error
1  Ae (e j  ) 0     p ,
E A ( )  
j
0

A
(
e
) s     .
e

Chapter 7: Filter Design Techniques
31
General Frequency Selective Filters
DSP
• A general multiband impulse response can be written as
N mb
hmb  n     Gk  Gk 1 
k 1
sin k  n  M / 2 
  n  M / 2
• Window methods can be applied to multiband filters
• Example multiband frequency response
– Special cases of
• Bandpass
• Highpass
• Bandstop
Chapter 7: Filter Design Techniques
32
DSP
7.4 Optimum Approximation of FIR
Filters
Chapter 7: Filter Design Techniques
33
DSP
Optimum Filter Design
• Filter design by windows is simple but not optimal
– Like to design filters with minimal length
• Optimality Criterion
– Window design with rectangular filter is optimal in one sense
• Minimizes the mean-squared approximation error to desired response
• But causes large error around discontinuities
hd  n 0  n  M
h  n  
else
 0
1
2 
2

 H  e   H  e 
j
j
d
2
d

– Alternative criteria can give better results
• Minimax: Minimize maximum error
• Frequency-weighted error
• Most popular method: Parks-McClellan Algorithm
– Reformulates filter design problem as function approximation
Chapter 7: Filter Design Techniques
34
DSP
Function Approximation
• Consider the design of type I FIR filter
– Assume zero-phase for simplicity
– Can delay by sufficient amount to make causal
    h ne
he n  he  n
Ae e
– Assume L=M/2 an integer
j
L
e
 jn
n  L
   h 0   2h ncosn
Ae e
j
L
e
e
n 1
• After delaying the resulting impulse response
hn  he n  M / 2  hM  n
 
 
H e j  A e e j e  jM / 2
• Example approximation mask
– Low-pass filter
Chapter 7: Filter Design Techniques
35
DSP
Polynomial Approximation
• Using Chebyshev polynomials
cos  n   Tn  cos   where Tn  x   cos  n cos 1 x 
• Express the following as aL sum of powers L
k
Ae  e j   he  0   2he  n  cos  n    ak  cos  
n 1
• Can also be represented as
   P  x
Ae e
j
x  cos 
k 0
L
where P  x    ak x k
• Parks and McClellan fix p, s, and L
k 0
– Convert filter design to an approximation problem
• The approximation error is defined as
E    W    H d  e j   Ae  e j 
–
–
–
–
W() is the weighting function
Hd(ej) is the desired frequency response
Both defined only over the passpand and stopband
Transition bands are unconstrained
Chapter 7: Filter Design Techniques
36
Lowpass Filter Approximation
DSP
• The weighting function for
lowpass filter is
 2

W   1
 1
0    p
s    
• This choice will force the
error to = 2 in both bands
• Criterion used is minmax
min
max E 
he n:0 nL 
F
• F is the set of frequencies
the approximations is made
over
Chapter 7: Filter Design Techniques
37
DSP
Alternation Theorem
• Fp denote the closed subset
– consisting of the disjoint union of closed subsets of the real axis x
•
•
•
•
r
The following is an order polynomial P  x    ak x k
k 0
Dp(x) denotes given desired function that is continuous on Fp
Wp(x) is a positive function that is continuous on Fp
The weighted error is given as
rth
E p  x   Wp  x   D p  x   P  x  
• The maximum error is defined as
E  max E p  x 
xFp
• A necessary and sufficient condition that P(x) be the unique rth
order polynomial that minimizes E is that Ep(x) exhibit at least
(r+2) alternations
• There must be at least (r+2) values xi in Fp such that x1<x2<…<xr+2
E p  xi   E p  xi 1  
E for i  1,2,...,(r  2)
Chapter 7: Filter Design Techniques
38
DSP
Example
Not
alternations
• Examine polynomials P(x)
• that approximate
1 for  1  x  0.1
0 for 0.1  x  1
• Fifth order polynomials shown
• Which satisfy the theorem?
Not
alternations
Chapter 7: Filter Design Techniques
39
DSP
Optimal Type I Lowpass Filters
• In this case the P(x) polynomial is the cosine polynomial
L
P  cos     ak  cos  
k
k 0
• The desired lowpass filter frequency response (x=cos)
1 cos  p    1
Dp  cos    
0 1    cos s
• The weighting function is given as
1/ K
Wp  cos    
 1
cos  p    1
1    cos s
• The approximation error is given as
E p  cos    Wp  cos    Dp  cos    P  cos   
Chapter 7: Filter Design Techniques
40
Typical Example Lowpass Filter Approximation
DSP
• 7th order approximation
Chapter 7: Filter Design Techniques
41
Properties of Type I Lowpass Filters
DSP
• Maximum possible number of alternations of the error is L+3
• Alternations will always occur at p and s
• All points with zero slope inside the passpand and all points
with zero slope inside the stopband will correspond to
alternations
– The filter will be equiripple except possibly at 0 and 
Chapter 7: Filter Design Techniques
42
Flowchart of Parks-McClellan Algorithm
Chapter 7: Filter Design Techniques
DSP
43
DSP
Butterworth Lowpass Filters
• Passband is designed to be maximally flat
• The magnitude-squared function is of the form
H c  j  
1
2
1   j / j c 
sk   1
1/ 2 N
Hc  s  
2
2N
 jc   c e j / 2 N  2 k  N 1
1
1   s / j c 
2N
for k  0,1,...,2N-1
Chapter 7: Filter Design Techniques
44