# Lecture 9: LTI filter design

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Lecture 9: LTI IIR filter design: Analog
Instructor:
Dr. Gleb V. Tcheslavski
Contact:
gleb@ee.lamar.edu
Office Hours:
Room 2030
Class web site:
http://ee.lamar.edu/gleb/ds
p/index.htm
ELEN 5346/4304
DSP and Filter Design
Fall 2008
2
Considerations
First of all, design = determination of filter coefficients!
We are interested in BIBO systems! Usually, frequency response (its magnitude)
is given by the specifications. Our goal is to find the filter coefficients while
satisfying specs. Additional requirements on phase distortions (i.e. group delay
and, therefore, phase characteristic) may give other constrains.
Two ways to design a digital filter
1. Directly digital
2. From analog filters
M
H a (S )  c
S  
m 1
N
S  p 
k 1
ELEN 5346/4304
DSP and Filter Design
m
Fall 2008
k

?G ( z )
3
Specs…
1   p  Ha ( j)  1   p
Ha ( j)   s
Peak passband ripple:
 p  20 lg 1   p dB
(9.3.1)
Minimum stopband attenuation:
s  20lg  s dB
(9.3.2)
Maximum passband attenuation:
amax  20lg
ELEN 5346/4304
DSP and Filter Design
Fall 2008


1   2  20lg 1  2 p 
(9.3.3)
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Butterworth LPF
H B ( ) 
1
2
1 
2
  
p
2N

1
1    c 
2N
(9.4.1)
Here, N is an order of the filter; and c – cutoff frequency –
corresponds to the -3 dB point.
First 2N - 1 derivatives of (9.4.1) are zero at  = 0. As a result, a Butterworth
LPF is said to have maximally flat magnitude at  = 0.
This filter is completely determined by its order N and the cutoff frequency.
The gain:
G     10 lg H B   
The gain at DC is 0 dB and at the cutoff frequency is at -3 dB.
(9.4.1) decreases monotonically with frequency.
ELEN 5346/4304
DSP and Filter Design
Fall 2008
(9.4.2)
5
Butterworth LPF
H ( s) H ( s) 
1
1 
2
 s

2

2 N
p

1
1   s 
2

(9.5.1)
2 N
c
c2   1 s 2 N  0s 2 N   1 1 c2e j 2 l
N
Analog poles:
j
2N
N
j
2
sk  e e c e
j
2
sk  e c e
j
k
N
j (2 k 1)
;k  0,1,...N  1

2N
(9.5.2)
(9.5.3)
;k  0,1, ...N  1
(9.5.4)
Poles of (9.5.1) occur on a circle of radius c at equally spaced points.
An all pole filter.
ELEN 5346/4304
DSP and Filter Design
Fall 2008
6
Butterworth LPF
1
at   s :
  s2
2
N
1   2  s  p 
1     
2
s
2
s
2

s
p 
2N
2N
1   s2
 2   2   2   s  p 
s
2N
2
 2    s  p 

N 
log   
log   s  p 
1
1
at p 

2N
1  2
1    p c 
1
1
at s 

2N
A2
1    s c 
ELEN 5346/4304
DSP and Filter Design
(9.6.1)
(9.6.2)
(9.6.3)
(9.6.4)
(9.6.5)
(9.6.6)
(9.6.7)
We can solve both (9.6.6) and (9.6.7) for c
and then use the average value.
Fall 2008
7
Chebyshev LPF
There are two types of Chebyshev filter:
Type I: equiripple in PB
monotonic in SB
1
2
(9.7.1)
HCh,1 () 
1  T
2
2
N
  
Type II: equiripple in SB
monotonic in PB
1
2
H Ch,2 () 
p
 T  s  p  
1   2

 TN   s   
2
2
N
(9.7.2)
Here, the Chebyshev polynomial:
1

cos  N cos x  , x  1
TN ( x)  

cosh  N cosh x  , x  1
ELEN 5346/4304
DSP and Filter Design
Fall 2008
(9.7.3)
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Chebyshev LPF
The Chebyshev polynomials can also be generated by recursion:
TN 1 ( x)  2xTN ( x)  TN 1 ( x),N  1, 2,...
where T0(x) = 1 and T1(x) = x.
The order:
Type I – all pole
ELEN 5346/4304
DSP and Filter Design
N
cosh 1   
cosh
1

Poles on the ellipse
Fall 2008
s
p 
Type II – zero-pole
(9.8.1)
(9.8.2)
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Elliptic (Cauer) LPF
H e () 
1
2
1  U
2
2
N
  
p
Here UN is the Jacobian elliptic function (tabulated).
This filter is equiripple in PB and SB – zero-pole filter. For the given order,
elliptic filter has “the fastest transition” from PB to SB.
Explicit expression for the
filter order includes evaluation
of elliptic integral, therefore,
the order is usually
determined experimentally.
Elliptic filter is optimal in
terms of approximation error
but its phase response is
worse than for B. and C.
ELEN 5346/4304
DSP and Filter Design
Fall 2008
(9.9.1)
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Bessel LPF
H Bs ( s) 
1
BN ( s)
(9.10.1)
N
BN ( s)   ak s k
(9.10.2)
(2 N  k )!
;k  0,1,...N
N k
2 k !( N  k )!
(9.10.3)
Nth order Bessel polynomial:
k 0
ak 
where:
BN (s)  (2N 1)BN 1 (s)  s 2 BN 2 (s);B0 (s) 1;B1 (s)  s 1
Alternatively:
Very ill defined in PB
Linear phase in PB –
to be destroyed later
ELEN 5346/4304
DSP and Filter Design
Fall 2008
(9.10.4)
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