CHAPTER 7 filter design techniques
7.0 introduction
7.1design of discrete-time IIR filters from continuous-time filters
7.1.1
filter design by impulse invariance
7.1.2
filter design by bilinear transform
7.1.3
not low pass filter design and other method
7.2 design of FIR filters by windowing.
7.3 summary
7.0 introduction
ideal frequency selective filter
band-stop filter：notch filter；quality factor of band-pass
filter=pass-band width/center frequency
Figure 7.1
impulse response of ideal low-pass filter，noncausal，unrealizable
hd (n) 
1
c
2  

H d (e
c
j
)e
jn
d 
sin( c n)
n
, n  ...... 
Specifications for filter design ：given in frequency domain
phase：linear?
magnitude：given by a tolerance scheme
analog or digital，absolute or relative
absolute specification
magnitude response of
equivalent analog system
passband
tolerance
3dB cutoff
frequency
1/
Figure 7.2
2
stopband
tolerance
passband
cutoff
frequency
monotonous descent
cc

stopband
cutoff
frequency
relative specifications：maximum magnitude in passband is normalized to 1, viz. 0dB
 p  20 * log10（1   p） 0 maximum attenuation in passband
 s  20 * log10  s  0
minimum attenuation in stopband
3dB cutoff frequency：
| H (e
j c
) | 1 /
 20 log10 | H (e
2
j c
) | 3dB
magnitude response of equivalent analog system:
H eff


j
H (e
) |  T
|  |


T
( j)  


0
|  |

T

digital specification, finally:
 p   pT , s  sT
Specifications for bandpass and bandstop filters：
up and down passband cutoff frequency，
up and down stopband cutoff frequency
Design steps：
（1）decide specifications according to application
（2）decide type according to specification：generally , if the
phase is required , choose FIR.
（3）approach specifications using causal and stable discretetime system：
viz. design H(z0) or h[n]，nonuniform
（4）choose a software or hardware realization structure, take
effects of limited word length into consideration
H(z) or h[n]
7.1
design of discrete-time IIR filters
from continuous-time filters
ala log y low
pass 
 digital
A/ D
 frequency transform
ala log y high
low
pass

pass 
 digital
A/ D
high
pass
attention：original analog filter and equivalent analog filter is different,
their frequency response is not always the same
7.1.0 introduction of analogy filter
7.1.1filter design by impulse invariance
7.1.2 filter design by bilinear transform
7.1.0 introduction of analog filter
comparison:
1.wave
2.the same order,
increase performance
3.increase design
complexity
(A)（C）small aliasing in the impulse invariance design technique
BW design formula: specificationsystem function
magnitude frequency function: | H c ( j) |2 
1  ( /  c )


N  log  p


take the specifications into system
function and get the results from
equation group:
c 
1
p
OR
s k  c e
N 1
H c (s)  
get system function：


 10 10  1 
s 
s

 10 10  1 


k 0
1
s
sk

2





1
s
10
 2 k 1 1 
j
 
2
N
2

1
1
s

10

  p 10
 2N
 1
10


 H c ( s ) H c ( s ) |s  j
p
c 
1
confirm the poles of system function:
（in the left half plane）
2N
 2N
 1

, k  0      N  1
c
N
N 1
 (s  s
k 0
k
)
EXAMPLE
design a low pass analogy filter:
 p  2 1000rad / s,  s  2  2000rad / s, p  1dB, s  15dB
[N,Wc]=buttord(2000*pi,4000*pi,1,15, 's')
[Bs,As]=butter(N,Wc, 's')
[H,W]=freqs(Bs,As);
plot(W/2/pi,20*(log10(abs(H))))
axis([1000,2000,-16,0])
grid on
OUTPUT:
N= 4
Wc = 8.1932e+003
Bs = 1.0e+015 *
0
0
0
0 4.5063
As = 1.0e+015 * 0.0000 0.0000 0.0000 0.0014 4.5063
b0  b1s  b2 s  b3 s  b4 s
2
H c ( s) 
3
4
a0  a1s  a2 s  a3 s  a4 s
2
3
4
Or：
[N,Wc]=buttord(2000*pi,4000*pi,1,15, 's')
[z,p,k]=butter(N,Wc, 's'
)
Output:
z = Empty matrix: 0-by-1
p = 1.0e+003 *
-7.5695 + 3.1354i
-7.5695 - 3.1354i
-3.1354 + 7.5695i
-3.1354 - 7.5695i
k = 4.5063e+015
H c ( s) 
k
( s  p0 )( s  p1 )( s  p2 )( s  p3 )
EXAMPLE
design a low pass cheby analogy filter:
 p  2 1000rad / s,  s  2  2000rad / s, p  1dB, s  15dB
[N,Wc]=cheb1ord(2000*pi,4000*pi,1,15, 's')
[Bs,As]=cheby1(N,1,Wc, 's')
[H,W]=freqs(Bs,As);
plot(W/2/pi,20*(log10(abs(H))))
axis([0,4000,-30,0])
grid on
EXAMPLE
design a low pass cheby analogy filter:
 p  2 1000rad / s,  s  2  2000rad / s, p  1dB, s  15dB
[N,Wc]=cheb2ord(2000*pi,4000*pi,1,15, 's')
[Bs,As]=cheby2(N,15,Wc, 's')
[H,W]=freqs(Bs,As);
plot(W/2/pi,20*(log10(abs(H))))
axis([0,4000,-30,0])
grid on
EXAMPLE
design a high pass analogy filter:
 p  2  2000rad / s,  s  2 1000rad / s, p  1dB, s  15dB
[N,Wc]=buttord(4000*pi,2000*pi,1,15, 's')
[Bs,As]=butter(N,Wc, 'high', 's')
[H,W]=freqs(Bs,As);
plot(W/2/pi,20*(log10(abs(H))))
axis([0,4000,-16,0])
grid on
transformation from analog filter to digital filter:
H(s)H(z)，viz. mapping from S plane to Z plane, must satisfy:
the same frequency response（map imaginary axis to the unit circle）；
causality and stability is preserved（map poles from left half plane to the
inside circle）。
S plane  Z plane
imaginary axis
the unit circle
H a (s )
H ( z)
poles in left half plane inside the
unit circle
7.1.1 filter design by impulse invariance
according to ：
h[ n]  Td hc [ nTd ]
h[ n ]T h [ nT ]
d c
d
H c ( s )  hc (t ) 

 h[ n]  H ( z )
conversion formula：
N 1
H c ( s) 
Ak
ss
k 0
k
N 1
H ( z )  H c (s)
1
s  sk


Td
1 e
sk Td
z
1
Td Ak
 1 e
k 0
s k Td
z
1
Relationship between poles（causal and stable):
relation between frequencies:
  Td ,     ,    
S plane
Z plane
jΩ
3π/Td
π/Td
-π/Td
-3π/Td
sk  e
s k Td
relation between frequency response: when aliasing is small, the frequency response is
the same.

2
j
H (e
)
H
k  
c(
j  jk
)
Td


Td
strongpoint：linear frequency mapping; shortcoming：aliasing in frequency
response。
restriction in application：can not used in high-pass and bandstop filter
Design steps：
(1)[equivalent
analog
  ' T ,  ' T
p
S
specificat ion 
p S
 ,digital
 p  p / Td , S  S / Td
digital specificat ion  prototype
specificat ion ]
analog specificat ion
(2)design Hc (s)
（3）
H ( z )  H c (s)
1
s  sk

Td
1 e
sk Td
z
1
about Td:
independent of T;
do not influence aliasing;
arbitrary value，generally, take 1（attention（1）and（3）have the same value）.
EXAMPLE
design a BW
digital lowpasss
filter , need :
 p  0.2 ,  s  0.4 ,  p  1dB,  s  12dB,
wp=0.2*pi;
ws=0.4*pi
ap=1;
as=12
Td=1;
Wp=wp/Td;
[N,Wc]=buttord(Wp,Ws, ap , as, 's')
[Bs,As]=butter(N,Wc, 's')
[Bz,Az]=impinvar(Bs,As,1/Td)
[H,W]=freqs(Bs,As);
plot(W/pi,20*(log10(abs(H))), 'r*')
hold on
[H,w]=freqz(Bz,Az);
plot(w/pi,20*(log10(abs(H))))
axis([0.2,0.4,-20,0])
grid
Ws=ws/Td
7.1.2 filter design by bilinear transform
design thought：
S 平面
jΩ
S1 平面
z 平面
jΩ1
Im(z)
+π/T
σ
Re(z)
σ1
-π/T
Ω=（-∞，∞）
ω=（-π，π）
Ω1=（-π/T，π/T）
design formula：
H ( z )  H c ( s) |
1
2 1 z
s 
Td 1 z 1
left half s-plane  inside the unit circle in the z-plane（causality and stability are
preserved）；
imaginary axis of the s-plane the unit circle in z-plane (the same frequency
response), one-by-one mapping
relation between frequencies:
 Td

  2 arctan 

 2

 

tan 
Td
 2
2
relation between frequency response:
H (e
j
)  H c ( j)

 
tan 
Td
2
2
prewarp :

p 
2

tan
 p 



Td
2



s 
   2 tan


 s
Td
 2 

strongpoint：no aliasing；
shortcoming：nonlinear。
restriction in application：
can not used in
differentiator.
design steps:
(1)
  ' T ,  ' T
p
S
[equivalent analog system specificat ion 
p S
 digital specificat ion ]
p 
2
tan( p / 2 ),  S 
2
tan( s / 2 )
digital specificat ion         specificat ions of prototype analog filter
Td
Td
(2) Hc (s)
(3)
H ( z )  H c ( s) |
1
2 1 z
s 
Td 1 z 1
about Td:
independent of T；
arbitrary value，generally, take 1（attention（1）and（3）have the same value）.
EXAMPLE
design a lowpass filter , need
 p  0.2613  ,  s  0.4018  , p  0.75 dB,  s  20 dB,
wp=0.2613*pi;
ws=0.4018*pi;
ap=0.75;
as=20;
Td=1;
Ws=2/Td*tan(ws/2); Wp=2/Td*tan(wp/2)
[N,Wc]=buttord(Wp,Ws,ap,as,’s’)
[Bs,As]=butter(N,Wc, ‘s’)
[Bz,Az]=bilinear(Bs,As,1/Td)
[H,W]=freqs(Bs,As);
plot(W/pi,20*(log10(abs(H))),’Rx‘)
hold on
[H,w]=freqz(Bz,Az);
plot(w/pi,20*(log10(abs(H))))
ylabel(‘虚线：模拟滤波器幅度[dB] 实线：数字滤波器幅度[dB]’)
xlabel(‘虚线：模拟角频率[*π弧度/秒] 实线:数字角频率[*π弧度]’)
axis([0.25,0.5,-20,-0.45])
grid
OR
wp=0.2613*pi;
ws=0.4018*pi;ap=0.75;
[N,wc]=buttord(wp/pi,ws/pi,ap,as)
[Bz,Az]=butter(N,wc)
[H,w]=freqz(Bz,Az);
plot(w/pi,20*(log10(abs(H))))
as=20
EXAMPLE
design a highpass
filter , need
 p  0.6 ,  s  0.5 , p  1dB, s  12dB,
[N,Wc]=buttord(0.6,0.5,ap,as)
[Bz,Az]=butter(N,Wc, 'high')
[H,w]=freqz(Bz,Az);
plot(w/pi,20*(log10(abs(H))))
grid
EXAMPLE
 0  0.5rad ,
 pl  0.45rad , pu  0.55rad : 3dB
 sl  0.4rad , su  0.6rad : 10dB
[N,wc]=buttord([0.45 0.55],[0.4 0.6],3,10) %双线性变换法
[B,A]=butter(N,wc)
[H,w]=freqz(B,A);
plot(w/pi,20*(log10(abs(H))))
ylabel(‘20log|H(ejω)| [dB]’)
xlabel(‘数字角频率[*π弧度]’)
axis([0.4,0.6,-10,0]);
grid on
Output:
N= 2
wc = 0.4410
B = 0.0271
A = 1.0000
0.5590
0 -0.0541
0 1.4838
0 0.0271
0 0.5920
7.1.3 IIR summary
1.design steps
（1）equivalent analog system specificat ion
 p  ' p T , S  'S T
 p   p / Td ,  S  S / Td
p 
2
Td
tan(  p / 2),  S 
2
Td
digital specificat ion
specificat ions of prototype analog filter
tan(  s / 2)
（2）H a ( s)
1
(3) H a ( s )
s  sk
s

2
Td
1 e

sk Td
1 z
1
z
1
H（z )
Td 1  z 1
2. impulse invariance:
frequency axis is linear many-to-one mapping , aliasing in frequency response,
inapplicable to high-pass filter.
bilinear transformation：
frequency axis is one-to-one mapping with aberrance, no aliasing in frequency
response, inapplicable to differentiator
7.1.4 frequency conversion
Frequency conversion in analog domain：
lowpass  lowpass ：s  s /  c
lowpass  highpass ：s   c / s
lowpass  bandpass ：s  ( s   c1 c 2 ) /[ s ( c 2   c1 )]
2
lowpass  bandstop ：s  [ s ( c 2   c1 )] /( s   c1 c 2 )
2
 c 2 : up cutoff frequency,  c1：down cutoff frequency
Frequency conversion in digital domain：
lowpass  lowpass ：z
1
lowpass  highpass ：z

1
z
1

1
1 z 

z
1
1
1 z 
z
lowpass  bandpass ：z
1


2

k 1
k 1
z
lowpass  bandstop ：z
1


2

k 1
k 1
2k
k 1
z
2

2k
k 1
z
2

z
1

2k
k 1
z
1

2k
k 1
k 1
k 1
z
1
1
k 1
k 1
z
1
1
 , k is dependent of filter cutoff frequency before and after conversion
7.2 design of FIR filters by windowing
7.2.1 design ideas
7.2.2 properties of commonly used windows
7.2.3 effect to frequency response
7.2.4 design step
7.2.1 design ideas
hd (n)  IFT ( H d (e
j
H（
e
d
))
j
)
2
M 1
w[ n ]
h[n]  hd [n]  wM [n]
(d )W (e
j
)
( f ) H (e
j
sin[  ( M  1) / 2]
sin(  / 2)
)
1
2
H d (e
j
)  W (e
j
)
Gibbs phenomenon: frequency response oscillate at wc.
transition width of filer’smainlobe width of window spectrum
length and shape of window’s
passband-stopband error of filter’ssidelobe amplitude of window
spectrum window’s shape
The smaller mainlobe width, the narrower transition width of filter’s;
The smaller sidelobe amplitude of window spectrum, the smaller
relative error of filter’s.
M increases mainlobe width of window spectrum minishes filter
oscillate more quickly, transition width minishes, relative amplitude of
oscillation is preserved.
In order to improve relative amplitude of oscillation, we need to
change the shape of window’s.
7.2.2 properties of commonly used windows
0nM
1，
(1)rec tan gular : w[n]  
0， other 2n
M
， 0n

M
2

2n M

( 2)bartlett (triangular ) : w[ n]  2 
， nM
M 2

other
 0，


  2n 
0.5  0.5 cos
 0nM
(3) hanning : w[ n]  
 M ， other

0


 2n 
0.54  0.46 cos
 0nM
( 4) ham min g : w[ n]  
 M ， other

0


 2n 
 4n 
0.42  0.5 cos 
  0.08 cos
 0nM
(5)blackman : w[ n]  
 M 
 M ， other

0

A general designation: Blackman window family, except triangular window
Figure 7.21
1
 
 I   (1  [(n  M ) /  M ]2 ) 2 

 0 
2  2 
(6)kaiser : w[n]   


I0 ( )

0

Figure 7.24(a)
0nM
other
(a)-(e) attenuation of sidelobe increases,
width of mainlobe increases.
M=20
As B increases,
attenuation of
sidelobe increases,
width of mainlobe
increases.
B=6
As N increases：
attenuation of
sidelobe is preserved
width of mainlobe
decreases.
Figure 7.24(b)(c)
Transition width is a
little less than mainlobe
width
Table 7.1
7.2.3 effect to frequency response
attenuation of window spectrum’s sidelobe attenuation of filter’s stopband
width of window spectrum’s mainlobetransition width of filter’s
check the table for Blackman window;
formulas of Kaiser window:
0.1102( A  8.7)


  0.584( A  21) 0.4  0.07886( A  21)

0

 
A  50
21  A  50
A  21
A8
2.285M
A：attenuation of stopband, Δω：transition width
(5.50)
7.2.4 design step
1. Write the ideal impulse response：
 c  ( p   s ) / 2
 jM

j
j ( )
H d (e )  H d ( )e
 e 2
 0


1
j
jn
hd [n] 
H

2 

d
(e
)e
pass
band
stop
band
d
2.Comfirm the shape of window’s based on attenuation of stopband：
  min(  p ,  s )
A  20 log 10 
（1）check the table for Blackman window
（2）calculation for Kaiser window
0.1102( A  8.7)


  0.584( A  21) 0.4  0.07886( A  21)

0

A  50
21  A  50
A  21
3.Comfirm length of window’s based on transition width（M is even):
 |  s   p |
(1) check the table for Blackman window: compute
the length of window’s based on width of mainlobe:
(2) calculation for Kaiser window ：
M 
M 
D
2
A8
2.285
4.Cut the ideal impulse response
h[n]  hd [n]  w[n]
j
5. validate H (e )，correct c , M , （Kaiser•）
 , , A
We can think that the 4 specifications is
c
transformed into：
The same with IIR, maybe the specification is equivalent analog one.
Blackman family
Example 1
design a FIR high
pass
filter , need :
 p  0.7 , s  0.54 , p  3dB, s  40dB
(1)  c  ( p   s ) / 2  0.62
hd [n] 

1
2



H d (e
sin[  (n  M / 2)]
 (n  M / 2)
j

)e
jn
d
sin[ 0.62 (n  M / 2)]
0.62 ( n  M / 2)
(2)   min(  ,  )  min( 1  10 
p
s
p
/ 20
,10
, n  ......
 s / 20
)  0.01, A  40dB
hamming
(3) |    | 8 / 2  (0.7  0.54) , M  25  26
p
s
M
(4)
 sin[ (n  13)] sin[0.62 (n  13)]  
2n 
h[n]  


0
.
54

0
.
46
cos(
)  R27 [n]
 
 (n  13)
25 
  (n  13)
 
（5）validate the frequency response in MATLAB
h=fir1(26,0.62, 'high',hamming(27))
H=fft(h,512);
plot([0:511]/256,20*log10(abs(H)))
axis([0.5,0.7,-50,0]);
grid on
（6）correct
h=fir1(24,0.665, 'high',hamming(25))
H=fft(h,512);
plot([0:511]/256,20*log10(abs(H)))
axis([0.5,0.7,-80,0]);
grid on
h=
0.0001
-0.0538
-0.0538
0.0004
0.0023
0.0014
0.0358
-0.0040
-0.0040
0.1288
0.0009
0.0023
0.0004 0.0104 -0.0170 0.0009 0.0358
-0.2727 0.3357 -0.2727 0.1288 0.0014
-0.0170 0.0104
0.0001
Example 2
Kaiser family
design a FIR high
pass
filter , need :
 p  0.7 , s  0.54 , p  3dB, s  40dB
Solution1:
(1)  c  ( p   s ) / 2  0.62
hd [ n ] 
1

H


2

d
(e
j
)e
jn
d 
(2)   min(  ,  )  min( 1  10 
p
s
p
sin[  (n  M / 2)]
 (n  M / 2)
/ 20
,10
 s / 20

sin[ 0.62 (n  M / 2)]
0.62 (n  M / 2)
)  0.01, A  40dB
  0.5842(40  21) 0.4  0.07886(40  21)  3.3953
(3) M 
40  8
2.285(0.7  0.54 )
 27.86  28
(4) h[n]  sin[  (n  14)]  sin[ 0.62 (n  14)]  w[n]
 (n  14)
, n  ......
（5）validate and correct in MATLAB
h=fir1(18,0.662, 'high', kaiser (19,3.2953));
H=fft(h,512);
k=0:511;
subplot(1,2,1);
plot(k/256,20*log10(abs(H))); grid on;
subplot(2,2,2);
plot(k/256,20*log10(abs(H)));
axis([0.5,0.55,-50,-0]);
grid on
subplot(2,2,4);
plot(k/256,20*log10(abs(H)));
axis([0.65,0.75,-5,0]);
grid on
Example 3
plot the response to signal
x[n]  10 cos(0.1 )  3sin(0.8 ) R20[n]
h=fir1(18,0.5, 'high',kaiser(19,6));
n=[0:100];
x=10*cos(0.1*pi*n)+3*sin(0.8*pi*n) ;
subplot(2,1,1); plot(n,x)
y=filter(h,1,x) ;
subplot(2,1,2); plot(n,y)
7.2.5 FIR summary
w[n]
shape
length
W (e
j
)
H (e
j
)
attenuation of sidelobe
attenuation of stopband
（the first row in the table）（ the third row in the table）
width of mainlobe
transition width
(the second row)
（the second row /2）
7.3 brief introduction of other filter design
techniques using matlab
1.
2.
3.
4.
IIR: minimum mean square error in frequency domain: invfreqz( )
FIR: frequency sampling ：fir2( )
FIR: minimum mean square error ：firls( )
FIR: Parks-McCellan/Remez arithmetic：
remez( )
7.4 summary
FIR digital filter
IIR digital filter
finite-length IR
infinite-length IR
no non-zero poles, stable
non-zero poles, stable?
realization: convolution /recursion recursion
realize linear phase easily
can not realize linear phase in deed
stable
higher order
maybe instable because of rounding
lower order because of recursion
FFT
no fast arithmetic
summary
7.1
7.2
7.3
design of IIR（continuous-time filter）
7.1.1 impulse invariance
7.1.2 bilinear transform
design of FIR（windowing）
7.2.1 design ideas
7.2.2 properties of commonly used windows
7.2.3 effect to frequency response
7.2.4 design step
comparison between IIR and FIR
requirements：
understand the principles of impulse invariance and bilinear
transform and their mapping characteristic, implication;
understand design ideas of windowing;
design various filters using MATLAB.
difficulties：
relationship among prototype analog filter, digital filter and
equivalent analog filter;
effects of shape and length of window’s to system characteristic；
why impulse invariance and bilinear transform can not be used
in high-pass filter design。
exercise and experiment
7.16
7.18
7.22(a)(b)
the second experiment
50
52
56
57
59
7.23