CHAPTER 7 filter design techniques

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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
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7.0 introduction
ideal frequency selective filter
band-stop filternotch filterquality factor of
band-pass filterpass-band width/center frequency
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Figure 7.1
impulse response of ideal low-pass
filter,noncausal,unrealizable
Specifications for filter design given in
frequency domain phaselinear?
magnitudegiven by a tolerance scheme
analog or digital,absolute or relative
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relative specificationsmaximum magnitude in
passband is normalized to 1, viz. 0dB
maximum attenuation in passband
minimum attenuation in stopband
3dB cutoff frequency
magnitude response of equivalent analog system
digital specification, finally
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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
specificationgenerally , if the phase is
required , choose FIR. (3)approach specifications
using causal and stable discrete-time
system viz. design H(z0) or hn,nonuniform (4)
choose a software or hardware realization
structure, take effects of limited word length
into consideration H(z) or hn
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7.1 design of discrete-time IIR filters
from continuous-time filters
attentionoriginal 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
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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
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BW design formula specification?system function
magnitude frequency function
take the specifications into system function and
get the results from equation group
OR
confirm the poles of system function (in the
left half plane)
get system function
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EXAMPLE
N,Wcbuttord(2000pi,4000pi,1,15,
's') Bs,Asbutter(N,Wc, 's') H,Wfreqs(Bs
,As) plot(W/2/pi,20(log10(abs(H)))) axis(1000,2
000,-16,0) grid on
OUTPUT N 4 Wc 8.1932e003 Bs
1.0e015 0 0 0
0 4.5063 As 1.0e015 0.0000 0.0000
0.0000 0.0014 4.5063
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Or  N,Wcbuttord(2000pi,4000pi,1,15,
's') z,p,kbutter(N,Wc, 's' ) Output z
Empty matrix 0-by-1 p 1.0e003 -7.5695
3.1354i -7.5695 - 3.1354i -3.1354
7.5695i -3.1354 - 7.5695i k 4.5063e015
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  design a low pass cheby analogy filter
EXAMPLE
N,Wccheb1ord(2000pi,4000pi,1,15,
's') Bs,Ascheby1(N,1,Wc, 's') H,Wfreqs(Bs
,As) plot(W/2/pi,20(log10(abs(H)))) axis(0,4000
,-30,0) grid on
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  design a low pass cheby analogy filter
EXAMPLE
N,Wccheb2ord(2000pi,4000pi,1,15,
's') Bs,Ascheby2(N,15,Wc, 's') H,Wfreqs(B
s,As) plot(W/2/pi,20(log10(abs(H)))) axis(0,400
0,-30,0) grid on
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  design a high pass analogy filter
EXAMPLE
N,Wcbuttord(4000pi,2000pi,1,15,
's') Bs,Asbutter(N,Wc, 'high',
's') H,Wfreqs(Bs,As) plot(W/2/pi,20(log10(a
bs(H)))) axis(0,4000,-16,0) grid on
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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)?
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7.1.1 filter design by impulse invariance
according to
conversion formula
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Relationship between poles(causal and stable)
relation between frequencies
S plane
Z plane
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relation between frequency response when
aliasing is small, the frequency response is the
same.
strongpointlinear frequency mapping
shortcomingaliasing in frequency
response? restriction in applicationcan not used
in high-pass and bandstop filter
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Design steps
(3)
about Td independent of T do not
influence aliasing arbitrary
value,generally, take 1(attention(1)and(3)have
the same value).
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EXAMPLE
wp0.2pi ws0.4pi ap1 as12 Td1 Wp
wp/Td Wsws/Td N,Wcbuttord(Wp,Ws, ap , as,
's') Bs,Asbutter(N,Wc, 's') Bz,Azimpi
nvar(Bs,As,1/Td) H,Wfreqs(Bs,As) plot(W
/pi,20(log10(abs(H))), 'r') hold
on H,wfreqz(Bz,Az) plot(w/pi,20(log10(abs
(H)))) axis(0.2,0.4,-20,0) grid
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7.1.2 filter design by bilinear transform
design thought
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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
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relation between frequency response
strongpointno aliasing shortcomingnonlinear? re
striction in application can not
used in differentiator.
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design steps (1) 
(3)
about Td independent of T
arbitrary value,generally, take
1(attention(1)and(3)have the same value).
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EXAMPLE
wp0.2613pi ws0.4018pi ap0.75 as20 Td1
Ws2/Tdtan(ws/2) Wp2/Tdtan(wp/2) N,Wcbutt
ord(Wp,Ws,ap,as,s) Bs,Asbutter(N,Wc,
s) Bz,Azbilinear(Bs,As,1/Td) H,Wfreqs(Bs
,As) plot(W/pi,20(log10(abs(H))),Rx) hold
on H,wfreqz(Bz,Az) plot(w/pi,20(log10(abs
(H)))) ylabel(?????????dB
?????????dB) xlabel(???????p??/?
???????p??) axis(0.25,0.5,-20,-0.45) grid
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OR wp0.2613pi ws0.4018pi ap0.75 as20 N,w
cbuttord(wp/pi,ws/pi,ap,as) Bz,Azbutter(N,wc
) H,wfreqz(Bz,Az) plot(w/pi,20(log10(ab
s(H))))
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EXAMPLE
N,Wcbuttord(0.6,0.5,ap,as) Bz,Azbutter(N,Wc,
'high') H,wfreqz(Bz,Az) plot(w/pi,20(lo
g10(abs(H)))) grid
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EXAMPLE
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N,wcbuttord(0.45 0.55,0.4 0.6,3,10)
?????? B,Abutter(N,wc) H,wfreqz(B,A) plot(
w/pi,20(log10(abs(H)))) ylabel(20logH(ej?)
dB) xlabel(?????p??) axis(0.4,0.6,-10,0)
grid on
Output N 2 wc 0.4410 0.5590 B
0.0271 0 -0.0541 0
0.0271 A 1.0000 0 1.4838
0 0.5920
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7.1.3 IIR summary
1.design steps
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
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7.1.4 frequency conversion
Frequency conversion in analog domain
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Frequency conversion in digital domain
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7.2 design of FIR filters by windowing
7.2.1 design ideas7.2.2 properties of commonly
used windows7.2.3 effect to frequency
response7.2.4 design step
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7.2.1 design ideas
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Gibbs phenomenon frequency response oscillate at
wc. transition width of filers?mainlobe width
of window spectrum ?length and shape of
windows passband-stopband error of
filters?sidelobe amplitude of window spectrum
?windows shape The smaller mainlobe width, the
narrower transition width of filters The
smaller sidelobe amplitude of window spectrum,
the smaller relative error of filters. 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 windows.
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7.2.2 properties of commonly used windows
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A general designation Blackman window family,
except triangular window
Figure 7.21
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(a)-(e) attenuation of sidelobe increases, width
of mainlobe increases.
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As B increases, attenuation of sidelobe
increases, width of mainlobe increases.
M20
B6
As N increases attenuation of sidelobe is
preserved, width of mainlobe decreases.
Figure 7.24(b)(c)
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Transition width is a little less than mainlobe
width
Table 7.1
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7.2.3 effect to frequency response
attenuation of window spectrums sidelobe
?attenuation of filters stopband width of window
spectrums mainlobe?transition width of filters
check the table for Blackman window formulas of
Kaiser window

Aattenuation of stopband, ??transition width
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7.2.4 design step
1. Write the ideal impulse response
2.Comfirm the shape of windows based on
attenuation of stopband
(1)check the table for Blackman
window (2)calculation for Kaiser window
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3.Comfirm length of windows based on transition
width(M is even)
(1) check the table for Blackman window compute
the length of windows based on width of
mainlobe
(2) calculation for Kaiser window
4.Cut the ideal impulse response
5.
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Example 1
Blackman family
(1)
(2)
hamming
(3)
(4)
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(5)validate the frequency response in MATLAB
hfir1(26,0.62, 'high',hamming(27)) Hfft(h,512)
plot(0511/256,20log10(abs(H))) axis(0.5,
0.7,-50,0) grid on
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(6)correct
hfir1(24,0.665, 'high',hamming(25)) Hfft(h,51
2) plot(0511/256,20log10(abs(H))) axis
(0.5,0.7,-80,0) grid on
h 0.0001 0.0023 -0.0040 0.0004
0.0104 -0.0170 0.0009 0.0358 -0.0538
0.0014 0.1288 -0.2727 0.3357 -0.2727
0.1288 0.0014 -0.0538 0.0358 0.0009
-0.0170 0.0104 0.0004 -0.0040
0.0023 0.0001
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Example 2
Kaiser family
Solution1
(1)

(2)
(3)
(4)
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(5)validate and correct in MATLAB
hfir1(18,0.662, 'high', kaiser
(19,3.2953)) Hfft(h,512) k0511 subplot(1,2,
1) plot(k/256,20log10(abs(H))) grid
on subplot(2,2,2) plot(k/256,20log10(abs(H)))
axis(0.5,0.55,-50,-0) grid
on subplot(2,2,4) plot(k/256,20log10(abs(H)))
axis(0.65,0.75,-5,0) grid on
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Example 3
plot the response to signal
hfir1(18,0.5, 'high',kaiser(19,6)) n0100
x10cos(0.1pin)3sin(0.8pin)
subplot(2,1,1) plot(n,x) yfilter(h,1,x)
subplot(2,1,2) plot(n,y)
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7.2.5 FIR summary
shape attenuation of
sidelobe attenuation of stopband (the
first row in the table)( the third row in the
table) length width of mainlobe transition
width (the second row) (the
second row /2)
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7.3 brief introduction of other filter design
techniques using matlab

• 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( )

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7.4 summary
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summary
7.1 design of IIR(continuous-time filter) 7.1.1
impulse invariance 7.1.2 bilinear
transform 7.2 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 7.3 comparison
between IIR and FIR
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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 windows to system characteristic
why impulse invariance and bilinear
transform can not be used in high-pass filter
design?
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exercise and experiment
7.16 7.18 7.22(a)(b) 7.23 the second
experiment 50 52 56 57 59