Filter Design Fall11
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Transcript of Filter Design Fall11
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Digital Filter Design
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(1) Frequency selective filters: spectral shapers
lowpass highpass bandpass bandstop
1. Introduction -- Digital Filter Design
FIR: - windowing
- equiripple design
IIR : - mapping from analog filters
- impulse invariance
(2)Filter Design Techniques
BGL/SNU
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4. FIR Filter Design by Windowing
- Given a desired frequency response
evaluate
specification.giventheinfallspectrumfrequency
resultingthethatsuchor,segment offinite
Therefore, take apractical.notsolong,infinitelyis
However,coefficients.filterdesiredtheis- Then,
,)( jd eH
deeHnh njjdd )(
2
1][
This process of getting out of is
called Windowing
][nh ][nhd
(1) Design Concept
][nhd
][nhd
][nhd ][nh
,)( jeH
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4
2. Window based method:
-30 -20 -10 0 10 20 30-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25Ideal LPFc=/4
Shifting
-30 -20 -10 0 10 20 30-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25Ideal LPFc=/4-shifted
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25LPF by Window Design Technique
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5
Digital Filter Specifications
For example the magnitude response of a
digital lowpass filter may be given as indicated
below
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Figure 7.2
magnitude response of
equivalent analog system
monotonous descent
2/1
c
passband
tolerance
c
stopband
tolerance
passband
cutoff
frequency
stopband
cutoff
frequency
3dB cutoff
frequency
absolute specification
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7
Digital Filter Specifications
In the passband we require that
with a deviation
In the stopband we require that
with a deviation
1)( jeG
0)( jeGs
pp0
s
pp
j
p eG
,1)(1
ssjeG ,)(
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8
Digital Filter Specifications
Filter specification parameters
- passband edge frequency
- stopband edge frequency - peak ripple valuein the passband
- peak ripple valuein the stopband
p
s
s
p
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9
Digital Filter Specifications
Practical specifications are often given in
terms of loss function (in dB)
Peak passband ripple
dB
Minimum stopband attenuationdB
)(log20)( 10
jeGG
)1(log20 10 pp
)(log20 10 ss
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(3)Filter Specification(LPF)
|)(| jeH
11
1
11
2
0p
s
In some IIR filter design
11
1
1 2 3
1
2
3
ripplepassband:1
ripplepassband:2
],0[bandpassp
],[bandtransitionsp
],[bandstop s
BGL/SNU
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1
decide specifications according to application2decide type according to specificationgenerally , if the phase is required ,
choose FIR.
3approach specifications using causal and stable discrete-time system
4choose a software or hardware realization structure, take effects of limited word
length into consideration
H(z) or h[n]
Design steps
Specifications for bandpass and bandstop filters
up and down passband cutoff frequency
up and down stopband cutoff frequency
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otherwise
Mnnwnwnhnh d
,0
0,1][],[][][
)2
sin(
)
2
)1(sin(
11)(
)()(2
1)()()(
2/
)1(
0
)(
M
ee
eeeW
deWeH
eWeHeH
Mjj
MjM
n
njj
jj
d
jj
d
j
(2) Rectangular Windowing
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0 2
1
2
M
sidelobepeak
lobemain
0 2
)( )( jeW)( j
d eH
)( jeH
)()(j
eW
j
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n
n
n
)( jeW
)( jeH
)( jd eH
cc
1
2
M
0 M
][nhd
][nw
][nh
)(e
][][][ nhnwnh Rd )()()( jj
R
j
d eHeWeH
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Window Design Techniques
Rectangular Window
otherwise,0
10,1)(
Mnnw
Exact transition width = s-
p= 1.8/M
Min. stopband attenuation = 21dB
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Window Design Techniques
Bartlett Window
Exact transition width = s-
p= 6.1/M
Min. stopband attenuation = 25dB
otherwise,0
12
1,
1
22
2
10,
1
2
)( MnM
M
n
Mn
M
n
nw
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Window Design Techniques
Hann Window
Exact transition width = s-
p= 6.2/M
Min. stopband attenuation = 44dB
otherwise,0
10)],1
2cos(1[50
)( MnM-
n
.nw
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Window Design Techniques
Hamming Window
Exact transition width = s-
p= 6.6/M
Min. stopband attenuation = 53dB
MATLAB function: w=hamming (M)
otherwise,0
10)],1
2cos(46.0540
)( MnM-
n
.nw
0 5 10 15 20 25 30 35 40 450
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1Hamming Window: M=45
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Window Design Techniques
Blackman Window
Exact transition width = s-
p= 11/M
Min. stopband attenuation = 74dB
otherwise,0
10)],1
4cos(08.0)
1
2cos(5.0420
)( Mn
M-
n
M-
n
.nw
0 5 10 15 20 25 30 35 40 450
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1Blackman Window: M=45