Feb.2008DISP Lab1 FIR and IIR Filter Design Techniques FIR IIR Speaker: Wen-Fu Wang Advisor: Jian-Jiun Ding E-mail: r96942061@ntu.edu.tw Graduate Institute.
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Feb.2008 DISP Lab 1
FIR and IIR Filter Design Techniques
FIR 與 IIR 濾波器設計技巧 Speaker: Wen-Fu Wang 王文阜 Advisor: Jian-Jiun Ding 丁建均 教授 E-mail: r96942061@ntu.edu.tw Graduate Institute of Communication Engineering National Taiwan University, Taipei, Taiwan, ROC
Feb.2008 DISP Lab 2
Outline
Introduction IIR Filter Design by Impulse
invariance method IIR Filter Design by Bilinear
transformation method FIR Filter Design by Window function
technique
Feb.2008 DISP Lab 3
Outline
FIR Filter Design by Frequency sampling technique
FIR Filter Design by MSE Conclusions References
Feb.2008 DISP Lab 4
Introduction Basic filter classification We put emphasis on the digital filter
now, and will introduce to the design method of the FIR filter and IIR filter respectively.
Filter
Analog Filter
Digital Filter
IIR Filter
FIR Filter
Feb.2008 DISP Lab 5
Introduction IIR is the infinite impulse response
abbreviation. Digital filters by the accumulator, the
multiplier, and it constitutes IIR filter the way, generally may divide into three kinds, respectively is Direct form, Cascade form, and Parallel form.
Feb.2008 DISP Lab 6
Introduction IIR filter design methods include the
impulse invariance, bilinear transformation, and step invariance.
We must emphasize at impulse invariance and bilinear transformation.
Feb.2008 DISP Lab 7
Introduction IIR filter design methods
Continuous frequency band transformation
Impulse Invariancemethod
Bilinear transformation method
Step invariance method
IIR filter
Normalized analog lowpass filter
Feb.2008 DISP Lab 8
Introduction
The structures of IIR filter
Direct form 1
Direct form2
b0
b1
b2 b2
b1
b0
-a1
-a2
-a1
-a2
x(n) x(n)Y(n) Y(n)
1z
1z
1z
1z
1z
1z
Feb.2008 DISP Lab 9
Introduction
The structures of IIR filter
Cascade form
x(n) Y(n)b0
b1
b2
-a1
-a2
-c1
-c2
d1
d2
Parallel form
Y(n)x(n)
b1
b0
d1
d0
E
-c1
-c2
-a1
-a2
1z
1z
1z
1z
1z
1z
1z
1z
Feb.2008 DISP Lab 10
Introduction FIR is the finite impulse response
abbreviation, because its design construction has not returned to the part which gives.
Its construction generally uses Direct form and Cascade form.
Feb.2008 DISP Lab 11
Introduction FIR filter design methods include the
window function, frequency sampling, minimize the maximal error, and MSE.
We must emphasize at window function, frequency sampling, and MSE.
Window function technique
Frequency sampling technique
Minimize the maximal error
FIR filter
Mean square error
Feb.2008 DISP Lab 12
Introduction The structures of FIR filter
x(n) x(n)
b1
b2
b3
b4
b0Y(n) Y(n)
Direct form Cascade form
b1
b2
d1
d2
b0
1z
1z
1z
1z
1z
1z
1z
1z
Feb.2008 DISP Lab 13
IIR Filter Design by Impulse invariance method
The most straightforward of these is the impulse invariance transformation
Let be the impulse response corresponding to , and define the continuous to discrete time transformation by setting
We sample the continuous time impulse response to produce the discrete time filter
( )ch t( )cH s
( ) ( )ch n h nT
Feb.2008 DISP Lab 14
IIR Filter Design by Impulse invariance method
The frequency response is the Fourier transform of the continuous time function
and hence
'( )H
*( ) ( ) ( )c cn
h t h nT t nT
1 2'( ) ( )c
k
H H j kT T
Feb.2008 DISP Lab 15
IIR Filter Design by Impulse invariance method
The system function is
It is the many-to-one transformation from the s plane to the z plane.
1 2( ) | )sT cz e
k
H z H s jkT T
Feb.2008 DISP Lab 16
IIR Filter Design by Impulse invariance method
The impulse invariance transformation does map the -axis and the left-half s plane into the unit circle and its interior, respectively
j
Re(Z)
Im(Z)
1
S domain Z domain
sTe
j
Feb.2008 DISP Lab 17
IIR Filter Design by Impulse invariance method
is an aliased version of
The stop-band characteristics are maintained adequately in the discrete time frequency response only if the aliased tails of are sufficiently small.
'( )H ( )cH j
0
'( )H
/T 2 /T
( )cH j
Feb.2008 DISP Lab 18
IIR Filter Design by Impulse invariance method
The Butterworth and Chebyshev-I lowpass designs are more appropriate for impulse invariant transformation than are the Chebyshev-II and elliptic designs.
This transformation cannot be applied directly to highpass and bandstop designs.
Feb.2008 DISP Lab 19
IIR Filter Design by Impulse invariance method
is expanded a partial fraction expansion to produce
We have assumed that there are no multiple poles
And thus
( )cH s
1
( )N
kc
k k
AH s
s s
1
( ) ( )k
Ns t
c kk
h t A e u t
1
( ) ( )k
Ns nT
kk
h n A e u n
11
( )1 k
Nks T
k
AH z
e z
Feb.2008 DISP Lab 20
IIR Filter Design by Impulse invariance method
Example:
Expanding in a partial fractionexpansion, it produce
The impulse invariant transformation yields a discrete time design with thesystem function
2 2( )
( )c
s aH s
s a b
1/ 2 1/ 2( )cH s
s a jb s a jb
( ) 1 ( ) 1
1/ 2 1/ 2( )
1 1a jb T a jb TH z
e z e z
Feb.2008 DISP Lab 21
IIR Filter Design by Bilinear transformation method
The most generally useful is the bilinear transformation. To avoid aliasing of the frequency
response as encountered with the impulse invariance transformation.
We need a one-to-one mapping from the s plane to the z plane.
The problem with the transformation is many-to-one. sTz e
Feb.2008 DISP Lab 22
IIR Filter Design by Bilinear transformation method
We could first use a one-to-one transformation from to , which compresses the entire s plane into the strip
Then could be transformed to z by with no effect from aliasing.
s 's
Im( ')sT T
's's Tz e
j
'
j
/T
/T
s domain s’ domain
Feb.2008 DISP Lab 23
IIR Filter Design by Bilinear transformation method
The transformation from to is given by
The characteristic of this transformation is seen most readily from its effect on the axis.
Substituting and , we obtain
s 's12
' tanh ( )2
sTs
T
js j ' 's j
12' tan ( )
2
T
T
Feb.2008 DISP Lab 24
IIR Filter Design by Bilinear transformation method
The axis is compressed into the interval for in a one-to-one method
The relationship between and is nonlinear, but it is approximately linear at small .
( , )T T
'
'
'
-
'/T
/T
Feb.2008 DISP Lab 25
IIR Filter Design by Bilinear transformation method
The desired transformation to is now obtained by inverting to produce
And setting , which yields
12' tanh ( )
2
sTs
T
2 'tanh( )
2
s TsT
s z
1' ( ) lns zT
2 lntanh( )
2
zsT
1
1
2 1( )1
z
T z
Re(Z)
Im(Z)
1
S domain Z domain
12
12
Ts
zTs
j
Feb.2008 DISP Lab 26
IIR Filter Design by Bilinear transformation method
The discrete-time filter design is obtained from the continuous-time design by means of the bilinear transformation
Unlike the impulse invariant transformation, the bilinear transformation is one-to-one, and invertible.
1 1(2/ )(1 )/(1 )( ) ( ) |c s T z z
H z H s
Feb.2008 DISP Lab 27
FIR Filter Design by Window function technique
Simplest FIR the filter design is window function technique
A supposition ideal frequency response may express
where
( ) [ ]j j nd d
n
H e h n e
1[ ] ( )
2j j n
d dh n H e e d
Feb.2008 DISP Lab 28
FIR Filter Design by Window function technique
To get this kind of systematic causal FIR to be approximate, the most direct method intercepts its ideal impulse response!
[ ] [ ] [ ]dh n w n h n
( ) ( ) ( )dH W H
Feb.2008 DISP Lab 29
FIR Filter Design by Window function technique
Truncation of the Fourier series produces the familiar Gibbs phenomenon
It will be manifested in , especially if is discontinuous.
( )H ( )dH
Feb.2008 DISP Lab 30
FIR Filter Design by Window function technique
1.Rectangular window
2.Triangular window (Bartett window)
1, 0[ ]
0,
n Mw n
otherwise
2 , 0 22[ ] 2 , 2
0,
n MnMn Mw n n MM
otherwise
Feb.2008 DISP Lab 31
FIR Filter Design by Window function technique
1.Rectangular window 2.Triangular window (Bartett window)
0 10 20 30 40 50 600
0.5
1
sequence (n)
T(n
)
Rectangular window
0 10 20 30 40 50 600
0.5
1
sequence (n)
T(n
)
Bartlett window
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-100
-50
0
50
100
pi unitsF
requ
ency
res
pons
e T
(jw)(
dB) Rectangular window
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-100
-50
0
50
100
pi units
Fre
quen
cy r
espo
nse
T(jw
)(dB
) Bartlett window
Feb.2008 DISP Lab 32
FIR Filter Design by Window function technique
3.HANN window
4.Hamming window
1 21 cos , 0
[ ] 2
0,
nn M
w n M
otherwise
20.54 0.46cos , 0
[ ]0,
nn M
w n Motherwise
Feb.2008 DISP Lab 33
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-100
-50
0
50
100
pi unitsF
requ
ency
res
pons
e T
(jw)(
dB) Hanning window
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-100
-50
0
50
100
pi units
Fre
quen
cy r
espo
nse
T(jw
)(dB
) Hamming window
0 10 20 30 40 50 600
0.5
1
sequence (n)
T(n)
Hanning window
0 10 20 30 40 50 600
0.5
1
sequence (n)
T(n)
Hamming window
FIR Filter Design by Window function technique
3.HANN window 4.Hamming window
Feb.2008 DISP Lab 34
FIR Filter Design by Window function technique
5.Kaiser’s window
6.Blackman window
20
0
2[ 1 (1 ) ]
[ ] , 0,1,...,[ ]
nI
Mw n n MI
2 40.42 0.5cos 0.08cos , 0
[ ]0,
n nn M
w n M Motherwise
Feb.2008 DISP Lab 35
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-100
-50
0
50
100
pi unitsF
requ
ency
res
pons
e T
(jw)(
dB) Blackman window
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-150
-100
-50
0
50
100
pi units
Fre
quen
cy r
espo
nse
T(jw
)(dB
) Kaiser window
5.Kaiser’s window 6.Blackman window
0 10 20 30 40 50 600
0.5
1
sequence (n)
T(n
)
Blackman window
0 10 20 30 40 50 600
0.5
1
sequence (n)
T(n
)
Kaiser window
FIR Filter Design by Window function technique
Feb.2008 DISP Lab 36
FIR Filter Design by Window function technique
( / )s M
Window Peak sidelobe level (dB)
Transition bandwidth
Max. stopband ripple(dB)
Rectangular -13 0.9 -21
Hann -31 3.1 -44
Hamming -41 3.3 -53
Blackman -57 5.5 -74
Feb.2008 DISP Lab 37
FIR Filter Design by Frequency sampling technique
For arbitrary, non-classical specifications of , the calculation
of ,n=0,1,…,M, via an appropriate approximation can be a substantial computation task.
It may be preferable to employ a design technique that utilizes specified values of directly, without the necessity of determining
' ( )dH
( )dh n
' ( )dH ( )dh n
Feb.2008 DISP Lab 38
FIR Filter Design by Frequency sampling technique
We wish to derive a linear phase IIR filter with real nonzero . The impulse response must be symmetric
where are real and denotes the integer part
( )h n
[ /2]
01
2 ( 1/ 2)( ) 2 cos( )
1
M
kk
k nh n A A
M
kA [ / 2]M
0,1,...,n M
Feb.2008 DISP Lab 39
FIR Filter Design by Frequency sampling technique
It can be rewritten as
where and Therefore, it may write
where
1/ 2 /
0/2
( )N
j k N j kn Nk
kk N
h n A e e
0,1,..., 1n N
1N M k N kA A
/ 2 /( ) j k N j kn Nk kh n A e e
1
0/2
( ) ( )N
kkk N
h n h n
0,1,..., 1n N
Feb.2008 DISP Lab 40
FIR Filter Design by Frequency sampling technique
with corresponding transform
where
Hence which has a linear phase
1
0/2
( ) ( )N
kkk N
H z H z
/
2 / 1
(1 )( )
1
j k N Nk
k j k N
A e zH z
e z
' ( 1)/2 sin / 2( )
sin[( / / 2)]j T N
k k
TNH A e
k N T
Feb.2008 DISP Lab 41
FIR Filter Design by Frequency sampling technique
The magnitude response
which has a maximum value at where
' sin / 2( )
sin[( / / 2)]k k
TNH A
k N T
kN A
/k sk N 2 /s T
Feb.2008 DISP Lab 42
FIR Filter Design by Frequency sampling technique
The only nonzero contribution to at is from , and hence that
Therefore, by specifying the DFT samples of the desired magnitude
response at the frequencies , and setting
'( )H
k ' ( )kH '( )k kH N A
' ( )dH k
' ( ) /k d kA H N
Feb.2008 DISP Lab 43
FIR Filter Design by Frequency sampling technique
We produce a filter design from equation (5.1) for which
The desired and actual magnitude responses are equal at the N frequencies
''( ) ( )k d kH H
k
Feb.2008 DISP Lab 44
FIR Filter Design by Frequency sampling technique
In between these frequencies, is interpolated as the sum of the responses , and its magnitude does not, equal that of
'( )H
' ( )kH ' ( )dH
Feb.2008 DISP Lab 45
FIR Filter Design by Frequency sampling technique
Example: For an ideal lowpass filter
from , we would choose
The frequency samples are indeed equal to the desired
' 1, 0,1,...,( )
0, 1,...,[ / 2]d k
k PH
k P M
' ( ) /k d kA H N
( 1) / ( 1), 0,1,...,
0, 1,...,[ / 2]
k
k
M k PA
k P M
' ( )kH
' ( )d kH
Feb.2008 DISP Lab 46
FIR Filter Design by Frequency sampling technique
The response is very similar to the result form using the rectangular window, and the stopband is similarly disappointing.
We can try to search for the optimum value of the transition sample would quickly lead us to a value of approximately , k p0.38( 1) /( 1)p
pA M
Feb.2008 DISP Lab 47
FIR Filter Design by MSE
: The spectrum of the filter we obtain
: The spectrum of the desired filter
MSE=
( )H f
( )dH f
2/
2/
21 s
s
f
f ds dffHfHf
0 0.1 0.2 0.3 0.4 0.5-0.5
0
0.5
1
1.5
Feb.2008 DISP Lab 48
FIR Filter Design by MSE
Larger MSE, but smaller maximal error
Smaller MSE, but larger maximal error
0 0.1 0.2 0.3 0.4-0.5
0
0.5
1
1.5
0 0.1 0.2 0.3 0.4-0.5
0
0.5
H(F) H(F) - H (F)d
0 0.1 0.2 0.3 0.4-0.5
0
0.5
1
1.5
0 0.1 0.2 0.3 0.4-0.5
0
0.5
H(F) H(F) - H (F) d
Feb.2008 DISP Lab 49
FIR Filter Design by MSE
1.
2/1
2/1
22/
2/
21 dFFHFRdffHfRfMSE df
f dss
s
dFFHFnns d
k
n
2/1
2/1
2
0
|| 2cos][
dFFHFnnsFHFnns d
k
nd
k
n
2/1
2/100
2cos][2cos][
1/2
1/20 0
[ ]cos 2 [ ]cos 2k k
n
s n n F s F dF
1/2 1/2 2
1/2 1/20
2 [ ]cos 2k
d dn
s n n F H F dF H F dF
Feb.2008 DISP Lab 50
FIR Filter Design by MSE 2. when n ,
when n = , n 0,
when n = , n = 0,
3. The formula can be repressed as:
02cos2cos2/1
2/1 dFFFn
2/12cos2cos2/1
2/1 dFFFn
12cos2cos2/1
2/1 dFFFn
dFFHdFFHFnnsnssMSE dd
k
n
k
n
2/1
2/1
22/1
2/101
22 2cos][22/][]0[
Feb.2008 DISP Lab 51
FIR Filter Design by MSE
4. Doing the partial differentiation:
5. Minimize MSE: for all n’s
2/1
2/12]0[2
]0[dFFHs
s
MSEd
2/1
2/12cos2][
][dFFHFnns
ns
MSEd
0][
ns
MSE
2/1
2/1]0[ dFFHs d
2/1
2/12cos2][ dFFHFnns d
[ ] [0]
[ ] [ ] / 2 for n=1,2,...,k
[ ] [ ] / 2 for n=1,2,...,k
[ ] 0 for n<0 and n N
h k s
h k n s n
h k n s n
h n
Feb.2008 DISP Lab 52
Conclusions
FIR advantage:1. Finite impulse response2. It is easy to optimalize3. Linear phase4. Stable FIR disadvantage:1. It is hard to implementation than IIR
Feb.2008 DISP Lab 53
Conclusions
IIR advantage:1. It is easy to design2. It is easy to implementation IIR disadvantage:1. Infinite impulse response2. It is hard to optimalize than FIR3. Non-stable
Feb.2008 DISP Lab 54
References [1]B. Jackson, Digital Filters and Signal
Processing, Kluwer Academic Publishers 1986 [2]Dr. DePiero, Filter Design by Frequency
Sampling, CalPoly State University [3]W.James MacLean, FIR Filter Design
Using Frequency Sampling [4] 蒙以正 , 數位信號處理 , 旗標 2005 [5]Maurice G.Bellanger, Adaptive Digital
Filters second edition, Marcel dekker 2001
Feb.2008 DISP Lab 55
References [6] Lawrence R. Rabiner, Linear Program
Design of Finite Impulse Response Digital Filters, IEEE 1972
[7] Terrence J mc Creary, On Frequency Sampling Digital Filters, IEEE 1972
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