Fir and iir filter_design

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 Invariance
method
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)
1
z−
1
z−
1
z−
1
z−
1
z−
1
z−

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
1
z−
1
z−
1
z−
1
z−
1
z−
1
z−
1
z−
1
z−

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.

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

Introduction
 The structures of FIR filter
x(n) x(n)
b1
b2
b3
b4
b0
Y(n) Y(n)
Direct form Cascade form
b1
b2
d1
d2
b0
1
z−
1
z−
1
z−
1
z−
1
z−
1
z−
1
z−
1
z−

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=

invariance method
 The frequency response is the
Fourier transform of the continuous
time function
and hence
'( )H ω
*
( ) ( ) ( )c c
n
h t h nT t nTδ
∞
=−∞
= −∑
1 2
'( ) ( )c
k
H H j k
T T
π
ω ω
∞
=−∞
 
= − 
 
∑

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 jk
T T
π∞
=
=−∞
 
= − 
 
∑

invariance method
 The impulse invariance
transformation does map the -axis
and the left-half s plane into the unit
circle and its interior, respectively
jω
Re(Z)
Im(Z)
1
S domain Z domain
sT
e
jω
σ

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 ω ( )cH jω
0 ω
'( )H ω
/Tπ 2 /Tπ
( )cH jω

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.

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
k
c
k k
A
H s
s s=
=
−
∑
1
( ) ( )k
N
s t
c k
k
h t A e u t
=
= ∑
1
( ) ( )k
N
s nT
k
k
h n A e u n
=
= ∑
1
1
( )
1 k
N
k
s T
k
A
H z
e z−
=
=
−
∑

invariance method
 Example:
Expanding in a partial fraction
expansion, it produce
The impulse invariant transformation
yields a discrete time design with the
system function
2 2
( )
( )
c
s a
H 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 T
H z
e z e z− + − − − −
= +
− −

IIR Filter Design by Bilinear
 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.sT
z e=

 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( ')s
T T
π π
− ≤ ≤
's
's T
z e=
σ
jω
'σ
jω
/Tπ−
/Tπ
s domain s’ domain

 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 's
12
' tanh ( )
2
sT
s
T
−
=
jω
s jω= ' 's jω=
12
' tan ( )
2
T
T
ω
ω −
=

 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
π π
− 'ω
ω
ω 'ω
'ω ω≈
-
ω
'ω
/Tπ
/Tπ−

 The desired transformation to is
now obtained by inverting
to produce
 And setting , which yields
12
' tanh ( )
2
sT
s
T
−
=
2 '
tanh( )
2
s T
s
T
=
s z
1
' ( )lns z
T
=
2 ln
tanh( )
2
z
s
T
=
1
1
2 1
( )
1
z
T z
−
−
−
=
+
Re(Z)
Im(Z)
1
S domain Z domain
1
2
1
2
T
s
z
T
s
+
=
−
jω
σ

 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 − −
= − +
=

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 n
d d
n
H e h n eω ω
∞
−
=−∞
= ∑
1
[ ] ( )
2
j j n
d dh n H e e d
π
ω ω
π
ω
π −
= ∫

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= g
( ) ( ) ( )dH W Hω ω ω= ∗

function technique
 Truncation of the Fourier series
produces the familiar Gibbs
phenomenon
 It will be manifested in ,
especially if is discontinuous.
( )H ω
( )dH ω

function technique
 1.Rectangular window
 2.Triangular window (Bartett window)
1, 0
[ ]
0,
n M
w n
otherwise
≤ ≤
= 

2 , 0
2
2[ ] 2 ,
2
0,
n Mn
M
n Mw n n M
M
otherwise
 ≤ ≤


= − < ≤



function technique
 1.Rectangular window
 2.Triangular window (Bartett window)
0 10 20 30 40 50 60
0
0.5
1
sequence (n)
T(n)
Rectangular window
0 10 20 30 40 50 60
0
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 units
FrequencyresponseT(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
Bartlett window

function technique
 3.HANN window
 4.Hamming window
1 2
1 cos , 0
[ ] 2
0,
n
n M
w n M
otherwise
π  
− ≤ ≤  =  


2
0.54 0.46cos , 0
[ ]
0,
n
n M
w n M
otherwise
π
− ≤ ≤
= 


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
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
Hamming window
0 10 20 30 40 50 60
0
0.5
1
sequence (n)
T(n)
Hanning window
0 10 20 30 40 50 60
0
0.5
1
sequence (n)
T(n)
Hamming window
function technique
 3.HANN window
 4.Hamming window

function technique
 5.Kaiser’s window
 6.Blackman window
2
0
0
2
[ 1 (1 ) ]
[ ] , 0,1,...,
[ ]
n
I
Mw n n M
I
β
β
− −
= =
2 4
0.42 0.5cos 0.08cos , 0
[ ]
0,
n n
n M
w n M M
otherwise
π π
− + ≤ ≤
= 


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
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
Kaiser window
 5.Kaiser’s window
 6.Blackman window
0 10 20 30 40 50 60
0
0.5
1
sequence (n)
T(n)
Blackman window
0 10 20 30 40 50 60
0
0.5
1
sequence (n)
T(n)
Kaiser window
function technique

function technique
( / )s Mω
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

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 ω
( )dh n
' ( )dH ω
( )dh n

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]
0
1
2 ( 1/ 2)
( ) 2 cos( )
1
M
k
k
k n
h n A A
M
π
=
+
= +
+
∑
kA [ / 2]M
0,1,...,n M=

sampling technique
 It can be rewritten as
where and
 Therefore, it may write
where
1
/ 2 /
0
/2
( )
N
j k N j kn N
k
k
k N
h n A e eπ π
−
=
≠
= ∑ 0,1,..., 1n N= −
1N M= + k N kA A −=
/ 2 /
( ) j k N j kn N
k kh n A e eπ π
=
1
0
/2
( ) ( )
N
k
k
k N
h n h n
−
=
≠
= ∑
0,1,..., 1n N= −

sampling technique
 with corresponding transform
where
 Hence
which has a linear phase
1
0
/2
( ) ( )
N
k
k
k N
H z H z
−
=
≠
= ∑
/
2 / 1
(1 )
( )
1
j k N N
k
k j k N
A e z
H z
e z
π
π
−
−
−
=
−
' ( 1)/2 sin / 2
( )
sin[( / / 2)]
j T N
k k
TN
H A e
k N T
ω ω
ω
π ω
− −
=
−

sampling technique
 The magnitude response
which has a maximum value
at where
' sin / 2
( )
sin[( / / 2)]
k k
TN
H A
k N T
ω
ω
π ω
=
−
kN A
/k sk Nω ω= 2 /s Tω π=

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 ω
kω ω= '
( )kH ω
'( )k kH N Aω =
'
( )dH ω
kω
'
( ) /k d kA H Nω= ±

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 Hω ω=
kω

sampling technique
 In between these frequencies, is
interpolated as the sum of the
responses , and its magnitude
does not, equal that of
'( )H ω
'
( )kH ω
'
( )dH ω

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 P
H
k P M
ω
=
= 
= +
'
( ) /k d kA H Nω= ±
( 1) / ( 1), 0,1,...,
0, 1,...,[ / 2]
k
k
M k P
A
k P M
 − + =
= 
= + '
( )kH ω
'
( )d kH ω

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 p≠0.38( 1) /( 1)p
pA M= − +

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

 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

 1. ( ) ( ) ( ) ( )∫∫ −−
−
−=−=
2/1
2/1
22/
2/
21
dFFHFRdffHfRfMSE d
f
f ds
s
s
( ) ( ) dFFHFnns d
k
n
∫ ∑−
=
−=
2/1
2/1
2
0
|| 2cos][ π
( ) ( ) ( ) ( ) dFFHFnnsFHFnns d
k
n
d
k
n
∫ ∑∑−
==






−





−=
2/1
2/1
00
2cos][2cos][ ππ
( ) ( )
1/2
1/2
0 0
[ ]cos 2 [ ]cos 2
k k
n
s n n F s F dF
τ
π τ π τ
−
= =
= ∑ ∑∫
( ) ( ) ( )
1/2 1/2
2
1/2 1/2
0
2 [ ]cos 2
k
d d
n
s n n F H F dF H F dFπ
− −
=
− +∑∫ ∫

 2. when n ≠ τ,
when n = τ, n ≠ 0,
when n = τ, n = 0,
 3. The formula can be repressed as:
( ) ( ) 02cos2cos
2/1
2/1
=∫−
dFFFn τππ
( ) ( ) 2/12cos2cos
2/1
2/1
=∫−
dFFFn τππ
( ) ( ) 12cos2cos
2/1
2/1
=∫−
dFFFn τππ
( ) ( ) ( )dFFHdFFHFnnsnssMSE dd
k
n
k
n
∫∫ ∑∑ −−
==
+−+=
2/1
2/1
22/1
2/1
01
22
2cos][22/][]0[ π

 4. Doing the partial differentiation:
 5. Minimize MSE: for all n’s
( )∫−
−=
∂
∂ 2/1
2/1
2]0[2
]0[
dFFHs
s
MSE
d ( ) ( )∫−
−=
∂
∂ 2/1
2/1
2cos2][
][
dFFHFnns
ns
MSE
dπ
0
][
=
∂
∂
ns
MSE
( )∫−
=
2/1
2/1
]0[ dFFHs d ( ) ( )∫−
=
2/1
2/1
2cos2][ dFFHFnns dπ
[ ] [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
=
+ =
− =
= ≥

Conclusions
 FIR advantage:
1. Finite impulse response
2. It is easy to optimalize
3. Linear phase
4. Stable
 FIR disadvantage:
1. It is hard to implementation than IIR

Conclusions
 IIR advantage:
1. It is easy to design
2. It is easy to implementation
 IIR disadvantage:
1. Infinite impulse response
2. It is hard to optimalize than FIR
3. Non-stable

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

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

Fir and iir filter_design

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