Chapter7 Filter Design Techniques
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Transcript Chapter7 Filter Design Techniques
Biomedical Signal processing
Chapter 7 Filter Design Techniques
Zhongguo Liu
Biomedical Engineering
School of Control Science and Engineering, Shandong
University
2015/4/13
1
Zhongguo Liu_Biomedical Engineering_Shandong Univ.
Chapter 7 Filter Design Techniques
7.0 Introduction
7.1 Design of Discrete-Time IIR Filters
From Continuous-Time Filters
7.2 Design of FIR Filters by Windowing
7.3 Examples of FIR Filters Design by the
Kaiser Window Method
7.4 Optimum Approximations of FIR Filters
7.5 Examples of FIR Equiripple
Approximation
7.6 Comments on IIR and FIR DiscreteTime Filters
2
Filter Design Techniques
7.0 Introduction
3
7.0 Introduction
Frequency-selective filters pass only
certain frequencies
Any discrete-time system that modifies
certain frequencies is called a filter.
We concetrate on design of causal
Frequency-selective filters
4
Stages of Filter Design
The specification of the desired
properties of the system.
The approximation of the specifications
using a causal discrete-time system.
The realization of the system.
Our focus is on second step
Specifications are typically given in the
frequency domain.
5
Frequency-Selective Filters
Ideal lowpass filter
H lp e
jw
1,
w wc
0, wc w
sin wc n
hlp n
, n
n
H e
jw
1
2
6
wc
0
wc
2
Frequency-Selective Filters
Ideal highpass filter
0,
w wc
H hp e
1, wc w
sin wc n
hhp n n
, n
n
jw
H e jw
1
2
7
wc
0
wc
2
Frequency-Selective Filters
Ideal bandpass filter
H bp e
jw
1, wc1 w wc2
0, others
H e jw
1
8
wc2
wc1
0
wc1
wc2
Frequency-Selective Filters
Ideal bandstop filter
H bs e
jw
0, wc1 w wc2
1, others
H e jw
1
9
wc2
wc1
0
wc1
wc2
Linear time-invariant discrete-time system
If input is bandlimited and sampling frequency
is high enough to avoid aliasing, then overall
system behave as a continuous-time system:
H e jT ,
H eff j
0,
T
T
continuous-time specifications are converted to discrete
time specifications by:
w
jw
H e
H eff j , w
w T
T
10
Example 7.1 Determining Specifications
for a Discrete-Time Filter
Specifications of the continuous-time filter:
1. passband 1 0.01 H eff j 1 0.01 for 0 2 2000
2. stopband H eff j 0.001 for 2 3000
4
H e jT ,
H eff j
0,
11
T 10 s
T
T
1
2 f max 2
2T T
2 5000
Example 7.1 Determining Specifications
for a Discrete-Time Filter
Specifications of the continuous-time filter:
1. passband 1 0.01 H eff j 1 0.01 for 0 2 2000
2. stopband H eff j 0.001 for 2 3000
1 0.01
2 0.001
p 2 (2000)
s 2 (3000)
12
4
T 10 s
1
2 f max 2
2T T
2 5000
Example 7.1 Determining Specifications
for a Discrete-Time Filter
T 104 s
T
Specifications of the
discrete-time filter in
1 0.01
2 0.001
p 0.4
p 2 (2000) s 2 (3000)
13
s 0.6
Filter Design Constraints
Designing IIR filters is to find the
approximation by a rational function of z.
The poles of the system function must lie
inside the unit circle(stability, causality).
Designing FIR filters is to find the
polynomial approximation.
FIR filters are often required to be linearphase.
14
Filter Design Techniques
7.1 Design of Discrete-Time IIR
Filters From Continuous-Time Filters
15
7.1 Design of Discrete-Time IIR Filters
From Continuous-Time Filters
The traditional approach to the design
of discrete-time IIR filters involves the
transformation of a continuous-time
filter into a discrete filter meeting
prescribed specification.
16
Three Reasons
1. The art of continuous-time IIR filter
design is highly advanced, and since
useful results can be achieved, it is
advantageous to use the design
procedures already developed for
continuous-time filters.
17
Three Reasons
2. Many useful continuous-time IIR
design method have relatively simple
closed form design formulas.
Therefore, discrete-time IIR filter
design methods based on such
standard continuous-time design
formulas are rather simple to carry
out.
18
Three Reasons
3. The standard approximation methods
that work well for continuous-time
IIR filters do not lead to simple
closed-form design formulas when
these methods are applied directly to
the discrete-time IIR case.
19
Steps of DT filter design by transforming a
prototype continuous-time filter
The specifications for the continuoustime filter are obtained by a
transformation of the specifications for
the desired discrete-time filter.
Find the system function of the
continuous-time filter.
Transform the continuous-time filter
to derive the system function of the
discrete-time filter.
20
Constraints of Transformation
to preserve the essential properties of the
frequency response, the imaginary axis of the
s-plane is mapped onto the unit circle of the
jw
z-plane.
s j z e
Im
Im
s plane
Re
21
z plane
Re
Constraints of Transformation
In order to preserve the property of
stability, If the continuous system has poles
only in the let half of the s-plane, then the
discrete-time filter must have poles only
inside the unit circle.
Im
s plane
Re
22
Im
z plane
Re
7.1.1 Filter Design by Impulse
Invariance
The impulse response of discrete-time
system is defined by sampling the impulse
response of a continuous-time system.
hn Td hc nTd
Relationship of
frequencies
He
if Hc j 0, Td
jw
w
2
H c j j
k
Td
k
Td
w
jw
then H e H c j , w
Td
w Td for w
23
relation between frequencies
Td , ,
w
2
H e H c j j
Td
k
Td
if Hc j 0, Td then H e jw H j w ,
c
T
d
j
Relationship of
frequencies
S plane 3 / Td
/ Td
- / Td
24
jw
Z plane
k
w
Aliasing in the Impulse Invariance
w
2
jw
H j T
He
if Hc j 0, Td
w
then H e H c j ,
Td
w
jw
25
k
c
d
j
k
Td
Review
st
periodic sampling
T:sample period; fs=1/T:sample rate
t
nT
n
Ωs=2π/T:sample rate
xs t xc t s t xc t t nT
n
x[n] xc (t ) |t nT xc (nT )
26
x nT t nT
n
c
Review
Relation between Laplace Transform
and Z-transform
Time domain:
x(t )
Complex frequency
domain:
s j
2f
27
Laplace transform
X ( s) x(t )e dt
st
j
0
s pl ane
X ( s) x(t )e dt
st
Since
So
s j
0
s j
j
s- pl ane
0
frequency domain :
X ( j) x(t )e
jt
dt
Fourier Transform
Fourier Transform
is the Laplace transform when s
have the value only in imaginary axis, s=jΩ
28
For discrete-time signal,
x(n) x(t ) (t nT ) x(nT ) (t nT )
n
n
the Laplace transform
L [ x(n)]
st
x(n)e dt
x(nT ) (t nT )e dt
z-transform
of discretetime signal
29
n
st
x(nT )e snT X (e sT )
n
x ( n) z X ( z )
n
n
令:z
e
sT
L
[ x(n)]
X ( z)
x(nT )e snT X (e sT )
n
x(n) z
n
let:z
n
Laplace transform
continuous time signal
z-transform
ze e
sT
so:
30
e
discrete-time signal
( j )T
T
r e
T
T
e e
jT
sT
relation
re
relation between
s and z
j
ze e
sT
( j )T
T
e e
jT
re
j
T 2 f f s
j
z re |r 1 e
j
X (e )
x ( n )e
n
j
j n
DTFT :
Discrete Time
Fourier Transform
j
S plane 3 / Ts
/ Ts
- / Ts
31
Z plane
Ts 2 f f s 2 f
0
2
f
/
2
s
:
0 s 2 f s
s 2s
3
Ts
Ts
3
Ts
32
f fs
0
0 2
2 4
j
s pl ane
fs
2
2 Ts
:
f
0
z plane
Im[ z ]
r
0
Re[ z ]
discrete-time filter design by impulse invariance
If input is bandlimited and fs>2fmax , :
H e jT ,
H eff j
0,
jw
w Td for w
w
then H e H c j ,
Td
jw
33
w
2
H e H c j j
k
Td
k
Td
if Hc j 0, Td
hn Td hc nTd
T
T
w
relation between frequencies
Td , ,
w
2
H e H c j j
Td
k
Td
if Hc j 0, Td then H e jw H j w ,
c
T
d
j
Relationship of
frequencies
S plane 3 / Td
/ Td
- / Td
34
jw
Z plane
k
w
Review
st
t nT
n
periodic sampling
T:sample period; fs=1/T:sample rate
Ωs=2π/T:sample rate
xs t xc t s t xc t t nT
n
x[n] xc (t ) |t nT xc (nT )
x nT t nT
n
c
2
S j
k s
T k
1
1
2
X s j
X c j * S j
X c j
k s d
2
2
T k
1
X c j k s
T k
35
2
S j
T
proof of
Review
k
k
s
T:sample period; fs=1/T:sample rate;Ωs=2π/T:sample rate
s(t)为冲击串序列,周期为T,可展开傅立叶级数
st
t nT
n
e
jk st
jk s t
a
e
k
n
F
2 ( ks )
2
S j
T
36
1 jk st
e
T n
k
k
s
periodic sampling
xs t xc t s t xc t t nT
X s ( j )
n
x nT t nT e
n
c
x[n] xc (t ) |t nT xc (nT )
j
X s ( j) X (e ) T X (e
X (e
jT
37
)
1
) X c j k s
T k
1
X (e ) X c
T k
j
jT
2 k
j
T
T
jt
n
dt
X (e j )
2
s
T
xc nT t nT
k
n
xc nT e jTn
xc nT e jn
1
X s j X c j k s
T k
if X (e
jT
) 0,
T
1
then X (e ) X c j
T
T
j
discrete-time filter design by impulse invariance
x[n] xc (t ) |t nT xc (nT )
h[n] hc (nT )
1
X (e ) X c
T k
2 k
j
T
T
1
X (e ) X c j
T
T
1
j
H (e ) H c
T k
2 k
j
T
T
1
H (e ) H c j
T
T
j
j
j
w
2
H e H c j j
hn Td hc nTd
Td
k
Td
if Hc j 0, Td
jw
w
then H e H c j ,
Td
jw
38
w
k
Steps of DT filter design by transforming
a prototype continuous-time filter
Obtain the specifications for continuoustime filter by transforming the specifications
for the desired discrete-time filter.
Find the system function of the continuoustime filter.
Transform the continuous-time filter to
derive the system function of the discretetime filter.
39
Transformation from discrete to
continuous
In the impulse invariance design
procedure, the transformation is
He
jw
w
H c j ,
Td
w
Assuming the aliasing involved in the
transformation is neglected, the
relationship of transformation is
w Td
40
Steps of DT filter design by transforming
a prototype continuous-time filter
Obtain the specifications for continuoustime filter by transforming the specifications
for the desired discrete-time filter.
Find the system function of the continuoustime filter.
Transform the continuous-time filter to
derive the system function of the discretetime filter.
41
Continuous-time IIR filters
Butterworth filters
Chebyshev Type I filters
Chebyshev Type II filters
Elliptic filters
42
Steps of DT filter design by transforming
a prototype continuous-time filter
Obtain the specifications for continuoustime filter by transforming the specifications
for the desired discrete-time filter.
Find the system function of the continuoustime filter.
Transform the continuous-time filter to
derive the system function of the discretetime filter.
43
Transformation from
continuous to discrete
N
Ak
H s
k 1 s sk
N
Ak e sk t , t 0
hc t k 1
0,
t0
N
hn Td hc t Td Ak e
k 1
N
sk nTd
un Td Ak e
k 1
N
Td Ak
H z
sk Td 1
1
e
z
k 1
pole :
s sk z e
sk Td
two requirements for transformation
44
un
sk Td n
Example 7.2 Impulse Invariance
with a Butterworth Filter
Specifications for the discrete-time filter:
0.89125 H e jw 1, 0 w 0.2
0.17783,
He
jw
0.3 w
let Td 1 w Td
Assume the effect of aliasing is negligible
0.89125 H c j 1, 0 0.2
H c j 0.17783,
45
0.3
Example 7.2 Impulse Invariance
with a Butterworth Filter
0.89125 H c j 1, 0 0.2
H c j 0.2 0.89125
H c j 0.17783,
H c j 0.3 0.17783
H c j
0.3
1
2
1 j j c
2N
1 j j c
2N
1
H c j
2N
0.2
1
1
0.89125
c
2N
0.2
46
0.3
0.3 1
1
0.17783
c
2
2
2
Example 7.2 Impulse Invariance
with a Butterworth Filter
H c j
H c j 0.2 0.89125
1
2
1 j j c
2N
2N
2
0.2
1
1
1.25893
c
0.89125
2N
0.2
0.25893
c
2N
2N
0.3 1 2
1
31.62204
c 0.17783
0.3
30.62204
c
N 5.8858, c 0.70470
47
H c j 0.3 0.17783
2N
3
118.26378
2
N 6, c 0.7032
Example 7.2 Impulse Invariance
with a Butterworth Filter
1
2
1
H c j
H c s H c s H c s
1 j j
2
c
sk j c 1
1 2N
1 s j c
j
ce
N 6, c 0.7032
H c s Plole pairs:
0.182 j 0.679,
0.497 j 0.497,
0.679 j 0.182
48
2N
2 N 2k N 1
,
2N
k 0,1, ,2 N 1
Example 7.2 Impulse Invariance
with a Butterworth Filter
H c s H c s H c s
2
H c s Plole pairs:
1
1 s j c
2N
c2 N
s c
2N
2N
0.182 j 0.679, 0.497 j 0.497, 0.679 j 0.182
cN 0.70326
0.12093
Hc s
s 0.182 j0.679 s 0.182 j0.679 s 0.497 j0.497
1
s 0.497 j0.497 s 0.679 j0.182 s 0.679 j0.182
H c s
49
0.12093
s 2 0.3640s 0.4945 s 2 0.9945s 0.4945 s 2 1.3585s 0.4945
Example 7.2 Impulse Invariance
with a Butterworth Filter
0.12093
Hc s
s 0.182 j0.679 s 0.182 j0.679 s 0.497 j0.497
1
s 0.497 j0.497 s 0.679 j0.182 s 0.679 j0.182
N
Ak
k 1 s sk
N
H z
k 1
50
N
Td Ak
1 e
skTd
z
1
k 1
Ak
sk
1 e z
1
Td 1
0.2871 0.4466 z 1
2.1428 1.1455 z 1
1
2
1 1.2971z 0.6949 z
1 1.0691z 1 0.3699 z 2
1.8557 0.6303z 1
1 0.9972 z 1 0.2570 z 2
Basic for Impulse Invariance
To chose an impulse response for the
discrete-time filter that is similar in some
sense to the impulse response of the
continuous-time filter.
If the continuous-time filter is bandlimited,
then the discrete-time filter frequency
response will closely approximate the
continuous-time frequency response.
The relationship between continuous-time
and discrete-time frequency is linear;
consequently, except for aliasing, the shape
of the frequency response is preserved.
51
7.1.2 Bilinear Transformation
Bilinear transformation can avoid the
problem of aliasing.
Bilinear transformation maps
onto w
1
2 1 z
Bilinear transformation: s
Td 1 z 1
2
H z Hc
Td
52
1 z
Hc s
2
1
s
1
z
Td
1
1 z 1
1
1 z
7.1.2 Bilinear Transformation
2
s
Td
Td 2 s(1 z ) 1 z
1
1 Td 2 s]z 1 Td 2 s
1 Td 2s
z
1 Td 2s
1 z
1
1 z
1
s j
1
1 Td 2 j Td 2
z
1 Td 2 j Td 2
0 z 1 for any
0 z 1 for any
53
1
7.1.2 Bilinear Transformation
1 Td 2 j Td 2
z
1 Td 2 j Td 2
s j
1 j Td 2
z
1 j Td 2
Im
s plane
Re
54
0 z 1 for any
0 z 1 for any
j axis s j
jw 1 j Td 2
e
z 1
1 j Td 2
Im
z plane
Re
7.1.2 Bilinear Transformation
2
j
Td
jw
1 e
jw
1 e
e jw/2 (e jw/2 e jw/2 ) 2
jw/2 jw/2 jw/2
e
Td
(
e
e
)
2
tanw 2
Td
2
Td
w 2 tan1 Td 2
55
1
2 1 z
s
1
Td 1 z
2
2 j sin w 2
j tan w 2
2 cos w 2 Td
relation between frequency response of Hc(s), H(z)
H (e j ) H c ( j)
2
tan
Td
2
prewarp :
56
p
2
tan
p
Td
2
2 tan s
s Td
2
Comments on the Bilinear Transformation
It avoids the problem of aliasing encountered
with the use of impulse invariance.
It is nonlinear compression of frequency axis.
2
tanw 2
Td
j
S plane 3 / Td
w 2 tan Td 2
1
/ Td
- / Td
57
Z plane
Comments on the Bilinear Transformation
The design of discrete-time filters using
bilinear transformation is useful only when
this compression can be tolerated or
compensated for, as the case of filters
that approximate ideal piecewise-constant
magnitude-response characteristics.
H e
jw
1
2
58
wc
0
wc
2
Bilinear Transformation of
2
s
Td
1 z 1
1
1 z
2
tan w 2
Td
59
Td
2
tanw 2
Td
e
e
s
j
Comparisons of Impulse Invariance
and Bilinear Transformation
The use of bilinear transformation is
restricted to the design of approximations to
filters with piecewise-constant frequency
magnitude characteristics, such as highpass,
lowpass and bandpass filters.
Impulse invariance can also design lowpass
filters. However, it cannot be used to design
highpass filters because they are not
bandlimited.
60
Comparisons of Impulse Invariance
and Bilinear Transformation
Bilinear transformation cannot design
filter whose magnitude response isn’t
piecewise constant, such as
differentiator. However, Impulse
invariance can design an bandlimited
differentiator.
61
7.1.3 Example of Bilinear
Transformation
Butterworth Filter,
Chebyshev Approximation,
Elliptic Approximation
0.99 H e jw 1.01,
0.001,
He
62
jw
w 0.4
0.6 w
Example 7.3 Bilinear Transformation of a
Butterworth Filter
0.89125 H e jw 1, 0 w 0.2
2
tanw 2
Td
jw
H e 0.17783,
0.3 w
2
0.2
0.89125 H c j 1, 0 tan
Td
2
2
0.3
H c j 0.7783,
tan
Td
2
For convenience, we choose Td 1
H c j 2 tan0.1 0.89125,
H c j 2 tan0.15 0.17783,
63
0.01 0.016
Example 7.3 Bilinear Transformation of a
Butterworth Filter
H c j 2 tan0.1 0.89125,
H c j 2 tan0.15 0.17783,
H c j
2
1
1 j jc
2 tan 0.1
1
c
2N
2N
3 tan 0.15
1
c
64
1
0.89125
2N
0.01 0.016
2
1
0.17783
N 5.305
2
N 6,
c 0.766
Locations of Poles
H c j
1
2
1 j j c
sk j c 1
1 2N
2N
j
ce
N 6, c 0.766
H c s Plole pairs:
0.1998 j 0.7401,
0.5418 j 0.5418,
0.7401 j 0.1998
65
H c s H c s H c s
2
2 N 2k N 1
,
1
1 s j c
2N
k 0,1, ,2 N 1
Example 7.3 Bilinear Transformation of a
Butterworth Filter
2N
2
1
c
H c s H c s H c s
2N
2N
2N
s
c
1 s j c
H c s Plole pairs:
0.1998 j 0.7401, 0.5418 j 0.5418, 0.7401 j 0.1998
0.20238
H c s 2
s 0.3996s 0.5871 s 2 1.0836s 0.5871 s 2 1.4802s 0.5871
1 6
0.00073781 z
H z
1 1.2686z 1 0.7051z 2 1 1.0106z 1 0.3583z 2
1
1
2
1
z
1
2
s
1 0.9904z 0.2155z
1
66
Td 1 z
Ex. 7.3 frequency response of discrete-time filter
67
Example 7.4 Butterworth
Approximation (Hw)
0.99 H e jw 1.01,
0.001,
He
68
jw
w 0.4
0.6 w
order N 14
Example 7.4 frequency response
69
Chebyshev filters
C
Chebyshev filter (type I)
1
2
| H c ( j) |
2
2
1 VN ( / c )
1
1
1
VN ( x) cos(N cos x)
Chebyshev polynomial
Chebyshev filter (type II)
1
| H c ( j) |
2
2
1 [ VN ( / c )]1
c
1
2
70
c
Example 7.5 Chebyshev Type I , II
Approximation
1.01,
0.99 H e
jw
H e jw 0.001,
Type I
71
w 0.4
order N 8
0.6 w
Type II
Example 7.5 frequency response of Chebyshev
Type I
72
Type II
elliptic filters
E
Elliptic filter
1
| H c ( j) |
1 2U N2 ()
2
Jacobian elliptic function
1
1 1
2
p s
73
Example 7.6 Elliptic Approximation
0.99 H e jw 1.01,
0.001,
He
74
jw
w 0.4
0.6 w
order N 6
Example 7.6 frequency response of Elliptic
75
*Comparison of Butterworth, Chebyshev, elliptic filters: Example
-Given specification
0.99 | H (e j ) | 1.01
| H (e j ) | 0.001
| | 0.4
0.6 | |
(s )
1 0.01, 2 0.001 p 0.4 , s 0.6
-Order
B Butterworth Filter : N=14. ( max flat)
C Chebyshev Filter
: N=8. ( Cheby 1, Cheby 2)
E Elliptic Filter
: N=6
( equiripple)
76
-Pole-zero plot (analog)
B
C1
C2
E
C2
E
-Pole-zero plot (digital)
B
(14)
77
C1
(8)
-Group delay
-Magnitude
C1
20
B
B
E
C1
E
C2
5
C2
0.4
78
0.6
0.4
0.6
7.2 Design of FIR Filters by
Windowing
FIR filters are designed based on
directly approximating the desired
frequency response of the discretetime system.
Most techniques for approximating the
magnitude response of an FIR system
assume a linear phase constraint.
79
Window Method
An ideal desired
frequency
response
h ne
1
h n
H e e
2
H d e jw
n
d
wc
jw
d
H e jw
0
H lp e
d
1
jwn
wc
jwn
dw
jw
1,
w wc
0, wc w
sin wc n
hlp n
n
Many idealized systems are defined by
piecewise-constant frequency response with
discontinuities at the boundaries. As a result,
these systems have impulse responses that
are noncausal and infinitely long.
80
Window Method
The most straightforward approach to
obtaining a causal FIR approximation is to
truncate the ideal impulse response.
hd n, 0 n M
hn
otherwise
0,
hn hd nwn
1,
wn
0,
1
jw
H e
2
81
0nM
otherwise
H d e jw W e j w d
Windowing in Frequency Domain
Windowed frequency response
He
j
1
j
j
H
e
W
e
d
d
2
The windowed version is smeared version
of desired response
82
Window Method
If
wn 1 n
wne
W e jw
jwn
2
n
1
H e
2
jw
Hd e
j
W e
j w
83
1
4
5 10 15
k
2
2
0
w 2k
W e jw
4 2
15 10 5
6
wc
d H d e jw
H e jw
0
wc
Choice of Window
wn is as short as possible in duration. This
minimizes computation in the
implementation of the filter.
1,
wn
0,
0nM
otherwise
W e jw approximates an impulse.
W e
jw
w n e
n
jw M 1
1 e
jw
1 e
84
jwn
e
jwM 2
M
e
M 1
W e jw
jwn
n 0
sin w M 1 2
sin w 2
2
M 1
2
M 1
Window Method
If wn is chosen so that W e jw is concentrated
in a narrow band of frequencies around w 0
then H e jw would look like H d e jw , except
where H d e jw changes very abruptly.
He
jw
M 1
85
2
M 1
1
2
W e
2
M 1
jw
H d e jw W e j w d H d e jw
1
wc
H d e jw
0
wc
Rectangular Window
W e jw for the rectangular window has a
generalized linear phase.
M
M 1
W e
jw
e
jwM 2
sin w M 1 2
sin w 2
M
M 1
As M increases, the width of the “main lobe”
decreases. wm 4 M 1
While the width of each lobe decreases with
M, the peak amplitudes of the main lobe and
the side lobes grow such that the area under
each lobe is a constant.
M 1
86
2
M 1
2
M 1
Rectangular Window
H d e jw W e j w d will oscillate at
the discontinuity.
The oscillations occur more rapidly, but
do not decrease in magnitude as M
increases.
The Gibbs phenomenon can be
moderated through the use of a less
abrupt truncation of the Fourier series.
87
Rectangular Window
By tapering the window smoothly to zero
at each end, the height of the side lobes
can be diminished.
The expense is a wider main lobe and thus
a wider transition at the discontinuity.
88
7.2 Design of FIR Filters by Windowing Method
Review
To design an ilowpass FIR Filters
H e
1
jw
H lp e
jw
1,
0,
sin wc n
hlp n
n
w wc
wc w
sin wc n M 2
n M 2
h n hd n wn w n 1, 0 n M
He
jw
0, otherwise
1
2
M 1
89
H d e jw W e j w d
2
M 1
W e jw
2
M 1
wc
0
0
M0 2
wc
M
0
M 2
M
0
M 2
M
7.2.1 Properties of Commonly Used
Windows
Rectangular
1, 0 n M
wn
0, otherwise
Bartlett (triangular)
2n M , 0 n M 2
wn 2 2n M , M 2 n M
0,
otherwise
90
7.2.1 Properties of Commonly Used
Windows
Hanning
0.5 0.5 cos2 n M , 0 n M
wn
otherwise
0,
Hamming
0.54 0.46cos2 n M , 0 n M
wn
otherwise
0,
91
7.2.1 Properties of Commonly Used
Windows
Blackman
0.42 0.5 cos2 n M
wn 0.08cos4 n M ,
0nM
otherwise
0,
92
7.2.1 Properties of Commonly Used
Windows
93
Frequency Spectrum of Windows
(a) Rectangular, (b) Bartlett,
(c) Hanning, (d) Hamming,
(e) Blackman , (M=50)
94
(a)-(e) attenuation of sidelobe increases,
width of mainlobe increases.
7.2.1 Properties of Commonly Used
Windows
Table 7.1
smallest,the sharpest transition
biggest,high oscillations
at discontinuity
95
7.2.2 Incorporation of Generalized
Linear Phase
In designing FIR filters, it is desirable
to obtain causal systems with a
generalized linear phase response.
The above five windows are all
symmetric about the point M 2 ,i.e.,
wM n, 0 n M
wn
otherwise
0,
96
7.2.2 Incorporation of Generalized
Linear Phase
Their Fourier transforms are of the form
jw
jw jwM 2
W e We e e
jw
We e is a real and even functionof w
hn hd nwn : causal
if hd M n hd n
h n hd n wn
M 2
h M n h n : generalized linear phase
A e e
He
97
jw
jw
e
jwM 2
M
7.2.2 Incorporation of Generalized
Linear Phase
if hd M n hd n hn hd nwn
hM n hn : generalized linear phase
H e
jw
jA e e
jw
jwM 2
o
M 2
98
M
Frequency Domain Representation
if hd M n hd n
W e e
wn w M n
1
H e
2
jw
1
2
H e
j
d
j
Ae
e
jw
jw
jw
j M 2
jwM 2
e
j w
W e
d
e
99
We
H e e
Ae e jw e jwM
where
H d e jw He e jw e jwM 2
h n hd n wn
j w j w M 2
We e
d
e
2
1
2
H e
e
jw
j w
We e
d
Example 7.7 Linear-Phase Lowpass
Filter
The desired frequency response is
jwM 2
e
, w wc
jw
H lp e
0, wc w H e jw 1 0 w wp
1 wc jwM 2 jwn
jw
hlp n
e
e
dw
H
e
ws w
w
c
2
sin wc n M 2
for n
n M 2
hlp M n
100
sinwc n M 2
hn
wn
n M 2
M 2
magnitude frequency response
H e 1 0 w w
p 20log10 p
H e w w
jw
p
p
jw
s 20log10 s
w ws wp
ws
s
s
wp
H e jw 1 0.05 0 w 0.25
H e jw 0.1 0.15 w
s 20dB
p 20log10 0.05 26dB
w ws wp 0.1
101
7.2.1 Properties of Commonly Used
Windows
biggest,high oscillations
at discontinuity
102
smallest,the sharpest
transition
7.2.3 The Kaiser Window Filter
Design Method
2
I 0 1 n
w n
I0
0,
where M 2,
12
, 0nM
otherwise
u 2
I0 u 1
r
!
r 1
r
2
I0 u : zero order modified Bessel function of the first kind
two parameters :
shape parameter:
Trade side-lobe amplitude for main-lobe width
length : M 1,
103
M=20
=6
As increases, attenuation of
sidelobe increases, width of
mainlobe increases.
As M increases, attenuation of
sidelobe is preserved, width of
mainlobe decreases.
104
Figure 7.24
(a) Window shape, M=20,
(b) Frequency spectrum, M=20,
(c) beta=6
Table 7.1
Transition width is a little less than
mainlobe width
105
Comparison
If the window is tapered more, the side lobe
of the Fourier transform become smaller, but
the main lobe become wider.
Increasing M wile holding
constant causes the main
lobe to decrease in width,
but does not affect the
amplitude of the side lobe.
M=20
=6
M=20
106
Filter Design by Kaiser Window
1 0 w w
H e w w
He
jw
jw
s
ws
wp
107
w ws wp
A 20log10
p
Filter Design by Kaiser Window
2
I 0 1 n
w n
I0
0,
w ws wp
12
,
0nM
otherwise
A 20log10
0.1102 A 8.7 ,
A 50
0.4
0.5842 A 21 0.07886 A 21, 21 A 50
0.0,
A 21
M=20
108
A8
M
2.285 w
2
Example 7.8 Kaiser Window Design
of a Lowpass Filter
0.99 H e jw 1.01,
H e jw 0.001,
w 0.4
0.6 w
2
I 0 1 n
sin wc n
n
I0
h n
0,
otherwise
12
, 0 n M
where M 2 18.5
0.1102 A 8.7 ,
A 50
0.5842 A 210.4 0.07886 A 21, 21 A 50
0.0,
A 21
109
A8
M
2.285w
A 20log10
w ws wp
Example 7.8 Kaiser Window Design
of a Lowpass Filter
0.99 H e jw 1.01, w 0.4
0.001,
He
jw
0.6 w
step 1 :
wp 0.4 , ws 0.6 ,
1 0.01, 2 0.001, min 1 , 2 0.001
110
ws wp
0.5
step 2 :
cutoff frequency wc
step 3 :
w ws wp 0.2
A 20log10 60
0.5653
M 37
2
Example 7.8 Kaiser Window Design
of a Lowpass Filter
step 3:
w ws wp 0.2
A8
M
37
2.285w
A 20log10 60
0.5653
0.1102 A 8.7 ,
A 50
0.5842 A 210.4 0.07886 A 21, 21 A 50
0.0,
A 21
2 12
I 0 1 n
sin wc n
, 0 n M
n
I0
h n
0,
111
otherwise
where M 2 18.5
u 2
I0 u 1
r 1
r!
r
2
Ex. 7.8 Kaiser Window Design of a Lowpass
Filter
12
2
I 0 1 n
sin wc n
h n
n
I0
112
, 0 n M
7.3 Examples of FIR Filters Design
by the Kaiser Window Method
The ideal highpass filter with
generalized linear phase
H hp e
jw
0,
w wc
jwM 2
, wc w
e
Hhp e jw e jwM 2 Hlp e jw
sin n M 2 sin wc n M 2
hhp n
, n
n M 2
n M 2
hn hhp n wn
113
Example 7.9 Kaiser Window Design
of a Highpass Filter
Specifications:
, w w
1 H e 1 ,
He
jw
2
s
jw
1
1
wp w
where ws 0.35 , wp 0.5 , 1 21 0.021
By Kaiser window method
2.6, M 24
114
Example 7.9 Kaiser Window Design
of a Highpass Filter
Specifications:
, w w
1 H e 1 ,
He
jw
2
s
jw
1
1
wp w
where ws 0.35 , wp 0.5 , 1 21 0.021
By Kaiser window method
2.6, M 24
115
7.3.2 Discrete-Time Differentiator
Hdiff e jw jwe jwM 2 , w
cos n M 2 sin n M 2
hdiff n
, n
2
n M 2
n M 2
hn hdiff nwn
hn hM n: type III or type IV generalized linear phase
116
Example 7.10 Kaiser Window
Design of a Differentiator
Since kaiser’s formulas were
developed for frequency responses
with simple magnitude discontinuities,
it is not straightforward to apply them
to differentiators.
Suppose M 10 2.4
117
Group Delay
Phase:
M
w 5w
2
2
2
Group Delay:M
2
118
5 samples
Group Delay
Phase:
M
5
w w
2
2
2
2
Group Delay:M
5
samples
2
2
Noninteger delay
119
7.4 Optimum Approximations of FIR
Filters
Goal: Design a ‘best’ filter for a given M
In designing a causal type I linear phase
FIR filter, it is convenient first to consider
the design of a zero phase filter.
he n he n
Then insert a delay sufficient to make it
causal.
120
7.4 Optimum Approximations of FIR
Filters
he n he n
h ne
Ae e jw
L
n L
jwn
e
,
LM 2
L
Ae e jw he 0 2he ncoswn : real, even, periodic function
n 1
A causal system can be obtained from he n by
delayingit by L M 2 samples.
hn he n M 2 hM n
A e e
He
121
jw
jw
e
jwM 2
7.4 Optimum Approximations of FIR
Filters
Designing a filter to meet these specifications
is to find the (L+1) impulse response values
he n, 0 n L
Packs-McClellan algorithm is the dominant
method for optimum design of FIR filters.
In Packs-McClellan algorithm, L, wp , ws , and 1 2
is fixed, and 1 or 2 is variable.
122
7.4 Optimum Approximations of FIR
Filters
1
coswn Tn cosw cos n cos cosw
cosw0 T cosw cos0 cos cosw 1
cosw1 T cosw cos1cos cosw cosw
1
0
1
1
cosw2 T2 cos w 2 cos w 1
2
coswn Tn cos w
2cos wTn1 cos w Tn2 cos w
cosw3 2cos wcos2w cos w
2 cos w 2 cos2 w 1 cos w 4 cos3 w 3 cos w
123
7.4 Optimum Approximations of FIR
Filters
h 0 2h ncoswn a cos w
L
Ae e
L
k
jw
e
n 1
e
k 0
k
L
where Px ak x
k
k 0
Define an approxim ation error function
A e
E w W w H d e
jw
jw
e
where W w is the weighting function
124
Px x cos w
7.4 Optimum Approximations of FIR
Filters
Hd e
jw
1, 0 w w p
0, ws w
1 2
, 0 w wp
W w K 1
1, ws w
125
Minimax criterion
Within the frequency interval of the
passband and stopband, we seek a
frequency response Ae e jw that
minimizes the maximum weighted
approximation error of
Ew W wH e A e
jw
d
min
max Ew
he n :0 n L
126
wF
jw
e
Other criterions
H 1 min E w dw
he n :0 n L 0
2
H 2 min E w dw
he n :0 n L 0
H
127
min
max Ew
he n :0 n L
wF
Alternation Theorem
Let Fp denote the closet subset consisting of
the disjoint union of closed subsets of the
real axis rx.
k
Px ak x is an r th-order polynomial.
k 0
DP x denotes a given desired function of x
that is continuous on Fp
WP x is a positive function, continuous on Fp
The weighted error is EP x WP xDP x Px
The maximum error is defined as
E max EP x
128
xFP
Alternation Theorem
A necessary and sufficient condition that be
the unique rth-order polynomial that Px
minimizes E is that EP x exhibit at least
(r+2) alternations; i.e., there must exist at
least (r+2) values xi in FP such that
x1 x2 xr 2
EP xi EP xi1 E
and such that
for
i 1,2,, r 1
129
Example 7.11 Alternation Theorem
and Polynomials
Each of these polynomials is of fifth
order.
The closed subsets of the real axis x
referred to in the theorem are the
regions
1 x 0.1 and 0.1 x 1
WP x 1
130
7.4.1 Optimal Type I Lowpass
Filters
For Type I lowpass filter
L
Pcos w ak cos w
k
k 0
The desired lowpass frequency response
cos wp cos w 1 0 w wp
1,
D p cos w
0, 1 cos w cos ws ws w
Weighting function
1
,
cos w p cos w 1 0 w w p
W p cos w K
1, 1 cos w cos ws ws w
131
7.4.1 Optimal Type I Lowpass
Filters
The weighted approximation error is
EP cos w WP cos wDP cos w Pcos w
The closed subset
0 w wp
EP x
is
and ws w
or
coswp cosw 1 and 1 w cosws
132
7.4.1 Optimal Type I Lowpass
Filters
The alternation theorem states that a set of
coefficients ak will correspond to the filter
representing the unique best approximation
to the ideal lowpass filter with the ratio
fixed at K and with passband and stopband
edge wp and ws if and only if EP (cosw)
EP (cosw)
exhibits at least (L+2) alternations on
,
i.e., if and only if FP alternately equals plus
and minus its maximum value at least (L+2)
times.
Such approximations are called equiripple
approximations.
1
133
2
7.4.1 Optimal Type I Lowpass
Filters
The alternation theorem states that
the optimum filter must have a
minimum of (L+2) alternations, but
does not exclude the possibility of
more than (L+2) alternations.
In fact, for a lowpass filter, the
maximum possible number of
alternations is (L+3).
134
7.4.1 Optimal Type I Lowpass
Filters
Because all of the filters satisfy the
alternation theorem for L=7 and for
the same value of K 1 2 , it follows
that wpand/or ws must be different for
each ,since the alternation theorem
states that the optimum filter under
the conditions of the theorem is
unique.
135
Property for type I lowpass filters
from the alternation theorem
The maximum possible number of
alternations of the error is (L+3)
Alternations will always occur at wp and ws
All points with zero slop inside the passband
and all points with zero slop inside stopband
will correspond to alternations; i.e., the filter
will be equiripple, except possibly at w
and w 0
136
7.4.2 Optimal Type II Lowpass
Filters
For Type II causal FIR filter: hn 0 n M
The filter length (M+1) is even, ie, M is odd
Impulse response is symmetric
hM n hn
The frequency response is
e
He
jw
jwM 2
M 1 2
n 0
1
e
bncos w n
2
n 1
where bn 2hM 1 2 n, n 1,2, , M 1 2
jwM 2
137
M 1 2
M
2hncos w n
2
7.4.2 Optimal Type II Lowpass
Filters
M 1 2
n 1
M 1 2 ~
1
bncosw n cosw 2 b ncoswn
2
n 0
e
He
jw
jwM 2
cosw 2Pcos w
L
where Pw ak cos w
k
and L M 1 2
k 0
~
find ak b n bn bn 2hM 1 2 n
138
7.4.2 Optimal Type II Lowpass
Filters
For Type II lowpass filter,
Hd e
jw
1
, 0 w wp
DP cos w cosw 2
ws w
0,
cosw 2
, 0 w wp
W w WP cos w K
cosw 2, s w
139
7.4.3 The Park-McClellan Algorithm
From the alternation theorem, the optimum
filter Ae e jw will satisfy the set of equation
i 1
jw
jw
W w Hd e Ae e 1
140
2
L
1
x
x
x
1
1
1
2
L
1 x
x
x
2
2
2
2
L
1 x
x
x
L2
L2
L2
where xi cos wi
i 1,2,, L 2
1
W w1 a H e jw1
0
d
1
jw2
a
H
e
1
W w2 d
jwL 2
L2
H
e
1 d
W wL 2
7.4.3 The Park-McClellan Algorithm
Guessing a set of alternation frequencies
wi for i 1,2,, L 2
L2
141
jwk
b
H
e
k d
k 1
L2
bk 1
k 1 W wk
k 1
and
wl wp , wl 1 ws
L2
where
1
bk
, xi cos wi
i 1 xk xi
ik
7.4.3 The Park-McClellan Algorithm
L 1
Ae e jw Pcos w
d x x C
k 1
L 1
k
k
1
dk
bk xk xL 2
i 1 xk xi
ik
142
k
d x x
k 1
L 1
k
k
,
xk x cos wk
7.4.3 The Park-McClellan Algorithm
For equiripple lowpass approximation
10log10 1 2 13
M
2.324w
where w ws wp
Filter length: (M+1)
143
7.5 Examples of FIR Equiripple Approximation
7.5.1 Lowpass Filter
0.99 H e jw 1.01,
H e jw 0.001,
w 0.4
0.6 w
M 26
unweightedapproxim ation error
jw
1 Ae e , 0 w w p
E w
E A w
W w 0 Ae e jw , ws w
144
Comments
M=26, Type I filter
The minimum number of alternations
is (L+2)=(M/2+2)=15
7 alternations in passband and 8
alternations in stopband
The maximum error in passband and
stopband are 0.0116 and 0.0016,
which exceed the specifications.
145
7.5.1 Lowpass Filter
M=27, , Type II filter, zero at z=-1
w
The maximum error in passband and
stopband are 0.0092 and 0.00092,
which exceed the specifications.
The minimum number of alternations
is (L+2)=(M-1)/2+2=15
7 alternations in passband and 8
alternations in stopband
146
Comparison
Kaiser window method require M=38
to meet or exceed the specifications.
Park-McClellan method require M=27
Window method produce
approximately equal maximum error in
passband and stopband.
Park-McClellan method can weight the
error differently.
147
7.6 Comments on IIR and FIR
Discrete-Time Filters
What type of system is best, IIR or
FIR?
Why give so many different design
methods?
Which method yields the best result?
148
7.6 Comments on IIR and FIR
Discrete-Time Filters
149
Generalized
Linear Phase
Order
IIR
ClosedForm
Formulas
Yes
No
Low
FIR
No
Yes
High
7.2.1 Properties of Commonly Used
Windows
Their Fourier transforms are concentrated
around w 0
They have a simple functional form that
allows them to be computed easily.
The Fourier transform of the Bartlett
window can be expressed as a product of
Fourier transforms of rectangular windows.
The Fourier transforms of the other
windows can be expressed as sums of
frequency-shifted Fourier transforms of
rectangular windows.(Problem7.34)
150
Homework
Simulate the frequency response
(magnitude and phase) for
Rectangular, Bartlett, Hanning,
Hamming, and Blackman window with
M=21 and M=51
151
Chapter 5 HW
7.2, 7.4, 7.15,
152
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