This preview shows pages 1–4. Sign up to view the full content.
EE264
Digital Signal Processing
Lecture 12
Digital Filter Design
November 3, 2008
Ronald W. Schafer
Department of Electrical
Engineering
Stanford University
STANFORD UNIVERSITY, EE264
Administrative
• HW 5 due on Tuesday,
Nov. 4
by 5pm in EE264 drawer
on 2
nd
floor Packard.
• Midterm exam:
– I’m rusty! I’ll try to challenge you more on the final.
– Average 89, about half above 90
• READ: Chapter 7.
• Review Session: Thursdays 4:15  5pm Gates B03
(recording available online)
• Office Hours:
– RWS: Mon./Weds 1011, and 12:1512:45
– Raunaq: Mon. 57pm, Tues. 911am
– Rahim: Friday 46pm
• Grader:
Pegah Afshar, Ramin Miri
STANFORD UNIVERSITY, EE264
Overview of Lecture
• Discretetime filtering of continuoustime signals
• Setting the specifications
• IIR design by bilinear transformation
– Butterworth, Chebyshev and elliptic
approximations
• FIR design by windowing
– The Kaiser window
• FIR design by minimax approximation – the
ParksMcClellan algorithm
• Summary comparison
STANFORD UNIVERSITY, EE264
Digital Filtering
H
eff
(
j
Ω
)
=
H
(
e
j
Ω
T
),

Ω

<
π
/
T
0,

Ω

>
/
T
⎧
⎨
⎩
H
(
e
j
ω
)
=
H
eff
j
T
⎛
⎝
⎜
⎞
⎠
⎟
,

< π
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document STANFORD UNIVERSITY, EE264
Setting the Specifications
H
eff
(
j
Ω
)
=
H
(
e
j
Ω
T
),

Ω

<
π
/
T
0,

Ω

>
/
T
⎧
⎨
⎩
H
(
e
j
ω
)
=
H
eff
j
T
⎛
⎝
⎜
⎞
⎠
⎟
,

<
Ω
T
Ω
p
T
STANFORD UNIVERSITY, EE264
A Design Example
• The CT specifications are (
1/T=
2000 Hz):
• The corresponding DT specifications are:
0.99
≤

H
eff
(
j
Ω
) 
≤
1.01,

Ω

≤
2
(400)

H
eff
(
j
Ω
)
≤
0.001,
2
(600)
≤

Ω

≤
2
(1000)
0.99
≤

H
(
e
j
≤
1.01,


≤
0.4

H
(
e
j
≤
0.001,
0.6
≤


≤
i.e.,
Ω
p
=
2
(400) and
Ω
s
=
2
(600).
i.e.,
p
=Ω
p
T
=
0.4
and
s
s
T
=
0.6
.
STANFORD UNIVERSITY, EE264
IIR Filter Design
• Find B(z) and A(z) so that the filter meets the
specifications on the frequency response.
H
(
z
)
=
b
k
z
−
k
k
=
0
M
∑
1
−
a
k
z
−
k
k
=
1
N
∑
=
B
(
z
)
A
(
z
)
Rational function
approximation
H
(
e
j
)
=
H
(
z
)
z
=
e
j
STANFORD UNIVERSITY, EE264
Rader and Gold Paper
C. M. Rader and B. Gold,
Proceedings of IEEE,
Vol. 55,
pp. 149171, February, 1967.
STANFORD UNIVERSITY, EE264
Ben Gold and Charlie Rader
First Kilby
Medallists
1997
STANFORD UNIVERSITY, EE264
IIR Design Based on Analog Filters
1. From the specifications on the discretetime filter,
determine an analog filter
H
c
(
s
) such that
H
(
z
) obtained by
meets the specifications.
2. The digital filter is then
H
(
z
).
3. A large body of results on analog filter approximation
can
be used to determine
H
c
(
s
).
–
Butterworth
–
Chebyshev
–
Elliptic
s
=
2
T
d
1
−
z
−
1
1
+
z
−
1
⎛
⎝
⎜
⎞
⎠
⎟
; i.e.,
H
(
z
)
=
H
c
(
s
)
s
=
2
T
d
1
−
z
−
1
1
+
z
−
1
⎛
⎝
⎜
⎞
⎠
⎟
.
Bilinear Transformation
STANFORD UNIVERSITY, EE264
Bilinear Transformation Method
• We simply transform an analog filter
H
c
(
s
) into a digital
filter
H
(
z
) with the following complex mapping:
Note that
H
c
(
s
) is NOT the same as
H
eff
(
s
), it is just a
filter that we have available that works. More later.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview. Sign up
to
access the rest of the document.