UNIVERSITY OF ILLINOIS AT URBANACHAMPAIGN
Department of Electrical and Computer Engineering
ECE 410
Digital Signal Processing
Homework 13
Prof. Bresler, Prof. Jones
Wednesday, December 3, 2008
Problem 1
In this problem you will design a secondorder digital filter using the Bilinear Transformation
method that converts an analog filter
H
a
(
s
) into a digital filter
H
d
(
z
). The mapping you will use is
s
= 2
1

z

1
1 +
z

1
.
The analog filter
H
a
(
s
) has poles at

1 +
j
and

1

j
and zeros at +
j
and

j
.
(a) Draw a polezero diagram for the digital filter and specify the exact locations of the poles and
zeros.
(b) If the digital filter
h
d
is causal, is it stable? Why or why not?
(c) Given that there is only one value of
ω
0
∈
(0
, π
) for which
H
d
(
ω
0
) = 0, what is
ω
0
?
(d) Given that
H
d
(0) = 1, give an expression for
H
d
(
ω
).
(e) What is the difference equation for this filter in part (d)?
(f) Use the matlab command
freqz
to plot the magnitude and phase of the frequency response of
the filter. Check that it evaluates to zero at
ω
0
(from part (c)).
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 Fall '09
 Digital Signal Processing, Signal Processing, Butterworth Filter, Digital filter

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