Digital Signal Processing Reference
In-Depth Information
Exercise 7.11: Echo cancellation.
In data communication systems
over phone lines (such as voiceband modems), one of the major problems is
represented by
echos
. Impedance mismatches along the analog line create
delayed and attenuated replicas of the transmitted signal. These replicas are
added to the original signal and represent one type of distortion.
Assume a simple situation where a single echo is created; the transmitted
signal is
x
l
g
r
,
y
i
d
.
,
©
,
L
s
[
n
]
and, because of the echo, the received signal is
y
[
n
]=
x
[
n
]
−
α
x
[
n
−
D
]
where
α
is the attenuation factor (with 0
<α<
1) and
D
is the echo delay
(assume
D
is an integer).
(a) Write the transfer function
H
(
z
)
of the “echo system”, i.e. the system
which produces
y
[
n
]
from
x
[
n
]
.
(b) Sketch the pole-zero plot for
H
(
z
)
for
α
=
0.1 and
D
=
12 (for our pur-
poses, assume
12
0.1).
(c) Sketch the squared magnitude response
H
(
0.826
)
=
2
e
j
ω
)
(
.
Now assume we have a good estimate of
and
D
; we want to design a causal
echo cancellation filter, i.e. a filter with causal impulse response
g
α
[
n
]
such
that
y
[
n
]
∗
g
[
n
]=
x
[
n
]
.
(d) Write the expression for
G
(
z
)
.
(e) Sketch its pole-zero plot and its ROC for the same values of
α
and
D
as before.
(f ) What is the practical difficulty in implementing this echo cancellation
system?
Exercise 7.12: FIR filter design - I.
Is it a good idea to use a Type III FIR
to design a lowpass filter? Briefly explain.
Exercise 7.13: FIR filter design - II.
Suppose you want to design a linear
phase FIR approximation of a Hilbert filter. Which FIR type would you use?
Why? Discuss advantages and disadvantages.
