ECE 513  DIGITAL SIGNAL PROCESSING
Cranos M. Williams
EXAM 1: September 25, 2012
NAME:
STUDENT ID:
1.
System Transfer Functions: (20 Pts)
Given below are the difference equations for several linear timeinvariant systems. As
sume that all of the systems are causal and right sided. Perform the following for each
system:
•
Find the Ztransform,
H
(
z
), of the impulse response of the system.
•
Give the region of convergence of the system. Is the system stable?
•
If the system is unstable, write (ONLY) the zeros and/or poles of a stablized
system that would result in the same magnitude response.
(a)
y
(
n
) =
x
(
n
)

2
.
5
x
(
n

1) +
x
(
n

2) +
√
2
2
y
(
n

1)

1
4
y
(
n

2)
(b)
y
(
n
) =
x
(
n
)

2
·
√
2
x
(
n

1) + 2
x
(
n

2)
(c)
y
(
n
) =
x
(
n
) + 2
x
(
n

1) +
x
(
n

2) + 2
·
√
2
y
(
n

1)

4
y
(
n

2)
NOTE: The roots of a general polynomial
ax
2
+
bx
+
c
is given as ¯
x
1
,
2
=

b
±
√
b
2

4
ac
2
a
.
1
ECE 513  Exam 1:
Name:
Student ID:
2
1. (cont’d)
ECE 513  Exam 1:
Name:
Student ID:
3
2.
Filtering Long Sequences: (20 Pts)
We obtain 6
.
8 seconds of a real sampled speech
signal that was sampled at a sampling rate of 8000 samples per second.
We wish
to use the overlapsave filtering procedure with an FIR digital filter with 120 real
coefficients.
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 Digital Signal Processing, Signal Processing