ECE 539 Digital Signal Processing
Extra Credit
For Midterm Exam
Due: Thursday, March 29
th
, 2007
No late extra credit will be accepted.
NO COLLABORATIONS PLEASE!
You can come to my office hours to
discuss this.
Name:
Problem 1 /
5
Problem
2 /
2.5
Problem 3 /
2.5
Problem 4 /5
Total
/
15
2
Problem 1 (
5
points total)
1(a)(3 points)
Evaluate the continuous

time Fourier Transform of:
0 0
sin
t u t u t t
.
1(b)(3 points)
Evaluate the discrete

time Fourier Transf
orm of:
0 0
sin
n u n u n n
.
1(b)(3 points)
Evaluate the Z

Transform of:
0 0
cos
n u n u n n
and specify the ROC.
1(c)(1 points)
Comment (briefly) about the relationships among the answers in 1(a), 1(b)
and 1(c).
Problem 2 (
2.5
points
total)
2(a)(
1.
25 points)
Suppose that
H z
represents the impulse response of a stable system
(not necessarily causal) and it is given by
2
1
6
H z
z z
.
Find
h n
.
2(b)(
1.
25 points)
Compute the f
requency response for:
1 1 2
y n y n x n x n x n
.
Problem 3 (
2.
5 points)
Suppose that the impulse response of an LTI system (
L)
is
supported over a finite region. By this, we mean that:
0
h n
for
1 2
N n N
and
0
h n
for
1
n N
or
2
n N
.
In this case, it is easy to show that
L
is BIBO stable
if and only if
it has
an absolutely summable impulse response.
You are asked to prove this simple statemen
t directly. You will receive no credit if you
assume the more general statement that this holds for both finitely and infinitely
supported impulse responses. Also, you should not be using Z

transforms in
your solution.
3
Problem 4 (
5
points total).
Consid
er the simple sampling system with an anti

aliasing,
lowpass filter:
x
a
(
t
)
x
d
(
n
)
F
a
(
)
Antialiasing filter
(lowpass)
Ideal
sampler
T
s
For the anti

aliasing filter, we have an ideal low

pass filter:
c
c
a
F
;
0
;
1
)
(
Throughout this problem, please assume that we always keep the same analog cutoff
frequency. Also, n
ote that the sampling period maybe different than the
reconstruction time used to hold the signal.
4(a)(
0.5
point)
Suppose that
T
s
is sufficiently small. Please sketch the magnitude spectra
of:
the analog signal
the analog signal after lowpass filtering
t
he sampled signal:
j
d
X e
4(b)(
1.5
points)
H
d
(
e
j
)
digital filter
A/D
D/A
x
a
(
t
)
x
d
(
n
)
y
a
(
t
)
y
d
(
n
)
H
a
(
)
For the system described in 4(a), recall that the response
j
d
Y e
of the
digital filter
with
frequency response
j
d
H e
is:
j
d
Y e
=
s
T
1
m
X
a
s
T
m
2
F
a
2
s
m
T
j
d
H e
(with anti

aliasing).
Suppose that the frequency response
j
d
H e
is given by:
4
1 2 2 1
1, and 
0,otherwise.
j
d
H e
Sketch the
frequency response of
j
d
Y e
. Here, make sure to treat the case that
2
0
.
4(c)(
1.5
points)
Give an expression for the maximum sampling period for 4(b) so that the
output shape remains the same. Your expression
should be in terms of the sampling
period given in 4(a).
4(d
)(1.5
points)
For reconstructing the analog signal we use zero

order hold in the
following system.
Digital
compensating filter
y
d
(
n
)
d
(
e
j
)
y
a
(
t
)
Z.O.H.
we have:
d
(
e
j
) =
else
;
0
;
)
2
/
(
sin
)
2
/
(
For the system in 4(c), indicate how you would modif
y
j
d
H e
so that we will not need
to implement the digital compensating filter.
Also, specify the holdup time so that the entire system correctly operates as a digital
system for processing analog signals in the range of
c c
.
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