ECE 539 Digital Signal Processing Extra Credit For Midterm Exam Due: Thursday, March 29 , 2007 No late extra credit will be accepted.

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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
(


)
Anti-aliasing 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
   
.