ECE 312 Final Exam (Summer 2011)
This exam has 6 questions. All work must be done on this exam sheet. You are required to
do 5 problems.
If you do all 6, I will treat the 6th problem points as “bonus points” and will add them
(unweighted) to your overall course point total. You are NOT required to do all 6
problems.
Do you want to do a bonus problem?
(circle one)
YES
NO
If you circled YES, which problem number is your bonus problem?
1
2
3
4
5
6
(circle one)
1. Consider a LTI system with input x[n] and an impulse response
0.5n
hn  
 0
n  0,1,2
otherwise
a) (5 points) Find H(z).
b) (5 points) Find the unit step response (i.e., when x[n]=u[n])

(a)
H(z)  1  0.5z 1  0.25z 2 
z 2  0.5z  0.25
z2
n
(b) y[n]   h[k]

k 0
y[0]  1


Thus,
y[1]  1  0.5  1.5
y[k]  1  0.5  0.25  1.75 ; n  2
2. An aperiodic signal x(t) is sampled between 1  t  3 seconds. Samples are taken
every 10 milliseconds to construct an N-point DFT.
a) (3 points) What is the sample frequency in Hz?

100 Hz
b) (3 points) What is N?
200
c) (3 points) The DFT harmonic X[9] corresponds to what frequency in Hz?
(9)(100 Hz)/200 = 4.5 Hz
d) (1 point) One of yourfellow students says the computed DFT harmonics will
accurately indicate the spectrum of x(t) over any 2 second interval (e.g., 5  t  7
seconds). Do you agree?
NO

3. Consider
(a) Find X(z)
X(z)  z  z 3 
z4  1
z3
(b) Where are the poles and zeros located?

zeros: 1, -1, j, -j (I will accept 4 poles with |z|=1)
poles: 0,0,0
(c) What is the ROC?
0 z 
(d) Find X  . (A formula is sufficient)

Since the ROC contains the unit circle, the DTFT exists.
 e j 4  1
X( ) 
e j 3

4. A LTI system with input x[n] and output response y[n] is described by the
difference equation
y[n]  0.4 y[n  2]  x[n]  0.1x[n  1]
(a) (3 points) Find H(z)
 0.1z 1 z 2  0.1z
Y(z) 1
H(z) 


X(z) 1 0.4z 2 z 2  0.4

(b) (4 points) Draw the direct form II implementation
(c) (3 points) Does this system have a DTFT? (You must justify your answer to get
any points on this portion of the problem.)
The ROC is 0.4  z   , which means the ROC contains the unit circle. Thus a DTFT
exists.

5. Consider the three-point moving average FIR filter
y[n] 
1
x[n]  x[n  1]  x[n  2]
3
(a) (4 points) find the impulse response h[n]
h[n] 

1
[n]   [n  1]  [n  2]
3

(b) (6 points) show that the frequency response magnitude is H   

1  N
n


(Hint: you should make use of the fact that 
1 
n 0

N 1
1 1 e  j 3
3 1 e  j
1 e  j 3 / 2 (e j 3 / 2  e  j 3 / 2 )

3 e  j / 2 (e j / 2  e  j / 2 )
1
sin( 3 /2)
 e  j
3
sin(  /2)
H( ) 
Since e  j  1, H   



1 sin 3 /2
3 sin  /2


1 sin 3 /2
3 sin  /2
for   1)
6. A continuous time causal filter has the transfer function
H(s) 
2
s 1
Assume the impulse response of this filter is sampled at a rate Ts so that the
discrete time filter has an impulse response

h[n]  h(t) t nTs
(a) (5 points) Find the transfer function H(z) of the IIR filter
The inverse Laplace (from 
the attached tables) gives h(t)  2e t u(t). Therefore,
h[n]  h[nTs ]  2e nTs u[n]



n
2
Then H(z)  2  e nTs z n  2 e Ts z 1  Ts 1
1 e z
n 0
n 0


(b) (5 points) Find the difference equation describing the IIR filter
y[n]  e Ts y[n  1]  2x[n]
