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(ELEC3100)[2012](s)midterm~=8yjrcimck^_36048.pdf
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The Hong Kong University of Science and Technology
Department of Electronic and Computer Engineering

ELEC 3100 Signal Processing and Communications Midterm Examination 29/03/2012
Name:___________________________________
Student Number:___________________________
Email:___________________________________
1.
This is a close-book examination.

2.
This is a one hour and twenty minutes examination.

3.
Answer all questions of this examination paper in the space provided.

4.
Use proper notations and show all your calculation steps clearly.

5.
No mark will be given for unjustified answers.


Useful relations,

7 .n1.(z) z .1
z ==
.17
n=01. z z(z .1) 1j n1 .j n
oo
cos n = e + e
o2 2
Questions Max. Points Score
Question 1 15+2
Question 2 15
Question 3 15
Question 4 15+2
Total 60+4

P.1 of 7
Q1 Sampling: From CT to DT (15 marks + 2 marks)
For a continuous\time signal x(t)=cos(o t)= cos(2fo t), we get a discrete\time signal x[n] by sampling x(t) with a o
sampling frequency FT, where x[n]= cos(oTn)=cos( n)= cos(n).
F
T (a). Write down the units of fo,.o, FT and.. (4marks)
fo : 1/s or Hz o : rad/s FT : samples/sec .: rad/sample
(b). For fo = 100, what is the requirement on FT to avoid aliasing? (2 mark)
FT 200
(c). For the same setting as (b), what is the corresponding requirement on.? (2 marks)
=.o/FT . FT =.o/. FT 200 ..o/.. 200 ..o/200 =..
(d). For fo = 100, FT = 300, determine the fundamental period of x[n]? (4 marks) 2fo
cos( n)=cos(n).
F
T =2fo/FT= 2100/300= 2 /3 . N=3.
(e). For fo= 100 and FT = 1000, we want to use a N\point DFT to obtain the magnitude spectrum of x[n]. Determine a proper N value. (3 marks)
=2fo/FT= 2100 /1000 = 2 /10 . N=10. N=10 points.
(f) Justify your answer for question (e). (Bonus: 2 marks)
k = k*2 ,0kN\1. N
2
As = , k=1 & N=10.
10
P.2 of 7
Q2 Linear Convolution, Circular Convolution, DFT and FFT (15 marks)
Let x[n]={2,\3,4,1}, 0n3, denote the input sequence. Let h[n]={\3,5,\6,4}, 0n3, denote the impulse response of an LTI system. The output of the LTI system to x[n] is the linear convolution between x[n] and h[n]denoted

.
(a). Determine the output y[2]. (3 marks)

y[n]=x[n]*h[n]=

y[2]=x[0]h[2]+x[1]h[1]+x[2]h[0]=2(\6)+(\3)5+4(\3)
y[2]=\39.
(b). The length\4 circular convolution between x[n] and h[n] is defined as

~
Determine the circular convolution output y[2].(3 marks)
~y[2]=h[0]x[2]+h[1]x[1]+h[2]x[0]+h[3]x[3]=\35
~y[2]=.\35

(c). DFT generates discrete\frequency signals in the frequency domain and is widely utilized in spectrum analysis 2k
N.1 .jn and other signal processing operations. Given the definition of DFT as X[k] = x[n]e N ,0kN\1, n=0 determine X[2] for N=4. (2 marks) 2k
N.1 .jn
X[k] = x[n]e N , N=4, 0k3.
n=0
.j0 .j.j