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(ELEC3100)[2013](s)midterm~=8yjrcimck^_31327.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 11/04/2013
Name:____std_soln.______________________________
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,

1j n1 .j n
oo
cos n = e + e
o2 2
Questions Max. Points Score
Question 1 18
Question 2 20
Question 3 20
Question 4 22
Total 80

P.1 of 5
Q1. Sampling (18 marks)
A speech signal x(t) band\limited to 4kHz is sampled at 10kHz to obtain x[n]. (a). Determine the Nyquist rate. (2 marks)
Nyquist rate = 2fM = 8kHz.
(b). The signal x[n] is filtered by an ideal digital low\pass filter with cutoff frequency.c = 0.45.to obtain ya[n]. Determine whether the output ya[n] be contaminated if x(t) also includes an undesired sinusoidal signal z(t) at (i) 800Hz, and (ii) 7.5kHz, respectively. Give reason. (8 marks)

With sampling rate of 10kHz, consider the spectrum of z[n] in the range of 0 < 2 ,
(i).Yes.
z(t) of 800Hz , peaks of z[n] : =2(800/10k)=0.16..(rad/sample) and 1.84. (2.\0.16.=1.84..rad/sample)
. the undesired signal is not blocked.

(ii). No.
z(t) of 7.5kHz , peaks of z[n] : =2(7.5k/10k)=1.5..(rad/sample) and 0.5. (2.\1.5.=0.5..rad/sample)
. the undesired signal is blocked.

(c). The signal x[n] is filtered by an ideal digital band\pass filter whose pass\band extends over 0.06.0.6.to obtain yb[n]. Determine whether the output yb[n] be contaminated if x(t) also includes an undesired signal z(t) at (i) 800Hz, and (ii) 7.5kHz, respectively. Give reason. (8 marks)

With sampling rate of 10kHz, consider the spectrum of z[n] in the range of 0 < 2 ,
(i).Yes.
z(t) of 800Hz , peaks of z[n]: =2(800/10k)=0.16..(rad/sample) and 1.84. (2.\0.16.=1.84..rad/sample)
. the undesired signal is not blocked.

(ii). Yes.
z(t) of 7.5kHz , peaks of z[n]: =2(7.5k/10k)=1.5..(rad/sample) and 0.5. (2.\1.5.=0.5..rad/sample)
. the undesired signal is not blocked.

P.2 of 5
Q2 Linear Convolution, Circular Convolution, DFT and FFT (20 marks)
.1, 0 n 20 .1, 0 n 2
Given a finite\duration signal a[n] =. , and h[n] =. .
.0, otherwise .0, otherwise
(a). Find ya[n], ya[n]=a[n]*h[n], ya[n] is output of the linear convolution of a[n] and h[n]. (5 marks)
.1n = 0
.

2n = 1
.
.32 n 20

.
ya [n] =.2n = 21 .
1n = 22
.
.0 otherwise

.
.
(b). If we compute ya[n] by use of DFT & IDFT operations, what is the minimum number of zeros