Exam (elaborations) Digital Signal Processing Digital Signal Processing Comprehensive Exams 1-8 Answered_ 2024.

1




Exam 1 (Chapter 1 to Chapter 3)

Show all your work for each problem.

1. Given the following DSP block diagram with a sampling rate of 4000 Hz, fill in each
blank block with the following given units.

fs=4000 Hz
ADC
coding DSP




DAC y (t )
y ( n)


a. Anti-image lowpass filter
b. Anti-aliasing lowpass filter
c. Sampler and hold circuit
d. Hold circuit

Answer the following questions:
e. Nyquist frequency =
f. Maximum frequency of the input analog signal that the DSP can handle
without aliasing
g. Identify the block that will cause signal distortion.
(10 points)

Answers: (b), (c), (d), (a)
e. 2000 Hz
f. 2000 Hz
g. (d)

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2. Given the following labeled digital signals,
x ( n)
10.0
10
7.0711 7.0711


T  200
0.0 0.0 5 6 7
n (Discrete-time index)
1 0 1 2 3 4
5

7.0711 7.0711
10
10.0
t (Real time (micro-seconds))
0 T 2T 3T 4T 5T 6T 7T



Answer the following questions. (15 points)

a. Sampling period ______________________

b. Sampling rate ________________________

c. Nyquist limit ________________________

d. x(3)  ______________________

e. If x(n) is the sampled version of the analog sine wave: x(t )  10sin(t ) , what is the
frequency of the analog sine wave? ______________

Answers:
a. 200  s
b. 5000 Hz
c. 2500 Hz
d. x(3)  7.7011
e. 625 Hz

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3. Sketch each of the following sequences for n  0,1, 2,3 .
a. x (n)  6 (n  1)  10 (n  3)
b. x (n)  6u(n  2)
bg
c. x (n)  0.8 u(n)
n


d. x(n)  10 sin(0.25n)u(n)
(20 points)
Answers:

(a) x(0)  0 , x(1)  6 , x(2)  0 , x(3)  10
(b) x(0)  0 , x(1)  0 , x(2)  6 , x(3)  6
(c) x(0)  1.0 , x(1)  0.8 , x(2)  0.64 , x(3)  0.512
(c) x(0)  0.0 , x(1)  7.071 , x(2)  10.0 , x(3)  7.071


10 6

5
4
(a)




(b)




0
2
-5

-10 0
0 1 2 3 4 5 0 1 2 3 4 5
Sample number n Sample number n

1 10

5
(c)




(d)




0.5 0

-5

0 -10
0 1 2 3 4 5 0 1 2 3 4 5
Sample number n Sample number n

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4. Given a DSP system with a sampling rate f s  10000 Hz, that is, the sampling period
is T  100 us, and each input is shown below:
a. x(t )  10cos(400 t )u(t ) ,   ________ f  _________
b. v(t )  10cos(17000 t )u (t ) ,   _______ f  _________
(1) Plot spectrum for (a) and (b) for the range from -5000 Hz to 25000 Hz, respectively.

(2) Determine which one will be aliased. Explain.

(3) Using x(n)  x(t ) t nT , determine the digital (DSP) sequences for x (n) (a) and v (n)
(b) respectively.

x ( n)  _ _ _ _ _ _ _ _ _ _ _ _ _

v ( n)  _ _ _ _ _ _ _ _ _ _ _ _ _

(20 points)

Answers:

(a)   400 radians/ second, f  200 Hz
(b)   17000 radians/second, f  8500 Hz
1)
(a)
Xs( f )
10 /T



f kHz
0.2 0.2 9.8 10 10.2 19.8 20 20.2

(b)
Vs ( f )
10 / T


f kHz
21.5 8.5 1.5 1.5 8.5 10 11.5 18.5 20 21.5

2) (b) is aliased. The sampling theory condition is not satisfied.

3) T  1/ f s  1/10000  0.0001 second
x(n)  10cos(0.04n )u (n)
v(n)  10cos(1.7n )u (n)