BEng (Hons) in ELECTRICAL & ELECTRONIC ENGINEERING
DIGITAL SIGNAL PROCESSING
2002/2003
Question 1:
Given a finite continuous-time signal
x(t )  sin( 2t )
0  t 1
(a) Determine the sampled result x(n) (n starts from 0) with the sampling period
TS=0.25, and plot x(n).
[5 marks]
(b) Decompose x(n) as a sum of scaled and shifted unit samples using
x ( n) 

 x(k ) (n  k ) .
k  
[5 marks]
(c) Determine type of following system, i.e. homogeneous, additive, and/or linear.
y (n)  8 x(n  2)  4 x(n)  2
[5 marks]
(d) Consider another system with finite unit-sample response h(n) {h(0)=-1,
h(1)=1}, choose two different methods to find the response y1 (n) to the input
x(n)(i.e. calculate the convolution y1 (n)  x(n)  h(n) ).
[10 marks]
Page 1 of 7
BEng (Hons) in ELECTRICAL & ELECTRONIC ENGINEERING
DIGITAL SIGNAL PROCESSING
2002/2003
Question 2:
(a) Using the definition of z-transform, determine the z-transform X(z) of the
following sequence
x(n)   n u(n  1) ,
and shade its ROC in the z-plane.
[9 marks]
(b) Using the result of (a) and the property of z-transform, determine the ztransform Y(z) of the following sequence
1
y (n)  (2)  n u ( n  1)
n
[7 marks]
(c) Determine the partial fraction of the system function H(z) of the system in
figure1, use it and the result of (a) to evaluate the inverse z-transform
h(n)  Z 1 H ( z )
[9 marks]
x(n)
H ( z) 
6
6  13 z 1  6 z  2
2
z 
3
y(n)
Figure 1
Page 2 of 7
BEng (Hons) in ELECTRICAL & ELECTRONIC ENGINEERING
DIGITAL SIGNAL PROCESSING
2002/2003
Question 3:
Sampling a continuous-time signal xa (t ) for 1s generates a sequence of 4096 samples.
(a) What is the highest frequency in xa (t ) if it was sampled without aliasing?
[5 marks]
(b) If a 4096-point DFT of the sampled signal is computed, what is the frequency
spacing in hertz between the DFT coefficients?
[5 marks]
(c) Suppose that we are only interested in the DFT samples that correspond to
frequencies in the range 200  f  300 Hz. How many complex
multiplications are required to evaluate these values computing the DFT
directly, and how many are required if a decimation-in-time FFT is used?
[8 marks]
(d) How many frequency samples would be needed in order for the FFT algorithm
to be more efficient than evaluating the DFT directly?
[7 marks]
Page 3 of 7
BEng (Hons) in ELECTRICAL & ELECTRONIC ENGINEERING
DIGITAL SIGNAL PROCESSING
2002/2003
Question 4:
Consider the sequence
x(n)   (n)  2 (n  2)   (n  3)
(a) Find the four-point DFT, X (k ) , of x(n) .
[5 marks]
(b) Find the finite-length sequence q(n) that has a four-point DFT
Q(k )  W42 k X (k )
[6 marks]
(c) If y (n) is the four-point circular convolution of x(n) with itself, find y (n)
and the four-point DFT Y (k ) .
[7 marks]
(d) With h(n)   (n)   (n  1)  2 (n  3) , find the four-point circular
convolution of x(n) with h(n) .
[7 marks]
Page 4 of 7
BEng (Hons) in ELECTRICAL & ELECTRONIC ENGINEERING
DIGITAL SIGNAL PROCESSING
2002/2003
Question 5:
Consider the high-pass filter as shown in figure 2,
H (e j )
1

 34 
0
3
4



Figure 2
(a) Find the unit sample response, h(n).
[8 marks]
(b) Sketch the other three ideal frequency selective filters (i.e. low-pass, bandpass,
and bandstop filter) using the frequency response magnitude figure. Assume
for low-pass filter, the cutoff frequency is 0<  C <, and for bandpass and
bandstop filter, the cutoff frequencies are 1 and  2 (0< 1 <  2 <).
[9 marks]
(c) A new system is defined so that its unit sample response is h1 (n)  h(2n) .
Sketch the magnitude of the frequency response, H 1 (e j ) , of this system.
[8 marks]
Page 5 of 7
BEng (Hons) in ELECTRICAL & ELECTRONIC ENGINEERING
DIGITAL SIGNAL PROCESSING
2002/2003
Question 6:
Consider the interconnection of LSI systems shown in figure 3.
h2(n)
x(n)
+
h1(n)
y(n)
+
h3(n)
h4(n)
Figure 3
(a) Express the frequency response H (e j ) of the overall system in terms of
H 1 (e j ) , H 2 (e j ) , H 3 (e j ) and H 4 (e j ) .
[10 marks]
(b) Find the frequency response H (e j ) , if
h1(n)   (n)  2 (n  2)   (n  4)
h2 (n)  h3 (n)  (0.2) n u (n)
h4 (n)   (n  2)
[10 marks]
(c) For H 4 (e j ) , determine its magnitude H 4 (e j ) , phase  4 h ( ) , and  4 h ( ) .
[5 marks]
Page 6 of 7
BEng (Hons) in ELECTRICAL & ELECTRONIC ENGINEERING
DIGITAL SIGNAL PROCESSING
2002/2003
Appendix:
You might find the following formulas useful during the calculation:
1 aN
1 a
n 0

1
an 

1 a
n 0
N 1
an 
a 1
( N  1)a N 1  Na N  a
(1  a) 2
n 0

a
na n 
a 1

(1  a) 2
n 0
N 1
1
n  N ( N  1)

2
n 0
N 1
1
n 2  N ( N  1)( 2 N  1)

6
n 0
N 1
 na n 
Page 7 of 7