Microsoft Word FinalDSP 2014 TT doc Digital Signal Processing by Assoc Prof Dr Thuong Le Tien Final Examination, December 25th 2014 1 FINAL EXAMINATION Digital Signal Processing Date of Exam December[.]
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FINAL EXAMINATION Digital Signal Processing
Time allowed: 150 Minutes
Signature of the lecturer Signature of the EEE Dept
INSTRUCTION:
1 One page A4 of personal notes is allowed in the exam!
2 Answer 5 from 6 recommended questions PART 1: Answer three following questions (60 marks)
Question 1: (20 marks)
Consider the cascade of LTI discrete-time systems shown in Figure
The first system is described by the equation
, 5
0 , 0
, 5 0 ,
1 ) ( 1
j e H
And the second system is described by the equation y(n) = w(n) - w(n-1) The input to this system is x(n) = cos(0.6n) + 3(n-5) + 2
a What the frequency components of the x(n)?
b Sketch the magnitude response H1(ejw) then conclude about the kind of the filter of the first system (e.g LPF, HPF, BPF or BSF) then based on the frequency
components of the x(n), discuss about the output signal w(n) of the first system?
c Using the properties of LTI systems, determine the output y(n)?
Question 2: (20 marks)
The given signal x(n) = 4cos(n/2) + cos(n),
a Compute the 8-point DFT of the length-8 signal, x1(n), in which these samples are the first 8 samples of the given signal x(n) (e.g values of n from 0 to 7)
b Repeat the question 2.a for the first 10 samples of the signal x(n) using the wrapped signal approach
c Check the result in question 2.a by using the 8-point FFT
Question 3: (20 marks)
Given the signal
) 2 1 )(
1 (
5 0 1 )
(
2 1
1
Z Z
Z Z
H
a Find the available inverse-Z-Transform, discuss about the ROCs, Stability and Causality of the solutions
b Realize the system using the direct form 1 and canonical form for the causal case
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PART 2: Select two from three following questions
Question 4: (20 marks) A stable system shown in the figure has a1=1.5, a2=1 and b2=2
a Write the Input/output difference equation of the system
b Specify the Region of Convergence ROC
c Find the impulse response of the system
d Specify the response of system to the input x(n)={0,-1,-2,0,3} using the
convolution table
Question 5: (20 marks) The system T in the given figure is known to be time
invariant When the input to the system are x1(n), x2(n), and x3(n), the responses of the system are y1[n], y2[n], and y3[n], as shown
a Determine whether the system T could be linear
b If the input x[n] to the system is [n], what is the system response y[n]?
c What are all possible inputs x[n] for which the response of the system T can be determined from the given information alone?
Question 6: (20 marks)
An LTI system has impulse response h(n) = u(n)
a Determine y1(n), the response of the system to the input x1(n) e j(/2)n,
b Use the result of Question 6.a to help to determine y2(n), the response of the system
to the input x2(n) = cos(n/2)
c Determine y3(n), the response of the system to the input x3( n ) ej(/2)nu ( n )
Hint: Transform the h(n) into the Fourier domain then apply y(n)=x(n)H(j)
End
Good lucks!
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SOLUTIONS
Question 1:
First x(n) goes through a LPF with cut-off frequency 0.5 Since the cosine has a
frequency of 0.6, it will be filter out The delayed impulse will be filtered to a delayed sinc and the constant will remain unchanged We thus get:
) 6 (
)) 6 ( 5 0 sin(
3 )
5 (
)) 5 ( 5 0 sin(
3 ) ( 2
) 5 (
)) 5 ( 5 0 sin(
3
)
(
n
n n
n n
y n
n n
w
Question 2:
a X={5,-1,-3,-1,5,-1,-3,-1} then X={0,0,16,0,8,0,16,0}
b Wrapped signal: xw={10,-2,-3,-1,5,-1,-3,-1}
c FFT applied to the x(n) will be exactly the same as the result in 2.a
Question 3:
a Rewrite:
) 2 (!
) 2 1 ( ) 1 ( ) (
1 1
Z j
C Z
j
B Z
A Z
the ROC, stability and Causality
b Realize the direct form 1 and canonical form for the causal case
Question 4:
a I/O equation : y(n) = b2x(n-2) + a1y(n-1) + a2y(n-2) where a1=1.5, a2=1 and b2=2
b
) 5 0 1 )(
2 1 (
2 5
1 1
2 )
2 2
1
2
Z Z
Z Z
Z
Z z
System has two poles: 2 and -0.5 so 0.5<ROC<2]
c The Impulse response: h(n) = 2(n-2) + 1.5h(n-1) + h(n-2)
a Causal signal x(n)={0,-1,-2,0,3}; h(n)={0,0,2,3,6.5,… }(assumed for only first five terms for the IIR filter)
b then apply the convolution table, obtained y(n)={0,0,0,-2,1,-0.5,19,9,19.5}
Question 5:
a Notice that x1(n) = x2(n)+ x3(n+4) but y1(n) is not equal y2(n)+ y3(n+4) then the system is not linear
b Note that x3(n+4)=(n) therefore h(n)= 3(n+6)+2(n+5)
c Since the system is known TI but not Linear, so we can not use choices such as:
(n)= x1(n) - x2(n) and (n)= 1/2 x2(n+1) to determine the impulse response With the given information, we can only use shifted input
Question 6:
e j
H
1
1 )
2
2
1 )
c Similar for then y3(n)=x3(n).H(j) with n > or = 0