Solution for Q1 & 3 of Final Sem1 2010 2011

8/9/2019 Solution for Q1 & 3 of Final Sem1 2010 2011

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SOLUTIONS

COLLEGE OF ENGINEERING

PUTRAJAYA CAMPUS

FINAL EXAMINATION

SEMESTER I 2010/2011

PROGRAMME : Bachelor of Electrical & Electronic Engineering /

Bachelor of Electrical Power Engineering

SUBJECT CODE : EEEB 363

SUBJECT : Digital Signal Processing

DATE : October 2010

TIME : (3 Hours)

INSTRUCTIONS TO CANDIDATES:

1. This paper contains SIX (6) questions in FOUR (4) pages.

2. Answer all questions.

3.

Write all answers in the answer booklet provided.

4. Write answer to each question on a new page.

TH IS QUESTION PAPER CONSISTS OF 4 PRINTED PAGES INCLUDING TH IS

COVER PAGE.


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Digital Signal Processing EEEB363, Semester 1 2010/2011

________________________________________________________________________

Page 2 of 7

QUESTION 1 [20 marks]

(a) The impulse response of a LTI discrete-time system is given by h[n]=[1 -2 1 0]

for 30 n . If the input to this system is given by x[n]=[2 -1 -1 2] for 30 n ,

determine the output sequence of this system via linear convolution.

[6 marks]

(b) Determine even part and odd part of x[n].

[4 marks]

(c) Determine the total energy and average power of y[n].

]4 marks]

(d) What is the frequency response of the above system?

[2 marks]

(e)

Is the above system causal and stable? Explain and give your reasons.[4 marks]

ANSWER. (a)

Using Tabular Convolution to get at y[n]:-x[n] 2 -1 -1 2

h[n] 1 -2 1 0

2 -1 -1 2

-4 2 2 -4 0

2 -1 -1 2

0 0 0 0y[n] 2 -5 3 3 -5 2 0


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Page 3 of 7

(b) Writing x[n] as train of delayed unit impulses:-

],3[2]2[]1[][2][ nnnnn x

Even part and odd part of x[n].

marks][4

1] 0.5- 0.5- 0 0.5 0.5 [-1

]3[]2[21]1[

21]3[]2[

21]1[

21

])3[2]2[]1[][2]3[2]2[]1[][2(2

1][

])[][(2

1][

1] 0.5- 0.5- 2 0.5- 0.5- [1

]3[]2[2

1]1[

2

1]3[]2[

2

1]1[

2

1][2

])3[2]2[]1[][2]3[2]2[]1[][2(2

1][

])[][(2

1][

nnnnnn

nnnnnnnnn x

n xn xn xand

nnnnnnn

nnnnnnnnn x

n xn xn x

od

od

ev

ev

(c) The total energy and average power of y[n].

marks][4

0P(finite),76ESince

intervalfiniteoverenergyis|][|Ewhere

E12

1Lim

|][|12

1LimP

-:sequenceAperiodicof PowerAverage

762)5(33)5(2|][|E

-:lengthinfiniteoverEnergyTotal

yK y,

2

,

,k

2

k

2222222

K

K n

K y

K y

K

K n

y

n

y

n y

K

n y K

n y

(d)

2 j21)H(eis systemabovetheof responsefrequencythe

,)2()1(2-(n)0] 1 2- 1[][Since

j jee

nnnh

[2 marks]


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________________________________________________________________________

Page 4 of 7

(e) The above system is causal because h[n] is a right hand function.

The system is stable because summation of h[n] over an infinite duration is finite.

[4 marks]


Consider the system in Figure 2.

Figure 2The input of the system is

t t t t xa 400cos3)100cos(4)(

The frequency response of the system is shown in the following figure 3.

2

2

Figure 3: Frequency Response

If the frequency of the sampling is given as f s=500 samples/second,

(a) Determine if aliasing occurs. Explain.[4 marks]

(b) Sketch the spectrums of xa(t), x[n], y[n], and y(t). Make sure you identify the

important points.

[6 marks]

(c) Write the expressions for x[n] and y(t).

[5 marks]


C/D)(t x

a ][n x ][n y)(

je H D/C)(t y

( je H 1


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________________________________________________________________________

Page 5 of 7

(a) Compute the DFT of a real sequences x[n]={1 1 0 1} using the radix 2, DIT-FFT

method.

[6 marks]

(b) Compute the inverse DFT of H(k)={2 1-j 0 1+j} using the flow graph of the DIT-

FFT method arrived in question 3(a) above.

[5 marks]

(c) If the input to the LTI discrete time system in question 3(b) is the real sequence x[n]

as given in question 3(a), what is the DFT of the output of this system.

[4 marks]

ANSWER. 3(a).

Flow Graph of Radix 2 , 4 Point

DIT FFT

04W 1

0

4W

x[0]=1

x[1]=1

x[2]=0

x[3]=1

1]0[0 X

1]1[0 X

2]0[1 X

0]1[1 X

0

4W 1

1

1

3]0[ X

1]1[ X

1]2[ X

1]3[ X jW 14


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________________________________________________________________________

Page 6 of 7

1)(*01]1[]1[]3[

1)(*01]1[]1[]1[

11*21]0[]0[]2[

31*21]0[]0[]0[

01*11]3[]1[]1[

21*11]3[]1[]0[

11*01]2[]0[]1[

11*01]2[]0[]0[

)

2

sin()

2

cos(,1

.0,0

1,0,12

4log

2log

1

410

1

410

0

410

0

410

0

41

0

41

0

40

0

40

24/21

4

0

4

22

jW X X X

jW X X X

W X X X

W X X X

W x x X

W x x X

W x x X

W x x X

j jeeW W

r m

r N

m

j j

3(b)

], j-1 0 j1 [2=(k)H

], j+1 0 j-1 [2=H(k)

},[k]W*HX[k]

,}[k]W*H{ N

1h[n]

[k].*Xsequenceof point DFT- NcomputetoalgorithmFFTusecanWe

,W][*][

-:conjugatecomplextakingand N byside bothgMultiplyin

./...............10,W][1

][

-: bygivenIDFT

IDFT.thecomputetoalsoFFTuseCan

*

1 N

0k

kn

N

1 N

0k

*kn

N

1

0

kn*

1

0

kn-

Let

k H n Nh

IDFT EqnSynthesis N nk H N

nh

N

k

N

N

k

N


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________________________________________________________________________

Page 7 of 7

Flow Graph of Radix 2 , 4 Point

DIT FFT

0

4W 1

0

4W

H*[0]=2

H*[1]=1+j

H*[2]=0

H*[3]=1-j

2]0[0 X

2]1[0 X

2]0[1 X

j X 2]1[1

0

4W 1

1

1

2]0[ X

4)(*22]1[ j j X

0]2[ X

0)1)((*22]3[ j j X

jW 14

h[n]=X*[k]/N=[2/4 4/4 0 0]=[0.5 1 0 0]

3(c) Y[k]=DFT x[n] multiply with DFT h[n] where L>=N+M-1=4+4-1=7. This isequivalent to linear convolution for y[n] in the frequency domain.

Since length of x[n] and h[n] are both equal to 4 (N=M=4), we need to pad zeros to

them before we can compute DFT of length 8 ( e.g. using FFT radix 2).

Y[k]=DFT of {1 1 0 1 0 0 0 0}.DFT of {0.5 1 0 0 0 0 0 0}

For Solution of Question 2, 4, 5, 6 refer to separate files.

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Solution for Q1 & 3 of Final Sem1 2010 2011