07 Rr410201 Digital Signal Processing

8/9/2019 07 Rr410201 Digital Signal Processing

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Code No: RR410201 Set No. 1

IV B.Tech I Semester Regular Examinations, November 2007DIGITAL SIGNAL PROCESSING

(Electrical & Electronic Engineering)Time: 3 hours Max Marks: 80

Answer any FIVE QuestionsAll Questions carry equal marks

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1. (a) For each of the following systems, determine whether or not the system is

i. stable

ii. causal

iii. linear

iv. shift-invariant.

A. T [x(n)] = x(n − n0)

B. T [x(n)] = ex(n)

C. T[x(n)] = a x(n) + b.Justify your answer.

(b) A system is described by the difference equation y(n)-y(n-1)-y(n-2) = x(n-1). Assuming that the system is initially relaxed, determine it s unit sample

response h(n). [8+8]

2. (a) Let x(n) and X (e jw) represent a sequence and its transform. Determine, interms of X (e jw), the transform of each of the following sequences :

i. k x(n), k = any constant

ii. x(n − n0), n0 = a real integer

(b) By explicitly evaluating the transforms X (e jw), H (e jw)andY (e jw) correspond-ing to x(n), h(n) and y(n) specified in part (a) show that Y (e jw) = H (e jw)X (e jw)

[8+8]

3. (a) Distinguish between DFT and DTFT .

(b) Consider a sequence x(n) of length L. Consider its DTFT X d(w) is sampledand N is the number of frequency samples. Discuss the relation between Land N for inverse DTFT = inverse DFT comment on the aliasing problem.

(c) Compute the DFT of x(n) = {1, 0, 0, 0} and compare with X d(w). [4+6+6]

4. An 8 point sequence is given by x(n) = {2,2,2,2,1,1,1,1}. Compute 8 point DFT of x(n) by

(a) radix - 2 D I T F F T

(b) radix - 2 D I F FF TAlso sketch magnitude and phase spectrum. [16]

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5. (a) Explain how the analysis of discrete time invariant system can be obtainedusing convolution properties of Z transform.

(b) Determine the impulse response of the system described by the differenceequation y(n)-3y(n-1)-4y(n-2)=x(n)+2x(n-1) using Z transform. [8+8]

6. (a) Discuss about the pole locations for the digital Chebyshev filters.

(b) Compare the impulse invariance and bilinear transformation methods. [8+8]

7. (a) Draw the frequency response of N-point rectangular window.

(b) Design a fifth order band pass linear phase filter for the following specifications.

i. Lower cut-off frequency = 0.4 πrad/sec

ii. Upper cut-off frequency = 0.6 πrad/seciii. Window type = Hamming

Draw the filter structure. [4+12]

8. (a) Realize an FIR filter with impulse response h(n) given by h (n) =1

2

n[u (n) − u (n − 5)]

(b) A system is described by it’s transfer function H(Z) given by H (Z ) = 4 +3Z

Z − 1

2

− 1

Z − 1

4

] [8+8]

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1. (a) Prove that a discrete LTI system is stable if and only if ∞

n=−∞

|h(n)| < ∞

where h(n) is the unit sample response of the system.

(b) Check the following systems for linearity, causality, time invariance and sta-bility using appropriate tests.

i. y(n) = n e|x(n)|

ii. y(n) = ancos(2πn/N ) [8+8]

2. (a) Show that the frequency response of a discrete system is a periodic functionof frequency.

(b) Obtain the frequency response of the first order system with difference equa-tion y(0) = x(n)+10y(n-1) with initial condition y(-1) = 0 and sketch it com-ment about its stability.

(c) State and prove the frequency shifting property of Fourier transform. [5+6+5]

3. (a) Compute the discrete Fourier transform of each of the following finite lengthsequences considered to be of length N.

i. x(n) = δ(n)

ii. x(n) = δ(n − n0)where0 < n0 < N

iii. x(n) = an0 ≤ n ≤ N − 1

(b) Let x2(n) be a finite duration sequence of length N and x1(n) = δ(n − n0)where n0 < N . Obtain the circular convolution of two sequences. [8+8]

4. (a) Let x(n) be a real valued sequence with N-points and Let X(K) represent itsDFT , with real and imaginary parts denoted by X R(K ) and X 1(K ) respec-tively. So that X(K) = X R(K ) + JX 1(K ). Now show that if x(n) is real,X R(K ) is even and X 1(K ) is odd.

(b) Compute the FFT of the sequence x(n) = { 1, 0, 0, 0, 0, 0, 0, 0 } [8+8]



6. (a) Design a second order Digital low pass Butterworth filter using impulse invari-ance method. The sampling period is “T”. Determine the difference equationand realize the filter in all possible forms.

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(b) Compare Bi-linear and Impulse Invariance transformation methods. [12+4]

7. (a) What is the principle of designing FIR filters using windows.

(b) Using a rectangular window technique design a low pass filter with pass bandgain of unity, cut-off frequency of 1kHz and working at a sampling frequencyof 5 kHz. The length of the impulse response should be 7. [6+10]

8. (a) What are the advantages in cascade and parallel realisation of IIR systems

(b) The transfer function of a system is given by

H (Z ) =(1 + Z −1)

3

1 − 1

4Z −1

1 − Z −1 + 1

2Z −2

[6+10]

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1. (a) Find the impulse and step responses for the given system:y(n)+y(n-1) = x(n)-2x(n-1)

(b) Test the following systems for linearity, time invariance, causality and stability.

i. y(n) = a|x(n)|

ii. y(n) = sin(2nfπ/F )x(n) [8+8]

2. (a) Explain magnitude and phase spectrum with reference to frequency responseof system.

(b) Find the impulse response ,frequency , magnitude response and phase responseof the given second order systemy(n) − y(n − 1) + 3

16y(n − 2) = x(n) − 1

2x(n − 1) [6+10]

3. (a) Distinguish between DFT and DTFT .

(b) Consider a sequence x(n) of length L. Consider its DTFT X d(w) is sampledand N is the number of frequency samples. Discuss the relation between Land N for inverse DTFT = inverse DFT comment on the aliasing problem.

(c) Compute the DFT of x(n) = {1, 0, 0, 0} and compare with X d(w). [4+6+6]

4. (a) Implement the Decimation in frequency FFT algorithm of N-point DFT whereN-8. Also explain the steps involved in this algorithm.

(b) Compute the FFT for the sequence x(n) = { 1, 1, 1, 1, 1, 1, 1, 1 } [8+8]

5. (a) With reference to Z-transform, state the initial and final value theorem.

(b) Determine the causal signal x(n) having the Z-transform X (Z ) = Z 2

+Z

(Z − 1

2)2

(Z −1

4) .

[6+10]

6. Use the Bilinear transformation to convert the analog filter with system functionH (S ) = S + 0.1/(S + 0.1)2 + 9 into a digital IIR filters. Select T = 0.1 and comparethe location of the zeros in H(Z) with the locations of the zeros obtained by applyingthe impulse invariance method in the conversion of H(S). [16]

7. Design a low pass Finite Impulse Response filter that approximate the followingfrequency response:

H (f ) =

1 ; 0 ≤ f ≤ 1000 Hz0 ; elsewhere in the range 0 ≤ f ≤ f s/2when the sampling frequency is 8000 sps. The impulse response duration is to belimited to 2.5 msec. Draw the filter structure. [16]

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8. (a) Write the difference equations for FIR and IIR system and hence derive thetransfer function of FIR and IIR system.

(b) Realize the following system with minimum number of multipliers

H (Z ) = 0.5+0.75Z −1+0.8Z −2+0.9Z −3+2Z −4+0.9Z −5+0.8Z −6+0.75Z −7+0.5Z −8

[8+8]

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1. By explicitly evaluating the convolution sum, evaluate the convolution y(n)=(n)*h(n)of the sequences

h(n) = αn 0 ≤ n < N

0 elsewhereX (n) =

β n−n0, n0 ≤ n0 , n ≤ n0

[16]

2. (a) Prove the modulation and time shifting properties of discrete time Fouriertransform.

(b) A discrete system is given by following difference equationy(n)-5y(n-1) = x(n) + 4x(n-1)where x(n) is the input and y(n) is the out put. Determine its magnitude andphase response as a function of frequency.

[8+8]

3. (a) Define DFT of a sequence x(n). Obtain the relationship between DFT andDTFS.

(b) Consider a sequence x(n) = {2,−1, 1, 1} and T = 0.5 compute its DFT andcompare it with its DTDT. [8+8]

4. (a) Implement the decimation in time FFT algorithm for N=16.

(b) In the above Question how many non - trivial multiplications are required.[10+6]



6. Design a low pass digital filter with the following specifications;Maximum pass band attenuation αmax = −3db,for0 ≤ w ≤ 2π/10Minimum stop band attenuation αmax = −15db,for3π/10 ≤ w ≤ πUse following two methods:

(a) Impulse invariance method(b) Bilinear transformation method.

Assume T = 0.2 msec. [16]

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7. Design a band pass Finite Impulse Response filter that approximate the followingfrequency response:

H (f ) =

1; 160 ≤ f ≤ 2000 ; elsewhere in the range 0 ≤ f ≤ f s/2

when the sampling frequency is 8000 sps. Limit the duration of impulse responseto 2 msec. Draw the filter structure. [16]

8. (a) Describe how targets can be decided using RADAR

(b) Give an expression for the following parameters relative to RADAR

i. Beam width

ii. Maximum unambiguous range

(c) Discuss signal processing in a RADAR system. [4+6+4]

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