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Blekinge Institute of TechnologySchool of Engineering

Written examination in Course code: ET1303 ETC012Signal Processing ET1303 ETC012 Date: 20080604

Time schedule: 9.0014.00

Maximum number of points available is 60. The number of points required to passthe exam is at most 30 points.Teacher: Muhammad Gufran KhanExaminer: Jorgen Nordberg

Allowed items on exam: Calculator, Collection of formulas - Signal Processing,Tables of Mathematical formulas, like Tefyma and Beta.

Indicate your name on all papers handed in, and put a maximum ofONEquestion

on each sheet!.

Questions:

1. Answer briefly on the following theoretical questions, completeness in detail isnot required. (1 point/sub question)

(a) Isy(n) =x(2n) a causal or a non-causal system?

(b) Ify(n) =x1(n) x2(n) and its Z-transform is Y(z) =X1(z)X2(z), wherethe region-of-convergence (ROC) ofX1(z) andX2(z) is ROC1and ROC2,respectively. What is the ROC ofY(z) in terms ofROC1 and ROC2?

(c) Let Z-transform of a signal y(n) is Y(z). IfY(z) is multiplied with zk,what is its effect on the signal in time-domain?

(d) Describe the relation between Fourier transform and Z-transform.

(e) A discrete-time system can be represented with system function H(z).Give two other ways of representation of a discrete-time system.

(f) What is the location of poles of a finite length causal FIR filter in theZ-plane?

(g) In the window method of FIR filter design, how do you decide the windowtype?

(h) In IIR filter design using analogue filters as templates, Butterworth andChebyshev filters play important roles. These filter types differ regardingfilter ripple. Explain how?

(i) Give one advantage and one drawback of IIR filters as compared to FIRfilters.

(j) Describe the difference between Bartletts method and Welchs methodof spectrum estimation.

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2. Consider the following difference equation

y(n) =5

6y(n 1)

1

6y(n 2) + x(n) (1)

(a) Calculate the corresponding system function H(z) . . . . . . . . . . . . . . . . . ( 2 p )

(b) Calculate the corresponding impulse response h(n). . . . . . . . . . . . . . . . (2p)(c) Determine the pole-zero plot of the system. Is the system stable? . (2p)

(d) Suppose that signal

x(n) = (1

4)nu(n), (2)

where u(n) denotes the unit-step function, is used as input to the sys-tem described in equation (1). Use Z-transform and partial fraction todetermine the output signaly(n). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4p)

3. Design an FIR filter that completely blocks the frequencywo =/4, by suit-ably placing poles and zeros in the Z-plane. The constructed filter should yieldreal output given that the input signal is real.

(a) Draw poles and zeros of the filter in the Z-plane. Clearly state the mag-nitude and angle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2p)

(b) Calculate the system function H(z) and frequency response H(w). (2p)

(c) Roughly sketch the magnitude response |H(w)|. . . . . . . . . . . . . . . . . . . (2p)

(d) If the input signal to the filter is given by,

x(n) = 10 + 3Cos(

4n) + Sin(

3n +

2), < n

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