Concordia University, ECE Dept. Final Exam (closed-book) Instructor: Dr. Wei-Ping Zhu April 30, 2009

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ELEC 442: Introduction to Digital Signal Processing

Students are allowed 3 single-sided (or equivalent) 8.5×11 inch formula sheet. Anything can be written on it. Non-programmable calculators are also allowed.

Students must return the question paper at the end of examination.

1. LTI system (20 marks)

.2||,)21)(1(

1)( 11

131

>+−

−= −−

−

zzz

zzH

(a) Draw the pole-zero plot for the system function. (b) Find the impulse response of the system. (c) Write the difference equation for this system (d) Is the system stable? Causal? Why?

2. DFT (20 marks)

Let x(n)=cos(πn/4).

(a) Taking one period from x(n), we obtain an 8-point sequence x1(n), i.e., x1(n)=x(n) (n=0,1,…7). Determine and sketch the 8-point DFT X1(k).

(b) Taking two periods from x(n), we have a 16-point sequence x2(n), i.e., x2(n)=x(n) (n=0,1,…15). Determine and sketch the 16-point DFT X2(k).

(c) If x1(n) in part (a) is zero-padded to form another 16-point sequence x3(n), determine the 16-point DFT X3(k) for even k.

3. Sampling and processing of sinusoidal signals (20 marks) In the system shown in Fig.1, assume that

πωπωω ≤≤−= ,/)( TjeH j , and 1.0=T sec.

Fig. 1 H(ejω) C/D

xc(t) x(n)

T

yr(t) y(n) D/C

T (a) For each of the following inputs )(txc , determine )(nx , )(ny and )(tyr . (i) )6sin()( ttxc π= (ii) )12sin()( ttxc π= (b) Are the outputs )(tyr those you would expect from a differentiator? Why?

4. FIR filter (20 marks)

An FIR filter of length 5 is defined by a symmetric impulse response, i.e., h(n)=h(4-n), )40( ≤≤n . Let the input to this filter be a sum of DC and two cosine sequences, i.e.,

)cos()cos(1)( 2 nnnx ππ ++= .

Concordia University, ECE Dept. Final Exam (closed-book) Instructor: Dr. Wei-Ping Zhu April 30, 2009

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We would like the filter to pass only the mid-frequency component )cos( 2 nπ of the input. (a) Determine the impulse response h(n). (b) Determine the amplitude response and group delay of the filter. (c) Draw an implementation diagram with minimum number of multiplies for the filter.

5. Miscellaneous (20 marks)

(a) Given that z(t) is a product of x(t) and y(t), i.e., z(t)=x(t)y(t), where x(t) and y(t) are band-limited signals with highest frequencies fm and 2fm Hz, respectively. Determine the minimum sampling frequency for z(t) in order to avoid aliasing. Explain your answer.

(b) Consider an LTI system with the impulse response )10(,)( || <<= αα nnh , determine the output of the system when the input is unit step sequence.

(c) Assume a signal x(n) has the following spectrum,

.||

80

8||1

)(⎪⎩

⎪⎨

⎧

≤<

≤=

πωπ

πωωjeX

Determine and sketch the spectrum of )2

cos()2()( nnxny π= .

(d) The following system is to change the sampling rate by a noninteger factor. Express v(n) and y(n) in terms of x(n) as well as V(ejω) and Y(ejω) in terms of X(ejω).

Fig.3 5 )(ny )(nx 6 )(nv