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SIR SYED University of Engineering & Technology, Karachi

Electronic Engineering Department

TE321 Digital Signal Processing

Final Examination (Fall-2010) Batch 2008

Max. Marks: 60 December 8th

, 2010 Time Allowed: 3hrs.

Instruction:

Attempt all five questions. Be conceptual & be logical. This is a closed book exam. Any type of material i.e. notes, handout, photocopies,

printed papers, books are not allowed.

Do not waste more than 3 minutes on 1 mark. You may use calculator but sharing is strictly not permitted.

Q1. (a) Figure 1 shows the flow graph for an 8-point DIT-FFT algorithm. Let

[

]be the

sequence whose DFT is []. In the flow graph, [], [], [] & [] representseparate arrays that are indexed consecutively in the same order as indicated nodes.

(i) Specify how the elements of the sequence [] should be placed in the array[], = 0,1, ,7. Also, specify how the elements of the DFT sequence should beextracted from the array [], = 0,1, ,7. (3)

(ii) Determine the sequence [], = 0,1, ,7, if the output Fourier transform is[] = 1, = 0,1, ,7. (5)

Figure 1

(b) The FFT requires the multiplication of complex numbers: (1 +1 ) (2 +2) =1 +1. Write out its complex multiplication, and determine how many realmultiplies & real adds are required. (2)

(c) The butterfly in Figure 2 was taken from DIT-FFT with = 8. what are the possiblevalues of in its stage? (2)

Figure 2

Q2. (a) Sampling a continuous-time signal [] for 1 generates a sequence of 4096 samples.(i) What could be the highest frequency in [] if it was sampled without

aliasing? (2)

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(ii) If a 4096-point DFT of the sampled signal is computed, what is the frequencyspacing in hertz between the DFT coefficients? (3)

(b) The signal [] is sampled at a rate of . Now the signal is modulated and becomes[] = [](0). What would be the sampling rate for the modulated signal[]. (3)

(c) How many bits are needed in an A/D converter if we want a signal-to-quantizationnoise ratio of at least 90 dB? Assume that []is gaussian with a variance 2, and thatthe range of the quantizer extends from -3 , to 3 ; that is, = 3; (with thisvalue for , only about one out of every 1000 samples will exceed the quantizerrange). (4)

OR

(c) Assume a 4-bit ADC channel accepts analog input range from 0 to 5 volts, determine:

(i) Step size of the quantizer (ii) Quantization level when the analog voltage is 3.2 volts

(iii) Binary code produced by ADC (iv) Quantization error when analog i/p is 3.2 volts

Q3. (a) Let [] be the sequence [] = 2[] + [ 1] + [ 3]. The five-point DFT of[] is computed and the resulting sequence is squared: [] = 2[]. A five-pointinverse DFT is then computed to produce the sequence []. Find the sequence [] byusing DFT properties. (3)

(b) Find 10-point inverse DFT of[] = 1 + 2[], 0 9. (5)(c) Find N-point DFT of[] = [] [ 0], 0 < 0 < . (4)

OR

(c) Find N-point DFT of[] = 4 + 2 2 , = 0,1,2, , . (4)

Q4. (a) The output ()of a discrete-time LTI system is found to be 213

; || > 13

when the input

() is 1;|| > 1. Provided that () (5)(i) Find the impulse response ()of the system(ii) Find the output ()when input () is 11

2

(b) An ideal discrete-time highpass filter with the cut-off frequency = 2 was designedusing the bilinear transformation with = 1 . What was the cut-off frequency for the prototype continuous-time ideal highpass filter? (2)

(c) An analog Integrator is described by the system function () = 1/. A digitalintegrator with system function () can obtained by use of the bilineartransformation. That is,

() = 21+

1

11 = ()|=2 11 1+1 .Write the difference equation for the digital integrator relating the input [] to theoutput []. (1)

(d) An analog lowpass prototype is given as = 1+1. Find the equivalent Z-transforminput - output equation of a lowpass digital filter when sampled at 5 ms and cut-offfrequency is 200 rad/s. (4)

Q5. (a) Given the following difference equation with the input-output relationship of a certain

initially relaxed DSP system: () 0.4( 1) + 0.29( 2) = () +0.5( 1). (4)(i) Find the impulse response sequence () due to an impulse sequence () =() ( 1)(ii) Find output response of the system when a unit step function () is applied.

(b) Obtain the coefficients of an FIR lowpass filter to meet the specifications given belowusing window method: Passband edge frequency = 1.5kHz, Transition width = 0.5kHz,

stopband attenuation > 20dB & sampling frequency = 8kHz. Use () = 0.54 +0.46, where = 1 & normalized transition width = 0.9 (4)

(c) Write the generalized transfer function & difference equation of the designed FIR in

Q5 (b). (1)

(d) A causal LTI Discrete-Time System develops an output () = (0.4)()

0.3(0.4

)1

(

1). For an input

(

)

=(0.2

)

(

), develop a parallel formrealization of the system. (3)