Question Paper: Discrete Time Signal Processing : Question Paper Dec 2013 - Electronics & Telecomm. (Semester 7) | Mumbai University (MU)
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## Discrete Time Signal Processing - Dec 2013

### Electronics & Telecomm. (Semester 7)

TOTAL MARKS: 100
TOTAL TIME: 3 HOURS
(1) Question 1 is compulsory.
(2) Attempt any four from the remaining questions.
(3) Assume data wherever required.
(4) Figures to the right indicate full marks.
1 (a) Transfer functions of casual and stable digital filters are given below. State whether these filtes are
Minimum/ Maximum/ Mixed Phase filters
$$\left(i\right)\ H_1\left(Z\right)=\dfrac{\left(1-\frac{1}{2}z\right)\left(1-\frac{1}{4}z\right)}{\left(z-\frac{1}{3}\right)\left(z-\frac{1}{5}\right)}$$
$$\left(ii\right)\ H_2\left(Z\right)=\dfrac{\left(1-\frac{1}{2}z\right)\left(z-\frac{1}{4}\right)}{\left(z-\frac{1}{3}\right)\left(z-\frac{1}{5}\right)}$$
$$\left(iii\right)\ \ H_3(Z)=\dfrac{\left(z-\frac{1}{2}\right)\left(z-\frac{1}{4}\right)}{\left(z-\frac{1}{3}\right)\left(z-\frac{1}{5}\right)}$$
(5 marks)
1 (b) Compute DFT of the sequence X1,(n) = {1,2,4,2} using property and not otherwise compute DFT of x2(n) = {1+j, 2+2j, 4+4j, 2+2j} (5 marks) 1 (c) The impulse response of a system is h(n)=anu(n), a≠0. Determine a and sketch pole zero plot for this system to act as :-
(i) Stable low pass filter. (ii) Stable High pass filter.
(5 marks)
1 (d) Draw direct form structure for a filter with transfer function, H(z) = 1+3z-1 + 2z+-3 - 4z-4(5 marks) 2 (a) Consider a filter with impulse response, h(n)={0.5, 1, 0.5}. Sketch its amplitude spectrum. Find its response to the inputs
$$\left(i\right)\ \ x_1\left(n\right)=\cos{\left(\frac{n\pi{}}{2}\right)\ }$$
$$\left(ii\right)\ \ x_2\left(n\right)=3+2\delta{}\left(n\right)-4\cos?(\frac{n\pi{}}{2})$$
(10 marks)
2 (b) Determine circular convolution of x(n)= {1,2,1,4} and h(n)= {1,2,3,2} using time domain convolution and radix -2FFT. Also find circular correlation using time domain correlation. (10 marks) 3 (a) Explain overlap and add method for long data filtering. Using this method find output of a system with impulse response, h(n)= {1,1,1} and input x(n)= {1,2,3,3,4,5}. (10 marks) 3 (b) Compute Dft of a sequence, x(n)= {1,2,2,2,1,0,0,0} using DIF-FFT algorithm. Sketch its magnitude spectrum. (10 marks) 4 (a) Draw lattice filter realization for a filter with the following transfer function.
$$H(Z)=\frac{1}{1+\frac{13}{24}z^{-1}+\frac{5}{8}z^{-2}+\frac{1}{3}z^{-3}}$$
(10 marks)
4 (b) Design a low pass Buttre worth filter with order 4 and passband cut off frequency of 0.4π. Sketch pole zero plot. Use Bilinear transformation. Draw direct form II structure for the designed filter. (10 marks) 5 (a) Design a FIR Bandpass filter with the following specifications :-
Length : 9
stop band cut off frequency : 0.7π
Use Hanning window.
(10 marks)
5 (b) The transfer function of a filter has two poles at z=0, two zeroes at z= -1 and a dc gain of 8. Final transfer function and impulse response.
Is this a causal or noncausal filter?
Is this a linear phase filter?
If another zero is added at z=1 find transfer function and check whether it is a linear phase filter or not.
(10 marks)
6 (a) Transfer function of an FIR filter is given by H(z)=1-zN. Sketch pole zero plots for N=4 and N=5 prove that it is comb filter. (10 marks) 6 (b) Write about frequency sampling realization of FIR filters. (10 marks) 7 (a) Explain the process of decimation for reducing sampling rate of signal. (10 marks) 7 (b) Compare various windows used for designing FIR filters.(10 marks)