## 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 X_{1},(n) = {1,2,4,2} using property and not otherwise compute DFT of x_{2}(n) = {1+j, 2+2j, 4+4j, 2+2j} (5 marks)
**1 (c) ** The impulse response of a system is h(n)=a^{n}u(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-z^{N}. 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)