## Discrete Time Signal Processing - Dec 2011

### Electronics & Telecomm. (Semester 6)

TOTAL MARKS: 80

TOTAL TIME: 3 HOURS
(1) Question 1 is compulsory.

(2) Attempt any **three** from the remaining questions.

(3) Assume data if required.

(4) Figures to the right indicate full marks.
**1 (a)** Derive the Parsevel's Energy relation. State the significance of Parsevel's theorem.(5 marks)
**1 (b)** One of zeros of a Causal Linear phase FIR filter is at 0?5 e^{j?/3}. Show the locations of other zeros and hence find the transfer function and impulse response of the filter.(5 marks)
**1 (c)** A two pole pass filter has the system function $$H(z)=\dfrac{b_0}{(1-pz^{-1})^2}$$ Determine the values of b_{0} and P. such that the frequency response H(w) satisfies the condition $$H(0)=1\ and\ \left|H \left(\dfrac{\pi}{4}\right)\right|^2=\dfrac{1}{2}$$(5 marks)
**1 (d)** Consider the signal x(n) = a^{n}u(n), |a| <1 :-

(i) Determine the spectrum.

(ii) The signal x(n) is applied to a decimator that reduces the rate by a factor 2. Determine the output spectrum. (5 marks)
**2 (a)** An analog signal x_{a}(t) is band limited to the range 900 ? F ? 1100 Hz. It is used as an input to the system shown in figure. In this system, H(w) is an ideal lowpass filter with cut off frequency F_{C} =125 Hz

(i) Determine and sketch the spectra x(n), w(n), v(n) and y(n).

(ii) Show that it is possible to obtain y(n) by sampling x_{a}(t) with period T=4 milisecond

(10 marks)
**2 (b)** Derive and draw the FFT for N = 6 = 2.3 use DITFFT method. X(n) ={ 1 2 3 1 2 3 } Find x(k) using DITFFT for N= 6=2.3 (10 marks)
**3 (a) ** Design a digital Butterworth low pass filter satisfying the following specifications using bilinear transformations. (Assume T=15).

$$0.9=\left\{\begin{array}{l}\leq{}\left\vert{}H\left(e^{jw}\right)\right\vert{}\leq{}1;\\ \ \ \&0\leq{}W\leq{}\frac{\pi{}}{2} \\\leq{}\left\vert{}H\left(e^{jw}\right)\right\vert{}\leq{}0.2\ ;\ \ \\&\frac{3\pi{}}{4}\leq{}W<\pi{}\end{array}\right.$$(12 marks)
**3 (b) (i) ** If x(n) = { 1+5j, 2+6j, 3+7j, 4+8j }. Find DFT X(K) using DIFFFT. (4 marks)
**3 (b) (ii)** Using the result obtained in (i) not otherwise, Find DFT of following sequences :-

x_{1}(n) = { 1,2,3,4 } and x_{2}(n) = {5 6 7 8 }(4 marks)
**4 (a) (i) ** Obtain System Function.

(4 marks)
**4 (a) (ii) ** Obtain Difference Equation.

(2 marks)
**4 (a) (iii)** Find the impulse response of system

(3 marks)
**4 (a) (iv)** Draw pole-zero plot and comment on System Stability

(3 marks)
**4 (b) ** Derive the Expression for impulse invariance technique for obtaining transfer function of digital filter from analog filter. Derive the necessary equation for relationship between frequency of analog and digital filter. (8 marks)
**5 (a)** What do you mean by inplace computations in FFT algorithm ? (4 marks)
**5 (b)** Find number of real additions and multiplication required to find DFT for 82 point. Compare them with number of computations required if FFT algorithms is used. (4 marks)
**5 (c) ** Design a digital Chebyshev filter to satisfy the following constraints :-

$$0.707=\left\{\begin{array}{l}\leq{}\left\vert{}H\left(e^{jw}\right)\right\vert{}\leq{}1;\\ \ \ \&0\leq{}w\leq{}0.2\pi{} \\\leq{}\left\vert{}H\left(e^{jw}\right)\right\vert{}\leq{}0.1\ ;\ \ \\&0.5\leq{}w<\pi{}\end{array}\right.$$

Use bilinear transformation and assume T=1 second. (12 marks)
**6 (a) ** Given x(n) = n+1 and N=8, find DFT X(K) using DIFFFT algorithm (8 marks)
**6 (b)** Obatin Direct form I, Direct form II realization to second order filter given by -

y(n) = 2b cos w_{0}y(n-1) - b^{2}y (n-2) + x(n) - b cos w_{0}x(n-1). (8 marks)
**6 (c) ** Explain the concept of decimation by integer (M) and interpolation by integer factor (L). (4 marks)
**7 (a)** Write short note on set top box for digital TV receiver. (4 marks)
**7 (b)** Application of Signal Processing in Radar. (4 marks)
**7 (c) ** What is linear phase filter? What condition are to be satisfied by the impulse response of an FIR system in order to have a linear phase? Define phase delay and group delay. (8 marks)
**7 (d) ** Short note on Frequency Sampling realization of FIR filters. (4 marks)