1 (a)
Derive the Parsevels Energy relation.State the significance of Parsevels theorem.

5 M

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 M

1 (c)
A two pole pass filter has the system function \[H\left(z\right)=\frac{b_0}{{\left(1-pz^{-1}\right)}^2}\] Determine the values of b

_{0}and P. such that the frequency response H(w) satisfies the condition \[H\left(0\right)=1\ and\{\left\vert{}H\left(\frac{\pi{}}{4}\right)\right\vert{}}^2=\frac{1}{2}\]
5 M

1 (d)
Consider the signal x(n) = a

(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.

^{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 M

2 (a)
An analog signal x

(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) 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 M

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 M

3 (a)
Design a digital Butterworth low pass filter satisfying the following specifications using bilinear transformations. (Assume T=15).

\[\begin{cases} 0.9 &\le \left|H(e^{j\omega})\right| \le 1; & 0 \le \omega \le \dfrac{\pi}{2}\\ & \ \ \ \ \left|H(e^{j\omega})\right| \le 0.2; & \dfrac{3\pi}{4} \le \omega \le \pi \end{cases}\]

\[\begin{cases} 0.9 &\le \left|H(e^{j\omega})\right| \le 1; & 0 \le \omega \le \dfrac{\pi}{2}\\ & \ \ \ \ \left|H(e^{j\omega})\right| \le 0.2; & \dfrac{3\pi}{4} \le \omega \le \pi \end{cases}\]

12 M

3 (b) (i)
If x(n) = { 1+5j, 2+6j, 3+7j, 4+8j }. Find DFT X(K) using DIFFFT.

4 M

3 (b) (ii)
Using the result obtained in (i) not otherwise, Find DFT of following sequences :-

x

x

_{1}(n) = { 1,2,3,4 } and x_{2}(n) = {5 6 7 8 }
4 M

4 (a) (i)
Obtain System Function.

4 M

4 (a) (ii)
Obtain Difference Equation.

2 M

4 (a) (iii)
Find the impulse response of system

3 M

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

3 M

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 M

5 (a)
What do you mean by inplace computations in FFT algorithm ?

4 M

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 M

5 (c)
Design a digital Chebyshev filter to satisfy the following constraints :-

\[\begin{cases} 0.707 &\le \left|H(e^{j\omega})\right| \le 1; & 0 \le \omega \le 0.2\pi\\ & \ \ \ \ \left|H(e^{j\omega})\right| \le 0.1; & 0.5 \le \omega \le \pi \end{cases} \]

Use bilinear transformation and assume T=1 second.

\[\begin{cases} 0.707 &\le \left|H(e^{j\omega})\right| \le 1; & 0 \le \omega \le 0.2\pi\\ & \ \ \ \ \left|H(e^{j\omega})\right| \le 0.1; & 0.5 \le \omega \le \pi \end{cases} \]

Use bilinear transformation and assume T=1 second.

12 M

6 (a)
Given x(n) = n+1 and N=8, find DFT X(K) using DIFFFT algorithm

8 M

6 (b)
Obatin Direct form I, Direct form II realization to second order filter given by -

y(n) = 2b cos w

y(n) = 2b cos w

_{0}y(n-1) - b^{2}y (n-2) + x(n) - b cos w_{0}x(n-1).
8 M

6 (c)
Explain the concept of decimation by integer (M) and interpolation by integer factor (L).

4 M

7 (a)
Write short note on set top box for digital TV receiver.

4 M

7 (b)
Application of Signal Processing in Radar.

4 M

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 M

7 (d)
Short note on Frequency Sampling realization of FIR filters.

4 M

More question papers from Digital Signal Processing & Processors