## Discrete Time Signal Processing - May 2015

### 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)** One of zeros of a causal linear phase FIR filter is at 0.5 e^{-ix/s}. Show the locations of the zeros and hence find the transfer function and impulse response of the filter.(5 marks)
**1 (b)** Determine Zeros of the following FIR system and indicate when the system is minimum phase maximum phase and mixed phase.

1. H(z)=6+Z^{-1}+Z^{-1}

2. H[z]=1-Z^{-1}-6Z^{-2}(5 marks)
**1 (c)** Find the number of complex multiplication and complex additions required to find DFT for 32 point sequence. Compare them with number of computation required if FFT algorithm is used.(5 marks)
**1 (d)** What is linear phase filters. Define group delay and phase delay.(5 marks)
**2 (a)** Derive Radix-2 Decimation in Time Fast Fourier-Transform and draw its signal flow graph.(10 marks)
**2 (b)** X[k]={36, -4+ j 9.656, -4 + j4, -4 +j1.656, -4, -4+j1.656, -4-4j4, -4 - j9.656} Find x[n] using IFFT algorithm (use DD IFFT).(10 marks)
**3 (a)** An 8 point sequence x[n]={1,2,3,4,5,6,7,8}

i) Find X[k] using DIF-FFT algorithm

ii) Let x_{1}[n]={5,6,7,8,1,2,3,4} using appropriate DFT property and result of part (i) determine X_{1}[k].(10 marks)
**4 (a)** Design a Chetryshev I bandstop digital filter with the following specifications:

Passband range: 0 to 275 Hz and 2KHz to ?

Stopband range: 550 to 1000 Hz

Sampling frequency: 8KHz

Passband attenuation: 1dB

Stopping attenuation: 15dB Use BLT and assume T=1sec.(10 marks)
**4 (b)** Design a Butterworth filter satisfying the following constraints: $$\begin {align*}
0.75 \le &|H(w)| \le 1 & for \ 0 \le w \le \pi /2 \ \ \\
& |H(w)|\le 0.2 & for \ 3 \pi /4 \le w \le \pi
\end{align*} $$ Use Bilinear Transformation Method.(10 marks)
**5 (a)** Design FIR digital highpass filter with a frequency response $$ \begin {align*}
H(w)&=1 &\pi /4 \le |w|\le \pi \\ &=0 &|w| \le \pi /4 \ \ \ \ \ \ \ \
\end{align*} $$ Use Hamming window: N=7(10 marks)
**5 (b)** With a neat diagram describe frequency sampling realization of FIR filters.(10 marks)
**6 (a)** An FIR filter is given by the difference equation $$ y[n] = 2x[n] + \dfrac {4}{5} x [ n-1] + \dfrac {3}{2} x [ n-2] + \dfrac {2}{3} x [n-3] $$ Determine the lattice form.(10 marks)
**6 (b)** Using linear convolution find y[n] for the sequence x[n]={1,2,-1,2,3,-2,-3,-1,1,2,-1} and h[n]={1,2}. Compare the result by solving the problem using overlap save method.(10 marks)

### Write Short Notes On:

**7 (a)** Digital Resonator.(5 marks)
**7 (b)** Parseval's Energy theorem and its significance.(5 marks)
**7 (c)** Goertzel Algorithm.(5 marks)
**7 (d)** Application of signal processing in RADAR(5 marks)