## Discrete Time Signal Processing - May 2015

### 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)** Sketch the frequency response and identify the following filters based on their pass band $$ i) \ h [n] = \left \{ 1, - \dfrac {1}{2} \right \} \\ ii) \ H[z] = \dfrac {z^{-1}-a}{1-az^{-1}} $$(5 marks)
**1 (b)** Justify DFT as a linear transformation.(5 marks)
**1 (c)** Explain the frequency warping in Bilinear transformation.(5 marks)
**1 (d)** What is multi rate DSP? Where it is required?(5 marks)
**2 (a)** An analog filter has transfer function $$ H(s) = \dfrac {s+0.1}{(s+0.1)^2 +16} $$ Determine the transfer function of digital filter using bilinear transformation. The digital filter should be have specification $$ \omega_r = \dfrac {\pi}{2} $$(8 marks)
**2 (b)** Explain the effects of coefficient quantization in FIR filters.(8 marks)
**2 (c)** The first point of eight point DFT of real valued sequence are {0.25, 0.125-j0.3018, 0, 0.125-j0.518, 0}.

Determine the remaining three points.(4 marks)
**3 (a)** $$ x[n] = \left\{\begin{matrix}
1, &0 \le n \le 3 \\0,
&4 \le n \le 7
\end{matrix}\right. $$ i) Find DFT X[k]

ii) Using the result obtained in (i) find the DFT of the following sequences. $$
x_1 [n] = \left\{\begin{matrix}
1, &n=0 \ \ \ \ \\0,
&1 \le n \le 4 \\ 1,
& 5 \le n \le 7
\end{matrix}\right. \ and \ x_2[n] = \left\{\begin{matrix}
0, &0\le n \le 1 \\ 1,
&2 \le n \le 5 \\ 0,
& 6 \le n \le 7
\end{matrix}\right. $$(10 marks)
**3 (b)** Implement a two stage decimator for the following specifications. Sampling rate of the input signal=20,000 Hz,

M=100

Pass band=0 to 40 Hz, Pass band ripple=0.01,

Transition band=40 to 50 Hz, Stop band ripple=0.002(10 marks)
**4 (a)** By means of FFT-IFFT technique compute the linear convolution of x[n]={2,1,2,1} and h[n]={1,2,3,4}.(8 marks)
**4 (b)** Consider the following specification for a low pass filter

0.99 ≤ H(e^{jω})1≤1.01

0≤ω≤0.3 Π and

1 H(e^{jω}) 1 ≤ 0.01

0.5 Π ≤ 1 ω 1 ≤ Π

Design a linear phase FIR filter to meet these specification using the window design method.(8 marks)
**4 (c)** Identify whether the following system is minimum phase, maximum phase, mixed phase.

i) H_{1}(z)=6+z^{-1}-z^{-2}

ii) H_{2}=1 - z^{-1} - 6z^{-2}.(4 marks)
**5 (a)** Design digital FIR filter for following specification. Use hamming window and assume M=7.
(10 marks)
**5 (b)** Design digital low pass IIR Butterworth filter for the following specifications:

pass band ripple ≤dB|dB

pass band edge: 4 KHz

stop band attenuation ? 40 dB

stop band edge: 6 KHz

sample rate: 24 KHz

Use bilinear transformation.(10 marks)
**6 (a)** Write a short note on

i) Dual tone Multi-frequency Signal Detection

ii) Different methods for digital signal Synthesis.(12 marks)
**6 (b)** The transfer function of digital causal system is given as follows: $$ H (z) = \dfrac {1-z^{-1}}{1-0.2z^{-1}-0.15z^{-2}} $$ Draw cascade form, parallel form realization.(8 marks)