## Digital Signal Processing - Dec 2014

### Electronics & Telecom Engineering (Semester 5)

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.

### Answer any one question from Q1 and Q2

**1 (a)** An analog signal is given as x(t) = sin(10 πt) + 2sin(20 πt) + 2cos(30 πt).

i) What is the Nyquist rate of this signal?

ii) If the signal is sampled with sampling frequency of 20 Hz, what is the discrete time signal obtained after sampling ?(6 marks)
**1 (b)** For a discrete time sequence x(n) = {1 2 3 4}, DFT is given by X(k) = {10 ? 2+2j ? 2 ? 2?2j}. Compute the DFT of x^(n) = {3 4 1 2} using circular time shift property of DFT.(6 marks)
**1 (c)** In the impulse response of the system is: h(n)=[(0.5)^{n} + n(0.2)^{n}] u(n).

i) Compute the transfer function

ii) Obtain the difference equation of the system.(8 marks)
**2 (a)** A signal x(t) = sin( ω t ) of frequency 50 Hz is sampled using a sampling frequency of 80 Hz. Obtain the recovered signal if ideal reconstruction is used.(6 marks)
**2 (b)** State and prove Parseval's theorem for the following sequence : x(n) = {1 2 3 4}.(8 marks)
**2 (c)** Find the Z transform of: $$ i) \ \ x(n)= e^{\left ( - \dfrac {n}{40} \right )} u(n) \ Draw \ the \ pole \ zero\ diagram \ for \ X(z) \\ ii) \ \ x(n) = \left ( - \dfrac {1}{5} \right )^n u(n) + 5 \left ( \dfrac {1}{2} \right )^{-n} u(-n-1) $$(6 marks)

### Answer any one question from Q3 and Q4

**3 (a)** Design a digital Butterworth filter that satisfies the following constraint using Bilinear transformation. Assume T=1 sec. $$ \begin {align*} 0.9 & \big \vert H(e^{j\omega})\big \vert \le 1 & 0 \le \omega \le \dfrac {\pi}{2} \\ & \big \vert H(e^{j\omega}) \big \vert \le 0.2 & 3 \dfrac {\pi}{4}\le \omega \le \pi \end{align*} $$(11 marks)
**3 (b)** Convert the analog filter with system function. $$ H_a (s) = \dfrac {s+0.2}{(s+0.2)^2 + 9} $$ into aa digital IIR filter by mean of Impulse Invariant technique. Assume T=1 sec.(6 marks)
**4 (a)** Design a digital Butterworth filter that satisfies the following specification using Bilinear transformation.

Sampling frequency | 8 KHz |

Passband | 0-500 Hz |

Passband ripple | 3 dB |

Stopband | 2-4 KHz |

Stopband ripple | 20 dB |

**4 (b)**Obtain direct form II and cascade realizations for the system:

y(n) = ? 0.1y (n - 1) + 0.2y(n - 2) + 3x(n) + 3.6x(n - 1) + 0.6x(n - 2)(6 marks)

### Answer any one question from Q5 and Q6

**5 (a)** Design a bandpass FIR filter using Hamming window for M = 11. $$ \begin {align*}
H(e^{j\omega}) & =1 & \dfrac {\pi}{4}\le \omega \le \dfrac {3 \pi}{4} \\ &=0 & otherwise \end{align*} $$(11 marks)
**5 (b)** A signal having values in the range [-1, +1], is quantized using 8 bits, with MSB as sign bit

i) Determine the quantization step size.

ii) Calculate the quantization noise power.(3 marks)
**5 (c)** What is Gibb's phenomenon? How it is reduced?(3 marks)
**6 (a)** Using frequency sampling method, design a FIR filter for N = 7 $$ \begin {align*}
H(e^{j\omega}) & =1 & 0\le \omega \le \dfrac {\pi}{2} \\ &=0 & \dfrac {\pi}{2} \le \omega \le \pi \end{align*} $$(9 marks)
**6 (b)** Show that the symmetric FIR filter has linear phase response.(8 marks)

### Answer any one question from Q7 and Q8

**7 (a)** Draw the block diagram of a system for sampling rate conversion by a non-integer factor and explain the operation of each block with the help of relevant diagrams and mathematical expressions. Can the positions of the decimator and interpolator be interchanged? Justify your answer.(10 marks)
**7 (b)** Explain the factors that influence the selection of a digital signal processor.(6 marks)
**8 (a)** Sampling rate is to be reduced from 96 KHz to 1 KHz. Highest frequency of interest is 450 Hz. δ_{p}=0.01, δ_{s}=0.001. Design a two stage decimator with decimating factors as 32 and 3.(8 marks)
**8 (b)** Write note on:

i) MAC unit

ii) Pipelining.(8 marks)