## Discrete Time Signal Processing - Dec 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)** The first five points of eight point DFT of real valued signal are {0.25, 0.125 -j0.3018, 0, 0.125 -j0.0150, 0}. Determine the remaining three points.(5 marks)
**1 (b)** Sketch the frequency response and identify the following filters based on their passband. $$ i) \ h(n) = \left \{1, - \dfrac {1}{2} \right \} \\
ii) \ H(z) = \dfrac {z^{-1}-a}{1-az^{-1}} $$(5 marks)
**1 (c)** What is multirate DSP? State its applications.(5 marks)
**1 (d)** An analog filter has transfer function $ H(s)= \dfrac {S+0.1}{(S+0.1)^1 + 16} $

Determine transfer function of digital filter using bilinear transformation. The digital filter should have a specification of Ω, = π/2.(5 marks)
**2 (a)** Compute DFT of sequence r(n)={1, 2, 2, 2, 1, 0, 0, 0} using DIT-FFT algorithm.(10 marks)
**2 (b)** Explain the effects coefficients quantization in FIR filters.(10 marks)
**3 (a)** Implement a two stage decimator for the following specification:

Sampling rate of the input signal = 20,000Hz,

Decimating factor M=100,

Passband=0 to 40Hz,

Passband ripple = 0.01,

Transition band = 40 to 50Hz.

Stop band ripple = 0.002.(10 marks)
**3 (b)** If x(n)={1+2j, 3+4j, 5+6j, 7+8j}. Find DFT X(k) using DIF-FFT algorithm.(10 marks)
**4 (a)** Explain upsampling process in detail and derive for input-output relationship in time domain and frequency domain.(10 marks)
**4 (b)** Obtain cascade and parallel realization structures for the system described by y(n)= -0.1y(n-1) + 0.72y(n-2) + 0.7x(n) - 0.252 x(n-1).(10 marks)
**5 (a)** Design a FIR digital filter using windo method for following specifications $$ \begin {align*} H(e^{j\omega}) &= e^{-j3\omega} & 0\le |\omega | \le \dfrac {3\pi}{4} \\ &=0 & \text{otherwise } \ \ \ \end{align*} $$ Use Hamming window of length 7.(10 marks)
**5 (b)** Design a digital low pass IIR Butterworth filter for the following specification

Passband ripple | : ≤1 dB |

Passband edge | : 4 KHz |

Stopband attenuation | : 40 dB |

Stop edge | : 8 KHz |

Sampling Rate | : 24 KHz |

Use bilinear transformation.(10 marks)

### Write a short note on.

**6 (a) (i)** Dual tone multi frequency signal detection.(5 marks)
**6 (a) (ii)** Different methods for digital signal synthesis.(5 marks)
**6 (b)** Determine the zeros of the following FIR systems and indicate whether the system is minimum phase, maximum phase or mixed phase, $$ i) \ H_1 (z)= 6+z^{-1}- 6z^{-2} \\
ii) \ H_2 (z)= 1-z^{-1} - 6z^{-2} \\
iii) \ H_3 (z) = 1-\dfrac {5}{2}z^{-1} - \dfrac {3}{2}z^{-2}\\
iv) \ H_4 (z) = 1-\dfrac {5}{2}z^{-1} - \dfrac {2}{3}z^{-2} $$ Comment on stability of minimum and maximum phase system.(10 marks)