MU Digital Signal Processing - December 2015 Exam Question Paper

MU Instrumentation Engineering (Semester 6)
Digital Signal Processing
December 2015

Total marks: --
Total time: --

INSTRUCTIONS
(1) Assume appropriate data and state your reasons
(2) Marks are given to the right of every question
(3) Draw neat diagrams wherever necessary

Answer the following (Any Four):

1 (a) State and prove the convolution property of z-transform.

5 M

1 (b) Draw the single butterfly of 2-Radix DIT & DIF FFT algorithm.

5 M

1 (c) State and prove the complex conjugate property of DFT.

5 M

1 (d) The transfer function of analog filter is

H (S) = \frac{S + 2}{(S + 1) (S + 3)}

$H(S) = \dfrac {S+2}{(S+1)(S+3)}$ design a digital IIR filters by means of IIT. Assume T=0.1 Sec.

5 M

1 (e) Determine IDFT of x(k)=[6, -1, -j, 0, -1+j] by using DIT FFT algorithm.

5 M

2 (a) If x₁(n)=[1, 2, 0, 3] and x_{(n)=[1, 2, 3, 2]. Find the DFT of both the sequence by using DFT only once (not otherwise).}

10 M

2 (b) Derive and draw the FFT for N=6=2.3 using DIT FFT algorithm.

10 M

3 (a) If x₁(n)=[1, 2, 3, 5] and x₂(n)=[2, 4, 2, 3]. Obtain x₁(n) ⊗ x₂(n) by using DIF FFT algorithm.

10 M

3 (b) Prove the relation between the analog and digital filter by means of Bilinear Transformation Technique.

10 M

4 (a) Determine the output of a Linear FIR Filter whose impulse response h(n)={3, 2, 1} x(n)={1, 0, 2, 1, 0, -2, -1, 0, 3, 1} using over lap add method.

10 M

4 (b) Determine the frequency response of the system h(n)=aⁿu(n) magnitude and phase response of it.

10 M

5 (a)

y (n) = x (n) + \frac{1}{4} x (n - 1) + \frac{1}{6} y (n - 1) + \frac{1}{6} y (n - z)

$y(n)=x (n) + \dfrac {1}{4} x (n-1) + \dfrac {1}{6} y (n-1)+ \dfrac {1}{6}y(n-z)$ Realize the system by using Direct form-I cascade and parallel form realization.

10 M

5 (b) Design a digital Butterworth filter that satisfy the following constraint using Bilinear Transformation. Assume T=1 Sec

\begin{aligned} 0.9 =\leq & | H (e^{j ω}) | \leq 1 & 0 \leq ω \leq % & | H (e^{j ω}) | \leq 0.2 & 3 π / 4 \leq ω \leq π \end{aligned}

$\begin {align*} 0.9= \le &|H(e^{j\omega})| \le 1 &0\le \omega \le \% \ \ \ \ \ &|H(e^{j\omega})| \le 0.2 & 3\pi/4 \le \omega \le \pi \end{align*}$

10 M

6 (a) A pass filter is to be designed which following frequency response

\begin{aligned} H d (e^{j ω}) = e^{- j 3 ω} - & \frac{3 π}{4} \leq ω \leq \frac{3 π}{4} & \frac{3 π}{4} \leq | ω | \leq π \end{aligned}

$\begin {align*} Hd(e^{j\omega}) =e^{-j3\omega} -&\dfrac {3\pi}{4} \le \omega \le \dfrac {3\pi}{4} \ &\dfrac {3\pi}{4} \le |\omega | \le \pi \end{align*}$ Determine the filter coefficient h(n) by using Hamming window.

10 M

6 (b) A one state decimator is characterised by the following decimator factor = 3 Antialising filter coefficient
h(0) = -0.06 = h(4)
h(1) = 0.30 = h(3)
h(2) = 0.62
Given the data x(n) with a successive [6, -2, -3, 8, 6, 4, -2]. Calculate and list filtered output w(n) and the output of the decimator y(n).

10 M

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