 MORE IN Digital Signal Processing
VTU Electrical and Electronic Engineering (Semester 6)
Digital Signal Processing
May 2016
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

1(a) Compute the N-point DFT of x[n] = an for 0≤n≤N-1. Also find the DFT of the sequence x[n] = 0.5n u[n]; 0≤n≤3.
7 M
1(b) Find the DFT of a sequence $$x[n]=\left\{\begin{matrix} 1\ \text{for}\ 0\leq n\leq 3\\ 0\ \text{otherwise} \end{matrix}\right.$$
For N = 8. Plot magnitude of the DFT x(k).
10 M
1(c) If $$x[n]\xleftarrow[N]{DFT}x(k)$$ then prove that DFT {x(k)} = N x(-l).
3 M

2(a) The first values of an 8 point DFT of a real value sequence is {28, -4.966j, 4+4j, -4+1.66j, -4}. Find the remaining values of the DFT.
4 M
2(b) Obtain the circular convolution of x1[n] = [1, 2, 3, 4] with [1, 1, 2, 2].
6 M
2(c) A long sequence x[n] is filtered though a filter with impulse response h(n) to yeild the output y[n]. If x[n] = {1, 4, 3, 0, 7, 4, -7, -7, -1, 3, 4, 3}, h(n) = {1,2} compute y[n] using overlap add technique. Use only a 5 point circuilar convolution.
10 M

3(a) Prove the symmetry and periodicity property of a twiddle factor.
4 M
3(b) Develop an 8 point DIT - FFT algorithm. Draw the signal flow graph. Show all the intermediate results on the signal flow graph.
12 M
3(c) What is FFT algorithm? State their advantages over the direct computation DFT.
4 M

4(a) Find 4 point circular convolution of x[n] and h[n] using radix 2 DIF FFT algorithm x[n] = [1, 1, 1, 1] and h[n] = [1, 0, 1, 0].
8 M
4(b) Calculate the IDFT of x(k) = {0, 2.828 - j2.828, 0, 0, 0, 0, 0, 2.82 + j 2.82} using inverse radix 2 DIT FFT algorithm.
12 M

5(a) the transfer function of an analog filter is given as $$H_a(s)=\dfrac{1}{(s+1)(s+2)}.$$ obtain H(z) using impulse invariant method. Take sampling frequency of 5 samples/sec.
5 M
5(b) Obtain H(z) using impulse invariance method for following analog filter $$H_a(s)=\dfrac{1}{(s+0.5)(s^2+0.5s+2)}.$$ Assume T = 1sec.
10 M
5(c) Convert the analog filter into a digital filter whose system function is $$H(s)=\dfrac{2}{(s+1)(s+3)}$$ using bilinear transformation, with T = 0.1 sec.
5 M

6(a) Design a Digital Butterworth filter using the bilinear transformation for the following specifications : $$\begin{matrix} 0.8\leq |H(e^{jw})|\leq 1\ \text{for}\ 0\leq w\leq 0.2\pi\\ \ \ \ \ \ \ \ \ \ \ |H(e^{jw})|\leq 0.2\ \text{for}\ 0.6\pi\leq w\leq \pi \end{matrix}$$
12 M
6(b) Determine the order of a Chebyshev digital low pass filter to meet the following specifications. In the passband extending from 0 to 0.25π a ripple of not more than 2dB is allowed. In the stop band extending form 0.4π to π, attenuation can be more than 40dB. Use bilinear transformation method.
8 M

7(a) The frequency response of a filter is given by $$H(e^{jw})=jw;-\pi\leq w\leq \pi.$$ Design the FIR filter, using a rectangular window function. Take N = 7.
12 M
7(b) The desired frequency response of the low pass FIR filter is given by $H_d(e^{jw})=H_d(w)=\left\{\begin{matrix} e^{-j3w}; & |w|<\dfrac{3\pi}{4}\\ 0; & \dfrac{3\pi}{4}<|w|<\pi \end{matrix}\right.$
Determine the frequency response of the FIR filter if the hamming window is used with N = 7.
8 M

8(a) A FIR filter is given by y[n] = x[n] + 2/5 x(n-1 + 3/4 x(n-2) + 1/3 x(n-2). Draw the direct and linear form realization.
10 M
8(b) Obtain the direct form II and cascade realization of the following function. $H(z)=\dfrac{8z^3-4z^2+11z-2}{(z-0.25)(z^2-z+0.5)}$
10 M

More question papers from Digital Signal Processing