STUPIDSID
1 (a)
Define DFT. Derive the relationship of DFT to the z-transform.
5 M
1 (b)
An analog signal is sampled at 10 KHz and the DFT of 512 samples is computed. Determine the frequency spacing between the spectral samples of DFT.
3 M
1 (c)
Consider the finite length sequence x(n)=?(n)-2?(n-5): Find (i) The 10 point DFT of x(n) (ii) The sequence y(n) that has a DFT \[ y(K) = e ^{\frac {J4\pi}{10}K} \] X(K) where X(K) is the 10 point DFT of x(n) and W(K) is the 10 point DFT of u(n)-u(n-6).
12 M
2 (a)
Determine the circular convolution of the sequence x(n)={1,2,3,1} and h(n)={4,3,2,2} using DFT and IDFT equations.
8 M
2 (b)
Let X(K) be a 14 point DFT of a length 14 real sequence x(n). The first 8 samples of X(K) are given by: X(0)=12, X(1)=1+J3, X(2)=3+j4, x(3)=1-J5, X(4)=-2+J2, X(5)=6+J3, X(6)=2-J3, X(7)=10. Determine the remaining samples of X(K). Also evaluate the following functions without computing the IDFT. \[ i) \ x(0) \ \ \ ii) \ x(7) \ \ \ (iii) \ \sum^{13}_{n=0} x(n) \ \ \ (iv) \ \sum^{15}_{n=0}\big\vert x (n) \big\vert^2 \]
12 M
3 (a)
Consider a FIR filter with impulse response. h(n)={3,2,1,1}. If the input is x(n)={1,2,3,3,2,1,-1,-2,-3,5,6,1,2,1}. Using the overlap save method and 8 point circular convolution.
10 M
3 (b)
What are FFT algorithms? Prove the (i) Symmetry and (ii) Periodicity property of the twiddle factor Wn.
6 M
3 (c)
How many complex multiplications and additions are required for computing 256 point DFT using FFT algorithms?
4 M
4 (a)
Find the DFT of the sequence x(n)={1,2,3,4,4,3,2,1} using the decimation in frequency FFT algorithm and draw the signal flow graph. Show the outputs for each stage.
10 M
4 (b)
Given x(n)={1,0,1,0}, find x(2) using the Geortzel algorithm.
5 M
4 (c)
Write a note on Chirp z-transform algorithm.
5 M
5 (a)
Given that \[ \bigg \vert H(e^{7 \Omega})\bigg \vert ^{2} = \dfrac {1}{1+64 \Omega^6} \] determine the analog Butterworth low pass filter transfer function.
6 M
5 (b)
Design an analog Chebyshev filter with a maximum passband attenuation of 2.5 dB at Ωp=20 rad/sec and the stopband attenuation of 30 dB at Ωs=50 rad/sec.
10 M
5 (c)
Compare Butterworth and Chebyshev filters.
4 M
6 (a)
What are the conditions to be satisfied while transforming an analog filter to a digital HR filter? Explain how this is achieved in Bilinear transformation technique.
5 M
6 (b)
Design a Butterworth filter using the impulse invariance method for the following specifications: Take T=1 sec, \[\begin {align*}0.8 \le & \bigg \vert H(e^{jW}) \bigg \vert \le 1 & 0 \le W \le 0.2 \pi \\ & \bigg \vert H (jW) \bigg \vert \le 0.2 & 0.6 \pi \le W \le \pi \end{align*} \]
10 M
6 (c)
Determine H(z) for the given analog system function \[ H(s) = \dfrac {(s+a)}{(s+a)^2 +b^2} \] by using Matched z-transform.
5 M
7 (a)
A z-plane pole zero plot for a certain digital filter shown in Fig. Q7 (a). Determine the system function in the \[ H(z)= \dfrac {(1+a_1 z^{-1})(1+b_1 z^{-1} + b_2 z^{-2})}{(1+c_1 z^{-1})(1+d_1 z^{-1}+d_2 z^{-2})} \] giving the numerical values for parameters a1, b1, b2, c1, d1 and d2. Sketch the direct form II and Cascade realizations of the system.
10 M
7 (b)
A FIR filter is given by, \[ y(n)= x(n) + \dfrac {2}{5}x(n-1) + \dfrac {3}{4} x (n-2) - \dfrac {1}{3}x (n-3) \] Draw the direct form I and lattice structure.
10 M
8 (a)
Design a FIR filter (low pass) with a desired frequency response, \[ \begin {align*}
H_d (e^{jW}) &= e^{-j3w}; &-\dfrac {3 \pi}{4} \le \omega \le \dfrac {3 \pi}{4} \\ &=0; &\dfrac {3 \pi}{4} < \vert \omega \vert < \pi \end{align*} \] Use Hamming window with M=7. Also obtain the frequency response.
10 M
8 (b)
Design a linear phase low pass FIR filter with 7 taps and cut off frequency of Ωc=0.3Π rad. Using the frequency sampling method.
10 M
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