7 M

Answer any two question from Q2 (b) or Q2 (c)

2 (b) Compute the convolution of the following signal. \[ x[n]= \begin{matrix} \propto^n&-3\le n \le 5 \\0 & otherwise \end{matrix} \\ h[n]= \begin{matrix} 1 &0\le n\le 4 \\0 &otherwise \end{matrix} \]

7 M

2 (c) Define periodic and unpatriotic signal. Determine whether the given sequence is periodic or not, if periodic determine fundamental period. \[ x(n)=\sin \left ( \dfrac {6\pi n}{7} \right ) \]

7 M

Answer any two question from Q3 (a), (b) or Q3 (c), (d)

3 (a) Determine Z-transform and ROC of x[n]=2ⁿu(n)-3ⁿu(n)

7 M

3 (b) Determine the IDFT of x(k)={3,2+j,1,2-j}

7 M

3 (c) Find inverse Z-transform of \[ X(z) = \dfrac {z+3z^{-1}}{(1+z^{-1})\left (1+\frac {1}{2}z^{-1} \right ) \left ( 1- \frac {1}{4} z^{-1}\right )} \] also verify result for 0≤3n≤3.

7 M

3 (d) Given x(n)=aⁿu(n). Determine Fourier transform X(w) of x(n). Determine discrete Fourier transform X(k), How X(k) relate to X(w).

7 M

Answer any two question from Q4 (a), (b) or Q4 (c), (d)

4 (a) Find Convolution using DFT and IDFT method of sequence
x1(n)={1,1,2,2} x2(n)={1,2,3,4}.

7 M

4 (b) Use butterfly structure of DIF and Calculate output for each stage for given sequence. x[n] ={1,2,2,1,0,0,0,0}.

7 M

4 (c) Draw the butterfly diagram of 8 point Inverse decimation in time FFT and use it to obtain the original sequence x(n).
X(K)={20, -5.828 - j2.414, 0, -0.172 - j0.414,0, -0.172+j0.414,0, -5.828+j2.414}

7 M

4 (d) Explain any three properties of DFT. What is twiddle factor of the DFT?

7 M

Answer any two question from Q5 (a), (b) or Q5 (c), (d)

5 (a) Design a digital Chebyshev filter to satisfy the constraints. Using bilinear transformation and assuming T=1s. \[ 0.707 \le \begin{matrix} |H(e^{j\omega}) |\le1 \ \ & 0 \le \omega \le 0.2 \pi \\ |H(e^{j\omega}) |\le0.1 &0.5\pi \le \omega \le \pi \end{matrix} \]

7 M

5 (b) The desired response of a low pass filter is given determine Hd(e^jω) for M=7 using hamming window. \[ Hd (e^{j\omega}) = \begin{matrix} e^{-3j\omega} & -\frac {3\pi }{4} \le \omega \le \frac {3 \pi}{4} \\ 0& \frac {3\pi}{4} \le \omega \le \pi \end{matrix} \]

7 M

5 (c) Design digital filter using bilinear transformation for following analog transfer function. \[ H(s)=1/(s^2+\sqrt{2} s+1) \] Obtain transfer function H(z) of digital filter assuming 3db cutoff frequency w_p 150 Hz and sampling frequency 1.28 Khz.

7 M

5 (d) Determine coefficient of linear phase FIR filter of length N = 15 which has symmetrical unit sample response and a freq. response that satisfies condition. \[ H\left [\dfrac {2\pi k}{15} \right ]= \begin{matrix} 1 &k=0,1,2,3 \\0.4 &k=4 \\0 & k=5,6,7 \end{matrix} \]

7 M