1(a)
Compute the N-point DFT of x[n] = a

^{n}for 0≤n≤N-1. Also find the DFT of the sequence x[n] = 0.5^{n}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).

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 x

_{1}[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.

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

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