RGPV Digital Signal Processing - December 2015 Exam Question Paper

RGPV Electronics and Communication 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

1(a) Test, whether the system y[n]=x[n]cos(ω₀n) is linear or nonlinear.

2 M

1(b) Determine the range of values of and b for which the L.T.I. system with impulse response<br>h[n]=

$\left\{\begin{matrix} a^{n},n\geq 0 & \\ b^{n},n< 0& \end{matrix}\right.$ is stable

2 M

1(c) The discrete time system y[n]=ny[n-1]+x[n],n≥0 is at rest[i.e. y(-1)=0\. Test whether the system is L.T.I.

3 M

Solve any one question from Q.1(d) & Q.1(e)

1(d) Determine the zero input response of the system described by the 2^ndorder difference equation.
x[n]=3y[n-1]-4y[n-2]=0

7 M

1(e) Determine the particular solution of the difference equation.

$y[n]=\dfrac{5}{6}y[n-1]-\dfrac{1}{6}y[n-2]+x[n]$ When the forcing function is x[n]=2ⁿ u[n]

7 M

2(a) Determine the Z-transform of the signal,
x[n]={3,0,0,0,0,6,1,-4}

2 M

2(b)

Obtain inverse Z-transform using residue method of the signal.
$X[Z]=\dfrac{1}{\left ( Z-1 \right )\left ( Z-3 \right )}$

2 M

2(c) Determine the response of the system characterized by impulse response h[n]=(0.5)ⁿ u[n] to the input signal x[n]=2n u[n]

3 M

Solve any one question from Q.2(d) & Q.2(e)

2(d) Determine the unit step response of the system characterized by the difference equation
y[n]=06y[n-1]-0.08y[n-2]+x[n]

7 M

2(e) Find the linear convolution of x₁[n]and x₂[n] using z-transform
x₁[n]={1,2,3,4}↑ and x₂[n]={1,2,0,2,1}↑

7 M

3(a) Compute the DFT of the following finite length sequence of length N, (N is even)
x[n]

$\ \left\{\begin{matrix} 1, & 0\leq n\leq \dfrac{N}{2}-1\\ 0, & \dfrac{N}{2}\leq n\leq N-1 \end{matrix}\right.$

2 M

3(b) State and prove the linearity property of DFT.

2 M

3(c) Let x[n] be a real and odd periodic signal with period N=7 and Fourier coefficients a_k. Given that a₁₅=j,a₁₆=2j,a₁₇=3j.
Determine the values of a₀ ,a_-1,a₂ and a_-3.

3 M

Solve any one question from Q.3(d) & Q.3(e)

3(d) Prove that multiplication of two DFT's is equivalent to the circular convolution of their sequences in time domain.

7 M

3(e) Using graphical method, obtain a 5-point circular convolution of two discrete time signals defined as:
x[n]=(1.5)ⁿ u[n] , 0≤n≤2
y[n]=(2n-3)u[n] , 0≤n≤3

7 M

4(a) State the computational requirements of FFT.

2 M

4(b) What is decimation in-time FFT algorithm?

2 M

4(c) Explain the Goertzel algorithm.

3 M

Solve any one question from Q.4(d) & Q.4(e)

4(d) Develop a radix-3 DIT FFT algorithm for evaluating the DFT for N=9

7 M

4(e) Obtain DFT of a sequence

$x[n]\left \{ \dfrac{1}{2},\dfrac{1}{2},\dfrac{1}{2},\dfrac{1}{2}0,0,0,0 \right \}$ using decimation in-frequency FFT algorithm.

7 M

5(a) An analog filter has the following system function. Convert this filter into a digital filter using backward difference for the derivative.

$\ H(s)=\dfrac{1}{\left ( s+0.1 \right )^{2}+9}$

2 M

5(b) Convert the analog filter into a digital filter whose system function is

$\ H(s)=\dfrac{s+0.2}{\left ( s+0.2 \right )^{2}+9}$
Use the impulse invariant technique. Assume T=1s.

2 M

5(c) Convert the analog filter with system function is

$\ H(s)=\dfrac{s+0.2}{\left ( s+0.2 \right )^{2}+9}$ into a digital IIR filter using bilinear transformation. The digital filter should have a resonant frequency of

$\omega_r=\dfrac{\pi }{4}$

3 M

Solve any one question from Q.5(d) & Q.5(e)

5(d) Describe the Butter worth filters.

7 M

5(e) Describe Chebyshev filters.

7 M

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