JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD
JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD B.Tech III DIGITAL SIGNAL PROCESSING Previous Question Paper (March - 2021 - R15)
Answer any five questions. All questions carry equal marks.
Main
Determine the Z-transform of the following signals and find ROC:
i) x(n)=(-1/6)n u(n)+5(1/3)-n u(-n-2)
ii) x(n)=0.5(n2)(1/3)n-1 u(n-1)
Answer
i) Given x(n) = (-1/6)n u(n) + 5(1/3)-n u(-n-2). The Z-transform of a(n)u(n) is 1/(1-az-1) with ROC |z| > |a|, and the Z-transform of -a(n)u(-n-1) is 1/(1-az-1) with ROC |z| < |a|.
For the first term, X1(z) = 1 / (1 + (1/6)z-1) with ROC |z| > 1/6.
For the second term, let y(n) = 5(3)n u(-n-2). Using the property Z{an u(-n-1)} = -1/(1-az-1), we shift the sequence. The Z-transform is X2(z) = 5 * [z-1 / (1 - 3z-1)] = 5 / (z - 3) with ROC |z| < 3.
Combining these, X(z) = 1 / (1 + (1/6)z-1) + 5 / (z - 3) with ROC 1/6 < |z| < 3.
ii) Given x(n) = 0.5 n2 (1/3)n-1 u(n-1). Let y(n) = n2 an u(n). The Z-transform is Y(z) = [az-1(1+az-1)] / (1-az-1)3. Adjusting for the shift and constant, X(z) = 0.5 * 3 * Z{n2(1/3)n u(n)} * z-1. Substituting a = 1/3, X(z) = 1.5 * z-1 * [(1/3)z-1(1+(1/3)z-1)] / (1-(1/3)z-1)3. The ROC is |z| > 1/3.
Determine the stability for each of the following system:
i) x(n)=3n u(-n)
ii) x(n)=x(-n)+x(n-2)+x(2n-1)
Answer
i) For x(n) = 3n u(-n), the system is stable if the sum of the absolute values of the impulse response is finite (Σ |h(n)| < ∞). Here, Σn=-∞ to 0 |3n| = Σk=0 to ∞ (1/3)k. This is a geometric series with ratio 1/3 < 1, which converges to 1 / (1 - 1/3) = 1.5. Since the sum is finite, the system is stable.
ii) The expression x(n) = x(-n) + x(n-2) + x(2n-1) represents a difference equation. For a system to be stable, it must be BIBO stable. The term x(2n-1) indicates a time-varying or non-causal expansion/scaling operation. Specifically, the mapping of n to 2n-1 implies that the system does not satisfy the condition that every bounded input produces a bounded output for all time indices, as the scaling factor leads to sensitivity to high-frequency components or infinite growth in certain configurations. Thus, the system is unstable.
Determine the response of the system described by the differential equation: y(n)+y(n-1)-2y(n-2) = u(n-1)+2u(n-2) due to y(-1)=0.5, y(-2)=0.25 using Z-transform.
Answer
Sign in to view the complete answer paper.
Find the Direct form-II, cascade and parallel form realizations of a discrete-time system represented by the:
Answer
Sign in to view the complete answer paper.
Explain in detail about radix-2 DIF-FFT algorithm.
Answer
Sign in to view the complete answer paper.
Determine the linear convolution of following two sequences using DFT, the input x(n) and h(n) of a system are given by, x(n) = {-1,1,2,-2} and h(n)={0.5,1,-1,2}.
Answer
Sign in to view the complete answer paper.
Define DFT? And also explain any five properties of DFT.
Answer
Sign in to view the complete answer paper.
Using the properties of Fourier transforms, find the discrete Fourier transform of the following signals:
i) x(n)=sin(πn).e-2n.u(n)
ii) x(n)= e-3|n-2|
iii) x(n)= d/dt [n. e-2n.u(n)]
Answer
Sign in to view the complete answer paper.
Design a Chebyshev digital filter using the impulse invariant method. The specifications of the LPF is:
Answer
Sign in to view the complete answer paper.
Design a Butterworth digital filter using bilinear transformation. The specifications of the LPF is:
Answer
Sign in to view the complete answer paper.
Design an ideal high pass digital FIR filter with a frequency response H(ejω)= 1, π/4 ≤ ω ≤ π and 0, 0 ≤ ω ≤ π/4 by using Hanning window of length N=11.
Answer
Sign in to view the complete answer paper.
Develop an expression for the output y(n) as a function of input of the input x(n).
Answer
Sign in to view the complete answer paper.
Explain about round off errors in FFT algorithms.
Answer
Sign in to view the complete answer paper.