(SEM V) THEORY EXAMINATION 2024-25 DIGITAL SIGNAL PROCESSING

Subject Name: Digital Signal Processing

Subject Code: BEE057

Exam Type: Theory

Semester: V (Fifth Semester, B.Tech)

Maximum Marks: 70

Time: 3 Hours

Exam Year: 2024–25

SECTION A – Short Questions (2 × 7 = 14 Marks)

Each question carries 2 marks. All are compulsory.

Draw the sequence x(n)=u(n)−u(n−4)x(n) = u(n) - u(n-4)x(n)=u(n)−u(n−4).

Calculate the DTFT for x(n)=anu(n)x(n) = a^n u(n)x(n)=anu(n).

Enlist the advantages of multirate signal processing.

State the Nyquist theorem.

State the circular convolution property of DFT.

Why is the Kaiser window superior to other window functions?

What is the need for the FFT algorithm, and why is it called so?
 

SECTION B – Medium-Length Questions (7 × 3 = 21 Marks)

Attempt any three of the following:

Draw the cascade and parallel realization structures of
H(z)=(1−z−1)3(1−12z−1)(1−18z−1)H(z) = \frac{(1 - z^{-1})^3}{(1 - \frac{1}{2}z^{-1})(1 - \frac{1}{8}z^{-1})}H(z)=(1−21​z−1)(1−81​z−1)(1−z−1)3​.

Explain interpolation and decimation processes with an example.

Compute circular convolution of
x(n)={1,1,2,2}x(n) = \{1, 1, 2, 2\}x(n)={1,1,2,2} and y(n)={1,2,3,4}y(n) = \{1, 2, 3, 4\}y(n)={1,2,3,4} using the graphical method.

Design a high-pass filter using the Hamming window
(cut-off frequency = 1.2 rad/sec, order = 9).

Given x(n)={1,2,3,4,4,3,2,1}x(n) = \{1, 2, 3, 4, 4, 3, 2, 1\}x(n)={1,2,3,4,4,3,2,1}, find X(k)X(k)X(k) using DIT FFT algorithm.
 

SECTION C – Long/Analytical Questions (7 × 5 = 35 Marks)

Attempt one part from each question.

Q3

a. Realize H(z)=1+8z−1+6z−21+8z−1+12z−2H(z) = \frac{1 + 8z^{-1} + 6z^{-2}}{1 + 8z^{-1} + 12z^{-2}}H(z)=1+8z−1+12z−21+8z−1+6z−2​ using ladder structure.
b. Obtain the FIR linear phase and cascade realization of
H(z)=(1+12z−1+z−2)(1+14z−1+z−2)H(z) = (1 + \frac{1}{2}z^{-1} + z^{-2})(1 + \frac{1}{4}z^{-1} + z^{-2})H(z)=(1+21​z−1+z−2)(1+41​z−1+z−2).

Q4

a. Define sampling, and state and prove the sampling theorem.
b. Explain the process of reconstruction of a signal from its samples.

Q5

a. Explain properties of twiddle factors.
Find the 4-point DFT of x(n)=cos⁡(nπ4)x(n) = \cos(\frac{n\pi}{4})x(n)=cos(4nπ​) using matrix method.
b. Find output y(n)y(n)y(n) of a filter with
h(n)={2,2,1}h(n) = \{2, 2, 1\}h(n)={2,2,1} and x(n)={3,0,−2,0,2,1,0,−2,−1,0}x(n) = \{3, 0, -2, 0, 2, 1, 0, -2, -1, 0\}x(n)={3,0,−2,0,2,1,0,−2,−1,0}
using overlap-add method.

Q6

a. A digital filter with 3 dB bandwidth = 0.25π is to be designed from the analog filter
H(s)=Ωcs+ΩcH(s) = \frac{\Omega_c}{s + \Omega_c}H(s)=s+Ωc​Ωc​​.
Use bilinear transformation and obtain H(z)H(z)H(z).

b. For H(s)=1(s+1)(s+2)H(s) = \frac{1}{(s + 1)(s + 2)}H(s)=(s+1)(s+2)1​, determine H(z)H(z)H(z) using impulse invariant technique.

Q7

a. Given x(n)=2nx(n) = 2^nx(n)=2n and N=8N = 8N=8, find DFT of x(n) using the DIF FFT algorithm.
b. Explain the applications of the Wavelet Cosine Transform.
 

 Important Topics to Focus On

DFT properties and circular convolution

FFT algorithms (DIT & DIF)

FIR and IIR filter design (Hamming, Kaiser window methods)

Sampling theorem and signal reconstruction

Bilinear & Impulse Invariant Transformations

Twiddle factor and frequency domain analysis

Applications of Wavelet and Multirate DSP

Preparation Tips

Practice derivations of DTFT, DFT, and FFT.

Revise filter design steps (windowing and frequency sampling).

Understand difference between FIR & IIR filters.

Be clear with signal transformations — time shift, fold, and scaling.

Solve numerical problems on DFT, convolution, and z-domain realization.