DIGITAL SIGNAL PROCESSING (REC-503)
B.Tech (SEM-V) – AKTU
SECTION A
(Attempt all questions – 2 × 7 = 14 marks)
Q1 (a) Define linear convolution and state its physical significance.
Linear convolution is a mathematical operation used to determine the output of a linear time-invariant (LTI) system for a given input signal. It is defined as the sum of products of input samples and time-reversed impulse response samples. Physically, it represents how an input signal is modified by the system characteristics.
Q1 (b) What are the advantages and disadvantages of window methods?
Window methods are simple and easy to implement for FIR filter design. They provide linear phase response and stable filters. However, disadvantages include limited control over transition bandwidth and ripple characteristics, and increased filter length for sharp frequency responses.
Q1 (c) What are the advantages of representing a digital system in block diagram form?
Block diagram representation provides a clear visual understanding of system structure. It simplifies system analysis, helps in implementation, and allows easy modification and realization of digital systems.
Q1 (d) Write expressions for Blackman and Bartlett windows.
Bartlett window is defined as
w(n) = 1 − |(n − (N−1)/2)| / ((N−1)/2)
Blackman window is defined as
w(n) = 0.42 − 0.5 cos(2πn/(N−1)) + 0.08 cos(4πn/(N−1))
Q1 (e) If x(n) = {4,3,5,7,4,6} and up-sampling factor = 3, find the up-sampled output.
Up-sampling by factor 3 inserts two zeros between each sample.
The up-sampled sequence is
{4,0,0,3,0,0,5,0,0,7,0,0,4,0,0,6,0,0}
Q1 (f) If x(n) = {1,5,2,3}, find x((3 − n))₄.
The given operation represents circular time reversal and shifting.
The resulting sequence after circular operation is
{3,2,5,1}
Q1 (g) Write advantages and disadvantages of bilinear transformation.
Advantages include no aliasing and stability preservation.
Disadvantages include frequency warping and nonlinear frequency mapping.
SECTION B
(Attempt any three – 7 × 3 = 21 marks)
Q2 (a) Obtain the parallel form realization of the given system function.
Given:
H(z) = (1 + 0.25z⁻¹) / ((1 + 0.5z⁻¹)(1 + 0.5z⁻¹ + 0.25z⁻²))
The system function is decomposed into partial fractions. Each term is realized separately in parallel form. The output is obtained by summing outputs of individual subsystems, which simplifies implementation and improves numerical stability.
Q3 (a) Design a Butterworth low-pass analog filter for given specifications.
The Butterworth filter is designed by determining filter order using passband and stopband specifications. The cutoff frequency is calculated and transfer function is obtained using standard Butterworth polynomial. This filter provides maximally flat frequency response in passband.
Q3 (b) What is frequency warping effect? How is it overcome in bilinear transformation?
Frequency warping is the nonlinear mapping of frequency axis caused by bilinear transformation. It is overcome using pre-warping technique, where analog frequencies are adjusted before transformation to achieve correct digital cutoff frequencies.
SECTION C
(Attempt any two – one from each unit, 7 × 2 = 14 marks)
Q5 (a) A FIR filter has symmetry h(n) = h(M−1−n) for even M. Derive frequency response and show linear phase.
Due to symmetry in impulse response, the frequency response can be expressed as a cosine series. This results in a linear phase term, ensuring constant group delay. Hence, symmetric FIR filters exhibit linear phase property.
Q6 (a) Find the 8-point DFT of x(n) = 2ⁿ using DIT-FFT algorithm.
Using Decimation-In-Time FFT, the sequence is divided into even and odd indexed samples. Butterfly computations are performed stage-wise to obtain final 8-point DFT values efficiently with reduced computations.
Q7 (b) Explain decimation and interpolation with suitable example.
Decimation is the process of reducing sampling rate by factor M by discarding samples, followed by low-pass filtering to avoid aliasing.
Interpolation increases sampling rate by inserting zeros between samples and applying low-pass filtering. Both processes are used in multirate signal processing.
