(SEM V) THEORY EXAMINATION 2021-22 DIGITAL SIGNAL PROCESSING

DIGITAL SIGNAL PROCESSING (KEC-503)

B.Tech (Sem V) – Exam Notes & Solved Guide

SECTION A – Short Answer Type (2 Marks Each)

a. Recursive and Non-Recursive Systems

A recursive system depends on present and past outputs (feedback system). Example: IIR filters.
A non-recursive system depends only on present and past inputs (no feedback). Example: FIR filters.

b. Condition for Linear Phase FIR Filter (N = 5)

For a linear phase FIR filter with odd length (N = 5), the impulse response must be symmetric or anti-symmetric:
h(n) = h(N−1−n) or h(n) = −h(N−1−n).

c. Butterworth LPF vs Chebyshev LPF (Filter Order)

Butterworth filter has a smooth response but requires higher order.
Chebyshev filter allows ripples in passband or stopband but needs lower order for same specifications.

d. Chebyshev Polynomial C₃(x)

C₃(x) = 4x³ − 3x

e. Gibbs Phenomenon

Gibbs phenomenon refers to oscillations near discontinuities when a signal is approximated using a finite number of Fourier series terms. These oscillations do not disappear even if more terms are added.

f. Truncation Error & Round-off Error

Truncation error occurs when infinite values are cut to finite length.
Round-off error occurs due to finite word length in digital systems.
Example: rounding 3.14159 to 3.14.

g. DFT of Sequence [1, 2, 7, 3]

Using DFT definition:                                            X(k) = Σ x(n) e^(−j2πkn/N), N=4
(Students should show stepwise calculation in exam.)

h. Computations in 8-point DFT

Conventional DFT:                                               Complex multiplications = N² = 64

Complex additions = N(N−1) = 56

FFT (DIT):

Complex multiplications = (N/2)log₂N = 12       Complex additions = N log₂N = 24

i. Decimation

Decimation reduces the sampling rate by an integer factor.
Example: For decimation by 2, every alternate sample is removed.

j. Upsampling [1 2 3] by N = 3

Upsampled sequence:                                        [1 0 0 2 0 0 3 0 0]

SECTION B – Descriptive Answers

a. Ladder Structure Realization of H(z)

Given transfer function is expressed as a continued fraction. Each stage represents a ladder section consisting of delays, multipliers, and adders. Ladder structures are preferred due to low sensitivity and numerical stability.

b. Butterworth Digital Filter (Bilinear Transformation)

Steps include determining passband and stopband frequencies, finding analog Butterworth order, converting to digital using bilinear transformation, and obtaining H(z). Butterworth filters provide maximally flat response.

c. Limit Cycle Oscillations & Dead Band Effect

Limit cycle oscillations occur due to finite word length effects even with zero input.
Dead band effect occurs when small signals are lost due to quantization.

d. Circular Convolution (Graphical Method)

Circular convolution is performed by folding one sequence, shifting, multiplying, and summing. For x(n)=[1,2,3,4] and h(n)=[4,3,2,1], output length is 4.

e. QMF & Filter Banks

Quadrature Mirror Filters divide input signal into sub-bands. Analysis filter bank splits the signal, and synthesis filter bank reconstructs it. Aliasing cancellation ensures perfect reconstruction.

SECTION C – Long Answer Type

Linear Phase FIR System

Linear phase FIR filters preserve waveform shape. For symmetric impulse response, phase is linear.
For h(n) = [1/2, 1/3, 1/5, 1/3, 1/2], H(z) is obtained by Z-transform.

Direct Form-I & Direct Form-II Realization

Direct Form-I uses separate delay lines for input and output.
Direct Form-II reduces memory by combining delays.

Chebyshev Digital LPF Design

Chebyshev filters allow ripples to reduce order. Design steps include finding ripple factor, filter order, analog prototype, and transformation to digital.

Window Method FIR Design

Rectangular and Hamming windows are used to truncate ideal impulse response.
Hamming window reduces sidelobes and improves frequency response.

FFT Algorithms (DIT & DIF)

DIT splits sequence in time domain, DIF splits in frequency domain. Both reduce complexity from O(N²) to O(N log N).

DSP Processor Architecture

DSP processor consists of MAC unit, Harvard architecture, specialized addressing modes, and pipeline structure. Used in audio, image, and communication systems.

LMS Algorithm

Least Mean Square algorithm adapts filter coefficients to minimize error. It is simple, stable, and widely used in adaptive noise cancellation.