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

Subject Code: BEC503
Maximum Marks: 70
Time: 3 Hours
Paper ID: 310692

Question Paper Overview

SECTION A (2 × 7 = 14 Marks)

(Short and conceptual questions covering key DSP fundamentals)

a. What are recursive and non-recursive systems?
b. Explain the difference between canonic and non-canonic structures.
c. List the characteristics of Butterworth filters.
d. What is an Infinite Impulse Response (IIR) filter?
e. Explain the purpose of windowing in FIR filter design.
f. Explain the difference between linear and circular convolution.
g. Define decimation in Multirate DSP (MDSP).

SECTION B (Attempt any three × 7 = 21 Marks)

a. Explain the basic building blocks of a Digital Signal Processing system.
b. Discuss the Impulse Invariant Method of IIR filter design and its properties.
c. Design an FIR filter to meet the following specifications using Hanning Window:

Passband edge = 2 kHz

Stopband edge = 5 kHz

Stopband attenuation = 42 dB

Sampling frequency = 20 kHz
d. Compute the circular convolution of

x1(n)={1,2,1,2},x2(n)={3,2,1,4}x_1(n) = \{1, 2, 1, 2\}, \quad x_2(n) = \{3, 2, 1, 4\}x1​(n)={1,2,1,2},x2​(n)={3,2,1,4}

e. Explain the principle of Quadrature Mirror Filters (QMF) and their use in MDSP.
Discuss the advantages of sub-band coding.

SECTION C (Attempt one part from each question × 7 = 35 Marks)

Q3

(a) Obtain the parallel form realization for the system function

H(z)=2+z−1+4z−2(1+12z−1)(1+14z−1)H(z) = \frac{2 + z^{-1} + 4z^{-2}}{(1 + \frac{1}{2}z^{-1})(1 + \frac{1}{4}z^{-1})}H(z)=(1+21​z−1)(1+41​z−1)2+z−1+4z−2​

OR
(b) Explain the direct form realization of FIR systems.

Q4

(a) What is the frequency warping effect, and how is it overcome using the Bilinear Transform Method in IIR filter design?
OR
(b) Design a Butterworth Low-Pass Analog Filter with the following specifications:

Passband gain = 0.9

Passband frequency = 100 rad/sec

Stopband gain = 0.4

Stopband frequency = 200 rad/sec

Q5

(a) What is Gibbs phenomenon? Describe the concept of windowing in FIR filter design.
OR
(b) Define coefficient quantization error and quantization noise in digital filters.
Explain the effects of truncation and rounding.

Q6

(a) Write short notes on:
i. Butterfly computation
ii. Bit reversal
OR
(b) List the key properties of the DFT and explain each briefly.

Q7

(a) Explain the significance of Multirate DSP in modern communication systems.
OR
(b) Describe the difference between decimation and interpolation in MDSP.

Key Topics for Revision

1. Recursive vs Non-Recursive Systems

Recursive (IIR): Output depends on both present and past input/output samples.
Example: y(n)=0.8y(n−1)+x(n)y(n) = 0.8y(n-1) + x(n)y(n)=0.8y(n−1)+x(n)

Non-Recursive (FIR): Depends only on present/past input samples.
Example: y(n)=x(n)+0.5x(n−1)y(n) = x(n) + 0.5x(n-1)y(n)=x(n)+0.5x(n−1)

2. Canonic vs Non-Canonic Structures

3. Butterworth Filter Characteristics

Maximally flat magnitude response in passband.

Monotonic in both passband and stopband.

Smooth transition with no ripples.

Transfer function:

• ∣H(jω)∣2=11+(ωωc)2N|H(j\omega)|^2 = \frac{1}{1 + (\frac{\omega}{\omega_c})^{2N}}∣H(jω)∣2=1+(ωc​ω​)2N1​

4. IIR Filters

Infinite impulse response due to feedback.

Designed using analog prototypes (Butterworth, Chebyshev).

Realized via Direct, Cascade, or Parallel forms.

5. Windowing in FIR Design

Used to control truncation effects in the ideal impulse response.

Common windows: Rectangular, Hamming, Hanning, Blackman.

Reduces Gibbs phenomenon (oscillations near discontinuities).

6. Linear vs Circular Convolution

7. Decimation

Reduces the sampling rate by an integer factor MMM.      y(n)=x(Mn)y(n) = x(Mn)y(n)=x(Mn).

Used in multirate systems and sub-band coding.

8. Impulse Invariant Method

Converts analog filter → digital filter.                                 Preserves impulse response shape.

Mapping: s=1Tln⁡(z)s = \frac{1}{T}\ln(z)s=T1​ln(z)

9. Quadrature Mirror Filters (QMF)

Decompose signal into sub-bands (low & high frequency).

Used in sub-band coding and wavelet filter banks.       Advantage: Efficient compression & reduced aliasing.

10. Frequency Warping & Bilinear Transform

Warping: Non-linear frequency mapping during analog-to-digital conversion.

Solution: Bilinear transform substitutes

• s=2T1−z−11+z−1s = \frac{2}{T} \frac{1 - z^{-1}}{1 + z^{-1}}s=T2​1+z−11−z−1​

→ removes aliasing and ensures frequency mapping consistency.

11. Gibbs Phenomenon

Occurs due to truncation of infinite impulse response.      Causes oscillations near discontinuities.

Mitigated by applying window functions.

12. Quantization & Truncation

Coefficient Quantization: Finite precision introduces error.

Truncation/Rounding: Affects filter accuracy and introduces quantization noise.

13. Butterfly Computation & Bit Reversal

Butterfly: Basic unit in FFT computation; combines DFT outputs efficiently.

Bit Reversal: Reordering input indices for in-place FFT computation.

14. DFT Properties

Linearity                                      Time shifting

Frequency shifting                     Convolution property

Symmetry                                   Parseval’s theorem

15. Multirate DSP

Processes signals at multiple sampling rates.

Applications: Filter banks, speech coding, adaptive filtering, and communication systems.

Decimation: Reduces rate; Interpolation: Increases rate.

Exam Preparation Tips

Memorize key formulas (DFT, convolution, filter design).

Practice FIR filter design using different windows.

Revise Butterworth filter derivation and FFT computation steps.

Draw realization structures (Direct, Cascade, Parallel).

Understand quantization effects and multirate concepts with examples.