(SEM V) THEORY EXAMINATION 2018-19 DIGITAL SIGNAL PROCESSING

DIGITAL SIGNAL PROCESSING (REC-503)

B.Tech (SEM-V) – AKTU
                                                                                                       Time: 3 Hours  Total Marks: 70

SECTION A

(Attempt all questions in brief – 2 × 7 = 14 marks)

Q1 (a) What is the main disadvantage of direct form realization?

The main disadvantage of direct form realization is that it requires a large number of delay elements, which increases memory requirements. It is also more sensitive to coefficient quantization errors, making it less suitable for practical implementations.

Q1 (b) What is the wrapping effect?

Wrapping effect occurs in circular convolution when the length of the sequence is insufficient, causing time-domain aliasing. It results in overlapping of signal samples, leading to incorrect output unless zero padding is used.

Q1 (c) Compare FIR and IIR filter.

FIR filters are always stable and have linear phase but require higher order.
IIR filters have feedback, require lower order, but may be unstable and usually have nonlinear phase.

Q1 (d) What are the advantages of Kaiser window?

Kaiser window provides adjustable main-lobe width and side-lobe attenuation using a single parameter β. It offers flexibility and better control over filter characteristics compared to fixed windows.

Q1 (e) What is window and why is it necessary?

A window is a finite-length function used to truncate an infinite impulse response. It is necessary to reduce Gibbs phenomenon and to design practical FIR filters.

Q1 (f) What is down-sampling and up-sampling?

Down-sampling reduces the sampling rate by discarding samples.
Up-sampling increases the sampling rate by inserting zeros between samples followed by filtering.

Q1 (g) Define decimation.

Decimation is the process of reducing the sampling rate of a signal by an integer factor after applying an anti-aliasing filter.

SECTION B

(Attempt any three – 7 × 3 = 21 marks)

Q2 (a) Obtain the cascade form realization.

Given difference equation:
y(n) = y(n−1) − 1/2 y(n−2) + 1/4 y(n−3) + x(n) − x(n−1) + x(n−2)

The system function is obtained using Z-transform. The denominator is factorized and expressed as a product of second-order and first-order sections. Each section is realized separately and connected in cascade form, which improves numerical stability.

Q2 (b) Find the order and cut-off frequency of a digital filter with the following specification:

|H(eʲω)| ≥ 0.89, 0 ≤ ω ≤ 0.4π

The filter order is calculated using standard Butterworth magnitude response formula. The cut-off frequency is determined from the passband edge condition by solving the magnitude equation.

Q2 (c) Explain finite word length effects in digital filters.

Finite word length effects arise due to limited precision in representing numbers. These include coefficient quantization, round-off noise, limit cycles, and overflow. They affect accuracy and stability of digital filters.

SECTION C

(Attempt any one part from each question – 7 × 2 = 14 marks)

Q4 (a) Using bilinear transformation, design a Butterworth filter which satisfies:

0.8 ≤ |H(eʲω)| ≤ 1, 0 ≤ ω ≤ 0.2π
|H(eʲω)| ≤ 0.2, 0.6π ≤ ω ≤ π

Using bilinear transformation, digital frequencies are pre-warped into analog frequencies. The order of the Butterworth filter is determined from passband and stopband specifications. The analog filter is designed and transformed into digital form using bilinear transformation.

Q4 (b) What is the difference between Butterworth and Chebyshev filter? Explain the frequency transformation done.

Butterworth filter has maximally flat passband with no ripple.
Chebyshev filter has ripple in passband or stopband but provides sharper transition.

Frequency transformation converts low-pass prototype filters into high-pass, band-pass, or band-stop filters using standard transformation equations.

Q5 (a) Using rectangular window technique, design a low-pass filter with unity gain, cut-off frequency 1000 Hz, sampling frequency 5 kHz and impulse response length 7.

The ideal impulse response is obtained using sinc function. It is truncated using a rectangular window of length 7. The resulting FIR filter coefficients are calculated by multiplying the ideal response with window values.

Q6 (a) Find the linear convolution using circular convolution of the following sequences:

x(n) = {1, 2, 1}
h(n) = {1, 2}

Zero padding is applied to make both sequences of equal length. Circular convolution is performed and the result matches the linear convolution output.

Q6 (b) Find the 8-point DFT of x(n) = {1, 2, 3, 4, 4, 3, 2, 1} using DIF-FFT.

Using Decimation-In-Frequency FFT, the sequence is divided into even and odd parts. Butterfly operations are performed stage-wise. Final frequency-domain samples are obtained efficiently with reduced computations.