CE335-6-AU/AT

Undergraduate Examinations 2021

DIGITAL SIGNAL PROCESSING

Question 1

(a)       A digital communication link carries binary-coded words representing samples of an input signal

xa(t) = 3 cos 600πt + 2 cos 1800πt

The link is operated at 10,000 bits/s and each input sample is quantized into 1024 different voltage levels. (Hint: Number of bits/sample =  log21024 = 10)

i.       What is the sampling frequency, Nyquist rate and folding frequency for the signal xa(t)? [6%]

ii.       What are the frequencies in the resulting discrete-time signal x(n)?                                [7%]

iii.       What is the resolution?                                                                                                       [7%]

(b)       Consider a system with input x(t) and output y(t), with input-output relation y(t) = x4(t) for −∞Indicate and justify whether the system satisfies the following system properties: linearity, time invariance, causality. [8%]

Question 2

Consider the following discrete-time signal: x = [2, -1, 3] .

(a)       Find the third root of unity, W3.                                                                                             [2%]

(b)       Find the 3 × 3 DFT transformation matrix W.                                                                         [3%]

(c)       Use W to find the DFT ofx.                                                                                                   [3%]

(d)       Find the discrete-time signal y whose DFT is given by Y=[ 3, -j,j ] .                                    [3%]

(e)       Use x and Y given in this question to verify the circular convolution property of DTF by showing the equality between the time and frequency domain computation for x oy X.Y [7%]

Question 3

Consider a discrete-time filter with finite impulse response for which the input is denoted as x(n) and the output as y(n) that they are related through y(n)= -0.5x(n) - 0.45x(n-2) where n is the discrete-time index.

(a)       Find the transfer function of the filter. [3%]

(b)       Write an expression for the magnitude of the frequency response of this filter. [9%]

(c)       Draw the magnitude of the frequency response of this filter and mark the values of the magnitude in dB at frequencies of 0, 2/π, and π .         [8%]

Question 4

You are given a digital filter with transfer function

(a)       Find the relationship between the input x[n] and the output y[n] of the filter.                           [6%]

(b)       Find the poles and the zeros of the filter and sketch the zeros-poles diagram. Comment on the stability of the filter. [12%]

(c)       Rewrite the transfer function so that it is a cascade of an all-pass filter with transfer                [8%]

function Hap(z) and another filter with transfer function H1(z)., i.e. H(z) = Hap(z)H1(z). Sketch the zeros-poles diagram of the all-pass filter.

(d)       Draw the canonical implementation of the original filter with transfer function H(z). 

How many delay units are required?               [8%]