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%]