CE335-6-AU
Undergraduate Examinations 2020
DIGITAL SIGNAL PROCESSING
Question 1
(a) Consider an analogue signal x(t) = 2cos(2000πt) + sin(4000πt). [10%]
(i) Determine the sampling frequency that should be used to avoid aliasing. [3%]
(ii) Assume that the analogue signal x(t) is sampled to obtain a discrete-time signal x[n] = 2cos(2πn/5) + sin(4πn/5). Determine the sampling frequency used. [4%]
(iii) If the discrete signal x[n] is passed through an ideal low pass filter with cutoff frequency ωc = 3π/5, what is the resulting output signal? [3%]
(b) Consider a linear time invariant (LTI) system with an impulse response h[n] = δ[n − 2] − 3δ[n + 4] and that the system is triggered with an input signal x[n] = δ[n − 1] + 0.5δ[n + 5]. Calculate the output signal y[n]. [6%]
(c) In the Welch method for power spectral density (PSD) estimation, the segments’ overlapping length is M − D, where M is the length of each segment and D is the non-overlap length. If the signal length is N = 125, the number of segments is K = 3, and M = 75, what is the percentage of overlapping between the segments? What are the advantages and disadvantages of the Welch method? [10%]
(d) Determine whether the system y[n] = H[x[n]] = 2x[n] + u[n] is a linear time-invariant (LTI) system, where u[n] is the unit step function. Justify your response. [10%]
Question 2
(a) Consider a rectangular pulse signal x[n] = {1, 1, 1, 1}, n ∈ [0, 3]. [18%]
(i) Find the 4-point discrete Fourier transform. (DFT) of x[n]. [8%]
(ii) Pad the signal x[n] with 4 zeros and find the 8-point DFT of the padded signal. [8%]
(iii) Discuss the results. [2%]
(Hint: Use P N−1k=0 a
k =
1−aN1−a, a = 1 and Euler’s formulae cos(ω) = e
jω+
2
e−jω and sin(ω) = e
jω−e−jω2j)
(b) One of the fundamental properties of DFT is the spectrum symmetry according to which it is WN N−m = (WN m) ∗ , where WN = e −j2π/N . Prove that property. [6%]
(c) You are given an LTI system with an impulse response h[n] = (−0.6)nu[n] + 0.4
nu[n], where u[n] is the unit step function. Find the region of convergence of this system. [6%]
(Hint: Use P ∞k=0 a
k = 1 1−a
, |a| < 1.)
Question 3
You are given a digital filter with transfer function H(z) = 1+0.81z−2/(1+0.2z−1)(1−9z−2
).
(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%]
(Hint: az2 + bz + c = (z − z1)(z − z2), where z1 =
−b +
√
b2 − 4ac2a and z2 =
−b −
√
b2 − 4ac2a)
(c) Rewrite the transfer function so that it is a cascade of an all-pass filter with transfer 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. [8%]
(d) Draw the canonical implementation of the original filter with transfer function H(z). How many delay units are required? [8%]