KING'S COLLEGE LONDON UNIVERSITY OF LONDON 7CCSMDSP Fundamentals of Digital Signal Processing Past Paper PDF 2022

KING’S COLLEGE LONDON UNIVERSITY OF LONDON This paper is part of an examination of the College counting toward the award of a degree. Examinations are governed by the College Regulations under the authority of the academic board. MSc EXAMINATION 7CCSMDSP FUNDAMENTALS OF DIG...

KING’S COLLEGE LONDON UNIVERSITY OF LONDON This paper is part of an examination of the College counting toward the award of a degree. Examinations are governed by the College Regulations under the authority of the academic board. MSc EXAMINATION 7CCSMDSP FUNDAMENTALS OF DIGITAL SIGNAL PROCESSING JANUARY 2022 TIME ALLOWED: TWO HOURS ANSWER ALL QUESTIONS. ANSWER EACH QUESTION IN A SEPARATE ANSWER BOOK AND WRITE ITS NUMBER ON THE COVER. A LIST OF FOURIER AND Z-TRANSFORM PAIRS IS PROVIDED ON PAGES 5 AND 6. TURN OVER WHEN INSTRUCTED 2022 c King’s College London  7CCSMDSP Questions One – Five : Consider the discrete-time system S such that for an input signal x[n], the corresponding output y[n] is given by y[n] = S {x[n]} = ejω0 n x[n] + α where ω0 and α are real finite constants, 0 < |α| < ∞, 0 < |ω0 | < π. 1) Is the system stable? (5 marks) 2) Is the system causal? (5 marks) 3) Is the system linear? (5 marks) 4) Is the system time invariant? (5 marks) 5) Is there a function S(ejω ) such that the Fourier transforms of y[n] and x[n] can be related as Y (ejω ) = S(ejω )X(ejω )? (5 marks) (Total: 25 marks) See Next Page 1  7CCSMDSP Questions Six – Eight: Consider a system for discrete-time processing of continuous-time signals as shown in the top figure. Assume that the spectrum of the input signal xc (t) is as shown in the bottom figure and that it is band limited to ΩN = 1000π rad/s. 6) Which condition should the sampling interval T1 of the continuous-to-discrete time con- verter satisfy to ensure that there is no aliasing in the system? (5 marks) 7) Sketch X(ejω ), the Fourier transform of x[n], when T1 = 2ms. (10 marks) 8) Assume that the input signal is sampled with T1 = 2ms. Find yc (t) if T2 = T1 and ( jω e−jω , |ω| < 21 π H(e ) = 1. 0, 2 π ≤ |ω| < π (10 marks) (Total: 25 marks) 𝑥! 𝑡 𝑥𝑛 LTI 𝑦𝑛 𝑦! 𝑡 C/D 𝐻 𝑒 "# D/C 𝑇! 𝑇" 𝑋! 𝑗Ω 1 −Ω$ Ω$ Ω Figure 1: Figure for Questions Six – Eight. See Next Page 2  7CCSMDSP Question Nine – Eleven: Consider the causal linear time-invariant system given by the follow- ing difference equation: 5 1 y[n] = x[n] − x[n − 1] + y[n − 1] − y[n − 2] 6 6 where x[n] is the input to the system and y[n] is the corresponding output. a) Does this system have a stable inverse system? (10 marks) b) Find a causal inverse of this system. (10 marks) c) Does this system have a unique inverse? (5 marks) (Total: 25 marks) See Next Page 3  7CCSMDSP Question Twelve – Fourteen: Consider the sequences x1 [n] and x2 [n] given by x1 [n] = 2δ[n − 2] + 3δ[n − 3] + δ[n − 4] , x2 [n] = 2δ[n] + δ[n − 1] + δ[n − 2], and let x̃[n], n ∈ ZZ, be the sequence obtained by applying the N -point inverse DFT to X̃[k] = X1 [k]X2 [k], k = 0, 1,... , N − 1, where X1 [k] and X2 [k] are the N -point DFTs of x1 [n] and x2 [n], respectively. Note that x̃[n], n ∈ ZZ, is obtained by applying the inverse DFT to X̃[k] for n along the whole time axis n ∈ ZZ. a) What is the minimal N such that x̃[n] contains samples of the linear convolution between x1 [n] and x2 [n] with no aliasing? (10 marks) b) Let x[n] be the linear convolution between x1 [n] and x2 [n], and let N be the minimal size of the DFT such that x̃[n] contains samples of x[n] with no aliasing. What is the range of n such that x̃[n] = x[n]? (10 marks) c) Let N be the minimal size of the DFT such that x̃[n] contains samples of the linear convolution between x1 [n] and x2 [n] with no aliasing. Find the numerical values of x̃[n] for n = 0, 1,... , N − 1. (5 marks) (Total: 25 marks) See Next Page 4  7CCSMDSP Fourier Transform Pairs δ[n] ↔ 1, δ[n − n0 ] ↔ e−jωn0 P∞ 1, − ∞ < n < ∞ ↔ r=−∞ 2πδ(ω + 2rπ) 1 an u[n], |a| < 1 ↔ 1 − ae−jω 1 P∞ u[n] ↔ 1−e−jω + r=−∞ πδ(ω + 2rπ) 1 (n + 1)an u[n], |a| < 1 ↔ (1 − ae−jω )2 ( sin ωc n 1, |ω| < ωc ↔ X(ejω ) = πn 0, ωc < |ω| ≤ π ( 1, 0 ≤ n ≤ N sin(ω(N + 1)/2) −jωN/2 x[n] = ↔ e 0, otherwise sin(ω/2) ∞ ejω0 n ↔ X 2πδ(ω − ω0 + 2πr) r=−∞ ∞   δ(ω − ω0 + 2rπ)ejφ + δ(ω + ω0 + 2rπ)e−jφ X cos(ω0 n + φ) ↔ π r=−∞ See Next Page 5  7CCSMDSP z-transform Pairs δ[n] ↔ 1, ROC is the whole complex plane. 1 an u[n] ↔ , ROC: |z| > |a|. 1 − az −1 1 −an u[−n − 1] ↔ , ROC: |z| < |a|. 1 − az −1 1 u[n] ↔ , ROC: |z| > 1. 1 − z −1 1 −u[−n − 1] ↔ , ROC: |z| < 1. 1 − z −1 az −1 nan u[n] ↔ , ROC: |z| > |a|. (1 − az −1 )2 az −1 −nan u[−n − 1] ↔ , ROC: |z| < |a|. (1 − az −1 )2 1 − cos ω0 z −1 (cos ω0 n)u[n] ↔ , ROC: |z| > 1. 1 − 2 cos ω0 z −1 + z −2 sin ω0 z −1 (sin ω0 n)u[n] ↔ , ROC: |z| > 1. 1 − 2 cos ω0 z −1 + z −2 1 − r cos ω0 z −1 (rn cos ω0 n)u[n] ↔ , ROC: |z| > r. 1 − 2r cos ω0 z −1 + r2 z −2 r sin ω0 z −1 (rn sin ω0 n)u[n] ↔ , ROC: |z| > r. 1 − 2r cos ω0 z −1 + r2 z −2 Final Page 6