Discrete-Time Signals and Systems. Introduction to Discrete-Time Systems. What is a Signal? What is a System? Analog and Digital Signals.

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1 Discrete-Time Signls nd Systems Introduction to Discrete-Time Systems Dr. Deep Kundur University of Toronto Reference: Sections. -.4 of John G. Prokis nd Dimitris G. Mnolkis, Signl Processing: Principles, Algorithms, nd Applictions, 4th edition, 007. Dr. Deep Kundur (University of Toronto) Introduction to Discrete-Time Systems / 34 Dr. Deep Kundur (University of Toronto) Introduction to Discrete-Time Systems / 34. Signls, Systems nd Signl Processing Wht is Signl? Wht is System? nd Signls. Clssifiction of Signls Signl: ny physicl untity tht vries with time, spce, or ny other independent vrible or vribles Exmples: pressure s function of ltitude, sound s function of time, color s function of spce,... x(t) = cos(πt), x(t) = 4 t + t 3, x(m, n) = (m + n) System: physicl device tht performs n opertion on Exmples: nlog mplifier, noise cnceler, communiction chnnel, trnsistor,... y(t) = 4x(t), dy(t) dt + 3y(t) = dx(t) dt + 6x(t), y(n) y(n ) = 3 + x(n ) nlog = continuous-time + continuous mplitude digitl = discrete-time + discrete mplitude continuous-time discrete-time continuous mplitude discrete mplitude x(t) x(t) t x[n] x[n] n 0 n t Dr. Deep Kundur (University of Toronto) Introduction to Discrete-Time Systems 3 / 34 Dr. Deep Kundur (University of Toronto) Introduction to Discrete-Time Systems 4 / 34

2 nd Signls. Clssifiction of Signls nd Systems. Clssifiction of Signls s re fundmentlly significnt becuse we must interfce with the rel world which is nlog by nture. s re importnt becuse they fcilitte the use of digitl processing (DSP) systems, which hve prcticl nd performnce dvntges for severl pplictions. nlog system = nlog input + nlog output dvntges: esy to interfce to rel world, do not need A/D or D/A converters, speed not dependent on clock rte digitl system = digitl input + digitl output dvntges: re-configurbility using softwre, greter control over ccurcy/resolution, predictble nd reproducible behvior Dr. Deep Kundur (University of Toronto) Introduction to Discrete-Time Systems 5 / 34 Dr. Deep Kundur (University of Toronto) Introduction to Discrete-Time Systems 6 / 34. Clssifiction of Signls Deterministic vs. Rndom Signls Wht is pure freuency? Deterministic : ny tht cn be uniuely described by n explicit mthemticl expression, tble of dt, or well-defined rule pst, present nd future vlues of the re known precisely without ny uncertinty Rndom : ny tht lcks uniue nd explicit mthemticl expression nd thus evolves in time in n unpredictble mnner it my not be possible to ccurtely describe the the deterministic model of the my be too complicted to be of use. x (t) = A cos(ωt + θ) = A cos(πft + θ), t R nlog, A x (t) A nd < t < A = mplitude Ω = freuency in rd/s F = freuency in Hz (or cycles/s); note: Ω = πf θ = phse in rd Dr. Deep Kundur (University of Toronto) Introduction to Discrete-Time Systems 7 / 34 Dr. Deep Kundur (University of Toronto) Introduction to Discrete-Time Systems 8 / 34

3 Continuous-time Sinusoids Sinusoids x (t) = A cos(ωt + θ) = A cos(πft + θ),. For F R, x (t) is periodic t R i.e., there exists Tp R + such tht x (t) = x (t + T p ). distinct freuencies result in distinct sinusoids i.e., for F F, A cos(πf t + θ) A cos(πf t + θ) 3. incresing freuency results in n increse in the rte of oscilltion of the sinusoid i.e., for F < F, A cos(πf t + θ) hs lower rte of oscilltion thn A cos(πf t + θ) = A cos(ωn + θ) = A cos(πfn + θ), n Z discrete-time (not digitl), A x (t) A nd n Z A = mplitude ω = freuency in rd/smple f = freuency in cycles/smple; note: ω = πf θ = phse in rd Dr. Deep Kundur (University of Toronto) Introduction to Discrete-Time Systems 9 / 34 Dr. Deep Kundur (University of Toronto) Introduction to Discrete-Time Systems 0 / 34 Sinusoids Sinusoids = A cos(ωn + θ) = A cos(πfn + θ), n Z. is periodic only if its freuency f is rtionl number Note: rtionl number is of the form k k for k, k Z periodic discrete-time sinusoids: = cos( 4 7 πn), = sin( π 5 n + 3) periodic discrete-time sinusoids: = cos( 4 7 n), = sin( πn + 3) = A cos(ωn + θ) = A cos(πfn + θ), n Z. rdin freuencies seprted by n integer multiple of π re identicl or cyclic freuencies seprted by n integer multiple re identicl 3. lowest rte of oscilltion is chieved for ω = kπ nd highest rte of oscilltion is chieved for ω = (k + )π, for k Z subseuently, this corresponds to lowest rte for f = k (integer) nd highest rte for f = k+ (hlf integer), for k Z; see Figure.3.4 of text. Dr. Deep Kundur (University of Toronto) Introduction to Discrete-Time Systems / 34 Dr. Deep Kundur (University of Toronto) Introduction to Discrete-Time Systems / 34

4 Complex Exponentils Complex Exponentils e jφ = cos(φ) + j sin(φ) Euler s reltion cos(φ) = ejφ +e jφ sin(φ) = ejφ e jφ j Continuous-time: : A e j(ωt+θ) = A e j(πft+θ) A e j(ωn+θ) = A e j(πfn+θ) where j Dr. Deep Kundur (University of Toronto) Introduction to Discrete-Time Systems 3 / 34 Dr. Deep Kundur (University of Toronto) Introduction to Discrete-Time Systems 4 / 34 Periodicity: Continuous-time Periodicity: x(t) = x(t + T ), T R + A e j(πft+θ) j(πf (t+t )+θ) = A e e jπft e jθ = e jπft e jπft e jθ = e jπft e jπk = = e jπft, k Z T = k F k Z T 0 = F, k = sgn(f ) = x(n + N), N Z + A e j(πfn+θ) j(πf (n+n)+θ) = A e e jπfn e jθ = e jπfn e jπfn e jθ = e jπfn e jπk = = e jπfn, k Z f = k N k Z N 0 = k f, min k Z such tht k f Z + Dr. Deep Kundur (University of Toronto) Introduction to Discrete-Time Systems 5 / 34 Dr. Deep Kundur (University of Toronto) Introduction to Discrete-Time Systems 6 / 34

5 Uniueness: Continuous-time Uniueness: Let f = f 0 + k where k Z, For F F, A cos(πf t + θ) A cos(πf t + θ) except t discrete points in time. x (n) = A e j(πf n+θ) = A e j(π(f 0+k)n+θ) = A e j(πf 0n+θ) e j(πkn) = x 0 (n) = x 0 (n) Dr. Deep Kundur (University of Toronto) Introduction to Discrete-Time Systems 7 / 34 Dr. Deep Kundur (University of Toronto) Introduction to Discrete-Time Systems 8 / 34 Uniueness: Hrmoniclly Relted Complex Exponentils Therefore, dst-time sinusoids re uniue for f [0, ). For ny sinusoid with f [0, ), f 0 [0, ) such tht x (n) = A e j(πf n+θ) = A e j(πf 0n+θ) = x 0 (n). Exmple: A dst-time sinusoid with freuency f = 4.56 is the sme s dst-time sinusoid with freuency f 0 = = Exmple: A dst-time sinusoid with freuency f = 7 is the 8 sme s dst-time sinusoid with freuency f 0 = 7 + =. 8 8 Figure.4.5 of text Hrmoniclly relted s k (t) = e jkω0t = e jπkf0t, (cts-time) k = 0, ±, ±,... Scientific Designtion Freuency (Hz) k for F 0 = 8.76 C C C C C C C Dr. Deep Kundur (University of Toronto) Introduction to Discrete-Time Systems 9 / 34 Dr. Deep Kundur (University of Toronto) Introduction to Discrete-Time Systems 0 / 34

6 Hrmoniclly Relted Complex Exponentils Scientific Designtion Freuency (Hz) k for F 0 = 8.76 C C C C4 (middle C) C C C C Hrmoniclly Relted Complex Exponentils Wht does the fmily of hrmoniclly relted sinusoids s k (t) hve in common? Hrmoniclly relted s k (t) = e jkω0t = e jπ(kf0)t, (cts-time) k = 0, ±, ±,... fund. period: T 0,k = cyclic freuency = kf 0 period: T k = ny integer multiple of T 0 common period: T = k T 0,k = F 0 C C C3 C4 C5 C6 C7 C8 Dr. Deep Kundur (University of Toronto) Introduction to Discrete-Time Systems / 34 Dr. Deep Kundur (University of Toronto) Introduction to Discrete-Time Systems / 34 Hrmoniclly Relted Complex Exponentils Hrmoniclly Relted Complex Exponentils Cse: For periodicity, select f 0 = N where N Z: Hrmoniclly relted s k (n) = e jπkf0n = e jπkn/n, (dts-time) k = 0, ±, ±,... There re only N distinct dst-time hrmonics: s k (n), k = 0,,,..., N. s k+n (n) = e jπ(k+n)n/n = e jπkn/n e jπnn/n = e jπkn/n = e jπkn/n = s k (n) Therefore, there re only N distinct dst-time hrmonics: s k (n), k = 0,,,..., N. Dr. Deep Kundur (University of Toronto) Introduction to Discrete-Time Systems 3 / 34 Dr. Deep Kundur (University of Toronto) Introduction to Discrete-Time Systems 4 / 34

7 -to- Conversion -to- Conversion Smpler Quntizer Smpler Quntizer Smpling: conversion from cts-time to dst-time by tking smples t discrete time instnts E.g., uniform smpling: = x (nt ) where T is the smpling period nd n Z Dr. Deep Kundur (University of Toronto) Introduction to Discrete-Time Systems 5 / 34 Dr. Deep Kundur (University of Toronto) Introduction to Discrete-Time Systems 6 / 34 -to- Conversion -to- Conversion Smpler Quntizer Smpler Quntizer Quntiztion: conversion from dst-time cts-vlued to dst-time dst-vlued untiztion error: e (n) = x (n) for ll n Z Coding: representtion of ech dst-vlue x (n) by b-bit binry seuence e.g., if for ny n, x (n) {0,,..., 6, 7}, then the coder my use the following mpping to code the untized mplitude: Dr. Deep Kundur (University of Toronto) Introduction to Discrete-Time Systems 7 / 34 Dr. Deep Kundur (University of Toronto) Introduction to Discrete-Time Systems 8 / 34

8 -to- Conversion Smpling Theorem Smpler Quntizer If the highest freuency contined in n nlog x (t) is F mx = B nd the is smpled t rte Exmple coder: F s > F mx = B then x (t) cn be exctly recovered from its smple vlues using the interpoltion function g(t) = sin(πbt) πbt Note: F N = B = F mx is clled the Nyuist rte. Dr. Deep Kundur (University of Toronto) Introduction to Discrete-Time Systems 9 / 34 Dr. Deep Kundur (University of Toronto) Introduction to Discrete-Time Systems 30 / 34 Smpling Theorem Smpling Period = T = F s = Smpling Freuency Therefore, given the interpoltion reltion, x (t) cn be written s x (t) = x (t) = n= n= x (nt )g(t nt ) g(t nt ) where x (nt ) = ; clled bndlimited interpoltion. See Figure.4.6 of text. Dr. Deep Kundur (University of Toronto) Introduction to Discrete-Time Systems 3 / 34 -to- Conversion originl/bndlimited interpolted 0 n Common interpoltion pproches: bndlimited interpoltion, zero-order hold, liner interpoltion, higher-order interpoltion techniues, e.g., using splines In prctice, chep interpoltion long with smoothing filter is employed. Dr. Deep Kundur (University of Toronto) Introduction to Discrete-Time Systems 3 / 34

9 -to- Conversion -to- Conversion originl/bndlimited interpolted originl/bndlimited interpolted liner interpoltion zero-order hold -3T t -T -T 0 T T 3T -3T t -T -T 0 T T 3T Common interpoltion pproches: bndlimited interpoltion, zero-order hold, liner interpoltion, higher-order interpoltion techniues, e.g., using splines In prctice, chep interpoltion long with smoothing filter is employed. Dr. Deep Kundur (University of Toronto) Introduction to Discrete-Time Systems 33 / 34 Common interpoltion pproches: bndlimited interpoltion, zero-order hold, liner interpoltion, higher-order interpoltion techniues, e.g., using splines In prctice, chep interpoltion long with smoothing filter is employed. Dr. Deep Kundur (University of Toronto) Introduction to Discrete-Time Systems 34 / 34