《数字信号处理》课件.ppt

BGL/SNU Chapter2. DISCRETE-TIME SIGNALS AND SYSTEMS 2.0 Introduction 2.1 Discrete-Time Signals : Sequences 2.2 Discrete-Time Systems 2.3 Linear Time-Invariant Systems 2.4 Properties of Linear Time-Invariant Systems 2.5 Linear Constant-Coefficient Difference Equations 2.6 Frequency-Domain Representation 2.7 Representation of Sequences of the Fourier Transform 2.8 Symmetry Properties of the Fourier Transform 2.9 Fourier Transform Theorems 2.1.Discrete - Time Signals x[n]= x(t)|t=nT n : -1,0,1,2,… T: sampling period x(t) : analog signal i) unit impulse signal(sequence) d[n] = 1, n=0 0, n?0 ii) unit step sequence u[n] = 1, n?0 0, n?0 iii) exponential/sinusoidal sequence x[n]= Aej(won+?), Acos(won+?) - not necessarily periodic in n with period 2p/wo - periodic in n with period N (discrete number) for woN=2pk or wo = 2pk/N [note] x(t)= Ae j(wo t +?) is periodic in t with period T= 2p/wo (continuous value) iv) general expression x[n] = S x[k]d[n-k] 2.2Discrete-Time Systems T[ ] x[n] y[n] System : signal processor i) memoryless or with memory y[n] = f(x[n]), y[n]=f(x[n-k]) with delay ii) linearity x1[n]?y1[n] x2[n]?y2[n] a1x1[n] + a2x2[n] ?a1y1[n] + a2 y2[n] - e.g. T[a1x1[n] + a2x2[n]] = T[a1x1[n]] + T[a2x2[n]] = a1T[x1[n]] + a2T[x2[n]] iii) time-invariance x[n] ? y[n] ? x[n-n0] ? y[n-n0] - e.g., T[x [n-n0]] = T[x [n]] | n ? n-no - e.g., d[n] ? h[n] ? d[n-k] ? h[n-k] - counter-example : decimator T[ ] = x[Mn] iv) causality y[n] for n=n1, depends on x[n] for n??n1 only - counter-example : y[n] = x[n+1] - x[n] v) stability bounded input yields bounded output(BIBO) |x[n]| ? for all n ? |y[n]| ? for all n - counter-example : y[n] = S u[k] = 0, n0 n+1, n?0 unbounded ( no fixed value By exists that keeps y[n] ? By ? .) k=-? n 2.3 Linear Time-Invariant Systems LTI x[n] y[n] d[n] h[n] T[d[n]] : impulse response In general, let x[n] = S x[k]d[n-k] ? In general, let x[n] = S x[k]d[n-k]