电子科大信号系统英文版课件chapter3 fourierseries-b

,3 Fourier Series Representation of Periodic Signals,3.Fourier Series Representation of Periodic Signal,Jean Baptiste Joseph Fourier, born in 1768, in France.1807,periodic signal could be represented by sinusoidal series.1829,Dirichlet provided precise conditions.1960s,Cooley and Tukey discovered fast Fourier transform.,3 Fourier Series Representation of Periodic Signals,3.2 The Response of LTI Systems to Complex Exponentials,(1) Continuous time LTI system,h(t),x(t)=est,y(t)=H(s)est,( system function ),3 Fourier Series Representation of Periodic Signals,(2) Discrete time LTI system,hn,xn=zn,yn=H(z)zn,( system function ),3 Fourier Series Representation of Periodic Signals,(3) Input as a combination of Complex Exponentials,Continuous time LTI system:,Discrete time LTI system:,Example 3.1,3 Fourier Series Representation of Periodic Signals,3.3 Fourier Series Representation of Continuous-time Periodic Signals,(1) General Form,3.3.1 Linear Combinations of Harmonically Related Complex Exponentials,The set of harmonically related complex exponentials:,Fundamental period: T ( common period ),3 Fourier Series Representation of Periodic Signals,So, arbitrary periodic signal can be represented as,: Fundamental components,: Second harmonic components,: Nth harmonic components,( Fourier series ),Example 3.2,3 Fourier Series Representation of Periodic Signals,(2) Representation for Real Signal,Real periodic signal: x(t)=x*(t),So a*k=a-k,Let (A),3 Fourier Series Representation of Periodic Signals,Let (A),(B),3 Fourier Series Representation of Periodic Signals,3.3.2 Determination of the Fourier Series Representation of a Continuous-time Periodic Signal,( Orthogonal function set ),Determining the coefficient by orthogonality: ( Multiply two sides by ),3 Fourier Series Representation of Periodic Signals,Fourier Series Representation:,3 Fourier Series Representation of Periodic Signals,Example 3.3 3.5,x(t),ak,3 Fourier Series Representation of Periodic Signals,3.4 Convergence of the Fourier Series,Approximation error:,(1) Finite series,If, then the series is,convergent. ( xN(t) x(t) ),3 Fourier Series Representation of Periodic Signals,Condition 1: Absolutely Integrable,(2) Dirichlet condition,3 Fourier Series Representation of Periodic Signals,Condition 2: Finite number of maxima and minima during a single period,3 Fourier Series Representation of Periodic Signals,Condition 3: Finite number of discontinuity,3 Fourier Series Representation of Periodic Signals,1898, Albert Michelson , An American physicist Constructed a harmonic analyzer Observed truncated Fourier series xN(t) looked very much like x(t) Found a strange phenomenonJosiah Gibbs, An American mathematical physicist Given out a mathematical Explanation,(3) Gibbs phenomenon,3 Fourier Series Representation of Periodic Signals,3 Fourier Series Representation of Periodic Signals,Any continuity: xN(t1) x(t1) Vicinity of discontinuity: ripples peak amplitude does not seem to decreaseDiscontinuity: overshoot 9%,Gibbss conclusion:,3 Fourier Series Representation of Periodic Signals,3.6.1 Linear Combination of Harmonically Related Complex Exponentials,Periodic signal xn with period N: xn=xn+N,3.6 Fourier Series Representation of Discrete-time Periodic Signals,Discrete-time complex exponential orthogonal signal set:,3 Fourier Series Representation of Periodic Signals,Property of orthogonal signal set:,3 Fourier Series Representation of Periodic Signals,3.6.2 Determination of the Fourier Series Representation of Periodic Signals,Fourier series of periodic signal xn:,Determine the coefficients ak by orthogonality:,3 Fourier Series Representation of Periodic Signals,The equations of Fourier series:,( ak is periodic ),3 Fourier Series Representation of Periodic Signals,Example 3.10 3.12,xn,ak,3 Fourier Series Representation of Periodic Signals,(1) System function,3.8 Fourier Series and LTI System,Continuous time system:,Discrete-time system:,(2) Frequency response,Continuous time system:,Discrete-time system:,3 Fourier Series Representation of Periodic Signals,(3) System response,Continuous time system:,Discrete-time system:,Example 3.16 3.17,3 Fourier Series Representation of Periodic Signals,3.9 Filtering,3.9.1 Frequency-shaping filters,Example1: Equalizer,3 Fourier Series Representation of Periodic Signals,Example2: Image Filtering (Edge enhancement),3 Fourier Series Representation of Periodic Signals,3.9.2 Frequency-selective filters,Several type of filter : (1) Lowpass filter (2) Highpass filter (3) Bandpass filter,3 Fourier Series Representati