电子科大信号系统英文版课件Chapter3 FourierSeries-B

1、3 Fourier Series Representation of Periodic Signals3.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

2、 Fourier transform. 3 Fourier Series Representation of Periodic Signals3.2 The Response of LTI Systems to Complex Exponentials(1) Continuous time LTI systemh(t)x(t)=esty(t)=H(s)est( system function ) 3 Fourier Series Representation of Periodic Signals(2) Discrete time LTI systemhnxn=znyn=H(z)zn( sys

3、tem function ) 3 Fourier Series Representation of Periodic Signals(3) Input as a combination of Complex ExponentialsContinuous time LTI system:Discrete time LTI system:Example 3.13 Fourier Series Representation of Periodic Signals3.3 Fourier Series Representation of Continuous-time Periodic Signals(

4、1) General Form3.3.1 Linear Combinations of Harmonically Related Complex ExponentialsThe set of harmonically related complex exponentials:Fundamental period: T ( common period )3 Fourier Series Representation of Periodic SignalsSo, arbitrary periodic signal can be represented as: Fundamental compone

5、nts: Second harmonic components: Nth harmonic components( Fourier series )Example 3.23 Fourier Series Representation of Periodic Signals(2) Representation for Real SignalReal periodic signal: x(t)=x*(t)So a*k=a-kLet (A) 3 Fourier Series Representation of Periodic SignalsLet (A) (B) 3 Fourier Series

6、Representation of Periodic Signals3.3.2 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 SignalsFourier Series Represe

7、ntation:3 Fourier Series Representation of Periodic SignalsExample 3.3 3.5x(t)ak3 Fourier Series Representation of Periodic Signals3.4 Convergence of the Fourier SeriesApproximation error:(1) Finite seriesIf, then the series isconvergent. ( xN(t) x(t) )3 Fourier Series Representation of Periodic Sig

8、nalsCondition 1: Absolutely Integrable(2) Dirichlet condition3 Fourier Series Representation of Periodic SignalsCondition 2: Finite number of maxima and minima during a single period3 Fourier Series Representation of Periodic SignalsCondition 3: Finite number of discontinuity3 Fourier Series Represe

9、ntation of Periodic Signals1898, 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 pheno

10、menon3 Fourier Series Representation of Periodic Signals3 Fourier Series Representation of Periodic SignalsAny 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

11、Signals3.6.1 Linear Combination of Harmonically Related Complex ExponentialsPeriodic signal xn with period N: xn=xn+N3.6 Fourier Series Representation of Discrete-time Periodic SignalsDiscrete-time complex exponential orthogonal signal set:3 Fourier Series Representation of Periodic SignalsProperty

12、of orthogonal signal set:3 Fourier Series Representation of Periodic Signals3.6.2 Determination of the Fourier Series Representation of Periodic SignalsFourier series of periodic signal xn:Determine the coefficients ak by orthogonality:3 Fourier Series Representation of Periodic SignalsThe equations

13、 of Fourier series:( ak is periodic )3 Fourier Series Representation of Periodic SignalsExample 3.10 3.12xnak3 Fourier Series Representation of Periodic Signals(1) System function3.8 Fourier Series and LTI SystemContinuous time system:Discrete-time system:(2) Frequency responseContinuous time system

14、:Discrete-time system:3 Fourier Series Representation of Periodic Signals(3) System responseContinuous time system:Discrete-time system:Example 3.16 3.173 Fourier Series Representation of Periodic Signals3.9 Filtering3.9.1 Frequency-shaping filtersExample1: Equalizer3 Fourier Series Representation of Periodic SignalsExample2: Image Filtering (Edge enhancement)3 Fourier Series Representation of Periodic Signals3.9.2 Frequency-selective filtersSeveral type of filter : (1) Lowpass filter (2) Highpass filter (3) Bandpass filter3 Fourier Series Representati