信号与系统课件第3章周期信号的傅里叶级数表示详解.ppt

So, the Fourier series for the output: If we write Then we have 3.8 SUMMARY 5. The concepts of system function and frequency response. 1. Eigenfunction property of complex exponentials; 2. Fourier series representations for both continuous-time and discrete- time periodic signals ― i.e., weighted sum of harmonically related complex exponentials that share a common period; 3. Different characteristics of signals reflected in their Fourier series coefficients (properties of Fourier series); 4. An important conclusion: if a periodic signal is applied to an LTI system, then the output will be periodic with the same period, and each of the Fourier coefficients of the output is the corresponding Fourier coefficient of the input multiplied by a complex number whose value is a function of the frequency corresponding to that Fourier coefficient. Convergence of the Fourier series representation of a square wave: an illustration of the Gibbs phenomenon. Any continuity: xN(t1) ? x(t1) Vicinity of discontinuity: ripples peak amplitude does not seem to decrease Discontinuity: overshoot 9% Gibbs’s conclusion: 3.4 PROPERTIES OF CONTINUOUS-TIME FOURIER SERIES We generally use a shorthand notation to indicate the relationship between a periodic signal and its Fourier series coefficients, that is 3.4.1 Linearity If then 3.4.2 Time Shifting If then When a periodic signal is shifted in time, the magnitude spectrum remains unaltered. 3.4.3 Time Reversal If then Time reversal applied to a continuous-time signal results in a time reversal of the corresponding sequence of Fourier series coefficients. If x(t) is even : If x(t) is odd: 3.4.4 Time Scaling If then