MIT(麻省理工)信号与系统讲义-lecture5MIT(麻省理工)信号与系统讲义-lecture5.ppt

Signals and Systems Fall 2003 Lecture #5 18 September 2003 1. Complex Exponentials as Eigenfunctions of LTI Systems 2. Fourier Series representation of CT periodic signals 3. How do we calculate the Fourier coefficients? 4. Convergence and Gibbs’ Phenomenon Portrait of Jean Baptiste Joseph Fourier Image removed due to copyright considerations. Signals Systems, 2nd ed. Upper Saddle River, N.J.: Prentice Hall, 1997, p. 179. Desirable Characteristics of a Set of “Basic” Signals a. We can represent large and useful classes of signals using these building blocks b. The response of LTI systems to these basic signals is particularly simple, useful, and insightful Previous focus: Unit samples and impulses Focus now: Eigenfunctions of all LTI systems The eigenfunctions φk(t)and their properties (Focus on CT systems now, but results apply to DT systems as well.) eigenvalue eigenfunction Eigenfunction in →same function out with a “gain” From the superposition property of LTI systems: Now the task of finding response of LTI systems is to determine λk. Complex Exponentials as the Eigenfunctions of any LTI Systems eigenvalue eigenfunction eigenvalue eigenfunction DT: What kinds of signals can we represent as “sums” of complex exponentials? For Now: Focus on restricted sets of complex exponentials CT: DT: s = jw – purely imaginaly, i.e., signals of the form ejwt Z = ejw i.e., signals of the form ejwt Magnitude 1 CT DT Fourier Series and Transforms Periodic Signals Fourier Series Representation of CT Periodic Signals smallest such T is the fundamental period is the fundamental frequency Periodic with period T -periodic with period T -{ak} are the Fourier (series) coefficients -k= 0 DC -k= ? first harmonic -k= ? 2second harmonic Question #1: How do we find the Fourier coefficients? First, for simple periodic signals consisting of a few sinusoidal terms ? For real periodic signals, there ar