Signals and Systems Fall 2003 Lecture#5 1 8 September 2003 Complex Exponentials as Eigenfunctions of LTI Systems 234 Fourier Series representation of CT periodic signals How do we calculate the fourier coefficients Convergence and gibbs phenomenon
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 coefficient s ? 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
Signals & Systems, 2nd ed. Upper Saddle River, N.J.: Prentice Hall, 1997, p. 179. Portrait of Jean Baptiste Joseph Fourier Image removed due to copyright considerations
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
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 ok(t) and their properties (Focus on CT systems now, but results apply to dt systems as well. System envalue Eigenfunction in→> same function out with a‘gain” From the superposition property of lti systems (t)=∑kakk( kak ok Now the task of finding response of lti systems is to determine nk
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 a(t)=es h() T h(r) std H(s eigenvalue eigenfunction hn ∑ m=-0 ∑Mml2-n|2n H(z) eigenvalue eigenfunction
Complex Exponentials as the Eigenfunctions of any LTI Systems eigenvalue eigenfunction eigenvalue eigenfunction