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
H(8) 1)=∑(s k H DT 2k)2 H()=∑bzn l=∑0k2→=∑H(x)a
DT:
What kinds of signals can we represent as “sums” of complex exponentials For Now: Focus on restricted sets of complex exponentials CT ow- purely imaginary i. e, signals of the form eJwt Magnitude 1 DT i. e. signals of the form ejan ct dt fourier series and Transforms Periodic signals
What kinds of signals can we represent as “sums” of complex exponentials? For Now: Focus on restricted sets of complex exponentials CT & DT Fourier Series and Transforms CT: DT: ⇓ Magnitude 1 Periodic Signals
Fourier series representation of ct Periodic signals a(t=at+r) for all t smallest such T'is the fundamental period 2丌 is the fundamental frequency periodic with period T+w= hwo ()=∑ak=∑ akeJk2mt/T k periodic with period T fak are the Fourier (series) coefficients k=0 DC -k=±1 first harmonic k=±2 second harmonic
Fourier Series Representatio n of CT Periodic Signals ωo = 2 π T - s m a l l e s t s u c h T is the fundamental period - i s t h e fundamental frequency - periodic with period T - { a k} are the Fourier (series) coefficients - k = 0 D C - k = ±1 first harmonic - k = ±2 second harmonic
Question #1: How do we find the Fourier coefficients First, for simple periodic signals consisting of a few sinusoidal terms Ex: a(t cOS4mt+2sin8丌t Euler's relation e art e art (memorize!) 2丌2丌1 4丌T 4 0 0-no dc component a 21000
Question #1: How do we find the Fourier coefficients? First, for simple periodic signals consisting of a few sinusoidal terms 0 – no dc component 0 0 Euler's relation (memorize!)
For real periodic signals, there are two other commonly used forms for ct Fourier series a(t)=a0+>lak cos hot+Bk sin kwot) k=1 (t) ∑7kcs(kot+) Because of the eigenfunction property of ejor, we will usually use the complex exponential form in 6.003 a consequence of this is that we need to include terms for both positive and negative frequencies kwo t kwo t e Remember cos(swot) +e and sin(swot)
• For real periodic signals, there are two other commonly used forms for CT Fourier series: • Because of the eigenfunction property of ejωt, we will usually use the complex exponential form in 6.003. - A consequence of this is that we need to include terms for both positive and negative frequencies:
