* AKW Fall 2012 Chapter 3-4-5 ELEC 2100 Signals and Systems Module 2 Fourier Analysis: Frequency Domain Representation of Signals Part I – Chapter 3: CT and DT Fourier Series * AKW Fall 2012 Chapter 3-4-5 ? Complex Exponentials as Eigenfunction of LTI Systems ? System Function and Frequency Response ? Frequency Domain Representation of Signals Lecture 7 Chapter 3 – System Function and Frequency Response * AKW Fall 2012 Chapter 3-4-5 What is Analysis? ? Analysis is to pose plex subject into a collection of some basic parts . ? We may pose the same subject differently depending on the problem that we want to solve, our domain of interest, etc. ? In Chapter 2, we derived the convolution sum/integral for LTI systems by posing the input into a sum of shifted delta functions: ??????????????dtxtxknkxnx k)()()(;][][][????????????dthxtyknhkxny k)()()(;][][][ Regard the input signal as a superposition of shifted impulses Deduce that the output signal must be a superposition of responses to shifted impulses, as given by the convolution sum or convolution integral * AKW Fall 2012 Chapter 3-4-5 Fourier Analysis - C omplex Sinusoids as Basic Signal ? Why posing into complex sinusoid s? It is because complex sinusoids and complex exponentials in general, are eigenfunctions of LTI systems. ? What does an eigenfunction mean ? Consider what is the output when the input to an CT LTI system is e st: LTI system h(t)e sty(t )=? ? Chapters 3-8 is about Fourier analysis , which is the position of signals into sums of complex sinusoids (or real sinusoids) **** AKW Fall 2012 AKW Fall 2012 AKW Fall 2012 AKW Fall 2012 Chapter 3-4-5Chapter 3-4-5Chapter 3-4-5Chapter 3-4-5 ? To determine the output, we apply the convolution integral from Chapter 2: st s st tsesHdehe dehdtxhty)()( )()()()( )(??????????????????????????????s sttseee ???)(? The output is the plex exponential except for the multiplication by H(s ) !e st is called an eigenfunction b