EECE 31 Signals & Sysems Prof. Mark Fowler Noe Se #1 C-T Sysems: Convoluion Represenaion Reading Assignmen: Secion 2.6 of Kamen and Heck 1/11
Course Flow Diagram The arrows here show concepual flow beween ideas. Noe he parallel srucure beween he pink blocks (C-T Freq. Analysis) and he blue blocks (D-T Freq. Analysis). New Signal Models Ch. 1 Inro C-T Signal Model Funcions on Real Line Sysem Properies LTI Causal Ec Ch. 3: CT Fourier Signal Models Fourier Series Periodic Signals Fourier Transform (CTFT) Non-Periodic Signals Ch. 2 Diff Eqs C-T Sysem Model Differenial Equaions D-T Signal Model Difference Equaions Zero-Sae Response Ch. 5: CT Fourier Sysem Models Frequency Response Based on Fourier Transform New Sysem Model Ch. 2 Convoluion C-T Sysem Model Convoluion Inegral Ch. 6 & 8: Laplace Models for CT Signals & Sysems Transfer Funcion New Sysem Model New Sysem Model D-T Signal Model Funcions on Inegers Zero-Inpu Response Characerisic Eq. D-T Sysem Model Convoluion Sum New Signal Model Powerful Analysis Tool Ch. 4: DT Fourier Signal Models DTFT (for Hand Analysis) DFT & FFT (for Compuer Analysis) Ch. 5: DT Fourier Sysem Models Freq. Response for DT Based on DTFT New Sysem Model Ch. 7: Z Trans. Models for DT Signals & Sysems Transfer Funcion New Sysem Model 2/11
Convoluion for C-T sysems We saw for D-T sysems: -Definiion of Impulse Response h[n] -How TI & Lineariy allow us o use h[n] o wrie an equaion ha gives he oupu due o inpu x[n] (Tha equaion is Convoluion) The same ideas arise for C-T sysems! (And he argumens o ge here are very similar so we won go ino as much deail!!) Impulse Response: h( is wha comes ou when δ( goes in δ( C-T LTI ICs = h( δ( If sysem is causal, h( = for < h( 3/11
Noe: In D-T sysems, δ[n] has a heigh of 1 In C-T sysems, δ( has a heigh of infiniy and a widh of zero -So, in pracice we can acually make δ[n] -Bu we canno acually make δ(!! How do we know or ge he impulse response h(? 1. I is given o us by he designer of he C-T sysem. 2. I is measured experimenally -Bu, we canno jus pu in δ( -There are oher ways o ge h( bu we need chaper 3 and 5 informaion firs 3. Mahemaically analyze he C-T sysem -Easies using ideas in Ch. 3, 5, 6, & 8 4/11
Our focus is here In wha form will we know h(? Now we can 1. h( known analyically as a funcion -e.g. h( = e -2 u( 2. We may only have experimenally obained samples: - h(nt) a n =, 1, 2, 3,, N-1 Use h( o find he zero-sae response of he sysem for an inpu Following similar ideas o he DT case we ge ha: x( h( y( CONVOLUTION y ( = x( λ) h( λ) dλ C-T LTI ICs = Noaion: y(=x(*h( 5/11
Example 2.14 Oupu of RC Circui wih Uni Sep Inpu The book considers a differen case x( is a pulse x( y( Problem: Find he zero-sae response of his circui o a uni sep inpu i.e., le x( = u( and find y( for he case of he ICs se o zero (for his case ha means y( - ) = ). We have seen ha his circui is modeled by he following Differenial Equaion: dy( 1 1 + y( = x( d RC RC So we need o solve his Diff. Eq. for he case of x( = u(. The previous slides old us ha we can use convoluion 6/11
Bu o do ha we need o know he impulse response h( for his sysem (i.e., for his differenial equaion)!!! In Chaper 6 we will learn how o find he impulse response by applying he Laplace Transform o he differenial equaion. The resul is: h( = 1 RC, e RC ) <, or h( = 1 RC e RC ) u( 7/11
For our sep inpu: 1v x( = u( x( y( = x(*h( This is he convoluion we need o do 1v x( = u( * 8/11
y ( = x( * h( = h( λ) x( λ) dλ 1 RC ) λ [ e u( λ) ] = u( λ dλ RC ) This makes he inegrand = whenever λ <. And i is 1 oherwise. This is he general form for convoluion Plug in given forms for h( and x( This makes he inegrand = whenever - λ < or in oher words whenever λ >. And i is 1 oherwise. (Noe ha if < hen he inegrand is for all λ ) So exploiing hese facs we see ha he only hing he uni seps do here is o limi he range of inegraion y 1, (1/ RC ) λ RC e ( = dλ, > So o find he oupu for his problem all we have o do is evaluae his inegral o ge a funcion of 9/11
This inegral is he easies one you learned in Calc I!!! 1 RC e RC ) λ dλ = 1 RC RC ) λ RC ) λ RC ) [ RCe ] = [ e ] = [ e ] [ e ] = 1 e RC ) 1 e y( =, RC ), > Once we can compue his kind of inegral in general we can find ou wha he oupu looks like for any given inpu! Oupu for RC = 1 y( (vols) Recall Time-Consan Rules: 63% afer 1 TC 1% afer 5 TCs (sec) 1/11
Big Picure Compare o Big Picure for DT Case Same! For a LTI C-T sysem in zero sae we no longer need he differenial equaion model -Insead we need he impulse response h( & convoluion New alernaive model! Differenial Equaion Convoluion & Impulse resp. Equivalen Models (for zero sae) 11/11