/9/ Coninuous Time Linear Time Invarian (LTI) Sysems Why LTI? Inroducion Many physical sysems. Easy o solve mahemaically Available informaion abou analysis and design. We can apply superposiion LTI Sysem
/9/ Impulse represenaion of Coninuous Time Sysems A signal can be represened as a sum of delas (impulses). n x x n rec n n x lim x n rec x d n x() =n To verify The well know sifing (sampling) propery The Impulse Response LTI Sysem Impulse response of a sysem is response of he sysem o an inpu ha is a uni impulse (i.e., a Dirac dela funcion in coninuous ime) Therefore, we know how o calculae he sysem oupu for any inpu, x() x x d x h d y() This operaion is called convoluion
/9/ Convoluion Concep Wha is y()? If we ake he limi as we ge % Dr.Ali Hussein Mqaibel clear all close all clc _sep=.; =:_sep:; h=exp( ).*Heaviside(); subplo(,,) plo(,h) xlabel ('ime,s') ylabel ('The impulse response') axis([ ]) y=h; x=[ zeros(,lengh(h) )].*Heaviside(); for i=:lengh(); subplo(,,) plo(,[zeros(,i) h(:end i)]) y=y+[zeros(,i) h(:end i)]; x(,i)=; hold on ylabel ('Componens') subplo (,,) bar(,x) xlabel ('ime,s') ylabel ('Inpu') axis([ ]) subplo(,,) plo(,y*_sep) xlabel ('ime,s') ylabel ('Oupu') axis([ ]) pause end Convoluion for Coninuous Time LTI Sysems Componens.5 4 5 6 7 8 9 Com ponens.5 4 5 6 7 8 9 Inpu Oupu 4 5 6 7 8 9 ime,s 4 5 6 7 8 9 ime,s In pu 4 5 6 7 8 9 ime,s O u p u 4 5 6 7 8 9 ime,s
/9/ Some Convoluion Properies Wha is he oupu if he inpu is impulse? For LTI sysem Recall some dela properies (no convoluion) conains a complee inpu oupu descripion.i.e. if he impulse response is known, he sysem response o any inpu can be found Example: Impulse Response of an Inegraor Wha is? Wha is he oupu if he inpu is ramp? ramp response Verify from he sysem equaion Common misake o forge Before you sar graphical convoluion visi: (The Joy of Convoluion) hp://www.jhu.edu/~signals/convolve/index.hml 4
/9/ Graphical Convoluion Mehods From he convoluion inegral, convoluion is equivalen o f f f f d Roaing one of he funcions abou he y axis Shifing i by Muliplying his flipped, shifed funcion wih he oher funcion Calculaing he area under his produc Assigning his value o f () * f () a Graphical Convoluion Example Convolve he following wo funcions: f() * - Replace wih in f()and g() Choose o flip and slide g() since i is simpler and symmeric g(-) Funcions overlap like his: g() f() - + + 5
/9/ Graphical Convoluion Example Convoluion can be divided ino 5 pars I. < Two funcions do no overlap Area under he produc of he funcions is zero II. < Par of g() overlaps par of f() Area under he produc of he funcions is ( ) d - + + 6 6 - + + g(-) f() g(-) f() Graphical Convoluion Example III. < Here, g() compleely overlaps f() Area under he produc is jus d 6 IV. < 4 Par of g() and f() overlap Calculaed similarly o < V. 4 g() and f() do no overlap Area under heir produc is zero g(-) f() - + + g(-) f() - + + 6
/9/ Graphical Convoluion Example Resul of convoluion (5 inervals of ineres): y( ) 6 f ( )* g( ) 6 4 y() 6 for for for for 4 for 4-4 How o Check he Convoluion Sar and end poins: Sar poin equal o he sum of he saring poins of he wo signals and he end poin equals o he sum of he end poins. Area under he curve=produc of he individual areas. f() g() 6 y() = = 4 7
/9/ Example : Graphical Convoluion 6 4 5 4 Find he convoluion beween and. Coninue Example : 8
/9/ If we are ineresed in he oupu only a a specific ime, we do no have o do full convoluion Example : See addiional Examples in he book.. You mus wrie by your hand! Properies of Convoluion Commuaive propery Associaive Propery Disribuive propery ) ) 9
/9/ Example: Impulse Response of Inerconneced Sysem Properies of Coninuous ime LTI Sysems The inpu oupu characerisics of a coninuous ime LTI sysem are compleely described by is impulse response Memory: In general LTI are dynamic because of inegraion over ime. The only way o ge he inegraion ou is if ideal amplifier An LTI sysem is memoryless if and only if Inverabiliy An LTI sysem is inverible only if we can find such ha No procedure is given o find
/9/ Coninue.. Properies of Coninuousime LTI Sysems Causaliy A signal ha is zero for is called a causal signal Because occurs a,. Alernaively = for Only for causal LTI Coninue... Properies of Coninuous ime LTI Sysems Sabiliy Since For a bounded inpu for all, he oupu is = i.e is absoluely inegrable BIBO Sabiliy es
/9/ Sabiliy Examples Example: Is he following sysem BIBO Sable?. Unsable BIBO Sabiliy es Inegraor Unsable Can be done by inspecion of he graph of Uni Sep Response ) If he inpu is, he oupu is noed as (some books hey use ) Relaion beween and Noe do no confuse wih in wriing! The uni sep response compleely describes he inpu oupu characerisics of an LTI sysem. ) Linear
/9/ Example: Sep Response from he Impulse Response, find he sep response. Verify: Sifing propery Pracice more convoluion examples. Consider boh Graphical and analyical (See old exams and quizzes)