Liear Time-Ivaria Sysems (LTI Sysems) Oulie Basic Sysem Properies Memoryless ad sysems wih memory (saic or dyamic) Causal ad o-causal sysems (Causaliy) Liear ad o-liear sysems (Lieariy) Sable ad o-sable sysems (Sabiliy) Time-Ivaria sysems (Time ivariace) Basic Sysem Properies Memoryless ad sysems wih memory (saic or dyamic) Causal ad o-causal sysems (Causaliy) Liear ad o-liear sysems (Lieariy) 4 Sable ad o-sable sysems (Sabiliy) 5 Time-Ivaria sysems (Time ivariace) Memoryless ad sysems wih memory (saic or dyamic): A sysem is called memoryless, if he oupu, y (, of a give sysem for each value of he idepede variable a a give ime depeds oly o he ipu value a ime A sysem has memory if he oupu a ime depeds i geeral o he pas values of he ipu x ( for some rage of he values of o A sysem wih memory reais or sores iformaio abou ipu values a imes oher ha he curre ipu value D-T sigal erms: The rasformaio does o deped o he previous samples of he sequece, i is memoryless D-T sysem Sysem ame Sysem Equaio Defiiio Descripio Ideal Amplifier/Aeuaor Iegraor y( k, y[ k where k is some real cosa y ( d Iegrae he values of he ipu sigal from all pas imes up o prese ime Memoryless Sysem wih memory
Coiuous ime sysems: a) y( 5si( cos( ): This sysem is memoryless 7 b) y( ) d For some geeral ipu fucio x (, his sysem is a sysem wih memory, because i depeds o all pas values of he ipu c) y( e d - cosider e Soluio: we use iegraio by pars, u du d dv e u v e v e vdu e e e y( e ( ) e ( So we have sysem wih memory Discree ime sysems : a y [ 5] This sysem i o memoryless, because he oupu value a depeds o he ipu values a 5 b y [ si( ) 5 - memoryless sysem Causal ad o-causal sysems: If he oupu of he sysem y ( a ay ime depeds oly o he ipu a prese ad/or previous imes, we say ha he sysem is causal, mahemaically his ca be represeed as y ( f (, ),) A ocausal sysem aicipaes he fuure values of he ipu sigal i some way All memoryless sysems are causal For real ime sysem where acually deoed ime causaliy is impora Causaliy is o a esseial cosrai i applicaios where is o ime, for example, image processig If we are doig processig o recorded daa, he also causaliy may o be required Coiuous ime sysems: a) Ideal Predicor: his sysem is give by he followig ipu-oupu relaioship y ( ), i is ocausal sysem, sice he value y ( of he oupu a ime depeds o he value x ( ) of he ipu a ime, so he oupu mus appear before he ipu sigal as show i figure -7 e d e e e
- y( - Figure -7: Ideal Predicor I geeral, y( k q), where q is a posiive real umber, is a ocausal sysem b) Ideal Time Delay : The Ideal Time Delay has he followig equaio y ( ) ad his sysem is causal Discree ime sysems : a) poi MA Filer: The poi MA Filer y [ ] ] ] is a causal sysem b) 9 poi MA Filer: The 9-poi MA Filer wih he followig defiiio: y[ 4] ] ] ] ] ] ] 4] 9 is a ocausal filer, sice he filer oupu a ime requires he fuure values x [ 4], ], ] ad x [ ] of he ipu c) The sysem defied by y [ k] is ocausal k d) y [ ] - is a causal, sice he oupu value a for he sysem described by y [ ] depeds o he previous values of e) y [ 5x [ ] - is a ocausal, sice he oupu value depeds o he ipu value Liear ad o-liear sysems (Lieariy): This is a impora propery of he sysem We will see laer ha if we have sysem which is liear ad ime ivaria he i has a very compac represeaio A operaor T is called liear if he followig relaioships hold: T Or T ( ax bx ( a Tx ( b Tx ( a y ( b y ( for C-T sigals [ ax bx [ a Tx [ b Tx [ a y [ b y [ for D-T sigals To explai hese relaioships, suppose ha T acs o wo ipu sigals x ( ) ad x ( ) o produce he followig sigals:
( T ( T y - The respose of he sysem o he ipu x ( ) ad ) y - The respose of he sysem o he ipu x ( ) Ad suppose ha a ad b are wo cosas To ge he lieariy propery, a liear sysem has he impora propery of superposiio: superposiio If a ipu cosiss of weighed sum of several sigals ( a b ), he oupu is also weighed sum of he resposes of he sysem o each of hose ipu sigals ( a y b y ( )) ( The superposiio propery cosiss of wo pars: Addiiviy: The respose o { } is { y( y( } Homogeeiy: The respose o a x ( ) is a y ( ), where a is ay real umber if we are cosiderig oly real sigals ad a is ay complex umber if we are cosiderig complex valued sigals This meas ha if a sysem is homogeeous, he he scaled ipu gives a scaled oupu (some scalig facors) From figure -8, we see ha we ca decompose complicaed sigal x ( io a sum of simpler sigals x ( ) ad x (, ad he rea each of hese sigals hrough he sysem x ( Liear Sysem y( T x ( x ( ) y Tx ( ) Liear Sysem ( a bx ( Liear Sysem y( ay( by ( Figure -8: Lieariy Sysem lieariy checks To deermie ha he sysem is liear, use he followig seps (see also he figure -8): Form he sum ay( by( cosiderig wo ipu-oupu relaioship y ( ) ad y ( ) Cosruc he respose, ax bx ( T (, of he ipu: a b Check for equaliy he respose of sep wih he respose of sep, if hese wo resposes are equal, he he sysem is liear 4
Coiuous ime sysems: a) y( 5 Soluio: From sep, we cosider wo ipus ad oupu sigals muliplied by scalars Sice y( 5, we have y( 5, y( 5 ad he sum weighed by wo cosas is: ay( by( 5a 5b From sep, we use he sum ax bx ( ) as a ipu o he sysem, where ( ax bx ( 5 ax ( bx ( 5ax ( 5bx ( T ( From sep, hese equaios are equal, ad he he sysem is liear b) y( Rx (, where R is a cosa Soluio: From sep, we have: ay( by( arx ( brx ( From sep, we cosider he rasformaio acig o ax Usig T T ax ( bx ( R ax ( bx ( 5 ( bx ( ) x y x x y xy y we obai he followig equaio ax ( bx ( R ax ( bx ( R a x ( a bx ( x ( ab x ( x ( b x ( From sep, he sysem is o liear d x c) y( d Soluio: d x d x ay ( by( a b d d ow d a b d x ( d x ( T a b a b d d d i his case, he sysem is liear, as he resul of he firs sep ad secod sep are equal Discree ime sysems: a) y [ 5] ax bx [ ax [ 5] bx [ 5] ay [ by [ T [ herefore y [ 5] is liear sysem
Dr Qadri Hamarsheh b) y [ u[ k], k ax bx [ ax [ u[ k] bx [ u[ k] ay [ by [ T [ liear sysem Sable ad o-sable sysems (Sabiliy): There are several defiiios for sabiliy Here we will cosider bouded ipu boded oupu (BIBO) sabiliy A sysem is said o be BIBO sable if every bouded ipu produces a bouded oupu We say ha a sigal x [ is bouded if we ca fid a cosa M such ha for all, x [ M, ad we say ha he oupu sigal y [ is also bouded if we ca fid a cosa K such ha y [ K a) The movig average sysem defied by y [ k] is sable as k y [ is sum of fiie umbers ad so i is bouded b) The accumulaor sysem defied by y [ k] is usable If we ake k u[, he ui sep he y [ ], y[], y[], y[, so y[ grows wihou boud c) y [ 7 ], assume ha, M for some fiie M for all I his case M implies ha y[ 7M, so he sysem is sable d) y [ ], assume ha, M for some fiie M for all I his case, sice we have y [ ], were he oupu direcly depeds o, i grows wihou boud as icreases, so he sysem is o sable Time-Ivaria sysems (Time ivariace): A sysem is said o be ime ivariace (TI) if he behavior ad he srucure of he sysem do o chage wih ime (TI sysem respods exacly he same way o maer whe he ipu sigal is applied) Thus a sysem is said o be ime ivaria if a ime shif (delay or advace) i he ipu sigal, x ( ), leads o ideical delay or advace i he oupu sigal Mahemaically If y[ T{ } he y[ ] T{ ]} for ay Time ivariace esig seps (see figure -9): Fid he shifed oupu y ] of he sysem [ Fid he oupu of he shifed ipu y [ T{ ]} Compare sep ad sep, if hey are equal, he he sysem is ime ivariace 6
Ipu sigal Track Shif o he lef or righ Track ] Respose o he shifed sigal Track y[ T{ ]} - y d [ y[ y [ Track Respose o he ipu sigal Track y[ Shifed oupu y[ y[ ] Track Figure -9: Time Ivariace Tesig seps Coiuous ime sysems: Deermie wheher or o he sysem is a ime-ivaria for? y( To solve his ask, we go hrough he seps described above for ime ivariace esig: Sep: he resul of he shifed oupu: y( ) ( ) ) Sep : he oupu of he shifed ipu: y ( T{ )} ) Sep : Comparig he wo oupus we see ha hey are o equal, so his sysem is ime varyig Discree ime sysems: Deermie wheher or o he sysems are ime-ivaria? a y [ 7 7] Soluio: Sep: he resul of he shifed oupu: y [ ] 7 7] Sep : he oupu of he shifed ipu: y T{ ]} 7 7] Sep : Comparig he wo oupus we see ha hey are equal, so his sysem is ime ivaria b y[ 7 Soluio: Sep: he resul of he shifed oupu: y[ ] 7( ) ] Sep : he oupu of he shifed ipu: y T{ ]} 7 ] Sep : Comparig he wo oupus we see ha hey are o equal, so his sysem is ime varyig 7
c Accumulaor sysem : y [ k] k Soluio: Sep: he resul of he shifed oupu is give by: y [ ] Sep : he oupu of he shifed ipu: y { k k] T ]}, le x ], he he correspodig oupu is [ [ x k y [ k] k] k Sep : Comparig he wo oupus: The wo expressios are equal To ge ha, le us chage he idex of summaio by i k i he secod sum he we see ha y[ i] y[ ] i So his sysem is ime ivaria A sysem ca be liear wihou beig ime ivaria ad i ca be ime ivaria wihou beig liear 8