4/9/2012. Signals and Systems KX5BQY EE235. Today s menu. System properties
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1 Signals and Sysems hp:// KX5BQY EE35 oday s menu Good weeend? Sysem properies iy Superposiion! Sysem properies iy: A Sysem is if i mees he following wo crieria: If { x( )} y( ) and { x( )} y( ) Addiiviy hen { x( ) x( )} { x( )} { x( )} If {()} x y () hen { ax ( )} a{ x ( )} Scaling Sysem Response o a linear combinaion of inpus is he linear combinaion of he oupus. ogeher superposiion { ax () bx ()} ay () by ()
2 iy Order of addiion and muliplicaion doesn maer. Sysem s y Sysem (), y () combinaion ay () by () x (), x () combinaion Combo s = ax() bx() { ax( ) bx( )} Sysem iy Posiive proof Prove boh scaling & addiiviy separaely Prove hem ogeher wih combined formula Negaive proof Show eiher scaling OR addiiviy fail (mahemaically, or wih a couner example) Show combined formula doesn hold iy Proof Combo Proof Sep : find y i () Sep : find y_combo Sep 3: find {x_combo} Sep 4: If y_combo = {x_combo} Sysem s y(), y() Sysem combinaion ay () by () x (), x () Combo s combinaion ax() bx() { ax( ) bx( )} Sysem
3 iy Example Is linear? x() y()=cx() y () cx (); y () cx () ay () by () acx () bcx () c( ax () bx ()) { ax ( ) bx ( )} c( ax ( ) bx ( )) Equal iy Example Is linear? x() y()=(x()) y () ( x ()) ay() a( x()) { ax ( )} ( ax ( )) a( x ( )) No equal non-linear iy Example Is linear? x() y()=x()+5 y () x () 5 ay( ) a( x( ) 5) ax( ) 5a { ax( )} ax( ) 5 No equal non-linear 3
4 iy Example Is linear? y () x ( ) d y () x ( ) d y () x ( ) d ay () by () a x ( ) d b x ( ) d ( ax ( ) bx ( )) d { ax ( ) bx ( )} ( ax ( ) bx ( )) d = iy unique case How abou scaling wih 0? y () { x ()} ay() a{ x()} 0 ifa 0 { ax ( )} ay ( ) 0 if linear If {x()} is a linear sysem, hen zero inpu mus give a zero oupu A grea negaive es Non-iy Rules of humbs muliplying x() by anoher x() y()=g[x()] where g() is nonlinear piecewise definiion of y() in erms of values of x, e.g. x x () x () 0 y () x () x () x () 0 (alhough someimes o) 4
5 Superposiion Superposiion is If x( ) y( ) x() a x () y() a y () Weighed sum of inpus weighed sum of oupus Divide & conquer Superposiion example Graphically x () y () x () y () 3 x () x () -? y () y () y () -y () 4 Superposiion example Slighly aside (same sysem) x () y () x () Is i ime-invarian? i i? No idea: no enough informaion Single inpu-oupu pair canno es posiively y () 3 5 5
6 Superposiion example Unique case can be used negaively x () x () y () y () NO ime Invarian: Shif by shif by x ()=u() S y ()=u() NO Sable: Bounded inpu gives unbounded oupu 6 Summary: Sysem properies Causal: oupu does no depend on fuure inpu imes Inverible: can uniquely find sysem inpu for any oupu Sable: bounded inpu gives bounded oupu ime-invarian: ime-shifed inpu gives a ime-shifed oupu : response o linear combo of inpus is he linear combo of corresponding oupus Impulse response (Definiion) Any signal can be buil ou of impulses Impulse response is he response of any ime Invarian sysem when he inpu is a uni impulse Impulse Response h() 6
7 Briefly: recall superposiion Superposiion is If x ( ) y ( ) x() a x () y() a y () Weighed sum of inpus weighed sum of oupus Using superposiion Easies when: x () are simple signals (easy o find y ()) x () are similar for differen wo differen building blocs: Impulses wih differen ime shifs Complex exponenials (or sinusoids) of differen frequencies If x() y() x() a x () y() a y () Briefly: recall Dirac Dela Funcion () 0 for 0 () d x( ) ( ) x( ) ( ) x() -3) 3 Go a gu feeling here? x-3) 3 7
8 Building x() wih δ() Using he sifing properies: x ( ) ( ) d x ( ) ( ) d x( ) ( ) d x( ) Change of variable: x() ( ) d x( ) 0 0 From a consan o a variable 0 x( ) ( d ) x ( ) = x( ) ( ) d? 8
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