6/27/2012. Signals and Systems EE235. Chicken. Today s menu. Why did the chicken cross the Möbius Strip? To get to the other er um

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1 Signals and Sysems EE35 Chicken Why did he chicken cross he Möbius Srip? To ge o he oher er um Today s menu Sysem properies Lineariy Time invariance Sabiliy Inveribiliy Causaliy Los of examples! 1

2 Sysem properies Lineariy: A Sysem is Linear if i mees he following wo crieria: If T{ x1( )} y1( ) and T{ x( )} y( ) Then T{ x1( ) x( )} T{ x1( )} T{ x( )} If T {()} x y () Then T{ ax ( )} at{ x ( )} Sysem Response o a linear Sysem Response is he same no combinaion of inpus is he linear maer combinaion when you of run he he oupus. sysem. Time-invariance: A Sysem is Time-Invarian if i mees his crierion If T{()} x y() Then T{( x )} y( ) 0 0 Sysem properies Sabiliy: A Sysem is BIBO if i mees his crierion If x( ) M Then T{ x( )} y( ) L BIBO = Bounded inpu, bounded oupu If you know he oupu signal, hen you know exacly The sysem doesn wha he blow inpu up if signal given was. reasonable inpus. Inveribiliy: A Sysem is Inverible if i mees his crierion: If T{ x( )} y( ) Ti s.. Ti{ y( )} Ti{ T{ x( )}} x( ) You can undo he effecs of he sysem. Sysem properies Causaliy: A Sysem is Causal if i mees his crierion If T{x()}=y() hen y(+a) depends only on x(+b) where b<=a The oupu depends only on curren or pas values of he inpu. The oupu The sysem depends d does only on no he anicipae curren value he inpu. of he inpu. (I does no laugh before i s ickled!) Memory: A Sysem is Memoryless if i mees his crierion If T{x()}=y() hen y(+a) depends only on x(+a) (If a sysem is memoryless, i is also causal.)

3 Tes for Causaliy Sysem is causal if oupu depends only on pas and presen inpu signal 1) y() = 4x() causal (amplificaion) ) y() = x( 3) causal (delay) 3) y() = x( + 5) non-causal (i (ime-shif forward, y(0)=x(5)) 4) y() = x(3) non-causal (speed-up, y(1)=x(3)) 5) y() = ( + 5)x() causal (ramp imes x()) 6) y() = x(-) non-causal (ime reverse, negaive ime needs fuure, y(-1)=x(1)) Wha values of 0 would make T causal? T{ x( )} x( ) 0 y () x ( ) 0 causal if 0 0 Is T causal? YES T{()} x x() d Depends only on pas and presen signals 3

4 Wha values of a would make T causal? T{()} x x( a) d a 0 1 y () x( ) d 0 y () x ( ) d T NOT causal: x() s include =+1 Causal: Change variable, y() does no depend on fuure. y () x( ) d NOT causal: x() s include = Inveribiliy es Posiive es: find he inverse For some sysems, you need ools ha we ll learn laer in he quarer Negaive es: find an oupu ha could be generaed by wo differen inpus (noe ha hese wo differen inpus migh only differ a only one ime value) Each inpu signal resuls in a unique oupu signal, and vice versa Inverible 4

5 Inveribiliy Example Is T inverible? T{()} x x() y () a x () a NOT Inverible Inveribiliy Example Is T inverible? T{()} x e x () y () a e x () ln( a) x( ) x() ln( a) YES Inveribiliy Example 1) y() = 4x() ) y() = x( 3) 3) y() = x () 4) y() = x(3) 5) y() = ( + 5)x() 6) y() = cos(x()) inverible: T i {y()}=y()/4 inverible: T i {y()}=y(+3) NOT inverible: don know sign of x() inverible: T i {y()}=y(/3) NOT inverible: can find x(-5) NOT inverible: x=0, π,4 π, all give cos(x)=1 5

6 For posiive proof: show analyically ha a bounded inpu signal gives a bounded oupu signal (BIBO sabiliy) x( ) B T{ x( )} y( ) B 1 For negaive proof: Find one couner example, a bounded inpu signal ha gives an unbounded oupu signal Some good hings o ry: 1, u(), cos(), 0 Is i sable? v () Ri () i ( ) B v ( ) Ri ( ) R i ( ) RB B 1 1 Bounded inpu resuls in a bounded oupu STABLE! How abou his? y x () 10 () Le x () M for all y( ) 10 x ( ) 10 x ( ) 10M 6

7 How abou his, your urn? y () 5 x( ) d Couner example: x()=u() y()=5u()=5r() Inpu u() is bounded. Oupu y() is a ramp, which is unbounded. No BIBO sable How abou his, your urn? y () x ( ) y () x () y () x () y () x ()cos( /3) y () 1/ x () NOT NOT 7

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