h[n] is the impulse response of the discrete-time system:

Transcription

1 Definiion Examples Properies Memory Inveribiliy Causaliy Sabiliy Time Invariance Lineariy Sysems Fundamenals Overview Definiion of a Sysem x() h() y() x[n] h[n] Sysem: a process in which inpu signals are ransformed by he sysem or cause he sysem o respond in some way, resuling in oher signals as oupus. All of he sysems ha we will consider have a single inpu and a single oupu All of he signals ha we will consider are liewise univariae We will use he noaion x() y() o mean he inpu signal x() causes an oupu signal y() h() is he impulse response of he coninuous-ime sysem: δ() h() h[n] is he impulse response of he discree-ime sysem: δ[n] h[n] J. McNames Porland Sae Universiy ECE 222 Sysem Fundamenals Ver J. McNames Porland Sae Universiy ECE 222 Sysem Fundamenals Ver Scope of Sysems In his class we will primarily wor wih circuis as sysems In mos cases a volage or curren will be he inpu signal o he sysem Anoher curren or volage will be he oupu signal of he sysem However, our reamen applies o a much broader class of sysems Examples Circuis Moors Chemical processing plans Engines Spring-mass sysems Memory Memoryless: A sysem is memoryless if and only if he oupu y() a any ime 0 depends only on he inpu x() a ha same ime: x( 0 ). Memory indicaes he sysem has he means o sore informaion abou he inpu from he pas or fuure Capaciors and inducors sore energy and herefore creae sysems wih memory Resisors have no such mechanism and are herefore memoryless sysems: v() =Ri() J. McNames Porland Sae Universiy ECE 222 Sysem Fundamenals Ver J. McNames Porland Sae Universiy ECE 222 Sysem Fundamenals Ver

2 Example 1: Memoryless Sysems Deermine wheher each of he following sysems are memoryless. = n = x[] Example 1: Worspace J. McNames Porland Sae Universiy ECE 222 Sysem Fundamenals Ver J. McNames Porland Sae Universiy ECE 222 Sysem Fundamenals Ver x[n] h[n] Inveribiliy g[n] Inverible: A sysem is inverible if and only if disinc inpus cause disinc oupus. If he sysem is inverible, hen an inverse sysem exiss When he inverse sysem is cascaded wih he original sysem, he oupu is equal o he inpu Normally you can es for inveribiliy by rying o solve for he inverse sysem Alernaively, if you can find wo inpu signals, x 1 () x 2 () ha boh generae he same oupu, he sysem is no inverible x[n] Example 2: Inverible Sysems Deermine which of he following are inverible sysems. If he sysem has an inverse, sae wha i is. = n = x[] J. McNames Porland Sae Universiy ECE 222 Sysem Fundamenals Ver J. McNames Porland Sae Universiy ECE 222 Sysem Fundamenals Ver

3 Example 2: Worspace Causaliy Causal: A sysem is causal if and only if he oupu y() a any ime 0 depends only on values of he inpu x() a he presen ime and possibly he pas, << 0. These sysems are someimes (rarely) called nonanicipaive If wo inpus o a causal sysem are idenical up o some poin in ime, he oupus mus also be equal All analog circuis are causal All memoryless sysems are causal No all causal sysems are memoryless (very few are) Some discree-ime sysems are non-causal J. McNames Porland Sae Universiy ECE 222 Sysem Fundamenals Ver J. McNames Porland Sae Universiy ECE 222 Sysem Fundamenals Ver Example 3: Causal Sysems Deermine which of he following are causal sysems. y() = = = 5 x[n + ] Example 3: Worspace J. McNames Porland Sae Universiy ECE 222 Sysem Fundamenals Ver J. McNames Porland Sae Universiy ECE 222 Sysem Fundamenals Ver

4 Sabiliy BIBO Sable: A sysem is bounded-inpu bounded-oupu (BIBO) sable if and only if (iff) all bounded inpus ( x() < ) resul in bounded oupus ( y() < ). Informally, sable sysems are hose in which small inpus do no lead o oupus ha diverge (grow wihou bound) All physical circuis are echnically sable Ideal op amp circuis wihou negaive feedbac are usually unsable Examples: hermosa, cruise conrol, swing Couner-examples: savings accouns, invered pendulum (quesionable), chain reacions Example 4: Sysem Sabiliy Deermine which of he following are BIBO sable sysems. If he sysem is no BIBO sable, specify an inpu signal ha violaes his propery. y() = = = 5 x[n + ] J. McNames Porland Sae Universiy ECE 222 Sysem Fundamenals Ver J. McNames Porland Sae Universiy ECE 222 Sysem Fundamenals Ver Example 4: Worspace Time Invariance Time Invarian: A sysem is ime invarian if and only if x[n] implies x[n n 0 ] y[n n 0 ]. In words, a sysem is ime invarian if a ime shif in he inpu signal resuls in an idenical ime shif in he oupu signal Circuis ha have non-zero energy sored on capaciors or in inducors a =0are generally no ime-invarian Circuis ha have no energy sored are ime-invarian Memoryless does no imply ime-invarian: y() =f() x() In general, if he independen variable, or n, is included explicily in he sysem definiion, he sysem is no ime-invarian J. McNames Porland Sae Universiy ECE 222 Sysem Fundamenals Ver J. McNames Porland Sae Universiy ECE 222 Sysem Fundamenals Ver

5 Tesing for Time Invariance x() S y() D y( 0 ) x() D x( 0 ) S y d () To es for ime invariance, you should calculae wo oupu signals Firs, calculae he delayed oupu, y( 0 ) in response o he original signal Second, calculae he oupu due o he delayed inpu, y d (). If hese are equal for any inpu signal and delay 0, he sysem is ime-invarian. Oherwise, i is no. Example 5: Time Invariance Deermine which of he following are ime-invarian sysems. If he sysem is no ime invarian, specify an inpu signal ha violaes his propery. y() =x(2) =x[ n] =nx[n +3] y() = = = 5 x[n + ] J. McNames Porland Sae Universiy ECE 222 Sysem Fundamenals Ver J. McNames Porland Sae Universiy ECE 222 Sysem Fundamenals Ver Example 5: Worspace Lineariy x() h() y() x[n] h[n] Consider any wo bounded inpu signals x 1 () and x 2 (). x 1 () y 1 () x 2 () y 2 () Linear: A sysem is linear if and only if a 1 x 1 ()+a 2 x 2 () a 1 y 1 ()+a 2 y 2 () for any consan complex coefficiens a 1 and a 2. J. McNames Porland Sae Universiy ECE 222 Sysem Fundamenals Ver J. McNames Porland Sae Universiy ECE 222 Sysem Fundamenals Ver

6 Lineariy Coninued a 1 x 1 ()+a 2 x 2 () a 1 y 1 ()+a 2 y 2 () a 1 x 1 [n]+a 2 x 2 [n] a 1 y 1 [n]+a 2 y 2 [n] There are wo relaed properies Addiive: x 1 [n]+x 2 [n] y 1 [n]+y 2 [n] Scaling: ax 1 [n] ay 1 [n] Scaling is also called he homogeneiy propery U1 Lineariy Coninued a x () U1 a u x u ()du U 0 a x [n] U 0 a y () a u y u ()du a y [n] Linear sysems enable he applicaion of superposiion If he inpu consiss of a linear combinaion of differen inpus, he oupu is he same linear combinaion of he resuling oupus This also wors for infinie sums (inegrals) J. McNames Porland Sae Universiy ECE 222 Sysem Fundamenals Ver J. McNames Porland Sae Universiy ECE 222 Sysem Fundamenals Ver Example 6: Lineariy Deermine which of he following are linear sysems. y() =x(2) =x[ n] =nx[n +3] y() = = = 5 x[n + ] Example 6: Worspace J. McNames Porland Sae Universiy ECE 222 Sysem Fundamenals Ver J. McNames Porland Sae Universiy ECE 222 Sysem Fundamenals Ver

7 Linear Time-Invarian (LTI) Sysems x() h() y() x[n] h[n] Asysemissaiobelinear ime invarian (LTI)ifiisboh linear and ime invarian All of he circuis we will wor wih are linear The circuis may no be ime invarian if here is some iniial energy sored in he circui Oherwise he circuis are LTI ECE 222 & ECE 223 will focus primarily on he properies, analysis, and design of LTI sysems J. McNames Porland Sae Universiy ECE 222 Sysem Fundamenals Ver