Lier Time-Ivrit Bsic Properties LTI The Commuttive Property The Distributive Property The Associtive Property Ti -6.4 / Chpter Covolutio y ] x ] ] x ]* ] x ] ] y] y ( t ) + x( τ ) h( t τ ) dτ x( t) * h( t) The Commuttive Property x ]* ] ]* x ] Let r- or -r; substitutig to covolutio sum: r x ]* ] x ] ] x r] r] ]* x ] Ti -6.4 / Chpter 3 Ti -6.4 / Chpter 4 The Commuttive Property x ] ] y] ] x ] y] The output of LTI system with iput x] d uit impulse respose ] is ideticl to the output of LTI system with iput ] d uit impulse respose x] The Distributive Property x ]* ( + ) x ]* + x ]* h ] The distributive property hs useful iterprettio i terms of system itercoectios > PARALLEL ITERCOECTIO Ti -6.4 / Chpter 5 Ti -6.4 / Chpter 6
The Distributive Property x ] h ] + y ] h ] x ] + h ] y ] y] y] The Associtive Property ( * ) x ]* ( x ]* )* As cosequece of ssocitive property the followig expressio is umbiguous y ] x ]* ]* ] Ti -6.4 / Chpter 7 Ti -6.4 / Chpter 8 The Associtive Property x ] h ]* ( * ) y ] x ]* y] ( x ]* )* y ]* y ] y ] x ] h ] h ] y] The Associtive Property The ssocitive property c be iterpreted s > SERIES (OR CASCADE) ITERCOECTIO OF SYSTES Ti -6.4 / Chpter 9 Ti -6.4 / Chpter The Associtive d Commuttive Property y ] x ]* ( * ) x ]* ( * ) x ] h ]* y ] ( x ]* )* y ]* y ] y ] x ] y ] The Properties of Cscde Coectio of The order of the systems i cscde c be iterchged The itermedite sigl vlues, w i ], betwee the systems re differet Ti -6.4 / Chpter Ti -6.4 / Chpter
The Cscde Coectio of y ] x ] h ] h ] y ] x ] h ] * x ] h ]* y ] y] y ] x ] y ] The Cscde Coectio of The properties of the cscde system deped o the sequetil order of cscded blocs The behvior of discrete-time systems with fiite wordlegth is sesitive to sigl vlues, w i ], betwee the blocs Wht is the optiml sequetil order of cscded blocs? Ti -6.4 / Chpter 3 Ti -6.4 / Chpter 4 Stbility for LTI Defiitio of stbility: A system is stble if d oly if every bouded iput produces bouded output Bouded-iput bouded-output (BIBO) stbility Stbility for LTI Cosider iput x] tht is bouded i mgitude x] < B for ll The output is give by the covolutio sum y ] ] x ] y ] ] x ] Ti -6.4 / Chpter 5 Ti -6.4 / Chpter 6 Stbility for LTI For bouded iput x-]<b y ] B ] h ] < for The output y] is bouded if the the impulse respose is bsolutely summble A SUFFICIET CODITIO FOR STABILITY! ll The Uit Step Respose The uit step respose is ofte used to chrcterize LTI systems s ] u ] * ] Ti -6.4 / Chpter 7 Ti -6.4 / Chpter 8 u ] ] u ]
Ivertible Cusl LTI ] Accumultor s] Iverse Accumultor s ] ] ] s ] s ] ] The step respose of DT LTI system is the ruigsum itsimpulse respose d the impulse respose of DT LTI system is the first differece of its step respose Cotiuous-time systems: Lier costt-coefficiet differetil equtios re used to describe wide vriety of systems d physicl pheome Discrete-time systems: Lier costt-coefficiet differece equtios re used to describe the sequetil behvior of my differet proceses Ti -6.4 / Chpter 9 Ti -6.4 / Chpter Lier Costt-Coefficiet Differece Equtios th-order costt-coefficiet differece equtio y ] The coditio for iitil rest: b x ] If x] for <, the y] for < With iitil rest the system is LTI d cusl Lier Costt-Coefficiet Differece Equtios Rerrgig d solvig for y] y ] b x ] y ] The output y] t time is expressed i terms of previous vlues of the iput d output Auxiliry coditios: I order to clculte y] we eed to ow y-],, y-] Ti -6.4 / Chpter Ti -6.4 / Chpter Lier Costt-Coefficiet Differece Equtios Lier Costt-Coefficiet Differece Equtios y ] b x ] Recursive equtio: y ] The output is specified recursively i terms of the iput d previous outputs I specil cse whe the differece equtio reduces to b y ] Ti -6.4 / Chpter 3 Ti -6.4 / Chpter 4 orecursive equtio: x ] Previously computed output vlues re ot eeded to compute the preset vlue of the output Auxiliry coditios re ot eeded! ] x ]
Lier Costt-Coefficiet Differece Equtios The impulse respose correspodig to the orecursive system is ] b,, otherwise The system specified by the orecursive equtio is ofte clled Fiite Impulse Respose (FIR) system Lier Costt-Coefficiet Differece Equtios Cosider the differece equtio y ] y ] x ] y ] x ] + y ] The impulse respose is obtied s respose to x]δ] : : y ] δ ] + y ] + y ] y ] δ ] + y ] + y ] Ti -6.4 / Chpter 5 Ti -6.4 / Chpter 6 Impulse Respose of First-Order System The impulse respose c be writte s h ],, ] u ] < Lier Costt-Coefficiet Differece Equtios Differece equtios with > re recursive d result i impulse resposeof ifiite legth The systems specified by recursive equtios re clled Ifiite Impulse Respose (IIR) systems I geerl, recursive differece equtios will be used i describig d lyzig discrete-time systems tht re lier, time-ivrit, d cusl, d cosequetly the ssumptio of iitil rest will usully be mde Ti -6.4 / Chpter 7 Ti -6.4 / Chpter 8 Bloc Digrm Represettio of Bsic elemets: Additio: ultiplictio: x ] x ] + x ] + x ] x ] x ] First Order Differece Equtio y ] + y ] bx ] y ] bx ] y ] b x ] + y] D y ] Uit dely: x ] D x ] Bloc digrm represettio of cusl discrete-time system ( first-order IIR digitl filter) Ti -6.4 / Chpter 9 Ti -6.4 / Chpter 3
Lier Costt-Coefficiet Differetil Equtios Cosider first order differetil equtio dy( t) + y( t) x( t ) where y(t) deotes the output of the system d x(t) is the iput Differetil equtios provide implicit specifictio of the system, i.e., the reltioship betwee the iput d output Lier Costt-Coefficiet Differetil Equtios I orderto obti explicit solutio, the differetil equtio must be solved ore iformtio is eeded th tht provided by the equtio loe, i.e., uxiliry coditios must be specified > A differetil equtio describes costrit betwee the iput d output of the system, but to chrcterize the system completely uxiliry coditios must be specified Ti -6.4 / Chpter 3 Ti -6.4 / Chpter 3 Lier Costt-Coefficiet Differetil Equtios The respose to iput x(t) will geerlly cosist of the sum of prticulr solutio, y p (t), to the differetil equtio - sigl of the sme form s the iput - i.e., the forced respose, d homogeeous solutio, y h (t), - solutio to the differetil equtio with the iput set to zero - i.e., the turl respose y (t) y p (t)+ y h (t) Lier Costt-Coefficiet Differetil Equtios Auxiliry coditios must be specified: Differet choices of uxiliry coditios leo differet reltioships betwee the iput d output For the most prt, the coditio of iitil rest will be used for systems described by differetil equtios, e.g., x(t) for t<, the coditio for iitil rest implies the iitil coditio y() Ti -6.4 / Chpter 33 Ti -6.4 / Chpter 34 Lier Costt-Coefficiet Differetil Equtios Lier Costt-Coefficiet Differetil Equtios Uder the coditio of iitil rest the system is lier time-ivrit (LTI) d cusl The coditio of iitil rest does ot specify zero iitil coditio t fixed poit of time, but rther djusts this poit i time so tht the respose is zero util the iput becomes ozero For exmple, if x(t) for t <t for cusl LTI system described differetil equtio,the y(t) for t <t d the iitil coditio y(t ) would be useo solve the output for t>t A geerl th-order lier costt-coefficiet differetil equtio is give by d y t b d ( ) x ( t ) where the order refers to the highest derivtive of y(t) I the csewhe y t b d x ( t ) ( ) y(t) is explicit fuctio of x(t) d its derivtives Ti -6.4 / Chpter 35 Ti -6.4 / Chpter 36
Lier Costt-Coefficiet Differetil Equtios For >, the output is specified implicitly by the iput The solutio of the equtio cosists of two prts: prticulr solutio d solutio of the homogeeous differetil equtio The solutios to the homogeeous differetil equtio d y ( t ) re referred to s turl resposes of the system Lier Costt-Coefficiet Differetil Equtios I order to determie the iput-output reltioship of the system completely, uxiliry coditios must be idetified Differet choices of uxiliry coditios result i differet iput-output reltioships The coditio of iitil rest: If x(t) for t<t, it is ssumed tht y(t) for t<t d, therefore, the respose for t>t c be clculted from dy( t ) d y( t ) y( t)... Uder the coditio of iitil rest, the system is cusl d LTI Ti -6.4 / Chpter 37 Ti -6.4 / Chpter 38