In order to simplify this discussioir, we h>t A T = N aaid allow some, but not all of tlrt.. FVI initial condition prohlcriis, these would be N initial coritlitioris y0.. Plugging ll-re
Trang 12.2 Block Dictarams I 9
(2.3)
T l i ~ greatest index N of cl xron-zero coefficieii( (1: R; cleterniirios wliat is isallrd the ordc~i o j f h e drflf rrntiul f p a t i o a In order to simplify this discussioir, we h>t
A T = N aaid allow some, but not all of tlrt corffic*icvits Nc; to hc cqunl to zero
clifrerent hrrearly ~ r i d t y f n d e r i t sci- lutioiih g ( t ) to (2.3) For il p~iiticular soliition, wp 11ccd t o gibe N conciiLiori5 FVI initial condition prohlcriis, these would be N initial coritlitioris y(0) Q(0) i j ( O ) ,
T h e dif€aerrritial eqiiation ( 2 3 ) tlcstril )es a continuous-time systeiir, if c ( f ) is t hc input signal, and y(1) is t’ttc outpiit sign;d In order to chsra
c w rcfm back t o Ueiinitions 3 ant1 5 a i i d also Figs 1.9 and 1.1 0 For now, RC?
igriove possibly given initial conditions; thcir iiifluence will l x (fisctiss~l in i
irr Chaptci 7
T t cm Thl ough mb-
s l i t u t i o n of variables t’ = t - z in ( 2 3 ) , it f o l l o ~ s iriiiiirtf that r ( l - z) leads to the solut,ion y(f - T) To show lincaritv IW consider the tn7o tliffererit i r i p u ( signals
X I ( t ) tmtl .x.a(-t) a n d the corresporitlirrg solutions y1 ( t ) and pd(t) Plugging ll-re linear c~4ucttion r j ( l ) 1 Ay1 ( t ) + Bsdjl) into ( 2 3 ) verifies that y i ( t ) = Aylji.) 1 L 3 ! / 2 ( ( )
is a soltition of- t h e di ritial c~jnaticiii, and thcretorc the uul,pixt sigrial of i hc> system
Every system that citii be nioctelled tising linear dillereritid rtpiations with coiist,anf coefficients (2.3) is thus an LT1 system This means we Imve foiintl our first method kor rrrodrlling such s w t pin5 in the folm ol a tlilftLient ial cqiic2tic)ii Tliis mrthotl fiilfills OUT initial requiieniexits
Foi a givcn fimction .r(t) t’rieic are up to
Lc-t 11% r i m ’ show that (2.3) repremits it tirw-i3ivari
Modelling of an LTI-system iiidq)eiidrnt fi on1 i t h realisation
T3cpiel;ciitatit.m of the input outpiit relationship, without det ails of thc sys- tem’h ir it ei no1 betimiour ,
ra
(wi rc.present inore irifbc)Imst ion than cliilcrcntial equation show riot only thc input a i d outpiit signals h t also i n t t ~ r i a l s t a t P s of a s
only t tic input-output rclat iomliip i a of iritcwst t h i the choice ol int vxrial st,ates