2.2 Block Dictarams I9 (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*icvitsNc; to hc cqunl to zero clifrerent hrrearly ~ridtyfnderitsciFoi a givcn fimction r(t) t’rieic are up to lutioiih g ( t ) to (2.3).For il p~iiticularsoliition, wp 11ccd to gibe N conciiLiori5 FVI initial condition prohlcriis, these would be N initial coritlitioris y(0) Q(0) i j ( O ) , The dif€aerrritial eqiiation ( ) 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 to Ueiinitions ant1 a i i d also Figs 1.9 and 1.10 For now, RC? igriove possibly given initial conditions; thcir iiifluence will l x (fisctiss~lin i irr Chaptci Lc-t 11%r i m ’ show that (2.3) repremits it tirw-i3ivari T t cm Thl ough mbs l i t u t i o n of variables t’ = t - z in ( ) , it f o l l o ~ iriiiiirtf s that r ( l - z) leads to the solut,ion y(f - T) To show lincaritv IW consider the tn7o tliffererit iripu( 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 ) 1Ay1( t ) Bsdjl) into ( ) verifies that y i ( t ) = Aylji.) L ! / ( ( ) is a soltition of- t h e di ritial c~jnaticiii,and thcretorc the uul,pixt sigrial of ihc> 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 + 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 system’h ir it ei no1 betimiour , (wi rc.present inore irifbc)Imstion than cliilcrcntial equation show riot only thc input a i d outpiit signals h t also i n t t ~ r i a sl 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