350 14 Discrete-Time LTI-Systeriis The second-order discrete system in Figure 13.1I is described by the followiilg matrices of the stat,e-space description: A = [ 10 : B = [ i ] , f ] , C=[O D=1 W e calculate the response of the system for ~ [ k=] 0, and the initial values y[0] and y(Z], using the formula (cornpare (7.64) - (7.66)): = -.A-I Y ( z )= G ( z ) z ~ , G ( z )= C(zE - A)-' 22 -1 11 z] The conriection between the given initial values y[O] arid yjl] mid the statc vector zo = z[/], which caiiseb the same output signal, can be wad froin Figure 14.11 21/11 = z2[1] = 4y[l] y[O], IE;lroin that we obt>ain yt4 = + I - (Y[OI z4 ~ ~ ) (14.32) and splitbirig into partial fractions Finally, the output signal is foiind by inverse z-transformation As we bid taken the initial state at XiJ = , thc calculated uiitput, 4gnal also starts a t k = 1, and delivers the expected value g[11 This can he confimed by insertion iril o the givw diffeience equation In Clinpter we introduced the iiripiilse rcsponse c2s a second irriportant characterist ic for contimious systems, in addition to thc system function Ii,s irnportancr can be expressed in three significant properties: