14 Discrete-Time LT1-Systcms 340 - As in (1.3), -4 aiid R can be any complex constants A discrete system S is famc-znvarznnl if for the rtv3ctioli y[k] to any input signal ~ [ k ] y[k] = S { z [ k ] } (14.2) t he relationsliip g [ k - K ] = S{lr[C- K j ) V K E (14.3) holtfs If the index k is in ad of time, a discrete position variable, for example, the systcm S is called ship-?n.ciarzcxnt In any case the shifl is oiily tlcfinied for integer valuw (if PG, as tlie digerence X* - K inlist give aii integer index of the series ~ [ k ] 14.3.1 Difference Equations As 1irietLI differential equations with const ant coefficients rorrwpond to miitinuous LTl-systenis, linear diffcrcnce equations with constant coefffcieiits cherarterim: discrete LTI-syslenis Exarnples for such difference equations arc ,yik] + -y[k-1] 1 ~ [ k-],pjk-l] + -y[k-’2] 3 = r[k] (34.4) = ~ [ k+]~ [ k - - l ] 7~[k-2] (14.5) The general form of itli N t l i - ~ ~ t l elinear r tiiffermce equation with constant coetficients a,, h, is In contrat to the rorresponding form (2.3) of differential rquatioii, a shift of the time index C by n values ocxiirs in place of the nth derivatives For a given input signal cr[k],the difference eyuatioii (14.6) has XI infiiilte number of solutions y[k] that caii iii general be formed from IV independent linear wiiiponents, hi1 imwmbiguous sctltition can be obt2tined with N conditions, given by initial vdiies instead rimtives, for example, gJ[O],y[-1], y[-2j, p &N t11 Using the superposition principle (14.2) and the condition for tineinvariance { 14.3), it can be confirmed that a systcAm described by the difference equation (14.6) is a discrete LTI-system