342 13 IJiscrete-Time LTI-Systmis initial states I oneffect output I Figure 14.2: Combination of the external and intcrnal tcrnis to forin tlic solrrtiori of a differential equation eigeiisequences ale completely analoguotis to those of cigenfiinctions of continuous LTI-systems in accordaiice with Definition 10 in Chapter 3.2 An eigenseries at tJie iiiput of a discrete LTX-system causes a response at the output that corresponds t o the input serieb with a constarit factor (see Figure 14.3) The p r o d that eigenseries of discrete LTI-systems are exponential series of the form c[k]= z k will be carried out exactly as for tlie eigenfniict,ions in Chapter 3.2.2 We consider a genrral LTII Figure 14.3: System S is excited by the eigcmcries c [ k ] system (Fignre 14.1) and only require that, it is linear (14.1) arid time-irivariant (14.2), (14.3) The iiiput signal should be an exponerrtial series a [ k ] = z k We want, to find thc corresponding oikpiit signal y[k] = S { } (14.9) F'roin the conditions of time-irivarittnce a i d linearity: y[k - /€I = S{r[X - 4)= s { z k - " } z - " s ( Z ~ } = :-"y[k] These difference equations are only fulfilled at thta same time for any a weiglitetl cxporiential series y[k] = x z h (14.10) K, if y[k] is (14.11) Equating x [ k ]= z k and y[k] = Xzh yields that tveiy exponential series t[kj Z X k , z €42 (14.12)