14.4 Cliararteristic Sequences arid System Functioiis of Discrete LTI-Systernd343 Figure 14.4: z k arc eigenseries of discrcte LTJ-systrems, H ( z ) is the eigenvalue is an eigenseries of ari LTI-sysstem The (*o1responding eigenvaliies X depend in general on z As with contiiiuoiis systems we call X = H ( z ) the s y s f m f u n c f m n n of R discretc LTJ-system ( function for contiiiuous s in fiinction for discrete hystenis Ff(s)has a region of roiivergence which contains only part of the roinplex plane Exciting t he syst em with a complex exponential sequence outside the region of convergence does iiot lcad to an output series with finite amplitude 11s with continuous systeins, the region of cwivergence of H ( z ) is oftcm riot exglicilly given Finally, it should also be nientioned t h a t a unilateral exporieribial series z [ k ]= Zk: € [ k ] is (1 4.13) not an eigensecluence of a disc*ret!esystem The system function H'jz) ran describe the system response to all input sigiials r[k],anti riot just eigensequences e [ k ] = z k l ' h e relationship between a genmal discrete input xeq~~erire z [ k ] and an exponential sequence is represented by the z-transform, which is given here as tlie inverse transform of the input sigrial n:[k]= y 2n3 dz X(z)zL- (14.14) and the output signal (14.15) of an LTI-systmi Thc output sequence y[k] is obtained as the system response to [k] y[kj = S { z [ k / )= Y ( ) = H ( z ) X ( z ) dz z (14.16) (I 4.17)