Wiley signals and systems e book TLFe BO 405

16 Stability and Feeclback Svsteriis 390 16.22 Discrete Systems A rorrcspondirrg result tiolds for discrete systems: The poles of the system fiinction H ( z ) of a causal arid stable discrete LTI-s,ybtern lie within the unit circle of the x-plane IIere we can also see the connection betweerl the locsation of tlie poles in Che complex frequency plane arid the system's characteristic frequencies in the timedomain (sec Exercise 16.5) The stability conditions can alsn be illustrated to aid understauding, as in Figure 13.6 The internal term of the system ~esponsehere only decays if thr corresponding poles lie within the unit circle Again, the initial states have rio influence if one waits long enough Example 16.3 Figarc 16 shows four pole-zero tliagrnnis of both contix~uous(left) a i d discrete (right) system We can bee inixnediately fiom tlie locatioix of the poles tlmt only the first and third conlirruous system (likewise discrete systeixi) are stable If, hom7ever, biliiteral impulsc responses are permitted - systems which are not causal - the first three continuous systems and tlie first, second and fourth disciete system are stable The ROC must be chosen so t h a t it lnrludes the imaginary axis of the s-plane,