IMPLEMENTING UNSTABLE FILTERS VIA INPUT SIGNAL REVERSAL - A STATE-VARIABLE PERSPECTIVE

Consider the response of an unstable LTI filter to an input over a tim e interval [0, T]. The total response consists of the signal response and the natural response of the filter, The latter grows exponentially so that an unstable filter is not usable, This note shows that by hol ding the final value of the filter fixed to a finite value (say, zero) , the natural response can be forced to be negligible over some initia l part of the time interval [0, T] so that the filter can be used, A m eans of implementing this idea is to reverse the input signal across t he interval [0, T] as suggested in [1]-[3]. The purpose of this brief is to present a state-variable perspective of this technique. An examp le is shown to implement a filter with no phase shifts.