Implementing unstable filters via input signal reversal-a state variable perspective

Consider the response of an unstable LTI filter to an input over a time 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. The authors show that by holding the final value of the filter fixed to a finite value (say, zero), the natural response can be forced to be negligible over some initial part of the time interval [0, T] so that the filter can be used. A means of implementing this idea is to reverse the input signal across the interval [0, T]. The purpose of this work is to present a state-variable perspective of this technique. An example is shown to implement a filter with no phase shifts.