General theory of time-reversed inversion for perfect reconstruction filter banks

It is shown how to invert a MIMO system with the transfer matrix E(z). Assuming that [det E(z)] does not have any zeros on the unit circle the time-reversal technique can be used to obtain a stable inverse system. Due to the time-reversal, proper initial conditions have to be chosen so that perfect reconstruction can be achieved. Using the state-space descriptions of linear systems, it is shown that the initial condition that guarantees perfect recovery is in fact the final condition of the original system. One important application of the technique is that for any analysis filters in an analysis/synthesis filter bank, a stable implementation of the synthesis filter bank can always be derived. Hence, constraints such as FIR filters and the paraunitary property are not necessary to assure stability of the synthesis filters