Structures for anticausal inverses and application in multirate filter banks

Anticausal or time-reversed inversion of digital filters has gained importance in recent years in the implementation of digital filter banks. Anticausal inversion has, in the past, been shown to be possible by using block processing with appropriate state initialization. With (A, B , C , D) denoting the state space description of a structure implementing a filter G(z), the anticausal inverse can be essentially regarded as a filter structure having an inverted state-space description, which we denote as (^ A, ^ B , ^ C , ^ D). It is usually not efficient to implement the state space equations given by (^ A, ^ B , ^ C , ^ D) directly because of excessive multiplier count. Rather, one seeks to find an efficient structure having the inverse description (^ A, ^ B , ^ C , ^ D). While this can be done by inspection in simple cases such as the direct-form structure, systematic procedures for other important structures have yet to be developed. In this correspondence, we derive anticausal inverse structures corresponding to several standard IIR filter structures such as the direct-form, cascade-form, coupled-form, and the entire family of IIR lattice structures including the tapped cascaded lattice. We introduce the notion of a causal dual, which we find convenient in the derivations. We show that the limit-cycle free property of the original structure is inherited by the causal dual in some but not all cases.