On inverses and quasi-inverses of linear time-varying discrete systems☆

Abstract This paper presents a solution to the general problem of finding inverses and quasi-inverses of time-varying linear discrete systems. The results are applicable to systems defined over any abstract field. Unlike fixed systems, a time-varying system does not necessarily possess an inverse, even if a delay in the output of the inverse is allowed. This inverse with delay is called a quasi-inverse. The necessary and sufficient conditions for the existence of a quasi-inverse are established. These conditions are difficult to apply, since they often require an infinite number of calculations. However, a simple necessary condition and a simple sufficient condition are derived. The existence of a quasi-inverse does not insure that the inverse system can be represented by a unique canonical form or a unique difference equation. Necessary and sufficient conditions for the existence of a unique inverse are shown. Under these same conditions, the canonical form of the inverse is the same as the canonical form of the inverse of a fixed system with the same structure. The necessary and sufficient conditions for the special case of a coding system where the system is described by a transmission matrix of finite dimension are given.