On Discrete-Time Linear Systems with Almost-Periodic Kernels

Abstract

The theory of discrete time-invariant linear causal systems is closely related to the theory of functions analytic and bounded in the open unit disk. Such systems are characterized by a Toeplitz kernel \(K(n,m)=r(n-m)\). The output at time \(n\) is \[ y(n)=\textstyle{\sup_0^n}r(n-m)x(m). \] The next step is to consider the class of periodic systems. Then, \(K(n+T,m+T)=K(n,m)\) (\(T\) being the period) and functions analytic and bounded in the open unit disk still play a role. They now are matrix-valued and have some symmetry constraints that reflect the periodicity. At the extreme end are the general time-varying systems, for which analytic functions are replaced by upper (or lower) triangular operators. The corresponding kernel \(K\) is then general. In the paper under review the authors consider an intermediate and very interesting class of systems, for which the above kernel \(K\) has some almost periodicity constraints. The authors study various issues for such systems, in particular stability properties.