Steady-state frequency response for periodic systems

This paper extends the concept of steady-state frequency response, well known in the theory of linear time-invariant (LTI) systems, to linear time-varying systems with periodic co-efficients, called periodic systems. It is shown that for an internally stable periodic system there exist complete orthogonal systems of real periodic functions {φn} and {ψn} called eigenfunctions, such that for the inputs φn every output of the system converges in steady state to σnψn, where σn are non-negative real numbers. The set of all such numbers is called the singular frequency response of the system. In the case of LTI systems, the singular frequency response turns out to be consisting of the magnitudes of the sinusoidal frequency responses of the system. The singular frequency response {σn} is shown to be the singular spectrum of a compact operator associated with the system and has all the characteristics of the magnitude frequency response of LTI systems. A state-space realization of this operator and its adjoint leads to an alternative formulation of inverse of the singular frequency response as eigenvalues arising from a boundary value problem with periodic boundary values.