With reference to periodic discrete-time systems, in this paper the concept of the periodic spectral interactor matrix operator (PSIMO) is introduced. Such an interactor enables one to transform the singular system into a nonsingular one with identical spectral properties. This concept is useful to characterize the family of periodic solutions of the singular periodic Riccati equation arising from the singular filtering problem. The PSIMO is also a synthetic tool to represent the invariant zeros structure at infinity (delays) of periodic systems in an operatorial fashion.