Block-shift invariance and block implementation of discrete-time filters

This paper describes the general class of linear, time-invariant multivariable systems that can be used in block implementations of time-invariant discrete-time filters. Explicit relations between the properties of the block processor and the properties of the implemented filter are derived, including an explicit expression for the matrix transfer function of the block processor in terms of the single-input, single-output filter transfer function. These properties and relations are independent of the form of realization of the block processor. It is shown that all irreducible state-space realizations of the block processor can be derived by a simple procedure from a simple realization of the required filter transfer function.