In this study, a matrix grid D is used to analyze and synthesize a multidimensional signal comprising an important and useful tool for sublatticed processing in the decomposition and reconstruction of a signal into its polyphase components. Also presented is a filter bank with polyphase components functioning freely as aliases when using a quincunx matrix to specify obligatory and sufficient conditions and to simplify the requisite procedures for determining the number of variables and equations for a perfect reconstruction of an analysis/synthesis system.