A necessary and sufficient condition for commutative PR orthogonal multifilter banks

Constructing multiwavelet-based filter banks (multifilters) is more difficult than constructing scalar wavelet filters, partly because the noncommutativity of matrix multiplication prevents a trivial extension of the scalar wavelet "flip construction". Commutative multifilters avoid this problem by allowing the flip construction to be used, thereby simplifying the design process. This paper presents a condition on the polyphase components of the analysis multifilters which is necessary and sufficient for the multifilters to achieve commutativity in addition to perfect reconstruction and orthogonality. This condition involves matrices of a form similar to that appearing in various other contexts, some of which are discussed. The paper includes simple examples of multifilters satisfying the condition obtained.