A design method of multidimensional linear-phase paraunitary filter banks with a lattice structure

A lattice structure of multidimensional (MD) linear-phase paraunitary filter banks (LPPUFBs) is proposed, which makes it possible to design such systems in a systematic manner. Our proposed structure can produce MD-LPPUFBs whose filters all have the region of support 𝒩(MΞ), where M and Ξ are the decimation and positive integer diagonal matrices, respectively, and 𝒩(N) denotes the set of integer vectors in the fundamental parallelepiped of a matrix N. It is shown that if 𝒩(M) is reflection invariant with respect to some center, then the reflection invariance of 𝒩(MΞ) is guaranteed. This fact is important in constructing MD linear-phase filter banks because the reflection invariance is necessary for any linear-phase filter. Since our proposed system structurally restricts both the paraunitary and linear-phase properties, an unconstrained optimization process can be used to design MD-LPPUFBs. Our proposed structure is developed for both an even and an odd number of channels and includes the conventional 1-D system as a special case. It is also shown to be minimal, and the no-DC-leakage condition is presented. Some design examples show the significance of our proposed structure for both the rectangular and nonrectangular decimation cases