Hermite reduction methods for generation of a complete class of linear-phase perfect reconstruction filter banks-Part I: Theory

Motivated by the possibility of extensions to two-dimensions, we address the problem constructing a linear-phase multiband perfect reconstruction finite impulse response filter bank by constructing the polyphase matrix associated with it. The equivalent problem of construction of linear phase, i.e., symmetric or antisymmetric compactly supported wavelets are, thus, also considered. Our approach rests on the fact that if any proper subset of the set of linear-phase analysis filters is almost arbitrarily specified, then the complete set of linear-phase analysis filters can always be constructed. The solution to this problem is obtained by solving the problem of completing an incompletely specified analysis polyphase matrix having the structure mandated by the linear-phase property. Symmetric versions of matrix reduction algorithms akin to the Hermite reduction algorithm well known in system theory are used in our method of construction. The technique closely follows the proof of (nonsymmetric) Quillen-Suslin theorem for the completion of multivariable polynomial matrices, and, thus, in addition, has the potential for extension to the multidimensional case. Examples are given to demonstrate the procedure.