Vector Space Projection Approach for Perfect Reconstruction Filter Banks Design

Abstract

A vector space projection for the analysis and design of general M-channel multirate filter banks is proposed in this paper. In this context, the input signals are interpreted as vectors in the measurable, square-summable real vector space I 2( L ). The operation of multirate signal filtering is modeled with a set of vector space projectors which projects signals onto a family of subspaces of I 2 ( L ). The projections onto the specific subspaces are performed by a set of linear transformations. The analysis filters and the synthesis filters of the filter banks are characterized by the bases which span the projection spaces. With these vector space operations, the design of multirate filter banks can be interpreted as the construction of the bases used to span the associated subspaces of I 2 ( L ). The perfect reconstruction of multirate filter banks can then be achieved by constructing those subspaces as the direct complementary subspaces of the input signal vector space, With this vector space description, the necessary and sufficient condition for perfect reconstruction can be derived according to the direct complementary property of subspaces, Both the biorthonormal system and the orthonormal system are formulated and discussed for perfect reconstruction finite impulse response filter banks design.