An insight into a redundant signal decomposition on bases in a vector space

Expansions of complex discrete-time signals onto a single basis in the given vector space provide an incomplete and in a way potentially misleading information about the signals in signal analysis applications. We use a simple example to illustrate the incompleteness of the information about the signal in the decompositions on a single basis. Then, we limit our attention to an oversampled decimated perfect reconstruction filter bank. It is illustrated that oversampled decimated filter banks can serve as a tool which provides more complete information about the signal and at the same time the filter banks can enjoy an efficient polyphase component implementation of maximally decimated, i.e. nonredundant, filter banks. In an oversampled decimated filter bank the relation between analysis filters given by H_k(z)=H₀(zW_M^k) does not in general imply that the filter bank formed by these filters is a uniform filter bank with center frequencies of the filters uniformly spaced on the frequency axis. A simple example which removes a possible surprise which accompanies this statement is provided. It is also shown that if the complete set of filters H_k(z)=H₀(zW_M^k) of the oversampled decimated filter bank can be derived from more than a single prototype then the efficient polyphase implementations of the oversampled decimated filter bank can be readily available. Relation of the considered oversampled decimated filter bank with a decomposition of the discrete-time signal onto several bases of the given vector space is emphasized throughout the paper