Filters and Filter Banks for Periodic Signals, the Zak Transform and Fast Wavelet Decomposition

In this paper we present a new approach to filtering and reconstruction of periodic signals. The tool that proves to handle these tasks very efficiently is the discrete Zak transform. The discrete Zak transform can be viewed as the discrete Fourier transform performed on the signal blocks. It also can be considered the polyphase representation of periodic signals. Fast filtering-decimation-interpolation-reconstruction algorithms are developed in the Zak Transform domain both for the undersampling and critical sampling cases. The technique of finding the optimal biorthogonal filter banks, i.e. those that would provide the best reconstruction even in the undersampling case, is presented. An algorithm for orthogonalization of non-orthogonal filters is developed. The condition for perfect reconstruction for the periodic signals is derived. The generalizations are made for the non-periodic sequences and several ways to apply the developed technique to the nonperiodic sequences are considere..