Transactions on Signal ProcessingFilters and Filter Banks for Periodic Signals , the ZakTransform and Fast Wavelet Decomposition

In this paper we present a new approach to ltering and reconstruction of periodic signals. The tool that proves to handle these tasks very eeciently 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 ltering-decimation-interpolation-reconstruction algorithms are developed in the Zak Transform domain both for the undersampling and critical sampling cases. The technique of nding the optimal biorthogonal lter banks, i.e. those that would provide the best reconstruction even in the undersampling case, is presented. An algorithm for orthogonalization of non-orthogonal lters 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 non-periodic sequences are considered. Finally the developed technique is applied to recursive lter banks and the discrete wavelet decomposition.