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The passive optical networks proposed by Marhic and Siegman to evaluate the discrete Fourier transform (DFT) of a linear array or a pixellated image (parallel input) are redesigned for the discrete wavelet transform or the wavelet packet (WP) decomposition. Using only asymmetric optical couplers, with a suitable power coupling ratio, it is shown that it is possible to optically implement Mallat's pyramidal decomposition scheme with a compactly supported wavelet basis function. The case of a serial input has a larger number of potential applications to optical communications systems or optical computing, and simple passive optical networks are presented that evaluate the DFT or the wavelet transform of a stream of optical digits, by using only Mach-Zehnder interferometers with output asymmetric couplers. In particular, an all-optical implementation of the WP-division demultiplexer for a waveform coded transmission system is given. The proposed networks are full-wavelength demultiplexers, as they split the spectrum of the incoming signal into a set of closed frequency channels; the filters have identical bandwidth and span the whole input spectrum. Using the WP-decomposition method, different subband filtering schemes can be designed, fabricating trees of quadrature mirror filters with the fiber or the integrated optics technology.