Polyphase decompositions and shift-invariant discrete wavelet transforms in the frequency domain

Given a signal and its Fourier transform, we derive formulas for its polyphase decomposition in the frequency domain and for the reconstruction from the polyphase representation back to the Fourier representation. We present two frequency-domain implementations of the shift-invariant periodic discrete wavelet transform (SI-DWT) and its inverse: one that is based on frequency-domain polyphase decomposition and a more efficient ‘direct’ implementation, based on a reorganisation of the à trous algorithm. We analyse the computational complexities of both algorithms, and compare them to existing time-domain and frequency domain implementations of the SI-DWT. We experimentally demonstrate the reduction in computation time achieved by the direct frequency domain implementation of the SI-DWT for wavelet filters with non-compact support.