Wavelet transform based fast approximate Fourier transform

We propose an algorithm that uses the discrete wavelet transform (DWT) as a tool to compute the discrete Fourier transform (DFT). The Cooley-Tukey FFT is shown to be a special case of the proposed algorithm when the wavelets in use are trivial. If no intermediate coefficients are dropped and no approximations are made, the proposed algorithm computes the exact result, and its computational complexity is on the same order of the FFT, i.e. O(Nlog(2) N). The main advantage of the proposed algorithm is that the good time and frequency localization of wavelets can be exploited to approximate the Fourier transform for many classes of signals resulting in much less computation. Thus the new algorithm provides an efficient complexity v.s. accuracy tradeoff. When approximations are allowed, under certain sparsity conditions, the algorithm can achieve linear complexity, i.e. O(N). The proposed algorithm also has built-in noise reduction capability.