Fast Approximate Fourier Transform via Wavelets 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 coeecients 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(N log 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 eecient complexity v.s. accuracy tradeoo. When approximations are allowed, under certain sparsity conditions, the algorithm can achieve linear complexity, i.e. O(N). It has been shown that the thresholding of the wavelet coeecients has near optimal noise reduction property for many classes of signals. We show that for the same reason, the proposed algorithm also reduces the noise while doing the approximation. If we need to compute the DFT of noisy signals, the proposed algorithm not only can reduce the numerical complexity, but also can produce cleaner results. In summary, we proposed a novel fast approximate Fourier transform algorithm using the wavelet transform. Since wavelets are the conditional basis of many classes of signals, the algorithm is very eecient and has builtin denoising capacity.