稀疏傅里叶变换（SFT）算法

 

Sparse Fast Fourier Transform : The discrete Fourier transform (DFT) is one of the most important and widely used computational tasks. Its applications are broad and include signal processing, communications, and audio/image/video compression. Hence, fast algorithms for DFT are highly valuable. Currently, the fastest such algorithm is the Fast Fourier Transform (FFT), which computes the DFT of an n-dimensional signal inO(nlogn) time. The existence of DFT algorithms faster than FFT is one of the central questions in the theory of algorithms.  A general algorithm for computing the exact DFT must take time at least proportional to its output size n. In many applications, however, most of the Fourier coefficients of a signal are small or equal to zero, i.e., the output of the DFT is sparse. This is the case for video signals, where a typical 8x8 block in a video frame has on average 7 non-negligible frequency coefficients (i.e., 89% of the coefficients are negligible). For sparse signals, the Ω(n)lower bound for the complexity of DFT no longer applies. If a signal has a small number k of non-zero Fourier coefficients the output of the Fourier transform can be represented succinctly using only k coefficients. Hence, for such signals, we can find DFT algorithms whose runtime is sublinear in the signal size, n.  We present here several new results for sparse Fourier transform: - An O(k log n)-time algorithm for the exactly k-sparse case. - An O(k log n log(n/k))-time algorithm for the general case. - An Ω(k log(n/k) loglog n) lower bound for sample complexity. Both algorithms improve over FFT, for any k = o(n). Moreover, if one assume that FFT is optimal, the algorithm for the exactly k-sparse case is optimal. Under the same assumption, the result for the general case is at most one loglog n factor away from the optimal runtime for the case of �large� sparsity k = n/log n.  This research is supported in part by NSF and DARPA. 本文来源：http://groups.csail.mit.edu/netmit/sFFT/