A Tutorial on Fast Fourier Sampling

This article describes a computational method, called the Fourier sampling algorithm. The algorithm takes a small number of (correlated) random samples from a signal and processes them efficiently to produce an approximation of the DFT of the signal. The algorithm offers provable guarantees on the number of samples, the running time, and the amount of storage. As we will see, these requirements are exponentially better than the FFT for some cases of interest.

[1]  G. Beylkin On the Fast Fourier Transform of Functions with Singularities , 1995 .

[2]  Douglas M. Hawkins,et al.  Robust Frequency Estimation using Elemental Sets , 1997 .

[3]  Alan V. Oppenheim,et al.  Discrete-Time Signal Pro-cessing , 1989 .

[4]  Yoram Bresler,et al.  FIR perfect signal reconstruction from multiple convolutions: minimum deconvolver orders , 1998, IEEE Trans. Signal Process..

[5]  Emmanuel J. Candès,et al.  Near-Optimal Signal Recovery From Random Projections: Universal Encoding Strategies? , 2004, IEEE Transactions on Information Theory.

[6]  Vladimir Rokhlin,et al.  Fast Fourier Transforms for Nonequispaced Data , 1993, SIAM J. Sci. Comput..

[7]  Mark A. Iwen,et al.  Empirical evaluation of a sub-linear time sparse DFT algorithm , 2007 .

[8]  Anna C. Gilbert,et al.  Improved time bounds for near-optimal sparse Fourier representations , 2005, SPIE Optics + Photonics.

[9]  Sudipto Guha,et al.  Near-optimal sparse fourier representations via sampling , 2002, STOC '02.

[10]  Chris Anderson,et al.  Rapid Computation of the Discrete Fourier Transform , 1996, SIAM J. Sci. Comput..

[11]  Eyal Kushilevitz,et al.  Learning decision trees using the Fourier spectrum , 1991, STOC '91.

[12]  Yishay Mansour,et al.  Randomized Interpolation and Approximation of Sparse Polynomials , 1992, SIAM J. Comput..

[13]  Graham Cormode,et al.  Combinatorial Algorithms for Compressed Sensing , 2006 .

[14]  D. Donoho,et al.  Uncertainty principles and signal recovery , 1989 .

[15]  David L Donoho,et al.  Compressed sensing , 2006, IEEE Transactions on Information Theory.

[16]  R. Vershynin,et al.  One sketch for all: fast algorithms for compressed sensing , 2007, STOC '07.