Empirical evaluation of a sub-linear time sparse DFT algorithm

In this paper we empirically evaluate a recently proposed Fast Approximate Discrete Fourier Transform (FADFT) algorithm, FADFT-2, for the first time. FADFT-2 returns approximate Fourier representations for frequency-sparse signals and works by random sampling. Its implementation is benchmarked against two competing methods. The first is the popular exact FFT implementation FFTW Version 3.1. The second is an implementation of FADFT-2’s ancestor, FADFT-1. Experiments verify the theoretical runtimes of both FADFT-1 and FADFT-2. In doing so it is shown that FADFT-2 not only generally outperforms FADFT-1 on all but the sparsest signals, but is also significantly faster than FFTW 3.1 on large sparse signals. Furthermore, it is demonstrated that FADFT-2 is indistinguishable from FADFT-1 in terms of noise tolerance despite FADFT-2’s better execution time.

[1]  Joel A. Tropp,et al.  Signal Recovery From Random Measurements Via Orthogonal Matching Pursuit , 2007, IEEE Transactions on Information Theory.

[2]  D. Donoho,et al.  Sparse MRI: The application of compressed sensing for rapid MR imaging , 2007, Magnetic resonance in medicine.

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

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

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

[6]  L. Greengard,et al.  Short Note: The type 3 nonuniform FFT and its applications , 2005 .

[7]  L. Greengard,et al.  The type 3 nonuniform FFT and its applications June - , 2005 .

[8]  J. Tropp,et al.  SIGNAL RECOVERY FROM PARTIAL INFORMATION VIA ORTHOGONAL MATCHING PURSUIT , 2005 .

[9]  Jeffrey A. Fessler,et al.  Nonuniform fast Fourier transforms using min-max interpolation , 2003, IEEE Trans. Signal Process..

[10]  Anna C. Gilbert,et al.  Sparse Gradient Image Reconstruction Done Faster , 2007, 2007 IEEE International Conference on Image Processing.

[11]  M. E. Muller,et al.  A Note on the Generation of Random Normal Deviates , 1958 .

[12]  Ingrid Daubechies,et al.  A Sparse Spectral Method for Homogenization Multiscale Problems , 2007, Multiscale Model. Simul..

[13]  A. Gilbert,et al.  Theoretical and experimental analysis of a randomized algorithm for Sparse Fourier transform analysis , 2004, math/0411102.

[14]  M. Rudelson,et al.  Sparse reconstruction by convex relaxation: Fourier and Gaussian measurements , 2006, 2006 40th Annual Conference on Information Sciences and Systems.

[15]  Emmanuel J. Candès,et al.  Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information , 2004, IEEE Transactions on Information Theory.

[16]  Steven G. Johnson,et al.  The Design and Implementation of FFTW3 , 2005, Proceedings of the IEEE.

[17]  S. Kirolos,et al.  Random Sampling for Analog-to-Information Conversion of Wideband Signals , 2006, 2006 IEEE Dallas/CAS Workshop on Design, Applications, Integration and Software.

[18]  J. Tukey,et al.  An algorithm for the machine calculation of complex Fourier series , 1965 .

[19]  S. Kirolos,et al.  Analog-to-Information Conversion via Random Demodulation , 2006, 2006 IEEE Dallas/CAS Workshop on Design, Applications, Integration and Software.

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