The Robust Sparse Fourier Transform (RSFT) and Its Application in Radar Signal Processing

The Sparse Fourier Transform (SFT), designed for signals that contain a small number of frequencies, enjoys low complexity, and thus is ideally suited for big data applications. In this paper, we propose Robust Sparse Fourier Transform, (RSFT), which is a modification of SFT that extends the SFT advantages to real world, noisy settings. RSFT can accommodate off-grid frequencies in the data. Furthermore, by incorporating Neyman–Pearson detection in the SFT stages, frequency detection does not require knowledge of the exact sparsity of the signal, and is robust to noise. We analyze the asymptotic performance of RSFT, and study the computational complexity versus detection performance tradeoff. We show that, by appropriately choosing the detection thresholds, the optimal tradeoff can be achieved. We discuss the application of RSFT on short-range ubiquitous radar signal processing and demonstrate its feasibility via simulations.

[1]  Yang Wang,et al.  A Multiscale Sub-linear Time Fourier Algorithm for Noisy Data , 2013, ArXiv.

[2]  H. Vincent Poor,et al.  MIMO Radar Using Compressive Sampling , 2009, IEEE Journal of Selected Topics in Signal Processing.

[3]  E.J. Candes,et al.  An Introduction To Compressive Sampling , 2008, IEEE Signal Processing Magazine.

[4]  Thomas Strohmer,et al.  High-Resolution Radar via Compressed Sensing , 2008, IEEE Transactions on Signal Processing.

[5]  Yiu-Tong Chan,et al.  Comparison of various periodograms for sinusoid detection and frequency estimation , 1999 .

[6]  Piotr Indyk,et al.  Simple and practical algorithm for sparse Fourier transform , 2012, SODA.

[7]  Yannis P. Tsividis.  Digital signal processing in continuous time: a possibility for avoiding aliasing and reducing quantization error , 2004, 2004 IEEE International Conference on Acoustics, Speech, and Signal Processing.

[8]  Piotr Indyk,et al.  Nearly optimal sparse fourier transform , 2012, STOC '12.

[9]  Piotr Indyk,et al.  Sample-Optimal Fourier Sampling in Any Constant Dimension , 2014, 2014 IEEE 55th Annual Symposium on Foundations of Computer Science.

[10]  Piotr Indyk,et al.  Recent Developments in the Sparse Fourier Transform: A compressed Fourier transform for big data , 2014, IEEE Signal Processing Magazine.

[11]  Toni Volkmer,et al.  Efficient Spectral Estimation by MUSIC and ESPRIT with Application to Sparse FFT , 2016, Front. Appl. Math. Stat..

[12]  Frédo Durand,et al.  Light Field Reconstruction Using Sparsity in the Continuous Fourier Domain , 2014, ACM Trans. Graph..

[13]  Murat Torlak,et al.  Automotive Radars: A review of signal processing techniques , 2017, IEEE Signal Processing Magazine.

[14]  Piotr Indyk,et al.  Sample-optimal average-case sparse Fourier Transform in two dimensions , 2013, 2013 51st Annual Allerton Conference on Communication, Control, and Computing (Allerton).

[15]  Zhao Song,et al.  A Robust Sparse Fourier Transform in the Continuous Setting , 2015, 2015 IEEE 56th Annual Symposium on Foundations of Computer Science.

[16]  Daniel Potts,et al.  Parameter estimation for exponential sums by approximate Prony method , 2010, Signal Process..

[17]  D. Potts,et al.  Sparse high-dimensional FFT based on rank-1 lattice sampling , 2016 .

[18]  P.P. Vaidyanathan,et al.  Compressed sensing in MIMO radar , 2008, 2008 42nd Asilomar Conference on Signals, Systems and Computers.

[19]  Rama Chellappa,et al.  Compressed Synthetic Aperture Radar , 2010, IEEE Journal of Selected Topics in Signal Processing.

[20]  Volkan Cevher,et al.  What’s the Frequency, Kenneth?: Sublinear Fourier Sampling Off the Grid , 2015, Algorithmica.

[21]  A.C. Gilbert,et al.  A Tutorial on Fast Fourier Sampling , 2008, IEEE Signal Processing Magazine.

[22]  Piotr Indyk,et al.  Faster GPS via the sparse fourier transform , 2012, Mobicom '12.

[23]  Volkan Cevher,et al.  What's the Frequency, Kenneth?: Sublinear Fourier Sampling Off the Grid , 2012, APPROX-RANDOM.

[24]  Omid Salehi-Abari,et al.  GHz-wide sensing and decoding using the sparse Fourier transform , 2014, IEEE INFOCOM 2014 - IEEE Conference on Computer Communications.

[25]  L. Kämmerer.  Reconstructing Multivariate Trigonometric Polynomials from Samples Along Rank-1 Lattices , 2014 .

[26]  Kannan Ramchandran,et al.  Fast and Efficient Sparse 2D Discrete Fourier Transform using Sparse-Graph Codes , 2015, ArXiv.

[27]  Mike Shuo-Wei Chen,et al.  A non-uniform sampling ADC architecture with embedded alias-free asynchronous filter , 2012, 2012 IEEE Global Communications Conference (GLOBECOM).

[28]  Moeness G. Amin,et al.  Compressed sensing technique for high-resolution radar imaging , 2008, SPIE Defense + Commercial Sensing.

[29]  Kannan Ramchandran,et al.  Fast sparse 2-D DFT computation using sparse-graph alias codes , 2016, 2016 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP).

[30]  Merrill Skolnik,et al.  Systems Aspects of Digital Beam Forming Ubiquitous Radar , 2002 .

[31]  F. Harris.  On the use of windows for harmonic analysis with the discrete Fourier transform , 1978, Proceedings of the IEEE.

[32]  Steven G. Johnson,et al.  FFTW: an adaptive software architecture for the FFT , 1998, Proceedings of the 1998 IEEE International Conference on Acoustics, Speech and Signal Processing, ICASSP '98 (Cat. No.98CH36181).

[33]  Athina P. Petropulu,et al.  RSFT: A realistic high dimensional sparse fourier transform and its application in radar signal processing , 2016, MILCOM 2016 - 2016 IEEE Military Communications Conference.

[34]  Petre Stoica,et al.  Spectral Analysis of Signals , 2009 .

[35]  Gonzalo R. Arce,et al.  Optimized Spectrum Permutation for the Multidimensional Sparse FFT , 2017, IEEE Transactions on Signal Processing.

[36]  Dina Katabi,et al.  Fast multi-dimensional NMR acquisition and processing using the sparse FFT , 2015, Journal of Biomolecular NMR.