Skip to main content
SpringerLink
Log in

Performance of the Multiscale Sparse Fast Fourier Transform Algorithm

  • Published:
Circuits, Systems, and Signal Processing Aims and scope Submit manuscript

Abstract

How to compute the sparse fast Fourier transform (sFFT) has been a critical topic for a long period of time. sFFT algorithms have faster runtimes and reduced sampling complexities by taking advantage of a signal’s inherent characteristics, namely, that a large number of signals are sparse in the frequency domain (e.g., sensors, video data, audio, medical images.). The first step of sFFT is frequency bucketization through window filters. sFFT algorithms using a flat filter are more convenient and efficient than other algorithms because the filtered signal is concentrated both in the time domain and frequency domain. To date, three sFFT algorithms using the flat filter (the sFFT1.0, sFFT2.0, and sFFT3.0 algorithms) have been proposed by the Massachusetts Institute of Technology (MIT) beginning in 2013. Since then, the sFFT4.0 algorithm using a multiscale approach method has not yet been implemented. This paper will discuss this algorithm comprehensively in theory and implement it in practice. It is proven that the performance of the sFFT4.0 algorithm depends on two parameters. The runtime and sampling complexity are directly correlated to the multiscale parameter and inversely correlated to the extension parameter. The robustness is in directly correlated to the extension parameter and in inverse ratio to the multiscale parameter. Compared with three similar algorithms and four different types of algorithms, the sFFT4.0 algorithm displays an improved runtime and sampling complexity that is ten to one hundred times better than the FFTW algorithm, although the robustness of the algorithm is moderate.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13

Similar content being viewed by others

Data Availability

The supporting data of this study are available from the corresponding author on request.

Notes

  1. The code is available at http://groups.csail.mit.edu/netmit/sFFT/.

  2. The code is available at https://github.com/urrfinjuss/mpfft.

  3. The code is available at https://www.iis.sinica.edu.tw/pages/lcs.

  4. The code is available at https://github.com/UCBASiCS/FFAST.

  5. The code is available at https://sourceforge.net/projects/aafftannarborfa/.

  6. The code is available at http://www.fftw.org/.

  7. https://github.com/ludwigschmidt/sft-experiments.

  8. https://github.com/zkjiang/-/tree/master/docs/sfftproject/experimentdata/.

  9. https://github.com/zkjiang/-/tree/master/docs.

References

  1. O. Abari, E. Hamed, H. Hassanieh, A. Agarwal, D. Katabi, A.P. Chandrakasan, V. Stojanovic, A 0.75-million-point fourier-transform chip for frequency-sparse signals, in Digest of Technical Papers—IEEE International Solid-State Circuits Conference (2014). https://doi.org/10.1109/ISSCC.2014.6757512

  2. V.S. Amin, Y.D. Zhang, B. Himed, Sequential time-frequency signature estimation of multi-component FM signals, in 2019 53rd Asilomar Conference on Signals, Systems, and Computers (IEEE, 2019), p. 1901–1905

  3. V.S. Amin, Y.D. Zhang, B. Himed, Sparsity-based time-frequency representation of FM signals with burst missing samples. Signal Process. 155, 25–43 (2019)

    Article  Google Scholar 

  4. R.G. Baraniuk, Compressive sensing [lecture notes]. IEEE Signal Process. Mag. 24(4), 118–121 (2007)

    Article  Google Scholar 

  5. G.L. Chen, S.H. Tsai, K.J. Yang, On performance of sparse fast Fourier transform and enhancement algorithm. IEEE Trans. Signal Process. 65(21), 5716–5729 (2017). https://doi.org/10.1109/TSP.2017.2740198

    Article  MathSciNet  MATH  Google Scholar 

  6. J. Chiu, Matrix probing, skeleton decompositions, and sparse Fourier transform. Thesis (Ph.D.). Massachusetts Institute of Technology, Department of Mathematics (2013). https://doi.org/10.1007/978-94-007-2739-7_22

  7. A. Christlieb, D. Lawlor, Y. Wang, A multiscale sub-linear time Fourier algorithm for noisy data. Appl. Comput. Harmon. Anal. 40(3), 553–574 (2016)

    Article  MathSciNet  Google Scholar 

  8. A.G. Farashahi, Wave packet transform over finite fields. Electron. J. Linear Algebra 30(1), 507–529 (2015)

    Article  MathSciNet  Google Scholar 

  9. A.G. Farashahi, Wave packet transforms over finite cyclic groups. Linear Algebra Appl. 489, 75–92 (2016)

    Article  MathSciNet  Google Scholar 

  10. A. Ghaani Farashahi, Cyclic wave packet transform on finite abelian groups of prime order. Int. J. Wavelets Multiresolut. Inf. Process. 12(06), 1450,041 (2014)

    Article  MathSciNet  Google Scholar 

  11. A.C. Gilbert, S. Guha, P. Indyk, S. Muthukrishnan, M. Strauss, Near-optimal sparse Fourier representations via sampling, in Conference Proceedings of the Annual ACM Symposium on Theory of Computing, vol. 2 (2002), p. 152–161. https://doi.org/10.1145/509931.509933

  12. A.C. Gilbert, M.J. Strauss, J.A. Tropp, A tutorial on fast Fourier sampling: how to apply it to problems 25(2), 57–66 (2008). https://doi.org/10.1109/MSP.2007.915000

  13. A.C. Gilbert, P. Indyk, M. Iwen, L. Schmidt, Recent developments in the sparse Fourier transform. IEEE Signal Process. Mag. 31, 1–21 (2014). https://doi.org/10.1109/MSP.2014.2329131

    Article  Google Scholar 

  14. H. Hassanieh, P. Indyk, D. Katabi, E. Price, Nearly optimal sparse Fourier transform, in Proceedings of the Annual ACM Symposium on Theory of Computing (2012), p. 563–577. https://doi.org/10.1145/2213977.2214029. arXiv:1201.2501

  15. H. Hassanieh, P. Indyk, D. Katabi, E. Price, Simple and practical algorithm for sparse Fourier transform. in Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms (2012), p. 1183–1194. https://doi.org/10.1137/1.9781611973099.93

  16. S.H. Hsieh, C.S. Lu, S.C. Pei, Sparse fast Fourier transform by downsampling, in ICASSP, IEEE International Conference on Acoustics, Speech and Signal Processing-Proceedings (2013), p. 5637–5641. https://doi.org/10.1109/ICASSP.2013.6638743

  17. S.H. Hsieh, C.S. Lu, S.C. Pei, Sparse fast Fourier transform for exactly and generally k-sparse signals by downsampling and sparse recovery. Eprint Arxiv (2014)

  18. P. Indyk, M. Kapralov, E. Price, (Nearly) sample-optimal sparse Fourier transform, in Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms (2014), p. 480–499. https://doi.org/10.1137/1.9781611973402.36

  19. M.A. Iwen, Combinatorial sublinear-time Fourier algorithms. Found. Comput. Math. 10(3), 303–338 (2010). https://doi.org/10.1007/s10208-009-9057-1

    Article  MathSciNet  MATH  Google Scholar 

  20. M.A. Iwen, Improved approximation guarantees for sublinear-time Fourier algorithms. Appl. Comput. Harmon. Anal. 34(1), 57–82 (2013). https://doi.org/10.1016/j.acha.2012.03.007

    Article  MathSciNet  MATH  Google Scholar 

  21. M.A. Iwen, A. Gilbert, M. Strauss, Empirical evaluation of a sub-linear time sparse DFT algorithm. Commun. Math. Sci. 5(4), 981–998 (2007). https://doi.org/10.4310/cms.2007.v5.n4.a13

    Article  MathSciNet  MATH  Google Scholar 

  22. Z. Jiang, J. Chen, B. Li, Empirical evaluation of typical sparse fast Fourier transform algorithms. IEEE Access 9, 97100–97119 (2021)

    Article  Google Scholar 

  23. M. Kapralov, Sample efficient estimation and recovery in sparse FFT via isolation on average, in Annual Symposium on Foundations of Computer Science—Proceedings, vol. 2017(1) (2017), p. 651–662. https://doi.org/10.1109/FOCS.2017.66

  24. M. Kapralov, A. Velingker, A. Zandieh, Dimension-independent sparse Fourier transform, in Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms (2019). https://doi.org/10.1137/1.9781611975482.168

  25. G.G. Kumar, S.K. Sahoo, P.K. Meher, 50 Years of FFT algorithms and applications. Circuits Syst. Signal Process. 38(12), 5665–5698 (2019). https://doi.org/10.1007/s00034-019-01136-8

    Article  Google Scholar 

  26. B. Li, Z. Jiang, J. Chen, On performance of sparse fast Fourier transform algorithms using the flat window filter. IEEE Access 8, 79134–79146 (2020)

    Article  Google Scholar 

  27. B. Li, Z. Jiang, J. Chen, On performance of sparse fast Fourier transform algorithms using the aliasing filter. Electronics (2021). https://doi.org/10.3390/electronics10091117

    Article  Google Scholar 

  28. A. López-Parrado, J. Velasco Medina, Efficient software implementation of the nearly optimal sparse fast Fourier transform for the noisy case. Ingeniería y Ciencia 11(22), 73–94 (2015). https://doi.org/10.17230/ingciencia.11.22.4

    Article  Google Scholar 

  29. S. Merhi, R. Zhang, M.A. Iwen, A. Christlieb, A new class of fully discrete sparse Fourier transforms: faster stable implementations with guarantees. J. Fourier Anal. Appl. 25(3), 751–784 (2019). https://doi.org/10.1007/s00041-018-9616-4arXiv:1706.02740

    Article  MathSciNet  MATH  Google Scholar 

  30. F. Ong, R. Heckel, K. Ramchandran, A fast and robust paradigm for fourier compressed sensing based on coded sampling. ICASSP, in IEEE International Conference on Acoustics, Speech and Signal Processing—Proceedings, vol. 2019 (2019), p. 5117–5121. https://doi.org/10.1109/ICASSP.2019.8682063

  31. C. Pang, S. Liu, Y. Han, High-speed target detection algorithm based on sparse Fourier transform. IEEE Access 6, 37828–37836 (2018)

    Article  Google Scholar 

  32. S. Pawar, K. Ramchandran, Computing a k-sparse n-length discrete Fourier transform using at most 4k samples and O(k log k) complexity, in IEEE International Symposium on Information Theory—Proceedings (2013), p. 464–468. https://doi.org/10.1109/ISIT.2013.6620269. arXiv:1305.0870

  33. S. Pawar, K. Ramchandran, A robust sub-linear time R-FFAST algorithm for computing a sparse DFT (2015), p. 1–35. arXiv:1501.00320

  34. S. Pawar, K. Ramchandran, FFAST: an algorithm for computing an exactly k-sparse DFT in O(k log k) time. IEEE Trans. Inform Theory 64(1), 429–450 (2018). https://doi.org/10.1109/TIT.2017.2746568

    Article  MathSciNet  MATH  Google Scholar 

  35. G. Plonka, K. Wannenwetsch, A deterministic sparse FFT algorithm for vectors with small support. Numeri. Algorithms 71(4), 889–905 (2016). https://doi.org/10.1007/s11075-015-0028-0arXiv:1504.02214

    Article  MathSciNet  MATH  Google Scholar 

  36. G. Plonka, K. Wannenwetsch, A sparse fast Fourier algorithm for real non-negative vectors. J. Comput. Appl. Math. 321, 532–539 (2017). https://doi.org/10.1016/j.cam.2017.03.019

    Article  MathSciNet  MATH  Google Scholar 

  37. G. Plonka, K. Wannenwetsch, A. Cuyt, W.S. Lee, Deterministic sparse FFT for M-sparse vectors. Numer. Algorithms 8, 9 (2018). https://doi.org/10.1007/s11075-017-0370-5

    Article  MathSciNet  MATH  Google Scholar 

  38. R. Roy, T. Kailath, Esprit-estimation of signal parameters via rotational invariance techniques. IEEE Trans. Acoust. Speech Signal Process. 37(7), 984–995 (1989)

    Article  Google Scholar 

  39. R. Schmidt, Multiple emitter location and signal parameter estimation. IEEE Trans. Antennas Propag. 34(3), 276–280 (1986)

    Article  Google Scholar 

  40. J. Schumacher, M. Püschel, High-performance sparse fast Fourier transforms. IEEE Workshop on Signal Processing Systems, SiPS: Design and Implementation (2014). https://doi.org/10.1109/SiPS.2014.6986055

  41. P. Stoica, R. Moses, Spectral Analysis of Signals (Prentice Hall Inc, Upper Saddle River, 2005)

    Google Scholar 

  42. C. Wang, CusFFT: a high-performance sparse fast fourier transform algorithm on GPUs. in Proceedings—2016 IEEE 30th International Parallel and Distributed Processing Symposium, IPDPS (2016), p. 963–972. https://doi.org/10.1109/IPDPS.2016.95

  43. S. Wang, V.M. Patel, A. Petropulu, Multidimensional sparse Fourier transform based on the Fourier projection-slice theorem. IEEE Trans. Signal Process. 67(1), 54–69 (2019). https://doi.org/10.1109/TSP.2018.2878546

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This work was supported by the Youth Program of National Natural Science Foundation of China under Grant 61703263.

Author information

Authors and Affiliations

  1. School of Mechanical and Electrical Engineering and Automation, Shanghai University, Shanghai, 200072, China

    Bin Li, Zhikang Jiang & Jie Chen

Authors
  1. Bin Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zhikang Jiang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jie Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jie Chen.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Li, B., Jiang, Z. & Chen, J. Performance of the Multiscale Sparse Fast Fourier Transform Algorithm. Circuits Syst Signal Process 41, 4547–4569 (2022). https://doi.org/10.1007/s00034-022-01989-6

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00034-022-01989-6

Keywords

Search

Navigation