Sparse Fourier transform

The sparse Fourier transform (SFT) is a kind of discrete Fourier transform (DFT) for handling big data signals. Specifically, it is used in GPS synchronization, spectrum sensing and analog-to-digital converters.:[1]

The fast Fourier transform (FFT) plays an indispensable role on many scientific domains, especially on signal processing. It is one of the top-10 algorithms in the twentieth century.[2] However, with the advent of big data era, the FFT still needs to be improved in order to save more computing power. Recently, the sparse Fourier transform (SFT) has gained a considerable amount of attention, for it performs well on analyzing the long sequence of data with few signal components.

Definition

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Consider a sequence xn of complex numbers. By Fourier series, xn can be written as

 

Similarly, Xk can be represented as

 

Hence, from the equations above, the mapping is  .

Single frequency recovery

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Assume only a single frequency exists in the sequence. In order to recover this frequency from the sequence, it is reasonable to utilize the relationship between adjacent points of the sequence.

Phase encoding

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The phase k can be obtained by dividing the adjacent points of the sequence. In other words,

 

Notice that  .

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An aliasing-based search

Seeking phase k can be done by Chinese remainder theorem (CRT).[3]

Take   for an example. Now, we have three relatively prime integers 100, 101, and 103. Thus, the equation can be described as

 

By CRT, we have

 

Randomly binning frequencies

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Spread all frequencies

Now, we desire to explore the case of multiple frequencies, instead of a single frequency. The adjacent frequencies can be separated by the scaling c and modulation b properties. Namely, by randomly choosing the parameters of c and b, the distribution of all frequencies can be almost a uniform distribution. The figure Spread all frequencies reveals by randomly binning frequencies, we can utilize the single frequency recovery to seek the main components.

 

where c is scaling property and b is modulation property.

By randomly choosing c and b, the whole spectrum can be looked like uniform distribution. Then, taking them into filter banks can separate all frequencies, including Gaussians,[4] indicator functions,[5][6] spike trains,[7][8][9][10] and Dolph-Chebyshev filters.[11] Each bank only contains a single frequency.

The prototypical SFT

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Generally, all SFT follows the three stages[1]

Identifying frequencies

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By randomly bining frequencies, all components can be separated. Then, taking them into filter banks, so each band only contains a single frequency. It is convenient to use the methods we mentioned to recover this signal frequency.

Estimating coefficients

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After identifying frequencies, we will have many frequency components. We can use Fourier transform to estimate their coefficients.

 

Repeating

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Finally, repeating these two stages can we extract the most important components from the original signal.

 

Sparse Fourier transform in the discrete setting

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In 2012, Hassanieh, Indyk, Katabi, and Price [11] proposed an algorithm that takes   samples and runs in the same running time.

Sparse Fourier transform in the high dimensional setting

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In 2014, Indyk and Kapralov [12] proposed an algorithm that takes   samples and runs in nearly linear time in  . In 2016, Kapralov [13] proposed an algorithm that uses sublinear samples   and sublinear decoding time  . In 2019, Nakos, Song, and Wang [14] introduced a new algorithm which uses nearly optimal samples   and requires nearly linear time decoding time. A dimension-incremental algorithm was proposed by Potts, Volkmer [15] based on sampling along rank-1 lattices.

Sparse Fourier transform in the continuous setting

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There are several works about generalizing the discrete setting into the continuous setting.[16][17]

Implementations

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There are several works based on MIT, MSU, ETH and Universtity of Technology Chemnitz [TUC]. Also, they are free online.

Further reading

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Hassanieh, Haitham (2018). The Sparse Fourier Transform: Theory and Practice. Association for Computing Machinery and Morgan & Claypool. ISBN 978-1-94748-707-9.

Price, Eric (2013). Sparse Recovery and Fourier Sampling. MIT.

References

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  1. ^ a b Gilbert, Anna C.; Indyk, Piotr; Iwen, Mark; Schmidt, Ludwig (2014). "Recent Developments in the Sparse Fourier Transform: A compressed Fourier transform for big data" (PDF). IEEE Signal Processing Magazine. 31 (5): 91–100. Bibcode:2014ISPM...31...91G. doi:10.1109/MSP.2014.2329131. hdl:1721.1/113828. S2CID 14585685.
  2. ^ Cipra, Barry A. (2000). "The best of the 20th century: Editors name top 10 algorithms". SIAM News. 33 (4).
  3. ^ Iwen, M. A. (2010-01-05). "Combinatorial Sublinear-Time Fourier Algorithms". Foundations of Computational Mathematics. 10 (3): 303–338. doi:10.1007/s10208-009-9057-1. S2CID 1631513.
  4. ^ Haitham Hassanieh; Piotr Indyk; Dina Katabi; Eric Price (2012). Simple and Practical Algorithm for Sparse Fourier Transform. pp. 1183–1194. doi:10.1137/1.9781611973099.93. hdl:1721.1/73474. ISBN 978-1-61197-210-8.
  5. ^ A. C. Gilbert (2002). "Near-optimal sparse fourier representations via sampling". Proceedings of the thiry-fourth annual ACM symposium on Theory of computing. S. Guha, P. Indyk, S. Muthukrishnan, M. Strauss. pp. 152–161. doi:10.1145/509907.509933. ISBN 1581134959. S2CID 14320243.
  6. ^ A. C. Gilbert; S. Muthukrishnan; M. Strauss (21 September 2005). "Improved time bounds for near-optimal sparse Fourier representations". In Papadakis, Manos; Laine, Andrew F; Unser, Michael A (eds.). Wavelets XI. Proceedings of SPIE. Vol. 5914. pp. 59141A. Bibcode:2005SPIE.5914..398G. doi:10.1117/12.615931. S2CID 12622592.
  7. ^ Ghazi, Badih; Hassanieh, Haitham; Indyk, Piotr; Katabi, Dina; Price, Eric; Lixin Shi (2013). "Sample-optimal average-case sparse Fourier Transform in two dimensions". 2013 51st Annual Allerton Conference on Communication, Control, and Computing (Allerton). pp. 1258–1265. arXiv:1303.1209. doi:10.1109/Allerton.2013.6736670. ISBN 978-1-4799-3410-2. S2CID 6151728.
  8. ^ Iwen, M. A. (2010-01-05). "Combinatorial Sublinear-Time Fourier Algorithms". Foundations of Computational Mathematics. 10 (3): 303–338. doi:10.1007/s10208-009-9057-1. S2CID 1631513.
  9. ^ Mark A.Iwen (2013-01-01). "Improved approximation guarantees for sublinear-time Fourier algorithms". Applied and Computational Harmonic Analysis. 34 (1): 57–82. arXiv:1010.0014. doi:10.1016/j.acha.2012.03.007. ISSN 1063-5203. S2CID 16808450.
  10. ^ Pawar, Sameer; Ramchandran, Kannan (2013). "Computing a k-sparse n-length Discrete Fourier Transform using at most 4k samples and O(k log k) complexity". 2013 IEEE International Symposium on Information Theory. pp. 464–468. doi:10.1109/ISIT.2013.6620269. ISBN 978-1-4799-0446-4. S2CID 601496.
  11. ^ a b Hassanieh, Haitham; Indyk, Piotr; Katabi, Dina; Price, Eric (2012). "Nearly optimal sparse fourier transform". Proceedings of the forty-fourth annual ACM symposium on Theory of computing. STOC'12. ACM. pp. 563–578. arXiv:1201.2501. doi:10.1145/2213977.2214029. ISBN 9781450312455. S2CID 3760962.
  12. ^ Indyk, Piotr; Kapralov, Michael (2014). "Sample-optimal Fourier sampling in any constant dimension". Annual Symposium on Foundations of Computer Science. FOCS'14: 514–523. arXiv:1403.5804.
  13. ^ Kapralov, Michael (2016). "Sparse fourier transform in any constant dimension with nearly-optimal sample complexity in sublinear time". Proceedings of the forty-eighth annual ACM symposium on Theory of Computing. STOC'16. pp. 264–277. arXiv:1604.00845. doi:10.1145/2897518.2897650. ISBN 9781450341325. S2CID 11847086.
  14. ^ Nakos, Vasileios; Song, Zhao; Wang, Zhengyu (2019). "(Nearly) Sample-Optimal Sparse Fourier Transform in Any Dimension; RIPless and Filterless". Annual Symposium on Foundations of Computer Science. FOCS'19. arXiv:1909.11123.
  15. ^ Potts, Daniel; Volkmer, Toni (2016). "Sparse high-dimensional FFT based on rank-1 lattice sampling". Applied and Computational Harmonic Analysis. 41 (3): 713–748. doi:10.1016/j.acha.2015.05.002.
  16. ^ Price, Eric; Song, Zhao (2015). "A Robust Sparse Fourier Transform in the Continuous Setting". Annual Symposium on Foundations of Computer Science. FOCS'15: 583–600. arXiv:1609.00896.
  17. ^ Chen, Xue; Kane, Daniel M.; Price, Eric; Song, Zhao (2016). "Fourier-Sparse Interpolation without a Frequency Gap". Annual Symposium on Foundations of Computer Science. FOCS'16: 741–750. arXiv:1609.01361.