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The '''sparse Fourier transform''' ('''SFT''') is a kind of [[Discrete Fourier transform|discrete Fourier transform]] (DFT) for handling [[Big data|big data]] signals. Specifically, it is used in [[Engineering:GPS|GPS]] synchronization, spectrum sensing and [[Analog-to-digital converter|analog-to-digital converter]]s.:<ref name="auto">{{Cite journal |doi = 10.1109/MSP.2014.2329131|title = Recent Developments in the Sparse Fourier Transform: A compressed Fourier transform for big data|journal = IEEE Signal Processing Magazine|volume = 31|issue = 5|pages = 91–100|year = 2014|last1 = Gilbert|first1 = Anna C.|last2 = Indyk|first2 = Piotr|last3 = Iwen|first3 = Mark|last4 = Schmidt|first4 = Ludwig|bibcode = 2014ISPM...31...91G|url = https://dspace.mit.edu/bitstream/1721.1/113828/1/Recent%20developments.pdf|hdl = 1721.1/113828| s2cid=14585685 |hdl-access = free}}</ref>
The [[Fast Fourier transform|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.<ref>{{ Cite journal | title = The best of the 20th century: Editors name top 10 algorithms | last1 = Cipra | first1 = Barry A. | date =2000 | series = SIAM news 33.4 }}</ref> 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==
Consider a sequence ''x''<sub>''n''</sub> of [[Complex number|complex number]]s. By [[Fourier series]], ''x''<sub>''n''</sub> can be written as
:<math>
x_n=(F^*X)_n=\sum_{k=0}^{N-1}X_k e^{j\frac{2\pi}{N}kn}.
</math>
Similarly, ''X''<sub>''k''</sub> can be represented as
:<math>
X_k=\frac{1}{N}(Fx)_k=\frac{1}{N}\sum_{n=0}^{N-1}x_n e^{-j\frac{2\pi}{N}kn}.
</math>
Hence, from the equations above, the mapping is <math>F:\mathbb C^N\to \mathbb C^N</math>.

===Single frequency recovery===
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====
The phase ''k'' can be obtained by dividing the adjacent points of the sequence. In other words,
:<math>
\frac{x_{n+1}}{x_n}=e^{j\frac{2\pi}{N}k} = \cos\left(\frac{2\pi k}{N}\right)+j \sin\left(\frac{2\pi k}{N}\right).
</math>
Notice that <math>x_n \in \mathbb C^N</math>.

====An aliasing-based search====
[[File:Aliasing-based search.png|thumb|An aliasing-based search]]
Seeking phase ''k'' can be done by [[Chinese remainder theorem]] (CRT).<ref>{{cite journal  |last1=Iwen |first1=M. A. |title=Combinatorial Sublinear-Time Fourier Algorithms |journal=Foundations of Computational Mathematics |date=2010-01-05 |volume=10 |issue=3 |pages=303–338 |doi=10.1007/s10208-009-9057-1|s2cid=1631513 }}</ref>

Take <math>k=104{,}134</math> for an example. Now, we have three relatively prime integers 100, 101, and 103. Thus, the equation can be described as
:<math>
k=104{,}134\equiv \left\{ \begin{array}{rl} 34 & \bmod 100, \\ 3 & \bmod 101, \\ 1 & \bmod 103. \end{array} \right.
</math>
By CRT, we have
:<math>
k=104{,}134\bmod (100\cdot101\cdot103)=104{,}134\bmod 1{,}040{,}300
</math>

====Randomly binning frequencies====
[[File:Randomlybinning.pdf|thumb|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.

:<math>
x_n'=X_k e^{j\frac{2\pi}{N}(c\cdot k+b)},
</math>
where ''c'' is scaling property and ''b'' is modulation property.

By randomly choosing ''c'' and ''b'', the whole spectrum can be looked like [[Discrete uniform distribution|uniform distribution]]. Then, taking them into [[Filter bank|filter bank]]s can separate all frequencies, including Gaussians,<ref>{{cite book |author1=Haitham Hassanieh |author2=Piotr Indyk |author3=Dina Katabi |author4=Eric Price |title=Simple and Practical Algorithm for Sparse Fourier Transform |pages=1183–1194 |date=2012 |doi=10.1137/1.9781611973099.93 |hdl=1721.1/73474 |isbn=978-1-61197-210-8 }}</ref> indicator functions,<ref>{{cite book |author1=A. C. Gilbert |others=S. Guha, P. Indyk, [[Biography:S. Muthukrishnan (computer scientist)|S. Muthukrishnan]], M. Strauss |title=Proceedings of the thiry-fourth annual ACM symposium on Theory of computing |chapter=Near-optimal sparse fourier representations via sampling |date=2002 |pages=152–161 |doi=10.1145/509907.509933|isbn=1581134959 |s2cid=14320243 }}</ref><ref>{{cite book|author1=A. C. Gilbert |author2=S. Muthukrishnan|author3= M. Strauss |editor-first1=Manos |editor-first2=Andrew F |editor-first3=Michael A |editor-last1=Papadakis |editor-last2=Laine |editor-last3=Unser |chapter=Improved time bounds for near-optimal sparse Fourier representations |title=Wavelets XI |volume=5914|pages=59141A|date=21 September 2005 |doi=10.1117/12.615931|series=Proceedings of SPIE|bibcode=2005SPIE.5914..398G|s2cid=12622592 }}</ref> spike trains,<ref>{{Cite book |doi = 10.1109/Allerton.2013.6736670|chapter = Sample-optimal average-case sparse Fourier Transform in two dimensions|title = 2013 51st Annual Allerton Conference on Communication, Control, and Computing (Allerton)|pages = 1258–1265|year = 2013|last1 = Ghazi|first1 = Badih|last2 = Hassanieh|first2 = Haitham|last3 = Indyk|first3 = Piotr|last4 = Katabi|first4 = Dina|last5 = Price|first5 = Eric|last6 = Lixin Shi|isbn = 978-1-4799-3410-2|arxiv = 1303.1209| s2cid=6151728 }}</ref><ref>{{cite journal |last1=Iwen |first1=M. A. |title=Combinatorial Sublinear-Time Fourier Algorithms |journal=Foundations of Computational Mathematics |date=2010-01-05 |volume=10 |issue=3 |pages=303–338 |doi=10.1007/s10208-009-9057-1|s2cid=1631513 }}</ref><ref>{{cite journal |author1=Mark A.Iwen |title=Improved approximation guarantees for sublinear-time Fourier algorithms |journal=Applied and Computational Harmonic Analysis |date=2013-01-01 |volume=34 |issue=1 |pages=57–82 |doi=10.1016/j.acha.2012.03.007 |language=en |issn=1063-5203|arxiv=1010.0014 |s2cid=16808450 }}</ref><ref>{{Cite book |doi = 10.1109/ISIT.2013.6620269|chapter = Computing a k-sparse n-length Discrete Fourier Transform using at most 4k samples and O(k log k) complexity|title = 2013 IEEE International Symposium on Information Theory|pages = 464–468|year = 2013|last1 = Pawar|first1 = Sameer|last2 = Ramchandran|first2 = Kannan|isbn = 978-1-4799-0446-4| s2cid=601496 }}</ref> and Dolph-Chebyshev filters.<ref name = "hikp12stoc">{{cite book |last1=Hassanieh |first1=Haitham |last2=Indyk |first2=Piotr |last3=Katabi |first3=Dina |last4=Price |first4=Eric |title=Proceedings of the forty-fourth annual ACM symposium on Theory of computing |chapter=Nearly optimal sparse fourier transform |date=2012 |pages=563–578 |doi=10.1145/2213977.2214029 |chapter-url=https://dl.acm.org/citation.cfm?id=2214029 |publisher=ACM |arxiv=1201.2501 |series=STOC'12 |isbn=9781450312455 |s2cid=3760962 }}</ref> Each bank only contains a single frequency.

==The prototypical SFT==
Generally, all SFT follows the three stages<ref name="auto"/>

===Identifying frequencies===
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===
After identifying frequencies, we will have many frequency components. We can use Fourier transform to estimate their coefficients.
:<math>
X_k'=\frac 1 L \sum_{l=1}^L x_n'e^{-j\frac{2\pi}{N}n'\ell}
</math>

===Repeating===
Finally, repeating these two stages can we extract the most important components from the original signal.
:<math>
x_n-\sum_{k'=1}^k X_k' e^{j\frac{2\pi}{N}k'n}
</math>

==Sparse Fourier transform in the discrete setting==
In 2012, Hassanieh, Indyk, Katabi, and Price <ref name=hikp12stoc /> proposed an algorithm that takes <math> O ( k \log n \log (n/k) ) </math> samples and runs in the same running time.

==Sparse Fourier transform in the high dimensional setting==
In 2014, Indyk and Kapralov <ref>{{cite journal |last1=Indyk |first1=Piotr |last2=Kapralov |first2=Michael |title=Sample-optimal Fourier sampling in any constant dimension |url=https://archive.org/details/arxiv-1403.5804 |journal= Annual Symposium on Foundations of Computer Science |date=2014 |pages=514–523 |arxiv=1403.5804 |series=FOCS'14 }}</ref> proposed an algorithm that takes <math> 2^{O(d \log d)} k \log n </math> samples and runs in nearly linear time in <math> n </math>. In 2016, Kapralov <ref>{{cite book |last1=Kapralov |first1=Michael |title=Proceedings of the forty-eighth annual ACM symposium on Theory of Computing |chapter=Sparse fourier transform in any constant dimension with nearly-optimal sample complexity in sublinear time |url=https://archive.org/details/arxiv-1604.00845 |date=2016 | arxiv=1604.00845 |series=STOC'16 |pages=264–277 |doi=10.1145/2897518.2897650 |isbn=9781450341325 |s2cid=11847086 }}</ref> proposed an algorithm that uses sublinear samples <math> 2^{O(d^2)} k \log n \log \log n </math> and sublinear decoding time <math> k \log^{O(d)} n </math>. In 2019, Nakos, Song, and Wang <ref>{{cite journal |last1=Nakos |first1=Vasileios |last2=Song |first2=Zhao | last3 = Wang | first3 = Zhengyu |title= (Nearly) Sample-Optimal Sparse Fourier Transform in Any Dimension; RIPless and Filterless |journal= Annual Symposium on Foundations of Computer Science |date=2019 | arxiv = 1909.11123 |series=FOCS'19 }}</ref> introduced a new algorithm which uses nearly optimal samples <math> O( k \log n \log k ) </math> and requires nearly linear time decoding time. A dimension-incremental algorithm was proposed by Potts, Volkmer <ref>{{cite journal |last1=Potts |first1=Daniel |last2=Volkmer |first2=Toni |title= Sparse high-dimensional FFT based on rank-1 lattice sampling |journal= Applied and Computational Harmonic Analysis|date=2016|volume=41 |issue=3 |pages=713–748 | doi= 10.1016/j.acha.2015.05.002}}</ref>   based on sampling along rank-1 lattices.

==Sparse Fourier transform in the continuous setting==

There are several works about generalizing the discrete setting into the continuous setting.<ref>{{cite journal |last1=Price |first1=Eric |last2=Song |first2=Zhao |title=A Robust Sparse Fourier Transform in the Continuous Setting |journal= Annual Symposium on Foundations of Computer Science |date=2015 |pages=583–600 |arxiv=1609.00896 |series=FOCS'15 }}</ref><ref>{{cite journal |last1 = Chen | first1 = Xue | last2 = Kane | first2 = Daniel M.| last3=Price |first3=Eric |last4=Song |first4=Zhao |title=Fourier-Sparse Interpolation without a Frequency Gap |url = https://archive.org/details/arxiv-1609.01361 |journal= Annual Symposium on Foundations of Computer Science |date=2016 |pages=741–750
 |arxiv=1609.01361 |series=FOCS'16 }}</ref>

==Implementations==
There are several works based on MIT, [[Organization:Michigan State University|MSU]], ETH and Universtity of Technology Chemnitz [TUC]. Also, they are free online.
* [https://sourceforge.net/projects/aafftannarborfa/ MSU implementations]
* [http://www.spiral.net/software/sfft.html ETH implementations]
* [http://groups.csail.mit.edu/netmit/sFFT/ MIT implementations]
* [https://github.com/davidediger/sfft GitHub]
* [https://www.tu-chemnitz.de/~tovo/software.php TUC implementations]

== Further reading ==
{{cite book |last=Hassanieh |first=Haitham |date=2018 | title=The Sparse Fourier Transform: Theory and Practice |publisher=Association for Computing Machinery and Morgan & Claypool | isbn=978-1-94748-707-9}}

{{cite book |last=Price |first=Eric |date=2013 | title=Sparse Recovery and Fourier Sampling |publisher=MIT  }}

== References ==
{{reflist}}

[[Category:Fourier analysis]]
[[Category:Big data]]

{{Sourceattribution|Sparse Fourier transform}}