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In applied mathematics, the '''non-uniform discrete Fourier transform''' ('''NUDFT''' or '''NDFT''') of a signal is a type of [[Fourier transform]], related to a [[discrete Fourier transform]] or [[discrete-time Fourier transform]], but in which the input signal is not sampled at equally spaced points or frequencies (or both). It is a generalization of the [[Discrete Fourier transform#Generalized DFT (shifted and non-linear phase)|shifted DFT]]. It has important applications in signal processing,<ref>{{cite book|last1=Bagchi|first1=Sonali|last2=Mitra|first2=Sanjit K.|title=The Nonuniform Discrete Fourier Transform and Its Applications in Signal Processing|date=1999|publisher=Springer US|location=Boston, MA|isbn=978-1-4615-4925-3}}</ref> [[magnetic resonance imaging]],<ref name=Fessler03>{{cite journal|last1=Fessler|first1=J.A.|last2=Sutton|first2=B.P.|title=Nonuniform fast fourier transforms using min-max interpolation|journal=IEEE Transactions on Signal Processing|date=February 2003|volume=51|issue=2|pages=560–574|doi=10.1109/TSP.2002.807005|bibcode=2003ITSP...51..560F|hdl=2027.42/85840|hdl-access=free}}</ref> and the numerical solution of partial differential equations.<ref>{{cite journal|last1=Lee|first1=June-Yub|last2=Greengard|first2=Leslie|title=The type 3 nonuniform FFT and its applications|journal=Journal of Computational Physics|date=June 2005|volume=206|issue=1|pages=1–5|doi=10.1016/j.jcp.2004.12.004|bibcode=2005JCoPh.206....1L}}</ref>

As a generalized approach for [[nonuniform sampling]], the NUDFT allows one to obtain frequency domain information of a finite length signal at any frequency. One of the reasons to adopt the NUDFT is that many signals have their energy distributed nonuniformly in the frequency domain. Therefore, a nonuniform sampling scheme could be more convenient and useful in many [[digital signal processing]] applications. For example, the NUDFT provides a variable spectral resolution controlled by the user.

==Definition==
The ''nonuniform discrete Fourier transform'' transforms a sequence of <math>N</math> complex numbers <math>x_0, \ldots, x_{N-1}</math> into another sequence of complex numbers <math>X_0, \ldots, X_{N-1}</math> defined by
{{NumBlk|:|<math>X_k = \sum_{n=0}^{N-1} x_n e^{-2\pi i p_n f_k}, \quad 0 \leq k \leq N-1,</math>|{{EquationRef|1}}}}
where <math>p_0, \ldots, p_{N-1} \in [0,1]</math> are sample points and <math>f_0, \ldots, f_{N-1} \in [0,N]</math> are frequencies. Note that if <math>p_n = n/N</math> and <math>f_k = k</math>, then equation ({{EquationNote|1}}) reduces to the [[discrete Fourier transform]]. There are three types of NUDFTs.<ref name="Greengard2004">{{cite journal|last1=Greengard|first1=Leslie|last2=Lee|first2=June-Yub|title=Accelerating the Nonuniform Fast Fourier Transform|journal=SIAM Review|date=January 2004|volume=46|issue=3|pages=443–454|doi=10.1137/S003614450343200X|bibcode=2004SIAMR..46..443G|citeseerx=10.1.1.227.3679}}</ref> Note that these types are not universal and different authors will refer to different types by different numbers.

* The nonuniform discrete Fourier transform of type I (NUDFT-I) uses uniform sample points <math>p_n = n/N</math> but nonuniform (i.e. non-integer) frequencies <math>f_k</math>. This corresponds to evaluating a [[generalized Fourier series]] at equispaced points. It is also known as NDFT<ref name="Plonka2019">{{cite book|last1=Plonka|first1=Gerlind |author1-link=Gerlind Plonka|last2=Potts|first2=Daniel |last3=Steidl|first3=Gabriele|author3-link=Gabriele Steidl |last4=Tasche|first4=Manfred |title=Numerical Fourier Analysis |publisher=Birkhäuser |date=2019 |isbn=978-3-030-04306-3 |doi=10.1007/978-3-030-04306-3}}</ref> or forward NDFT <ref name="PyNUFFT">{{cite web |last1=PyNUFFT Services |title=Basic use of PyNUFFT — PyNUFFT 2023.2.2 documentation |url=https://pynufft.readthedocs.io/en/latest/tutor/basic_use.html |website=pynufft.readthedocs.io |access-date=27 February 2024}}</ref><ref name="finufft">{{cite web |last1=The Simons Foundation |title=Mathematical definitions of transforms — finufft 2.2.0 documentation |url=https://finufft.readthedocs.io/en/latest/math.html#math |website=finufft.readthedocs.io |access-date=27 February 2024}}</ref>
* The nonuniform discrete Fourier transform of type II (NUDFT-II) uses uniform (i.e. integer) frequencies <math>f_k = k</math> but nonuniform sample points <math>p_n</math>. This corresponds to evaluating a Fourier series at nonequispaced points. It is also known as adjoint NDFT.<ref name="finufft" /><ref name="PyNUFFT" />
* The nonuniform discrete Fourier transform of type III (NUDFT-III) uses both nonuniform sample points <math>p_n</math> and nonuniform frequencies <math>f_k</math>. This corresponds to evaluating a [[generalized Fourier series]] at nonequispaced points. It is also known as NNDFT.

A similar set of NUDFTs can be defined by substituting <math>-i</math> for <math>+i</math> in equation ({{EquationNote|1}}).
Unlike in the uniform case, however, this substitution is unrelated to the inverse Fourier transform.
The inversion of the NUDFT is a separate problem, discussed below.

==Multidimensional NUDFT==
The multidimensional NUDFT converts a <math>d</math>-dimensional array of complex numbers <math>x_{\mathbf{n}}</math> into another <math>d</math>-dimensional array of complex numbers <math>X_{\mathbf{k}}</math> defined by
:<math>X_{\mathbf{k}} = \sum_{\mathbf{n}=\mathbf{0}}^{\mathbf{N}-1} x_{\mathbf{n}} e^{-2\pi i \mathbf{p}_{\mathbf{n}} \cdot \boldsymbol{f}_{\mathbf{k}}}</math>
where <math>\mathbf{p}_{\mathbf{n}} \in [0,1]^d</math> are sample points, <math>\boldsymbol{f}_{\mathbf{k}} \in [0,N_1] \times [0,N_2] \times \cdots \times [0,N_d]</math> are frequencies, and <math>\mathbf{n} = (n_1,n_2,\ldots,n_d)</math> and <math>\mathbf{k} = (k_1,k_2,\ldots,k_d)</math> are <math>d</math>-dimensional vectors of indices from 0 to <math>\mathbf{N}-1 = (N_1-1,N_2-1,\ldots,N_d-1)</math>. The multidimensional NUDFTs of types I, II, and III are defined analogously to the 1D case.<ref name="Greengard2004"/>

==Relationship to Z-transform==
The NUDFT-I can be expressed as a [[Z-transform]].<ref name="Marvasti2001">{{cite book|last1=Marvasti|first1=Farokh|title=Nonuniform Sampling: Theory and Practice|date=2001|publisher=Springer|location=New York|isbn=978-1-4615-1229-5|pages=325–360}}</ref> The NUDFT-I of a sequence <math>x[n]</math> of length <math>N</math> is
:<math>X(z_k)=X(z)|_{z=z_k}=\sum_{n=0}^{N-1}x[n]z_k^{-n},\quad k=0, 1, ..., N-1,</math>

where <math>X(z)</math> is the Z-transform of <math>x[n]</math>, and <math>\{z_i\}_{i=0, 1, ..., N-1}</math> are arbitrarily distinct points in the z-plane. Note that the NUDFT reduces to the DFT when the sampling points are located on the unit circle at equally spaced angles.

Expressing the above as a matrix, we get
:<math>\mathbf{X}=\mathbf{D}\mathbf{x}</math>

where
:<math>
\mathbf{X}=\begin{bmatrix}
 X(z_0)\\
 X(z_1)\\
 \vdots\\
 X(z_{N-1})
\end{bmatrix},\quad
\mathbf{x}=\begin{bmatrix}
 x[0]\\
 x[1]\\
 \vdots\\
 x[N-1]
\end{bmatrix},\text{ and}\quad
\mathbf{D}=\begin{bmatrix}
 1 & z_0^{-1} & z_0^{-2} & \cdots & z_0^{-(N-1)}\\
 1 & z_1^{-1} & z_1^{-2} & \cdots & z_1^{-(N-1)}\\
 \vdots & \vdots & \vdots & \ddots & \vdots\\
 1 & z_{N-1}^{-1} & z_{N-1}^{-2} & \cdots & z_{N-1}^{-(N-1)}
\end{bmatrix}.
</math>

===Direct inversion of the NUDFT-I===
As we can see, the NUDFT-I is characterized by <math>\mathbf{D}</math> and hence the <math>N</math> <math>{z_k}</math> points. If we further factorize <math>\det(\mathbf{D})</math>, we can see that <math>\mathbf{D}</math> is nonsingular provided the <math>N</math> <math>{z_k}</math> points are distinct. If <math>\mathbf{D}</math> is nonsingular, we can get a unique inverse NUDFT-I as follows:
:<math>\mathbf{x}=\mathbf{D^{-1}}\mathbf{X}</math>.

Given <math>\mathbf{X}\text{ and }\mathbf{D}</math>, we can use [[Gaussian elimination]] to solve for <math>\mathbf{x}</math>. However, the complexity of this method is <math>O(N^3)</math>. To solve this problem more efficiently, we first determine <math>X(z)</math> directly by polynomial interpolation:
:<math>\hat X [k]=X(z_k),\quad k=0, 1, ..., N-1</math>.

Then <math>x[n]</math> are the coefficients of the above interpolating polynomial.

Expressing <math>X(z)</math> as the [[Lagrange polynomial]] of order <math>N-1</math>, we get
:<math>X(z)=\sum_{k=0}^{N-1}\frac{L_k(z)}{L_k(z_k)}\hat X [k],</math>

where <math>\{L_i(z)\}_{i=0, 1, ..., N-1}</math> are the fundamental polynomials:

:<math>L_k(z)=\prod_{i\ne k}(1-z_iz^{-1}),\quad k=0, 1, ..., N-1</math>.

Expressing <math>X(z)</math> by the Newton interpolation method, we get
:<math>X(z)=c_0 + c_1(1-z_0z^{-1}) + c_2(1-z_0z^{-1})(1-z_1z^{-1}) + \cdots + c_{N-1}\prod_{k=0}^{N-2}(1-z_kz^{-1}),</math>

where <math>c_j</math> is the divided difference of the <math>j</math>th order of <math>\hat X [0], \hat X [1], ..., \hat X [j]</math> with respect to <math>z_0, z_1, ..., z_j</math>:
:<math>c_0 = \hat X [0],</math>

:<math>c_1 = \frac{\hat X [1]-c_0}{1-z_0z_1^{-1}},</math>

:<math>c_2 = \frac{\hat X [2]-c_0-c_1(1-z_0z^{-1})}{(1-z_0z_2^{-1})(1-z_1z_2^{-1})},</math>

::<math>\vdots</math>

The disadvantage of the Lagrange representation is that any additional point included will increase the order of the interpolating polynomial, leading to the need to recompute all the fundamental polynomials. However, any additional point included in the Newton representation only requires the addition of one more term.

We can use a lower triangular system to solve <math>\{c_j\}</math>:
:<math>\mathbf{L}\mathbf{c}=\mathbf{X}</math>

where
:<math>
\mathbf{X}=\begin{bmatrix}
 \hat X [0]\\
 \hat X [1]\\
 \vdots\\
 \hat X [N-1]
\end{bmatrix},\quad
\mathbf{c}=\begin{bmatrix}
 c_0\\
 c_1\\
 \vdots\\
 c_{N-1}
\end{bmatrix},\text{ and}\quad
\mathbf{L}=\begin{bmatrix}
 1 & 0 & 0 & \cdots & 0\\
 1 & (1-z_0z_1^{-1}) & 0 & \cdots & 0\\
 1 & (1-z_0z_2^{-1}) & (1-z_0z_2^{-1})(1-z_1z_2^{-1}) & \cdots & 0\\
 \vdots & \vdots & \vdots & \ddots & \vdots\\
 1 & (1-z_0z_{N-1}^{-1}) & (1-z_0z_{N-1}^{-1})(1-z_1z_{N-1}^{-1}) & \cdots & \prod_{k=0}^{N-2}(1-z_kz_{N-1}^{-1})
\end{bmatrix}.
</math>

By the above equation, <math>\{c_j\}</math> can be computed within <math>O(N^2)</math> operations. In this way Newton interpolation is more efficient than Lagrange Interpolation unless the latter is modified by
:<math>L_{k+1}(z) = \frac{(1-z_{k+1}z^{-1})}{(1-z_kz^{-1})}L_k(z),\quad k=0, 1, ..., N-1</math>.

==Nonuniform fast Fourier transform==
While a naive application of equation ({{EquationNote|1}}) results in an <math>O(N^2)</math> algorithm for computing the NUDFT, <math>O(N \log N)</math> algorithms based on the [[fast Fourier transform]] (FFT) do exist. Such algorithms are referred to as NUFFTs or NFFTs and have been developed based on oversampling and interpolation,<ref>{{cite thesis|type=PhD|last=Dutt|first=Alok|title=Fast Fourier Transforms for Nonequispaced Data|date=May 1993|publisher=Yale University|url=https://cpsc.yale.edu/sites/default/files/files/tr981.pdf}}</ref><ref>{{cite journal|last1=Dutt|first1=Alok|last2=Rokhlin|first2=Vladimir|title=Fast Fourier Transforms for Nonequispaced Data|journal=SIAM Journal on Scientific Computing|date=November 1993|volume=14|issue=6|pages=1368–1393|doi=10.1137/0914081|bibcode=1993SJSC...14.1368D }}</ref><ref>{{cite journal|last1=Potts|first1=Daniel|last2=Steidl|first2=Gabriele|author2-link= Gabriele Steidl |title=Fast Summation at Nonequispaced Knots by NFFT|journal=SIAM Journal on Scientific Computing|date=January 2003|volume=24|issue=6|pages=2013–2037|doi=10.1137/S1064827502400984|bibcode=2003SJSC...24.2013P }}</ref><ref>{{cite journal|last1=Boyd|first1=John P|title=A fast algorithm for Chebyshev, Fourier, and sinc interpolation onto an irregular grid|journal=Journal of Computational Physics|date=December 1992|volume=103|issue=2|pages=243–257|doi=10.1016/0021-9991(92)90399-J|bibcode=1992JCoPh.103..243B|url=https://deepblue.lib.umich.edu/bitstream/2027.42/29694/1/0000026.pdf|hdl=2027.42/29694|hdl-access=free}}</ref> min-max interpolation,<ref name=Fessler03/> and low-rank approximation.<ref>{{cite journal|last1=Ruiz-Antolín|first1=Diego|last2=Townsend|first2=Alex|title=A Nonuniform Fast Fourier Transform Based on Low Rank Approximation|journal=SIAM Journal on Scientific Computing|date=20 February 2018|volume=40|issue=1|pages=A529–A547|doi=10.1137/17M1134822|arxiv=1701.04492|bibcode=2018SJSC...40A.529R |url=https://repositorio.unican.es/xmlui/bitstream/handle/10902/13767/ANonuniform.pdf|hdl=10902/13767}}</ref> In general, NUFFTs leverage the FFT by converting the nonuniform problem into a uniform problem (or a sequence of uniform problems) to which the FFT can be applied.<ref name="Greengard2004"/> Software libraries for performing NUFFTs are available in 1D, 2D, and 3D.<ref name="finufft" /><ref name="PyNUFFT" /><ref>{{cite web|title=NUFFT page|url=https://cims.nyu.edu/cmcl/nufft/nufft.html|website=cims.nyu.edu}}</ref><ref>{{cite web|title=NFFT|url=http://www.nfft.org/|website=www.nfft.org|language=en}}</ref><ref>{{cite web|title=MikaelSlevinsky/FastTransforms.jl|url=https://github.com/MikaelSlevinsky/FastTransforms.jl|website=GitHub|language=en|date=2019-02-13}}</ref><ref>{{cite web|title=chebfun/chebfun|url=https://github.com/chebfun/chebfun|website=GitHub|language=en|date=2019-02-07}}</ref>

==Applications==
The applications of the NUDFT include:
* [[Digital signal processing]]
* [[Magnetic resonance imaging]]
* [[Numerical partial differential equations]]
* [[Semi-Lagrangian scheme|Semi-Lagrangian schemes]]
* [[Spectral methods]]
* [[Spectroscopy|Spectral analysis]]
* [[Digital filter|Digital filter design]]
* [[Antenna array|Antenna array design]]
* Detection and decoding of [[Dual-tone multi-frequency signaling|dual-tone multi-frequency (DTMF) signals]]

==See also==
* [[Discrete Fourier transform]]
* [[Fast Fourier transform]]
* [[Least-squares spectral analysis]]
* [[Lomb–Scargle periodogram]]
* [[Spectral estimation]]
* [[Unevenly spaced time series]]

==References==
{{Reflist}}

== External links==
* [http://homepages.inf.ed.ac.uk/rbf/CVonline/LOCAL_COPIES/PIRODDI1/NUFT/NUFT.html Non-Uniform Fourier Transform: A Tutorial].
* [http://www-user.tu-chemnitz.de/~potts/nfft/guide3/html/index.html NFFT 3.0 – Tutorial]
* [https://cims.nyu.edu/cmcl/nufft/nufft.html NUFFT software library]

[[Category:Fourier analysis]]
[[Category:Transforms]]
[[Category:Digital signal processing]]