# nufftn

*N*-D nonuniform fast Fourier transform

## Description

## Examples

### Nonuniform Sample Points

Create a 3-D signal `X`

sampled at unevenly spaced points `t`

in each dimension. Compute the nonuniform fast Fourier transform `Y`

.

t = [1:10 11:2:29]'; x = t; y = t'; z = reshape(t,[1 1 20]); X = cos(2*pi*0.01*x) + sin(2*pi*0.02*y) + cos(2*pi*0.03*z); Y = nufftn(X,{t,t,t});

## Input Arguments

`X`

— Input array

vector | matrix | multidimensional array

Input array, specified as a numeric vector, matrix, or multidimensional array.

**Data Types: **`double`

| `single`

| `int8`

| `int16`

| `int32`

| `uint8`

| `uint16`

| `uint32`

| `logical`

**Complex Number Support: **Yes

`t`

— Sample points

vector | matrix | cell array of vectors

Sample points, specified as a vector, matrix, or cell array of vectors for each
dimension of the input array `X`

.

When specified as a vector or matrix, the number of rows of `t`

must equal the number of elements in `X`

. When no query points are
specified, the transform is computed at *N* evenly spaced query points
in each dimension, where `N = ceil(numel(X).^(1/D))`

and
*D* is the number of columns in `t`

. The output
`Y`

is a *D*-dimensional array of length
*N* in each dimension.

When `t`

is specified as a cell array of *D*
vectors, the length of each vector must equal the length of the corresponding dimension
of `X`

.

**Data Types: **`double`

| `single`

`f`

— Query points

vector | matrix | cell array of vectors

Query points, specified as a vector, matrix, or cell array of vectors for each
dimension of the input array `X`

. When specified as a matrix,
`f`

must be an *M*-by-*k* array,
where *k* is greater than or equal to the number of dimensions
*D* defined by the sample points.

When `f`

is specified as a cell array of *D*
vectors, the length of each dimension of the output `Y`

is equal to the
length of the corresponding vector in the cell array.

To specify `f`

without specifying sample points, use
`nufftn(X,[],f)`

.

**Data Types: **`double`

| `single`

## Extended Capabilities

### Thread-Based Environment

Run code in the background using MATLAB® `backgroundPool`

or accelerate code with Parallel Computing Toolbox™ `ThreadPool`

.

This function fully supports thread-based environments. For more information, see Run MATLAB Functions in Thread-Based Environment.

## Version History

**Introduced in R2020a**

### R2022a: Improved performance with nonuniform sample points or query points

The `nufftn`

function shows improved performance when operating on
either nonuniformly spaced sample points or nonuniformly spaced query points.

For example, this code constructs a 32768-by-3 matrix of nonuniform sample points
`t`

and calculates the nonuniform discrete Fourier transform along each
dimension of a 32-by-32-by-32 array. It runs about 14.5x faster than in the previous
release:

function timingSamplePoints rng default t = rand(32^3,3); X = rand(32,32,32); tic; Y = nufftn(X,t); toc end

The approximate execution times are:

**R2021b:** 2.76 s

**R2022a:** 0.19 s

The code was timed on a Windows^{®} 10, Intel^{®}
Xeon^{®} CPU E5-1650 v4 @ 3.60 GHz test system by calling the
`timingSamplePoints`

function.

As another example, this code constructs a 65536-by-3 matrix of nonuniform query points
`f`

and calculates the nonuniform discrete Fourier transform along each
dimension of a 64-by-32-by-32 array. It runs about 42.6x faster than in the previous
release:

function timingQueryPoints rng default f = rand(64*32*32,3); X = rand(64,32,32); tic; Y = nufftn(X,[],f); toc end

The approximate execution times are:

**R2021b:** 4.26 s

**R2022a:** 0.10 s

The code was timed on a Windows 10, Intel
Xeon CPU E5-1650 v4 @ 3.60 GHz test system by calling the
`timingQueryPoints`

function.

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