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# fftn

N-D fast Fourier transform

Y = fftn(X)
Y = fftn(X,sz)

## Description

example

Y = fftn(X) returns the multidimensional Fourier transform of an N-D array using a fast Fourier transform algorithm. The N-D transform is equivalent to computing the 1-D transform along each dimension of X. The output Y is the same size as X.

example

Y = fftn(X,sz) truncates X or pads X with trailing zeros before taking the transform according to the elements of the vector sz. Each element of sz defines the length of the corresponding transform dimensions. For example, if X is a 5-by-5-by-5 array, then Y = fftn(X,[8 8 8]) pads each dimension with zeros resulting in an 8-by-8-by-8 transform Y.

## Examples

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You can use the fftn function to compute a 1-D fast Fourier transform in each dimension of a multidimensional array.

Create a 3-D signal X. The size of X is 20-by-20-by-20.

x = (1:20)';
y = 1:20;
z = reshape(1:20,[1 1 20]);
X = cos(2*pi*0.01*x) + sin(2*pi*0.02*y) + cos(2*pi*0.03*z);

Compute the 3-D Fourier transform of the signal, which is also a 20-by-20-by-20 array.

Y = fftn(X);

Pad X with zeros to compute a 32-by-32-by-32 transform.

m = nextpow2(20);
Y = fftn(X,[2^m 2^m 2^m]);
size(Y)
ans = 1×3

32    32    32

## Input Arguments

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Input array, specified as a matrix or a multidimensional array. If X is of type single, then fftn natively computes in single precision, and Y is also of type single. Otherwise, Y is returned as type double.

Data Types: double | single | int8 | int16 | int32 | uint8 | uint16 | uint32 | logical
Complex Number Support: Yes

Length of the transform dimensions, specified as a vector of positive integers. The elements of sz correspond to the transformation lengths of the corresponding dimensions of X. length(sz) must be equal to ndims(X).

Data Types: single | double | int8 | int16 | int32 | uint8 | uint16 | uint32 | logical

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### N-D Fourier Transform

The discrete Fourier transform Y of an N-D array X is defined as

${Y}_{{p}_{1},{p}_{2},...,{p}_{N}}=\sum _{{j}_{1}=0}^{{m}_{1}-1}{\omega }_{{m}_{1}}^{{p}_{1}{j}_{1}}\sum _{{j}_{2}=0}^{{m}_{2}-1}{\omega }_{{m}_{2}}^{{p}_{2}{j}_{2}}...\sum _{{j}_{N}}^{{m}_{N}-1}{\omega }_{{m}_{N}}^{{p}_{N}{j}_{N}}{X}_{{j}_{1},{j}_{2},...,{j}_{N}}.$

Each dimension has length mk for k = 1,2,...,N, and ${\omega }_{{m}_{k}}={e}^{-2\pi i/{m}_{k}}$ are complex roots of unity where i is the imaginary unit.