ifftn

Multidimensional inverse fast Fourier transform

Syntax

X = ifftn(Y)

X = ifftn(Y,sz)

X = ifftn(___,symflag)

Description

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

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

X = ifftn(___,symflag) specifies the symmetry of Y in addition to any of the input argument combinations in previous syntaxes. For example, ifftn(Y,'symmetric') treats Y as conjugate symmetric.

Examples

3-D Inverse Transform

You can use the ifftn function to convert multidimensional data sampled in frequency to data sampled in time or space. The ifftn function also allows you to control the size of the transform.

Create a 3-by-3-by-3 array and compute its inverse Fourier transform.

Y = rand(3,3,3);
ifftn(Y);

Pad the dimensions of Y with trailing zeros so that the transform has size 8-by-8-by-8.

X = ifftn(Y,[8 8 8]);
size(X)

ans = 1×3

     8     8     8

Conjugate Symmetric Array

For nearly conjugate symmetric arrays, you can compute the inverse Fourier transform faster by specifying the 'symmetric' option, which also ensures that the output is real.

Compute the 3-D inverse Fourier transform of a nearly conjugate symmetric array.

Y(:,:,1) = [1e-15*i 0; 1 0];
Y(:,:,2) = [0 1; 0 1];
X = ifftn(Y,'symmetric')

X = 
X(:,:,1) =

    0.3750   -0.1250
   -0.1250   -0.1250


X(:,:,2) =

   -0.1250    0.3750
   -0.1250   -0.1250

Input Arguments

Y — Input array

vector | matrix | multidimensional array

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

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

sz — Lengths of inverse transform dimensions

vector of positive integers

Lengths of inverse transform dimensions, specified as a vector of positive integers.

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

symflag — Symmetry type

'nonsymmetric' (default) | 'symmetric'

Symmetry type, specified as 'nonsymmetric' or 'symmetric'. When Y is not exactly conjugate symmetric due to round-off error, ifftn(Y,'symmetric') treats Y as if it were conjugate symmetric by ignoring the second half of its elements (that are in the negative frequency spectrum). For more information on conjugate symmetry, see Algorithms.

More About

N-D Inverse Fourier Transform

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

$$X_{p_1,p_2,\mathrm{...},p_N}={\displaystyle \sum _{j_1=1}^{m_1}\frac{1}{m_1}{\omega }_{m_1}^{p_1{j}_1}{\displaystyle \sum _{j_2=1}^{m_2}\frac{1}{m_2}{\omega }_{m_2}^{p_2{j}_2}\mathrm{...}{\displaystyle \sum _{j_N=1}^{m_N}\frac{1}{m_N}{\omega }_{m_N}^{p_N{j}_N}}}}{Y}_{j_1,j_2,\mathrm{...},j_N}.$$

Each dimension has length m_k 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.

Algorithms

• The ifftn function tests whether the multidimensional array Y is conjugate symmetric. If Y is conjugate symmetric, then the inverse transform computation is faster and the output is real.

  A function $$g(a,b,c,\mathrm{...})$$ is conjugate symmetric if $$g(a,b,c,\mathrm{...})=g^{*}(-a,-b,-c,\mathrm{...})$$. However, the fast Fourier transform of a multidimensional time-domain signal has one half of its spectrum in positive frequencies and the other half in negative frequencies, with the first row, column, page, and so on, reserved for the zero frequencies. For this reason, for example, a 3-D array Y is conjugate symmetric when all of these conditions are true:

  • Y(1,1,2:end) is conjugate symmetric, or Y(1,1,2:end) = conj(Y(1,1,end:-1:2))

  • Y(1,2:end,1) is conjugate symmetric, or Y(1,2:end,1) = conj(Y(1,end:-1:2,1))

  • Y(2:end,1,1) is conjugate symmetric, or Y(2:end,1,1) = conj(Y(end:-1:2,1,1))

  • Y(1,2:end,2:end) is conjugate centrosymmetric, or Y(1,2:end,2:end) = conj(Y(1,end:-1:2,end:-1:2))

  • Y(2:end,1,2:end) is conjugate centrosymmetric, or Y(2:end,1,2:end) = conj(Y(end:-1:2,1,end:-1:2))

  • Y(2:end,2:end,1) is conjugate centrosymmetric, or Y(2:end,2:end,1) = conj(Y(end:-1:2,end:-1:2,1))

  • Y(2:end,2:end,2:end) is conjugate centrosymmetric, or Y(2:end,2:end,2:end) = conj(Y(end:-1:2,end:-1:2,end:-1:2))

Extended Capabilities

C/C++ Code Generation

Generate C and C++ code using MATLAB® Coder™.

Usage notes and limitations:

• Symmetry type 'symmetric' is not supported.

• The sz argument must have a fixed size.

• For MEX output, MATLAB^® Coder™ uses the library that MATLAB uses for FFT algorithms. For standalone C/C++ code, by default, the code generator produces code for FFT algorithms instead of producing FFT library calls. To generate calls to a specific installed FFTW library, provide an FFT library callback class. For more information about an FFT library callback class, see coder.fftw.StandaloneFFTW3Interface (MATLAB Coder).

• For simulation of a MATLAB Function block, the simulation software uses the library that MATLAB uses for FFT algorithms. For C/C++ code generation, by default, the code generator produces code for FFT algorithms instead of producing FFT library calls. To generate calls to a specific installed FFTW library, provide an FFT library callback class. For more information about an FFT library callback class, see coder.fftw.StandaloneFFTW3Interface (MATLAB Coder).

• Using the Code Replacement Library (CRL), you can generate optimized code that runs on ARM^® Cortex^®-A processors with Neon extension. To generate this optimized code, you must install the Embedded Coder^® Support Package for ARM Cortex-A Processors (Embedded Coder). The generated code for ARM Cortex-A uses the Ne10 library. For more information, see Ne10 Conditions for MATLAB Functions to Support ARM Cortex-A Processors (Embedded Coder).

• Using the Code Replacement Library (CRL), you can generate optimized code that runs on ARM Cortex-M processors. To generate this optimized code, you must install the Embedded Coder Support Package for ARM Cortex-M Processors (Embedded Coder). The generated code for ARM Cortex-M uses the CMSIS library. For more information, see CMSIS Conditions for MATLAB Functions to Support ARM Cortex-M Processors (Embedded Coder).

GPU Code Generation

Generate CUDA® code for NVIDIA® GPUs using GPU Coder™.

Usage notes and limitations:

• Symmetry type 'symmetric' is not supported.

• The sz argument must have a fixed size.

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.

GPU Arrays

Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.

The ifftn function supports GPU array input with these usage notes and limitations:

• Unless symflag is 'symmetric', the output is always complex even if all imaginary parts are zero.

For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).

Distributed Arrays

Partition large arrays across the combined memory of your cluster using Parallel Computing Toolbox™.

This function fully supports distributed arrays. For more information, see Run MATLAB Functions with Distributed Arrays (Parallel Computing Toolbox).

Version History

Introduced before R2006a