ifft2

2-D inverse fast Fourier transform

Syntax

X = ifft2(Y)

X = ifft2(Y,m,n)

X = ifft2(___,symflag)

Description

X = ifft2(Y) returns the two-dimensional discrete inverse Fourier transform of a matrix using a fast Fourier transform algorithm. If Y is a multidimensional array, then ifft2 takes the 2-D inverse transform of each dimension higher than 2. The output X is the same size as Y.

X = ifft2(Y,m,n) truncates Y or pads Y with trailing zeros to form an m-by-n matrix before computing the inverse transform. X is also m-by-n. If Y is a multidimensional array, then ifft2 shapes the first two dimensions of Y according to m and n.

X = ifft2(___,symflag) specifies the symmetry of Y in addition to any of the input argument combinations in previous syntaxes. For example, ifft2(Y,'symmetric') treats Y as conjugate symmetric.

Examples

2-D Inverse Transform of Matrix

You can use the ifft2 function to convert 2-D signals sampled in frequency to signals sampled in time or space. The ifft2 function also allows you to control the size of the transform.

Create a 3-by-3 matrix and compute its Fourier transform.

X = magic(3)

X = 3×3

     8     1     6
     3     5     7
     4     9     2

Y = fft2(X)

Y = 3×3 complex

  45.0000 + 0.0000i   0.0000 + 0.0000i   0.0000 + 0.0000i
   0.0000 + 0.0000i  13.5000 + 7.7942i   0.0000 - 5.1962i
   0.0000 - 0.0000i   0.0000 + 5.1962i  13.5000 - 7.7942i

Take the inverse transform of Y, which is the same as the original matrix X, up to round-off error.

ifft2(Y)

ans = 3×3

    8.0000    1.0000    6.0000
    3.0000    5.0000    7.0000
    4.0000    9.0000    2.0000

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

Z = ifft2(Y,8,8);
size(Z)

ans = 1×2

     8     8

Conjugate Symmetric Matrix

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

Compute the 2-D inverse Fourier transform of a nearly conjugate symmetric matrix.

Y = [3+1e-15*i 5;
     5 3];
X = ifft2(Y,'symmetric')

X = 2×2

     4     0
     0    -1

Input Arguments

Y — Input array

matrix | multidimensional array

Input array, specified as a matrix or a multidimensional array. If Y is of type single, then ifft2 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

m — Number of inverse transform rows

positive integer scalar

Number of inverse transform rows, specified as a positive integer scalar.

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

n — Number of inverse transform columns

positive integer scalar

Number of inverse transform columns, specified as a positive integer scalar.

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, ifft2(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

2-D Inverse Fourier Transform

This formula defines the discrete inverse Fourier transform X of an m-by-n matrix Y:

$$X_{p,q}=\frac{1}{m}{\displaystyle \sum _{j=1}^m\frac{1}{n}{\displaystyle \sum _{k=1}^n{\omega }_m^{(j-1)(p-1)}{\omega }_n^{(k-1)(q-1)}}}{Y}_{j,k}$$

ω_m and ω_n are complex roots of unity:

$$\begin{array}{l}{\omega }_m=e^{2\pi i/m}\\ {\omega }_n=e^{2\pi i/n}\end{array}$$

i is the imaginary unit. p runs from 1 to m and q runs from 1 to n.

Algorithms

• The ifft2 function tests whether the matrix Y is conjugate symmetric. If Y is conjugate symmetric, then the inverse transform computation is faster and the output is real.

  A function $$g(a,b)$$ is conjugate symmetric if $$g(a,b)=g^{*}(-a,-b)$$. However, the fast Fourier transform of a 2-D time-domain signal has one half of its spectrum in positive frequencies and the other half in negative frequencies, with the first row and column reserved for the zero frequencies. For this reason, a matrix Y is conjugate symmetric when all of these conditions are true:

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

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

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

• 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.

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 ifft2 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