ifft

Inverse fast Fourier transform

Syntax

X = ifft(Y)

X = ifft(Y,n)

X = ifft(Y,n,dim)

X = ifft(___,symflag)

Description

X = ifft(Y) computes the inverse discrete Fourier transform of Y using a fast Fourier transform algorithm. X is the same size as Y.

• If Y is a vector, then ifft(Y) returns the inverse transform of the vector.

• If Y is a matrix, then ifft(Y) returns the inverse transform of each column of the matrix.

• If Y is a multidimensional array, then ifft(Y) treats the values along the first dimension whose size does not equal 1 as vectors and returns the inverse transform of each vector.

X = ifft(Y,n) returns the n-point inverse Fourier transform of Y by padding Y with trailing zeros to length n.

X = ifft(Y,n,dim) returns the inverse Fourier transform along the dimension dim. For example, if Y is a matrix, then ifft(Y,n,2) returns the n-point inverse transform of each row.

X = ifft(___,symflag) specifies the symmetry of Y. For example, ifft(Y,'symmetric') treats Y as conjugate symmetric.

Examples

Inverse Transform of Vector

The Fourier transform and its inverse convert between data sampled in time and space and data sampled in frequency.

Create a vector and compute its Fourier transform.

X = [1 2 3 4 5];
Y = fft(X)

Y = 1×5 complex

  15.0000 + 0.0000i  -2.5000 + 3.4410i  -2.5000 + 0.8123i  -2.5000 - 0.8123i  -2.5000 - 3.4410i

Compute the inverse transform of Y, which is the same as the original vector X.

ifft(Y)

ans = 1×5

     1     2     3     4     5

Padded Inverse Transform of Matrix

The ifft function allows you to control the size of the transform.

Create a random 3-by-5 matrix and compute the 8-point inverse Fourier transform of each row. Each row of the result has length 8.

Y = rand(3,5);
n = 8;
X = ifft(Y,n,2);
size(X)

ans = 1×2

     3     8

Conjugate Symmetric Vector

For nearly conjugate symmetric vectors, you can compute the inverse Fourier transform faster by specifying the 'symmetric' option, which also ensures that the output is real. Nearly conjugate symmetric data can arise when computations introduce round-off error.

Create a vector Y that is nearly conjugate symmetric and compute its inverse Fourier transform. Then, compute the inverse transform specifying the 'symmetric' option, which eliminates the nearly 0 imaginary parts.

Y = [1 2:4+eps(4) 4:-1:2]

Y = 1×7

    1.0000    2.0000    3.0000    4.0000    4.0000    3.0000    2.0000

X = ifft(Y)

X = 1×7 complex

   2.7143 + 0.0000i  -0.7213 + 0.0000i  -0.0440 - 0.0000i  -0.0919 + 0.0000i  -0.0919 - 0.0000i  -0.0440 + 0.0000i  -0.7213 - 0.0000i

Xsym = ifft(Y,'symmetric')

Xsym = 1×7

    2.7143   -0.7213   -0.0440   -0.0919   -0.0919   -0.0440   -0.7213

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 ifft 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

n — Inverse transform length

[] (default) | nonnegative integer scalar

Inverse transform length, specified as [] or a nonnegative integer scalar. Padding Y with zeros by specifying a transform length larger than the length of Y can improve the performance of ifft. The length is typically specified as a power of 2 or a product of small prime numbers. If n is less than the length of the signal, then ifft ignores the remaining signal values past the nth entry and returns the truncated result. If n is 0, then ifft returns an empty matrix.

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

dim — Dimension to operate along

positive integer scalar

Dimension to operate along, specified as a positive integer scalar. By default, dim is the first array dimension whose size does not equal 1. For example, consider a matrix Y.

• ifft(Y,[],1) returns the inverse Fourier transform of each column.

  

• ifft(Y,[],2) returns the inverse Fourier transform of each row.

  

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, ifft(Y,'symmetric') treats Y as if it were conjugate symmetric. For more information on conjugate symmetry, see Algorithms.

More About

Discrete Fourier Transform of Vector

Y = fft(X) and X = ifft(Y) implement the Fourier transform and inverse Fourier transform, respectively. For X and Y of length n, these transforms are defined as follows:

$$\begin{array}{l}Y(k)={\displaystyle \sum _{j=1}^{n}X}(j)W_n^{(j-1)\text{}(k-1)}\\ X(j)=\frac{1}{n}{\displaystyle \sum _{k=1}^{n}Y}(k)W_n{}^{-(j-1)\text{}(k-1)},\end{array}$$

where

$$W_n=e^{(-2\pi i)/n}$$

is one of n roots of unity.

Algorithms

• The ifft function tests whether the vectors in Y are conjugate symmetric. A vector v is conjugate symmetric when it equals conj(v([1,end:-1:2])). If the vectors in Y are conjugate symmetric, then the inverse transform computation is faster and the output is real.

Extended Capabilities

C/C++ Code Generation

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

Usage notes and limitations:

• Output is complex.

• Symmetry type 'symmetric' is not supported.

• For limitations related to variable-size data, see Variable-Sizing Restrictions for Code Generation of Toolbox Functions (MATLAB Coder).

• 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 Support Package for ARM Cortex-A Processors). 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 Support Package for ARM Cortex-A Processors).

• 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 Support Package for ARM Cortex-M Processors). 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 Support Package for ARM Cortex-M Processors).

GPU Code Generation

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

Usage notes and limitations:

• Output is complex.

• Symmetry type 'symmetric' is not supported.

• For limitations related to variable-size data, see Variable-Sizing Restrictions for Code Generation of Toolbox Functions (MATLAB Coder).

GPU Arrays

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

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).