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# fft

fast Fourier transform.

# ifft

Inverse fast Fourier transform.

### Syntax

X=fft(A [,sign] [,option]) X=fft(A,sign,selection [,option]) X=fft(A,sign,dims,incr [,option] )

### Arguments

- A
a real or complex vector or real or complex array (vector, matrix or N-D array).

- X
- a real or complex array with same shape as
`A`

. - sign
- an integer. with possible values
`1`

or`-1`

. Select direct or inverse transform. The default value is`-1`

(direct transform). - option
- a character string. with possible values
`"symmetric"`

or`"nonsymmetric"`

. Indicates if`A`

is symmetric or not. If this argument is omitted the algorithm automatically determines if`A`

is symmetric or not. See the Description part for details. - selection
- a vector containing index on
`A`

array dimensions. See the Description part for details. - dims
- a vector of positive numbers with integer values, or a
vector of positive integers. See the Description part for details.
Each element must be a divisor of the total number of elements of

`A`

.The product of the elements must be less than the total number of elements of

`A`

. - incr
- a vector of positive numbers with integer values, or a
vector of positive integers. See the Description part for
details.
`incr`

must have the same number of elements than`dims`

.Each element must be a divisor of the total number of elements of

`A`

.The

`incr`

elements must be in strictly increasing order.

### Description

This function realizes direct or inverse 1-D or N-D Discrete Fourier Transforms.- Short syntax
- direct
`X=fft(A,-1 [,option])`

or`X=fft(A [,option])`

gives a direct transform.- single variate
If

`A`

is a vector a single variate direct FFT is computed that is:(the

`-1`

argument refers to the sign of the exponent..., NOT to "inverse"),- multivariate
If

`A`

is a matrix or a multidimensional array a multivariate direct FFT is performed.

- inverse
`X=fft(A,1)`

or`X=ifft(A)`

performs the inverse normalized transform, such that`A==ifft(fft(A))`

.- single variate
- If
`A`

is a vector a single variate inverse FFT is computed - multivariate
If

`a`

is a matrix or or a multidimensional array a multivariate inverse FFT is performed.

- Long syntax for FFT along specified dimensions
`X=fft(A,sign,selection [,option])`

allows to perform efficiently all direct or inverse fft of the "slices" of`A`

along selected dimensions.For example, if

`A`

is a 3-D array`X=fft(A,-1,2)`

is equivalent to:and

`X=fft(A,-1,[1 3])`

is equivalent to:`X=fft(A,sign,dims,incr [,option])`

is a previous syntax that also allows to perform all direct or inverse fft of the slices of`A`

along selected dimensions.For example, if

`A`

is an array with`n1*n2*n3`

elements`X=fft(A,-1,n1,1)`

is equivalent to`X=fft(matrix(A,[n1,n2,n3]),-1,1)`

. and`X=fft(A,-1,[n1 n3],[1 n1*n2])`

is equivalent to`X=fft(matrix(A,[n1,n2,n3]),-1,[1,3])`

.

- Using option argument This argument can be used
to inform the fft algorithm about the symmetry of
`A`

or of all its "slices". An N-D array`B`

with dimensions`n1`

, ...,`np`

is conjugate symmetric for the fft if and only if`B==conj(B([1 n1:-1:2],[1 n2:-1:2],...,[1 np:-1:2]))`

.In such a case the result`X`

is real and an efficient specific algorithm can be used. - "symmetric" that value causes fft to treat
`A`

or all its "slices" conjugate symmetric. This option is useful to avoid automatic determination of symmetry or if`A`

or all its "slices" are not exactly symmetric because of round-off errors. - "nonsymmetric" that value causes fft not to take care of symmetry. This option is useful to avoid automatic determination of symmetry.
- unspecified If the option is omitted the fft algorithm automatically checks for exact symmetry.

- "symmetric" that value causes fft to treat
- Optimizing fft
Remark: fftw function automatically stores his last parameters in memory to re-use it in a second time. This improves greatly the time computation when consecutives calls (with same parameters) are performed.

It is possible to go further in fft optimization using get_fftw_wisdom, set_fftw_wisdom functions.

### Algorithms

This function uses the fftw3 library.

### Examples

1-D fft

//Frequency components of a signal //---------------------------------- // build a noised signal sampled at 1000hz containing pure frequencies // at 50 and 70 Hz sample_rate=1000; t = 0:1/sample_rate:0.6; N=size(t,'*'); //number of samples s=sin(2*%pi*50*t)+sin(2*%pi*70*t+%pi/4)+grand(1,N,'nor',0,1); y=fft(s); //s is real so the fft response is conjugate symmetric and we retain only the first N/2 points f=sample_rate*(0:(N/2))/N; //associated frequency vector n=size(f,'*') clf() plot(f,abs(y(1:n)))

2-D fft

---------------------------------- A = zeros(256,256); A(5:24,13:17) = 1; X = fftshift(fft(A)); set(gcf(),"color_map",jetcolormap(128)); clf;grayplot(0:255,0:255,abs(X)')

multiple fft

//simple case, 3 1-D fft at a time N=2048; t=linspace(0,10,2048); A=[2*sin(2*%pi*3*t)+ sin(2*%pi*3.5*t) 10*sin(2*%pi*8*t) sin(2*%pi*0.5*t)+4*sin(2*%pi*0.8*t)]; X=fft(A,-1,2); fs=1/(t(2)-t(1)); f=fs*(0:(N/2))/N; //associated frequency vector clf;plot(f(1:100)',abs(X(:,1:100))') legend(["3 and 3.5 Hz","8 Hz","0.5 and 0.8 Hz"],"in_upper_left") // 45 3-D fft at a time Dims=[5 4 9 5 6]; A=matrix(rand(1,prod(Dims)),Dims); y=fft(A,-1,[2 4 5]); //equivalent (but less efficient code) y1=zeros(A); for i1=1:Dims(1) for i3=1:Dims(3) ind=list(i1,:,i3,:,:); y1(ind(:))=fft(A(ind(:)),-1); end end

//Using explicit formula for 1-D discrete Fourier transform //------------------------------------------------ function xf=DFT(x, flag); n=size(x,'*'); //Compute the n by n Fourier matrix if flag==1 then,//backward transformation am=exp(2*%pi*%i*(0:n-1)'*(0:n-1)/n); else //forward transformation am=exp(-2*%pi*%i*(0:n-1)'*(0:n-1)/n); end xf=am*matrix(x,n,1);//dft xf=matrix(xf,size(x));//reshape if flag==1 then,xf=xf/n;end endfunction //Comparison with the fast Fourier algorithm a=rand(1,1000); norm(DFT(a,1) - fft(a,1)) norm(DFT(a,-1) - fft(a,-1)) timer();DFT(a,-1);timer() timer();fft(a,-1);timer()

### See also

- corr — correlation, covariance
- fftw_flags — set method for fft planner algorithm selection
- get_fftw_wisdom — return fftw wisdom
- set_fftw_wisdom — set fftw wisdom
- fftw_forget_wisdom — Reset fftw wisdom

### Bibliography

Matteo Frigo and Steven G. Johnson, "FFTW Documentation" http://www.fftw.org/#documentation

## Comments

Author :Jan Skoda posted the 19/04/2016 09:47Add a comment:Please login to comment this page.