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See the recommended documentation of this function
fft
fast Fourier transform.
ifft
fast Fourier transform.
Calling Sequence
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 ND array.
 X
 a real or complex array with same shape as
A
.  sign
 an integer. with possible values
1
or1
. Select direct or inverse transform. The default value is1
(direct transform).  option
 a character string. with possible values
"symmetric"
or"nonsymmetric"
. Indicates ifA
is symmetric or not. If this argument is omitted the algorithm automatically determines ifA
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 thandims
.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 1D or ND Discrete Fourier Transforms. Short syntax
 direct
X=fft(A,1 [,option])
orX=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)
orX=ifft(A)
performs the inverse normalized transform, such thatA==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" ofA
along selected dimensions.For example, if
A
is a 3D arrayX=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 ofA
along selected dimensions.For example, if
A
is an array withn1*n2*n3
elementsX=fft(A,1,n1,1)
is equivalent toX=fft(matrix(A,[n1,n2,n3]),1,1)
. andX=fft(A,1,[n1 n3],[1 n1*n2])
is equivalent toX=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 ND arrayB
with dimensionsn1
, ...,np
is conjugate symmetric for the fft if and only ifB==conj(B([1 n1:1:2],[1 n2:1:2],...,[1 np:1:2]))
.In such a case the resultX
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 ifA
or all its "slices" are not exactly symmetric because of roundoff 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 reuse 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
1D 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)))
2D 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)')
mupliple fft
//simple case, 3 1D 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 3D 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 1D 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:n1)'*(0:n1)/n); else //forward transformation am=exp(2*%pi*%i*(0:n1)'*(0:n1)/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
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