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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 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 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 1-D or N-D 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 3-D 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 N-D 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 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)')
mupliple 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
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