fft - fast Fourier transform.
Short syntax (one or two dimensional fft):
gives a direct transform (the -1 refers to the sign of the exponent..., NOT to "inverse"), that is
x(k)=sum over m from 1 to n of a(m)*exp(-2i*pi*(m-1)*(k-1)/n)
for k varying from 1 to n (n=size of vector a).
performs the inverse transform normalized by 1/n.
(fft(fft(.,-1),1) is identity).
When the first argument given to fft is a matrix a two-dimensional FFT is performed.
Long syntax (multidimensional FFT):
allows to perform an multidimensional fft.
If a is a real or complex vector implicitly indexed by x1,x2,..,xp i.e. a(x1,x2,..,xp) where x1 lies in 1..dim1, x2 in 1.. dim2,... one gets a p-dimensional FFT p by calling p times fft as follows
a1=fft(a ,-1,dim1,incr1) a2=fft(a1,-1,dim2,incr2) ...
where dimi is the dimension of the current variable w.r.t which one is integrating and incri is the increment which separates two successive xi elements in a.
In particular,if a is an nxm matrix, x=fft(a,-1) is equivalent to the two instructions:
a1=fft(a,-1,m,1) and x=fft(a1,-1,n,m).
if a is an hypermatrix (see hypermat) fft(a,flag) performs the N dimensional fft of a.
a=[1;2;3];n=size(a,'*'); norm(1/n*exp(2*%i*%pi*(0:n-1)'.*.(0:n-1)/n)*a -fft(a,1)) norm(exp(-2*%i*%pi*(0:n-1)'.*.(0:n-1)/n)*a -fft(a,-1))