two-dimension fast Fourier transform
y=fft2(x)
y=fft2(x,n,m)
:x a vector/matrix/array (Real or Complex) : :y a vector/matrix/array (Real or Complex) : :m integer, number of rows. : :n integer, number of columns. :
This functions performs the two-dimension discrete Fourier transform.
`y=fft2(x)`y and x have the same size
y=fft2(x,m,n): If m (respectively n) is less than the rows number (respectively columns) of x then the x rows number (resp. columns) is truncated, else if m (resp. n) is more than the rows number (resp. columns) of x then x rows are completed by zero (resp. columns) .
if x is a matrix then y is a matrix, if x is a hypermatrix then y is a hypermatrix, with the size of the first dimension of y is equal to m, the size of the second dimension of y is equal to n, the size of the ith dimension of y (for i>2, case hypermatrix) equal to the size of the ith dimension of x. (i.e size(y,1)=m, size(y,2)=n and size(y,i)=size(x,i) for i>2)
//Comparison with explicit formula
a=[1 2 3 ;4 5 6 ;7 8 9 ;10 11 12]
m=`size`_(a,1)
n=`size`_(a,2)
// fourier transform along the rows
for i=1:n
a1(:,i)=`exp`_(-2*%i*%pi*(0:m-1)'.*.(0:m-1)/m)*a(:,i)
end
// fourier transform along the columns
for j=1:m
a2temp=`exp`_(-2*%i*%pi*(0:n-1)'.*.(0:n-1)/n)*(a1(j,:)).'
a2(j,:)=a2temp.'
end
`norm`_(a2-fft2(a))