Creating a grid in Fourier-space

[Mizuti]

Hello,

I'm new here and not sure where to put this question, which is more of a mathematical question, but also involves some programming in python (mainly the use of numpy.fft).

I have a code which creates a square image with dimensions 4x4 arcsec running from -2 arcsec to +2 arcsec and is created on an 80x80 grid. To this I want to add another image. This second image is created through a FFT of an 80x80 grid and thus starts out in Fourier space. After the FFT, I want the image to have exactly the same dimensions in real space as the first image.

Because Fourier space represents the scales and the wavenumber is defined as k = 2pi/x (although in this case the numpy.fft uses the definition where I think k = 1/x), I thought the largest scale would have to have the smallest k-value and the smallest scale the largest k-value.

So if x_max = 2 (the dimensions in the x-direction of the first image) and dim_x = 80 (the number of columns in the grid):

k_x,max = 1/(2*x_max/dim_x)

k_x,min = 1/(2*x_max)

and let the grid in Fourier-space run from k_x,min to k_x,max (same for the y-direction)

I hope I explained this clearly enough, but I haven't been able to find any confirmation or explanation for this in the literature about FFT's and would really like to know if this correct.

Thanks in advance

 

[sunjin09]

Exact definition of FFT in the standard way can be found here
http://www.mathworks.com/help/techdoc/ref/fft.html

By definition the sample spacing of k is 2*pi/N, where N is the number of samples in the dimension along which fft is performed. It does not care what unit of length is used in the spatial domain.