I'm facing the following problem: I have a cubic domain of side $L=2\pi$; this domain is divided in a cubic grid, each side is divided in $N$ points, where $N$ is an even integer number. the coordinates of the grid points are:
$\boldsymbol{x}_{grid}=\frac{2\pi}{N}(i-1,\;j-1\,\;k-1) \; \text{for} \; i,j,k=0, \;N-1$
On each grid point i have a real valued white noise $\eta(\boldsymbol{x}_{grid})$ that is to say an independent standard Gaussian random number on each grid point.
What i want to do is to take the Fourier transform of this three dimensional array $\eta(\boldsymbol{x}_{grid})$ through the Fourier routine in mathematica.
Therefore i define a wavevector grid $\boldsymbol{k}_{grid}$ which is a three dimensional grid in the wavevector space. the points of this grids are:
$\boldsymbol{k}_{grid}=(i,j,k) \;\;\text{if}\;\; i||j||k \geq 0 \leq N/2$
while if either $i$, $j$ or $k$ are greater than $N/2$ the grid coordinate of $\boldsymbol{k}_{grid}=$ becomes respectively $i-N$, $j-N$ or $k-N$
in practice the wavevector grid goes from $0$ to $N/2$ then shifts back to $-N/2 +1$ clearly the last element is $-1$.
(for those who are interested i'm following section $5$ of this paper: http://www.sciencedirect.com/science/article/pii/S0021999114001867)
Now i want to look at the properties of the transformed random noise $\hat{\eta}(\boldsymbol{k}_{grid})$. I would like to know if what i get is correct because i'm not an expert of discrete fourier transforms.
In the wavevector space i get a random variable which satisfies the property $\hat{\eta}(\boldsymbol{k}_{grid})=\overline{\hat{\eta}(\boldsymbol{-k}_{grid}})$ where the overline refers to the complex conjugate. Moreover I get that the $\hat{\eta}(0,0,0)$ and $\hat{\eta}(N/2,N/2,N/2)$ are real valued, (i.e. their imaginary part is zero)
Are these properties correct for the Fourier transform of a real noise? Is it also true the vice-versa? (i mean that the inverse fourier transform of a complex noise, which satisfies the above properties in the wavevector space, is a real noise in real space).
I'm also interested in the relationship between the meaning of the transformed value at $\boldsymbol{k}_{grid}=(0,0,0)$ and $\boldsymbol{k}_{grid}=(N/2,N/2,N/2)$
In particular if a the discrete fourier transform of a real valued random process has the property: $\hat{\eta}(\boldsymbol{k}_{grid})=\overline{\hat{\eta}(\boldsymbol{-k}_{grid}})$ what happens for the points for which one of the index is $N/2$? as there's no $-N/2$ point index in the grid...
Thanks in advance for the answers...