Q:

domain of a discrete fourier transform

The task at hand is to plot a discrete Fourier transform of a function $f(x)$ defined at each element of an array of size $2N$.
The 1D Fourier transform can be approximated as a sum over discrete values:
$F(u) = \frac{1}{2N}\sum_{x=-N}^{N-1} \left(f(x) e ^ {-\frac{\pi i x u}{N}}\right)$
I'm confused by a few elements here.
What should the domain of $F(u)$ be?  When I perform the DFT I find that $F(u)$ is periodic with period 2N, with vertical lines of symmetry at integer multiples of $N$.  This is far from what I expect, given what I see when I google DFTs of the kinds of functions I'm trying.  Whatever function I try, I always seem to get plots of a similar form to this.  .  (Don't worry, I'm not here for debugging - if it's wrong just say and I'll go down the debugging avenue myself).  The plot is for $N=250$ and with $f(x) = 1$ for $-6<x<5$ and $0$ otherwise.
On a similar note, does it make sense to plot ANY frequency, or should I only be plotting integer multiples of some 'fundamental frequency'?  What would this fundamental frequency be?
I'm happy to provide any further details or plots if necessary. Thanks for any help!

A:

You must be carefull as the mathematics of DFT are not the same as for the fourier transform as an integral. However the fourier transform of the rectangular function is $sinc(x)$, which what you are showing here(altougth is the absolute value here). However because the DFT takes limited points, the first consecuence is that the DFT is periodic, as:
\begin{eqnarray}
F(u+2N) &=&\frac{1}{2N}\sum_{x=-N}^{N-1}f(x)e^{-πix(u+2N)/N}
\\
&=& \frac{1}{2N}\sum_{x=-N}^{N-1}f(x)e^{-πix(u)/N}e^{-πix(2N)/N}
\end{eqnarray}
And as x takes values 0,1,2... 2N ,  $e^{-πix(2N)/N}=e^{-2\pi i x}=1$ meaning that 
$F(u+2N)=F(u)$ in your definition. This also means that F(-u)=F(2N-u), so that plotting $u=0..2N$ is most of the times enougth. 
On another note, usually the DFT as defined in MATLAB and many other codes, for example, goes from u=1:N, or u=0:N-1, not 2N, but that relies more on your definition, is not that important.

