Advection - approximation - von Neumans analysis

$\displaystyle u_x = u_t ; 0 < x < 1, t > 0$
Bondaries
$\displaystyle u(0,t) = u(1,t) ; t > 0$
$\displaystyle u(x,0) = u(x) ; 0 < x < 1$

I'd like to preform a von Neumann analysis on approimation.

$\displaystyle \frac{u^{n+1}_j - u^n_j}{\Delta t} = \frac{u^{n+1}_{j+1}- u^{n+1}_{j-1}}{2 \Delta x} ; j = 0,1,... N-1 , n = 0,1...$

$\displaystyle u^{n}_j = u^{n}_{N+1} ; all j$

$\displaystyle u^{o}_j = f(x_j) ;j = 0,1,... N-1$

I did an analysis unsing Taylor expantion and got first order in space and first order in time. If I do a von Neumann I don't really know what how to start.

Should I do a Fourier transformation on the approximation or on the equation it self?