This likely has to do with folding of the results due to the Nyquist behavior (http://en.wikipedia.org/wiki/Nyquist_frequency)
You can see it occur in a Manipulate like this
Manipulate[F = Table[Piecewise[{{1, Abs[i] < x}}], {i, -10, 10, 1}];
A = Fourier[F]; GraphicsRow[{ListPlot[Abs[F]], ListPlot[Abs[A]]}],
{x, 1, 10, 1}]
or this
Manipulate[
F = Table[Piecewise[{{1, i < x}}], {i, -10, 10, 1}];
A = Fourier[F]; GraphicsRow[{ListPlot[Abs[F]], ListPlot[Abs[A]]}],
{x, 1, 10, 1}]