Manipulate[
x = Table[0,{i,31}];
x[[1]] = 1;
h :={1,1};
Do[
x = ListConvolve[h,x,1], {n,m}
];
{ListPlot[x, Filling->Axis, PlotRange->All],ListPlot[Abs[Fourier[x]], Filling->Axis]},
{m, 1, 30, 1}]

Manipulate[
x = Table[0,{i,31}];
x[[1]] = 1;
h :={1,-1};
Do[
x = ListConvolve[h,x,1], {n,m}
];
{ListPlot[x, Filling->Axis],ListPlot[Abs[Fourier[x]], Filling->Axis]},
{m, 1, 30, 1}]