f1=Sqrt[110]; dt=0.01;n=256;a=Table[Sin[2Pi f1 i dt],{i,n}];

ListLinePlot[a]

b=Fourier[a];

ListLinePlot[Log[10,Abs[b]],PlotRange->All]

w1=Table[1-Abs[(i-1-n/2)/(n/2)],{i,n}];

ListLinePlot[a w1]

c=Fourier[a w1];

ListLinePlot[Log[10,Abs[c]],PlotRange->All]

w2 = Table[Sin[(i-1)Pi/n]^2,{i,n}];

ListLinePlot[a w2]

d=Fourier[a w2];

ListLinePlot[Log[10,Abs[d]],PlotRange->All]

ListPlot[Table[{n,First[Timing[Fourier[Table[i^2,{i,n}]]]]},{n,2^18-5,2^18+5}],PlotRange->All]

Table[{n,First[Timing[Fourier[Table[i,{i,n}]]]]},{n,2^20-3,2^20+3}]

2^20

Manipulate [ dt=0.01;n=256;a=Table[Sin[2Pi f1 i dt],{i,n}];b=Fourier[a];ListLinePlot[Log[10,Abs[b]],PlotRange->{{0,250},{-2,1}}],{f1,0,200}]