Problem of Discrete Fourier?

Question

I tried Fourier Transform for a range of figures.

Firstly, I tried in this way.

data1=Table[1, {i,1,2,1}];
data2=Table[0, {i,1,1,1}];
data3=Join[data1, data2];
data=Flatten[Table[data3, {i,1,3}]];
ListPlot[data]

I tried Fourier transform according to its definition as follow:

ft1=Table[Sum[data[[m]] Exp[-i*m*k], {m,1,Length[data]}], {k,0,10 Pi,0.1}];
ListLinePlot[Abs[(ft1)^2], PlotRange -> All]

Secondly, I tried fourier transform by another way:

fdata[x_]:= 1 /; (x>0) && (x<=2)
fdata[x_]:= 0 /; (x>2) && (x<=3)
fdata[x_]:= 1 /; (x>3) && (x<=5)
fdata[x_]:= 0 /; (x>5) && (x<=6)
fdata[x_]:= 1 /; (x>6) && (x<=8)
fdata[x_]:= 0 /; (x>8) && (x<=9)
Plot[fdata[x], {x,0,9}]

ft2[k_]:=NIntegrate[fdata[x]Exp[-i*x*k],{x,0,9}];
Plot[Abs[(ft2[k])^2],{k,0,5Pi},PlotRange->All]

Then I got the result as follow:

The second result is the one should be. But what is wrong with the first method?Why the difference appears? Any suggestions R welcomed.

If it is the result of discrete data?How to avoid?

Such as, I plan to get the Fourier Transform of a large amount data as follow. (And I do not want to fit the curve by any functions because the difference will affect the result.)

data={1., 0.494297, 0.0554558, -0.00119684, 0.0347557, 0.0274166,0.0844178, 0.140589, 0.100378, 0.0305703, 0.00906179, 0.0165403,0.0333436, 0.0559314, 0.0647997, 0.0487751, 0.0278078, 0.0198066, 0.022007, 0.0273021, 0.0328905, 0.0344371, 0.0283518, 0.0199067,0.0149784, 0.0139971, 0.017503, 0.0188609, 0.0190994, 0.0164812, 0.0121582, 0.00943724, 0.00991142, 0.0110141, 0.0114948, 0.0092613, 0.00763611, 0.00498833, 0.0051873, 0.00499402, 0.00549799,0.00380389, 0.00191446, 0.00156621, 0.0015737, 0.00183529, 0.00192207, 0.00272739, 0.00171912, 0.00133939, 0.000664813, -0.000731862, 0.000542469, 0.00210733, 0.00157748, 0.000378449,-0.000413625, -0.000241379, -0.000524688, 0.00125707, 0.000893426, -0.000960053, -0.00141913, -0.000102416, -0.000507345, -0.00153032,-0.000878255, -0.00153944, -0.00212547, -0.00216173, -0.00205534, -0.00268722, -0.00284011, -0.00307404, -0.00385506, -0.00360069, -0.00226998, -0.00166825, -0.0022364, -0.0032332, -0.00312994,-0.00264376, -0.00222135, -0.00129998, -0.00118719, -0.00209173, -0.00368347, -0.00310659, -0.00122648, -0.000768231, -0.00177089,-0.0023162, -0.00259222, -0.00185519, 0.000111307, 0.000415605, 0.0000449796, -0.000621294, -0.00165163, -0.00163402, -0.000711393, -0.00110962, -0.00229118, -0.00360164, -0.00602596, -0.00820126, -0.00657783, -0.00431852, -0.00486986, -0.0060202, -0.00708399, -0.00858061, -0.00780665, -0.00629949, -0.00510292, -0.00752368, -0.00787104, -0.0106296, -0.0113897, -0.0107749, -0.0125735,-0.0143625, -0.0161466, -0.0163181, -0.0187493, -0.0200038, -0.0189971, -0.0182773, -0.0193625, -0.0191536, -0.0189072, -0.0165243, -0.0165929, -0.0167107, -0.0209827, -0.0244816, -0.0200228, -0.0224443, -0.0217907, -0.0182094, -0.0173637,-0.0290431};

ListLinePlot[data,PlotRange->All]

Then I did

ftEx=Table[Sum[data[[i]] Exp[-I*i*k], {i, 1,Length[data]}], {k, 0, 20, 0.01}];
ListLinePlot[Abs[ftEx], PlotRange -> All,Frame -> True]

I do not want all the peaks after the 1 on x-axis, shows as in the red circles.

This is the problem of discrete Fourier transform? Then how to avoid?

Answer 1

I'm pretty sure if you do ListLinePlot[data] you'll see your first data is actually this plot and not the square wave you expect:

Generate your second plot with a Piecewise:

Plot[Piecewise[{{1, 0 < x < 2}, {0, 2 < x < 3}, {1, 3 < x < 5}, {0, 
5 < x < 6}, {1, 6 < x < 8}, {0, 8 < x < 9}}], {x, 0, 9}, Exclusions -> None]

You can use that Piecewise function to generate a better sampling of the square wave:

dt=1/100; (* Your sample rate *)
Table[Piecewise[{{1, 0 < x < 2}, {0, 2 < x < 3}, {1, 3 < x < 5}, 
{0, 5 < x < 6}, {1, 6 < x < 8}, {0, 8 < x < 9}}], {x, 0, 9, dt}]]]

Now remember Fourier is a discrete transform, so you need to sample the function. Because you can do it analytically with a FourierTransform, I recommend avoiding the problems of discrete sampling altogether. If you want a continuous FT in terms of w, then try :

FourierTransform[
 Piecewise[{{1, 0 < x < 2}, {0, 2 < x < 3}, {1, 3 < x < 5},
 {0, 5 < x < 6}, {1, 6 < x < 8}, {0, 8 < x < 9}}], x, w]

(*Result: ((E^(I w) + E^(4 I w) + E^(7 I w)) Sqrt[2/\[Pi]] Sin[w])/w *)

To complicate things further, you may need to explore the FourierParameters option of Fourier and FourierTransform

We can now plot the spectrum square magnitude:

Plot[Abs[(((E^(I w) + E^(4 I w) + E^(7 I w)) Sqrt[2/\[Pi]] Sin[w])/w)]^2,
{w, 0, 20},PlotRange -> Full]