(*declaration of parameters*)
T = 2.0;
w = 0.05;
a = 2.0;
num = 1000.0;
A = 1.0;
Amp = 0.05;(*noise amplitude*)
(*declaration of continous function*)
pulse[x_] := A*(UnitStep[x + w*T] - UnitStep[x - w*T])
(*funciton sampling*)
funX = Table[i, {i, -T/2, T/2, T/(num - 1)}];
fun1 = pulse /@ funX + Amp*RandomReal[{-0.5, 0.5}, num];
fun2 = pulse /@ (a*funX) + Amp*RandomReal[{-0.5, 0.5}, num];
ListPlot[Transpose[{funX, fun1}], PlotRange -> All, Filling -> Axis,
 Frame -> True, FrameLabel -> {"Time [s]", "Amplitude [V]", "Pulse"},
 PlotLegends -> {"Pulse"}, ImageSize -> Large]

(*Fourier Transform*)
xf = With[{
    i = Function[{t0, t1, x0, x1},
      Evaluate@
       Integrate[(x0 + (x1 - x0)/(t1 - t0) *(t - t0))*
         Exp[-2*Pi*I*f*t], {t, t0, t1}]]},
      Compile[{{f, _Real}, {t, _Real, 1}, {x, _Real, 1}} ,
    Total@i[Most[t], Rest[t], Most[x], Rest[x]]]
   ];

freq = Table[j, {j, -40., 40.}]/T;
Xorig = xf[#, funX, fun1] & /@ freq;
Xscaled = xf[#, funX, fun2] & /@ freq;