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Multiple Frequencies

ListPlot[Table[
  Abs[FourierTransform[Sin[t] + Sin[15 t] + Cos[30 t], 
     t, \[Omega]] /. {DiracDelta -> DiscreteDelta, \[Omega] -> 
      f}], {f, 1, 100, 1}], Filling -> Axis, 
 PlotStyle -> PointSize[0.02]]

Sin[x]

Sin[x] UnitStep[x]

Sin[x] using FourierTransform

Sin[x] UnitStep[x] using FourierTransform

Sin[x]

ListLogLogPlot[
 Table[Abs[FourierTransform[Sin[t]  , t, \[Omega]]] /. 
   DiracDelta -> DiscreteDelta, {\[Omega], 0.1, 10, 0.1}], 
 PlotRange -> All, Filling -> Axis]

Sin[x] UnitStep[x]

ListLogLogPlot[
 Table[Abs[FourierTransform[Sin[t] UnitStep[t] , t, \[Omega]]] /. 
   DiracDelta -> DiscreteDelta, {\[Omega], 0.1, 10, 0.1}], 
 PlotRange -> All, Filling -> Axis]