1.2 Spectrum of a raise cosine pulse

Let (Subscript[ω, 0])=2 π (Subscript[f, 0]) where (Subscript[f, 0])=200 Thz. Let envelope s[t] be a half-raised cosine function as shown below. Find its Fourier spectrum and plot.

Partial Answer (given)

You ONLY have to make one plot, use whatever parameters you like. (Use the app to verify your answer)

A raised cosine power signal can be expressed as:  
                                               P[t]=(1/2) (1+Cos[((2π t)/τ)])  for t∈ [-(τ/2),(τ/2) ] and zero elsewhere.
                                               P[t]=((Cos[((π t)/τ)])^2)
Hence, the E field envelope is the squareroot of the power envelope (let phase=0), which is:
                                                s[t]=(Sqrt[P[t]]) =(Sqrt[((Cos[((π t)/τ)])^2)]) = Cos[((π t)/τ)]

Its Fourier transform is:

t=.; τ=.;Ω=.;
(∫_(-τ/2)^(τ/2))Cos[((π t)/τ)](E^(I Ω t))\[DifferentialD]t

Hence:                  a[ω]=(1/(Sqrt[2 π]))(∫_(-∞)^∞)(e^(i Ω t))s[t] dt =(((τ(Sqrt[2 π]))/(π^2)) ) ((Cos[((τ Ω)/2)])/(1-(((((τ Ω)/π)))^2))) ,

How to plot

Manipulate[
Grid[{{
Plot[(((τ(Sqrt[2 π]))/(π^2)) ) ((Cos[((τ Ω)/2)])/(1-(((((τ Ω)/π)))^2))),{Ω,-5 π,5 π}
,PlotRange-> {-0.25,2.5}
,PlotStyle->{Thickness[0.01],Hue[0.07,1,1]}
,Filling-> {1-> {0,Hue[0.13,1,1,0.5]}}
,Frame->True, GridLines->Automatic
,FrameStyle->White,GridLinesStyle->GrayLevel[0.5]
,Background->Black, ImageSize->300]
,
Plot[20Log[10.,Abs[(((τ(Sqrt[2 π]))/(π^2)) ) ((Cos[((τ Ω)/2)])/(1-(((((τ Ω)/π)))^2)))]],{Ω,-5 π,5 π}
,PlotRange->{-80.,5}
,PlotStyle->{Thickness[0.01],Hue[0.07,1,1]}
,Filling-> {1-> {Bottom,Hue[0.13,1,1,0.5]}}
,Frame->True, GridLines->Automatic
,FrameStyle->White,GridLinesStyle->GrayLevel[0.5]
,Background->Black, ImageSize->300]}}]

,{{τ,2.},0.5,10}]