Plotting power spectral density from transfer function

Question

I am new to signal processing. The equations below are given in $LaTeX$.

I have the following transfer function (from the Dryden Model) for the two-sided Power Spectral Density:

$$S(w) = \frac{\sigma_u ^2 \tau_u}{\pi} \frac{1}{1+(\tau_u w)^2}$$

I would like to plot the above as: Power Spectral Density, dB vs. w

Any help on this would be appreciated. (please use any numerical values for $\sigma_u, \tau_u$)

I also need to calculate the auto-correlation function by taking the inverse Fourier transform (call it R(s) ) of S(w).

Finally, I want to compute

$$S1(w) = \int_{0}^{\infty} R(s) \cos (ws) ds$$

I would really appreciate someone's kindest help and advice in this regard. Thank you!

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L := 50
Umc := 4.31
Su[ω_, σ_, τ_] := (σ^2 *τ/π )*1/(1 \
+ (τ *ω)^2)
Sw[ω_, σ_, τ_] := (σ^2 *τ/(2*π \
))*(1 + 3*(τ* ω)^2)/(1 + (τ*ω)^2)^2

Ru[s_] := InverseFourierTransform[Su[ω, 1, L/Umc], ω, s]

S2 [ω_] := Integrate[Ru[s]*Cos[2*ω*s], {s, 0, Infinity}]

Plot[S2[ω] , {ω, 0, 1}]

Answer 1

If you simply want a dB PK ω plot you can use the built-in BodePlot function.

s[ω_, σ_, τ_] := σ^2 τ/π 1/(1 + (τ ω)^2)

BodePlot[Tooltip[s[ω, 1, 2]], ImageSize -> 550, Frame -> True, 
            PlotStyle -> {Directive[Thick, ColorData[20, 1]], 
            Directive[Thick, ColorData[20, 9]]}, Frame -> False, 
            AspectRatio -> 1/2.25, GridLines -> Automatic, 
            GridLinesStyle -> Directive[GrayLevel[0.7], Dashed]]

I will leave the rest to you - Integrate and InverseFourierTransform should do the trick :)