# Plotting power spectral density from transfer function

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[\[Omega]_, \[Sigma]_, \[Tau]_] := (\[Sigma]^2 *\[Tau]/\[Pi] )*1/(1 \
+ (\[Tau] *\[Omega])^2)
Sw[\[Omega]_, \[Sigma]_, \[Tau]_] := (\[Sigma]^2 *\[Tau]/(2*\[Pi] \
))*(1 + 3*(\[Tau]* \[Omega])^2)/(1 + (\[Tau]*\[Omega])^2)^2

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

S2 [\[Omega]_] := Integrate[Ru[s]*Cos[2*\[Omega]*s], {s, 0, Infinity}]

Plot[S2[\[Omega]] , {\[Omega], 0, 1}]

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You can get your latex to show as Mathematica code like this: ToExpression["S(w) = \\ frac{\\ sigma_u ^2 \\ tau_u}{\\ pi} \\ frac{1}{1+(\\ tau_u w)^2}",TeXForm] and this gives !Mathematica graphics – Nasser Mar 22 '14 at 20:22
@Kuba Yes it is – Stoc Mar 22 '14 at 20:26
@Nasser thanks so much...I'll keep this in mind for next time – Stoc Mar 22 '14 at 20:26
Please do not post the same question on multiple sites (this is to avoid duplicated efforts). If you're doing this in Mathematica, please include the relevant code. – R. M. Mar 22 '14 at 20:37
@rm-rf sorry about that....i have deleted my other post..i have no idea as to which functions in mathematica can do this for me so unfortunately i have no code at the moment. – Stoc Mar 22 '14 at 20:46

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 :)

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thanks for the help...I am having a problem in plotting now..my function S2[w] outputs values for {w,0,1} but nothing shows up on the plot...i have no idea what I am doing wrong...i'll put the comment above...can you please take a look? sorry for the trouble. – Stoc Mar 23 '14 at 0:02
@Stoc Have you resolved the plotting issue ? – Sektor Mar 25 '14 at 9:17