# 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!

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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}]

• 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 Mar 22, 2014 at 20:22
• @Kuba Yes it is
– Stoc
Mar 22, 2014 at 20:26
• @Nasser thanks so much...I'll keep this in mind for next time
– Stoc
Mar 22, 2014 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.
– rm -rf
Mar 22, 2014 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, 2014 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 :)

• 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, 2014 at 0:02
• @Stoc Have you resolved the plotting issue ? Mar 25, 2014 at 9:17