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I have a pressure signal at two locations (r=1 and r=1.3). I found the power spectral density (PSD) and the root mean-square (RMS) of the signal. My question is why at r=1 the PSD is higher than the PSD at r=1.3 while the RMS at r=1 is less than RMS at r=1.3.

I mean is there any relation between PSD and RMS of the signal?

enter image description here

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  • $\begingroup$ the premise to your question is not completely correct. " at r=1 the PSD is higher than the PSD at r=1.3 [at some frequencies] while the RMS at r=1 is less than RMS at r=1.3." $\endgroup$ Oct 4, 2017 at 20:45
  • $\begingroup$ Thank you. That is correct. Just to be clear, the second plot represents the Root Mean Square of the pressure signal Pressure_rms. You mean the RMS at r=1 is less than at r=1.3 because the PSD at r=1 is higher that PSD atr=1.3 at some points. Is that correct? $\endgroup$
    – Math
    Oct 4, 2017 at 23:22
  • $\begingroup$ yes. at more frequencies the PSD at r=1.3 is higher that the PSD is at `r=1'. $\endgroup$ Oct 5, 2017 at 0:27

1 Answer 1

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instantaneous power is:

$$ p_x(t) = |x(t)|^2 $$

mean power is:

$$\begin{align} P_x &= \lim_{T \to \infty} \quad \frac{1}{T} \int\limits_{-\frac{T}{2}}^{+\frac{T}{2}} p_x(t) \, dt \\ \\ &= \lim_{T \to \infty} \quad \frac{1}{T} \int\limits_{-\frac{T}{2}}^{+\frac{T}{2}} |x(t)|^2 \, dt \\ \end{align}$$

the relation between Power Spectral Density to mean power is:

$$ P_x = \int\limits_{-\infty}^{\infty} S_x(f) \, df $$

the relation between Root Mean Square to mean power is:

$$ P_x = \lVert x(t) \rVert^2 $$

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  • $\begingroup$ No, $|x(t)|^2$ is instantaneous energy? $d/dt |x(t)|^2$ is power $\endgroup$ Feb 24, 2023 at 16:23
  • $\begingroup$ Sorry @OverLordGoldDragon , you missed the mark again. Just curious, are you an Electrical Engineer? $\endgroup$ Feb 24, 2023 at 17:21
  • $\begingroup$ Graduated as one, never considered myself one. I do know a thing about circuits that'll blow your mind though. At any rate, what'd I miss? $\endgroup$ Feb 24, 2023 at 17:54
  • $\begingroup$ So, you have a real-valued signal represented as a voltage, $v(t)$, and you apply that voltage to a resistor $R$. What is this quantity: $$ \frac{v^2(t)}{R} = v(t) \times \frac{v(t)}{R} = v(t) \times i(t) $$ ? $\endgroup$ Feb 24, 2023 at 22:13
  • $\begingroup$ While I'm sorting this out, what would be the instantaneous energy? $\endgroup$ Feb 25, 2023 at 6:05

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