Units For a Power Spectral Density (PSD)

[person123]

(Just as clarification on what I am doing, I am attempting to use MATLAB to create PSD plots of time series from pressure and accelerometer sensors. Wave gauge sensors might still be helpful for me. I did not discuss accelerometer sensors in the post but maybe I could figure it out if I get the basic idea).

At the very least, I would like to make sure I am correct that the Fourier Transform of a time series with some units $U$ ($U$ referring to any arbitrary unit), would be $U/Hz$ (so $Us$). If this is wrong, then I think I have a fundamental misunderstanding.

I'm trying to understand Power Spectral Distributions; I think my confusion can largely be reduced to a question of units. I am particularly concerned with the PSD created from time series of wave height and pressure, but I think a general understanding of a PSD would be useful. From what I understand, the units for the y-axis of a PSD depends on the time series from which the PSD was created. On this Stack Overflow thread, someone states that a PSD of $V^2/Hz$ can be used when the time series is voltage. I think this make sense because $P=V^2/R$.

I don't understand how I would extend this to other cases. I understand that there should be some way of relating the amplitude of the time series to the power. I don't quite understand how much freedom I have when doing this. Could I for example, just declare the following: $$E=\rho g H^2$$ where $H$ is the wave height? In that case $E$ would be some quantity which in some is sense associated with energy. As energy is dimensionally equivalent to power divided by frequency, and $\rho$ and $g$ are constants, could I simply square the Fast Fourier Transform and just let the y-axis be $H^2$ with the understanding that there's a clear relation to power?

In the case of pressure sensors, I am a bit more confused. I do understand that sound waves are pressure waves and the PSDs for sound waves use dB as the y axis. This is computed by $\log_{10} (P/P_0)$ (multiplied by 10 but I believe this is a detail in this context). Could I do something similar in my case?

 

[boneh3ad]

Do you know the equations defining Fourier transforms and the power spectral density? The units fall directly out of those equations.

 

[person123]

Do you know the equation defining Fourier transforms and the power spectral density? The units fall directly out of those equations.

When I started this thread I did not. However, I ended up asking my graduate mentor this question, and I believe his answer directly relates to your question. He said the equation was the following: $$PSD=Y Y^* L \Delta t$$ $Y$ is the Fourier Transform, $L$ is the sample length, and $\Delta t$ is the sample rate. He also said that the units of the Fourier Series itself (for pressure say) would simply be $Pa$. Therefore, the PSD would be $Pa^2/Hz$ I believe. This is what I used in my code. I still don't really have a true understanding of why you do this, but I think my understanding is good enough to create functions based on this. If you or anyone else could give me (or refer me to) an intuitive understanding of how this relates to power, I would greatly appreciate it.

 

[boneh3ad]

That looks like you are at least on the right track. Consider that you have some time series, $y(t)$. At this point, it has arbitrary units. As you've stated, it has a Fourier transform defined as
$$Y(f) = \mathscr{F}[y(t)](f) = \int\limits_{-\infty}^{\infty}y(t)\exp[-2\pi i f t]\;dt.$$
The power spectral density of $y(t)$ is then defined as
$$S_{yy}(f) = \lim_{T\to\infty}\dfrac{1}{T}E[Y(f)Y^*(f)],$$
where $E[\cdot]$ is the expectation operator and $^*$ denotes a complex conjugate.

So then let's look at the units assuming our signal is a raw voltage. The units of $Y(f)$ are
$$[Y(f)] = [\mathrm{V}\cdot \mathrm{s}]$$
since the exponential term is unitless, $y(t)$ has units of volts, and $dt$ has units of seconds. The units of the power spectrum are therefore
$$[S_{yy}(f)] = \left[\dfrac{1}{\mathrm{s}}\right][\mathrm{V}\cdot \mathrm{s}][\mathrm{V}\cdot \mathrm{s}] = [\mathrm{V}^2\cdot \mathrm{s}] = \left[ \dfrac{\mathrm{V}^2}{\mathrm{Hz}} \right].$$
If your measured signal $y(t)$ happens to be a different unit, just substitute that for volts. When it's a voltage signal, it's pretty clear how the power spectral density is related to actual physical power: $S_{yy}$ is literally the density of fluctuating power per frequency of the voltage signal. If you have a calibrated quantity (e.g. Pascals), it's related to the actual physical power by a calibration curve. Otherwise, you needn't always relate it to a physical power. You can just interpret it as the signal power instead of actual, physical Watts.

A good source:
[1] Bendat JS, Piersol AG. 2011. "Random Data." 4th Ed. Wiley.