I consider a generic signal $X(t)$. We could imagine to fix ideas $X(t)=\sqrt{R}I(t)$ ($R$ being a resistor and $I$ current flowing through it)

The energy of this signal is defined as $E=\int_{-\infty}^{+\infty} |X(t)|^2 dt$. The motivation of calling this "signal energy" can be seen from my signal example (it would give the heat dissipated in the resistor).

From Parseval theorem, we have: $\int_{-\infty}^{+\infty} |X(t)|^2 dt=\int_{-\infty}^{+\infty} |X(\omega)|^2 d \omega$.

From this, we can define the energy spectral density $E_{xx}(\omega)=|X(\omega)|^2$

Unfortunately, for many signal (if they are not square integrable for instance), $X(\omega)$ does not exists. This is the reason why in signal processing people introduced the power spectral density as being:

$$S_{xx}(\omega)=\lim_{T \to +\infty} \mathbb{E}[|X_T(\omega)|^2]$$

Where $X_T(\omega)$ is a windowed Fourier transform:

$$X_T(\omega)=\frac{1}{\sqrt{T}}\int_{-T/2}^{+T/2}dt X(t) e^{-i\omega t} $$

Then, what people call the noise power spectral density does not exactly match what we would like initially (it would be easier to work with $|X(\omega)|^2$), but this new definition works for many more signals. Furthermore there are some very usefull theorem on this quantity like the Wiener–Khinchin theorem.

My question:

I am okay to create a new definition for noise power if it makes math doable and provides good theorem. My troubles start when I try to put physics label on it.

In theoretical derivations, when it exists, we usually have access to $\mathbb{E}(|X(\omega)|^2)$. For example, I take the important example for my purpose which is the Johnson-Nyquist noise. In this case, we have ($V$ being the voltage around a resistor at thermal equilibrium at temperature $T$):

$$\mathbb{E}(|V(\omega)|^2)=4 R k_b T$$

And we call this quantity the power spectral density. It can be seen on the wikipedia page, or on page 70 here. However it is not the power spectral density (it doesn't come from a windowed calculation). But anyway we usually do "as if" it was the case and then we use all the results from signal processing to derive some relations. Also in this document (I could find other), we use this assimilation $S_{xx}(\omega) \leftrightarrow \mathbb{E}(|X(\omega)|^2)$ to derive some properties.

Why are people doing this ? Is it kind of an "handwaving" things that wouldn't be too far from reality so that we can assimilate when the quantity exist $S_{xx}(\omega)$.

In summary: why when $\mathbb{E}(|X(\omega)|^2)$ does exist, physisict associate it to the noise power spectral density even if the two quantities do not have the same definition ? I would like to have a source explaining it if possible.

  • $\begingroup$ I always thouight that some sort of windowed quantity was the more physical, because, when we measure the noise, we only connect our measuring apparatus for a finite time. $\endgroup$ – mike stone Jul 10 '20 at 16:51
  • $\begingroup$ @mikestone thank you for your comment. I would probably agree with you on this (but my question is different from this point). Thanks. $\endgroup$ – StarBucK Jul 10 '20 at 17:04
  • $\begingroup$ maybe if you changed the notation it would not be so confusing, $X(t)$ vs. $X(\omega)$ or $T(ime)$ vs.$T(emperature)$ $\endgroup$ – hyportnex Jul 10 '20 at 18:37
  • $\begingroup$ @hyportnex I'm not sure to see where it is confusing ? I added a precision just before the $k_b T$ to say that this $T$ is temperature if it makes things more clear. But for the rest I think the notation are pretty clear ? $\endgroup$ – StarBucK Jul 10 '20 at 19:12
  • $\begingroup$ My answer on Signal Processing Stack Exchange may be helpful: dsp.stackexchange.com/a/65992/27128 $\endgroup$ – Jagerber48 Jul 10 '20 at 19:18

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