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Say we have a noise power spectral density that has the following form (I think the question is the same for more general forms as well) $$ S(f) = Af^\alpha, $$ In experiments people usually discuss the magnitude of the noise at $1\,\text{Hz}$ and give a number for that, for example $$ S(1\,\text{Hz}) = C \,\, \frac{u^2}{\text{Hz}} $$ where $C$ is a constant number, and $u$ is the unit for the underlying measured noisy quantity (e.g., Volts or flux quantum).

Questions

  1. What is the intuition for understanding the meaning of magnitude of the noise power spectral density at $1\,\text{Hz}$? How that relates to the deviations of the underlying stochastic process?
  2. What if we look at the magnitude of the noise power spectral density at say $1\,\text{kHz}$? What does that mean?
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  • $\begingroup$ hm I've never seen anyone discuss the noise PSD at 1 Hz. That seems to be a very specific special case, and it's only special for your specific $S(f)$, so your question really doesn't generalize to other forms! $\endgroup$ Jun 25 at 21:33
  • $\begingroup$ So, when you set $f=1$ in your $S(f)$, what do you get? This really seems like a basic exercise in "how to use a formula". $\endgroup$ Jun 25 at 21:34
  • $\begingroup$ As I mentioned in my question, you are given a constant number with appropriate units, which should work for any form of $S(f)$. $\endgroup$
    – Mostafa
    Jun 25 at 21:44
  • $\begingroup$ yes, but your claim is "I think the question is the same for more general forms as well" and that's what I don't agree to. Again, have you inserted $f=1$ (no units) in your $S(f)=Af^\alpha$ and realized what you get? $\endgroup$ Jun 25 at 21:46
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You're missing the point, a bit here:

In your formula

$$S(f)=Af^\alpha,$$

you forgot to actually state "what" $\alpha$ and $f$ are supposed to be, but from the formula itself it's quite clear that they have to be real numbers: if either $f$ or $\alpha$ "contained" any units, then the result of $Af^\alpha$ would have different physical units (not magnitude, units!) for different $f$. That makes physically no sense.

So, what you can insert as $f$ in your formula can never be $1\,\text{Hz}$; it must be a real number, not a real number times a unit!

Therefore, the question

What is the intuition for understanding the meaning of magnitude of the noise power spectral density at 1Hz?

is a bit off. $f=1$ is what you mean, and quite elementary,

$$S(f=1) = A1^\alpha = A,$$

so that's the whole significance (I hoped to point that out in my comment and it seems I failed); you get $A$, with all its physical units.

What if we look at the magnitude of the noise power spectral density at say 1kHz? What does that mean?

So, we cannot insert 1 kHz into $S(f)$, but keeping the consistency to above, 1 kHz corresponds to $f=1000$.

So, that has no "special" meaning (like any other $f$). It just gives us the value of the power spectral density at that frequency. So, "what does that mean" is: it's what the PSD is at 1 kHz, nothing more, nothing less.

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