I have seen some devices which specify their internal noise in RMS or Peak-to-peak value. How does one calculate RMS or p2p values for white noise?

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    $\begingroup$ The answer depends on what you know about the noise process. Do you know its probability density function? Or do you have a recorded signal? $\endgroup$ – Deve Nov 14 '14 at 19:27
  • $\begingroup$ @Deve I have a recorded signal (noise) and I want to measure peak to peak and RMS of that signal (noise) $\endgroup$ – zud Nov 25 '14 at 13:14

three things to think about:

  1. "white" describes the power spectrum, but to know the peak values, you need to know the probability density function (p.d.f.) of the noise process. often, with "white noise", we are assuming "normal" or "gaussian" p.d.f.

  2. even for a finite power (which means a finite variance for the p.d.f.), a gaussian random variable has infinite peak value (just that it hits whatever arbitrary high value with decreasing probability). so for gaussian, you need to define like a "1% likelihood peak value".

  3. true white noise (which does not exist in reality) has infinite power and infinite variance. this is because true white noise has infinite bandwidth. so, in reality, we deal with bandlimited "white" noise, so the bandwidth might be a necessary parameter to know, before you can think about its r.m.s. (which is the square root of the variance, assuming no DC component) and peak.


Let $r_k$ be the sequence of $N$ measured noise samples. Then

  1. Peak-to-peak value $$ r_\mathrm{pp} = \operatorname{max}(r_k) - \operatorname{min}(r_k) $$
  2. RMS value $$ r_\mathrm{RMS} = \sqrt{\frac{1}{N}\sum_k r_k^2} $$

But Robert's answer should be taken seriously, especially the comment about peaks of Gaussian random variables. If your noise source is really Gaussian, the probability of a certain peak value increases with $N$. You should also take into account the limited range of values of your measurement device.


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