Upper RMS Estimate of Noise Waveform - John Dunn, Consultant, Ambertec, P.E., P.C.

Imagine a noise voltage waveform of ±100 (name your units as mV or volts or whatever) and further imagine that this noise waveform has a flat probability distribution function (PDF) over its ±100 range and then hold that thought.

The PDF of a ±100 triangular waveform is also flat over its ±100 range and we recall from previous calculations (We are just so smart!) that the RMS of that triangular waveform of ±100 equals 100/sqrt(3).

Since the two functions, the noise and the triangular, have the same PDF, they also have the same RMS value.

A few examples of the RMS calculation bear this out: 

 
Of course, any real world noise waveform won't have the flat PDF that we've described for this waveform here, but rather will tend to taper off toward the peaks and will therefore have a lower RMS value than the value we've just found.

Therefore, the 100/sqrt(3) number is a good upper value estimate for the RMS value of a noise waveform and that might just be a handy fact to have in hand from time to time.

 

 

Comments

[George Storm]

I'm clearly missing something here, as I can't see any reason that there should be any relation between the triangular waveform and the topic under discussion. You could argue with equal justification a similar relationship with the peaks of a waveform that consisted of two equal sinusoids - and this would give you a lower-bound factor of two.
The old-boys' physiologically-based oscilloscope method for Johnson noise (turning the level of an oscilloscope up so the trace looked bright, and estimating the RMS as 2/7 of the apparent peak level on the oscilloscope) was remarkably accurate. That would make 1/sqrt(3) an upper bound - but one that was a factor of two above the actual value - and so not generally very useful.
And if we are talking about measuring interference rather than random noise, the RMS can actually be larger than 1/sqrt(3) times the peak.

[George Storm]

Erratum: "lower bound" should of course read "upper bound"

[Peter Kay]

In the audio world I have always used
White RMS noise = (peak to peak)/6.
Pink RMS noise = (peak to peak)/4
Your measurement is indicate noise is closer to (P-P)/3

[John D.]

The key to this estimate is that the probability distribution function is flat over the range of the variable. This is true for any sawtooth or triangular waveform and is also true for a random waveform when they all share the same range as shown here.

The white RMS and pink RMS are both smaller than the upper bound which is precisely the point.

The graphics shown came from some code in GWBASIC. If anyone would like to have it, I'll gladly send it along. Just e-mail me at [email protected] and I'll reply.