Gaussian distribution of crest factor of Gaussian noise I have a confusing problem.
I've produced a collection of waveforms, comprised of $n$ samples from a Gaussian distribution. All waveforms are composed of the same number of samples, picked from a Gaussian with the same values for $\mu$ and $\sigma$ (it's actually a natural source, however it's fairly consistent).
So, when I plot the sample values on a histogram for a single waveform, I get a Gaussian distribution (horizontal axis being sample value, vertical axis being # of samples). Similarly, when I plot the SNR for each sample (horizontal axis being SNR), where noise is defined as the RMS of the waveform, I get a Gaussian distribution.
Now, I'm looking at the SNR of whole waveforms. I define the SNR of a waveform to be the maximum value of the waveform, minus the minimum value of the waveform, quantity divided by the RMS of the waveform. So, $\frac{max-min}{RMS}$. I compute this value for each waveform, and make a histogram with the horizontal axis being # of waveforms, and the vertical axis being SNR, as defined.
Somewhat suprisingly, I get a very clear Gaussian distribution here, too. I guess I was expecting that $\frac{max-min}{RMS}$ would be roughly the same for all waveforms, and therefore I'd get a nice sharp peak somewhere in the histogram.
Why is this? I can't make up my mind if it's surprising, or expected.
 A: I'm no stats expert, but here's some insight from what I've learned in the world of signal processing for random vibration:
The distributions of the maximum and minimum of the waveform will each follow an Extreme Value Distribution--namely the Gumbel distribution when the random variable is drawn from a Gaussian distribution. The location and scale parameters of the Gumbel distribution will be related to the length of waveform $n$, process mean $\mu$, and process standard deviation $\sigma$.
The RMS of the waveform is also a random variable. For a gaussian process distribution, it will follow a chi distribution, and is related to the true standard deviation of the Gaussian process.
Putting it all together, the maximum and minimum values for a single waveform are both randomly chosen from independent Gumbel distributions mirrored about the process mean, and the RMS from a chi distribution, so your $\frac{max-min}{RMS}$ statistic is another random variable, resulting from the combination of each of these independent statistics. While the parameters of the Gumbels and chi are dependent, the samples drawn from them are independent.
The result you are producing is the range of the waveform (positive) divided by the RMS (positive), so this quantity is strictly positive and would not be Gaussian.
