# How to measure the amplitude of noise?

I'm trying to set up an analogue to digital converter to read electrical noise. The noise is kinda Gaussian with a very very very rough amplitude of the order of 2Vp-p. I'm using a 10 bit range ADC that spans 2.5V, and have the opportunity to alter the amplitude of the signal up or down.

My problem is the Gaussian bit. This means that there is no limit to the range of the signal, just that signals spanning much more than 2V become increasingly rarer. Some times they might span 3Vp-p on rare occasion. More on even rarer occasions.

How do I set the input amplitude? Do I go low and only use 5 bits most of the time ensuring that I will never miss an out lying reading until the Universe cools? Or do I go high, use 9 bits then often miss higher amplitude signals? Err...

I'm not sure if this is actually an electrical or statistics question. However, this must be a common situation in signal acquisition so I'm hoping for some best practice advice. How would a commercial industrial acquisition device be configured specifically for this use case? I can only think of stupid analogies such as an accurate scientific beaver weighing device. How would I get the best use from my 10 bits to ensure that I can weigh all beavers with a good resolution? Please try to see this from a commercial perspective so that you're actually trying to design and sell these devices.

P.S. Having written the title lastly, my problem seems to boil down to "How to measure the amplitude of noise?"

I'm glad to see you said the noise was 'kinda Gaussian'. If you had asserted that the noise was Gaussian, I would have been busy warning the map is not the territory, the noise can do what it likes, however Gaussian is a good approximation to many noise processes.

As you so correctly point out, you'll need to wait until the universe cools to see all the samples. So you need to investigate the signal, and hope that it's stationary within the time of your analysis, and stationary into the future if you want to use your analysis to set a maximum gain.

The thing to do is calculate a CCDF (google, wikipedia) of your input signal, with a wide enough range so there's no observed overload in your observation time. This will lead very quickly to a good estimate of how many samples you will fail to catch for any given maximum range, and so underestimate the signal power if measured with the new range.

You will be able to do calculations like 'if you want to measure with 95% probability to 0.01% power accuracy, then you have to set an x.sigma peak signal'. Of course, that's based on signal you've already seen. If you want to estimate the effect of signals you haven't seen, that will always be an extrapolation, which is always uncertain.

As a hint, if the noise looks guassian(ish), and you can identify a probable sigma, three sigma is often large enough for some people, though demonstrably too small for others, six sigma is only twice as large, and will satisfy all but the theorists and the 'one in 10^12' disk drive error researchers.

Amplitude of Gaussian noise is defined as the width of its PDF - probability density function. Estimation of this function is usually done on the basis of several thousands ADC samples, obviously with proper sampling rate (to obey Nyquist-Shannon-Kotelnikov-etc. sampling theorem) that exceeds the particular noise bandwidth. Then DSP (digital signal processing) makes a best fit to the Gaussian curve, and calculates its width. For more details, look at practical guide from Analog Devices.

• Well that's the thing isn't it? The width of any Gaussian PDF is infinite hence my question. – Paul Uszak Oct 9 '16 at 2:20
• No, the width is defined at a certain level of PDF relative to its peak, at "1-sima", "2-sigma", whatever. Did you have a chance to read the guide? – Ale..chenski Oct 9 '16 at 2:24
• So how many sigmas would a commercial (beaver weighing) device be designed for? It's the trade off between accuracy and range that I guess I'm struggling with. Your first Analogue Devices example takes 512 samples. I could take 512 million to be all inclusive. Should I? – Paul Uszak Oct 9 '16 at 2:31
• You can, but it might be unnecessary. All depends on how much time you have, and accuracy you want to achieve. For USB3, for example, estimation of PDF is done on a basis several hundred thousands samples, and PDF width is estimated at 10^-12 level. And I have no idea what the beaver weighing device is. – Ale..chenski Oct 9 '16 at 2:51