Is the noise of op-amps and ADCs Gaussian, possibly with zero mean?

Your title question about the "Gaussianity" brings up a lot of thorny issues, since noise in electronic components is quite a difficult topic from a physical, fundamental point of view.

TL;DR:

From a designer perspective:

• Noise is assumed to have zero mean because even if it had non-zero mean that mean value would just be an added DC component which can be filtered-out or compensated. And it would be indistinguishable from other sources of offset (e.g. circuit asymmetries).

• In most applications, even noise-sensitive ones, you don't really care about the exact distribution of noise. Most of the time the total noise is assumed to be Gaussian since it can be shown that this is a good (and sometimes excellent) approximation of the reality. And it is a very convenient approximation, since Gaussian distributions have all sorts of good mathematical properties and are well understood.

• The assumption of Gaussianity is at the basis of the definition of noise RMS and peak-to-peak values. In fact these two values are related by a formula which assumes the noise process is Gaussian. In particular, the pp value is usually defined as $V_{pp} = 6 V_{RMS}$ (sometimes the conversion factor is 6.6).
  
  It can be shown, under the Gaussianity assumption, that the noise value at any given time has a >99.7% probability of being inside the interval $\bigl[\frac {-V_{pp}}{2}, \frac{V_{pp}}{2}\bigr]$.
  
  Note, however, that in theory a Gaussian noise process can produce values that are arbitrarily big, since the Gaussian distribution tails extend to infinity on both sides. But that's just a theoretical argument. In practice, since any device has physical limits, arbitrarily high noise values are impossible. This notwithstanding, values somewhat outside that interval are indeed possible, however with low probability.
  
  In other words, for all practical design purposes, you can consider the noise process amplitude bounded by the $\pm\frac{V_{pp}}{2}$ interval. If you really have a design in your hands that could misbehave if the noise level went out of that interval for a (say) picosecond once in a (say) year, you are really in the realm of ultra-sensitive, extreme measurement stuff, e.g. tunnel-effect microscopy, high-physics experiments dealing with neutrinos or gravitational waves or such weird stuff. In these cases you would need much more than a datasheet and some noise-oriented design guidelines and techniques!

• Actual RMS and peak-to-peak values found in datasheets are measured and may or may not agree perfectly with the assumption of a Gaussian distribution. This is not a problem in design, since you should always design with appropriate safety margins in mind, and any deviation from "true" Gaussianity" will be swamped by those margins.

You may find these documents interesting:

Noise Analysis in Operational Amplifier Circuits (Texas Instruments)

Fundamentals of Precision ADC Noise Analysis (Texas Instruments)

What an Electronics Engineer Needs to Know About Noise

Noise in Electronics Engineering: Distribution, Noise RMS and Peak-to-Peak Value, and Power Spectral Density

Now a bit longer analysis.

First some background:

1. The most common Gaussian noise sources are thermal noise sources (e.g. "resistors"). Any conducting physical medium produces Gaussian noise due to atomic and molecular thermal motions. This is also called Johnson noise and it is "approximately" white noise. "approximately" because its actual noise bandwidth is extremely high, so for electronics purposes is like "true" (mathematical) white noise.

2. Any superimposition of a sufficiently high number of independent noise sources produces a (approximately) Gaussian distribution. This is a consequence of a fundamental theorem of stochastic process theory, related to the law of large numbers in statistics.
   
   It's the same thing that goes on during a rainy day. When the rainstorm begins, you can hear just the single raindrops falling on a hard surface ("plock, plock"). Then the frequency of "plocks" increases and also their numbers until you just hear the "uniform" noise of rain (which is Gaussian).

3. There may be additional noise-generating mechanisms at work during electronic devices operation. For example there is flicker noise, whose density is higher as frequency goes down, so it's particularly problematic in DC applications. There are also shot noise and burst noise.
   
   In a complex circuit, like the one found in ICs, all these sources combine together in complicated ways and generate the output noise. Keep also in mind that an IC, even if it behaves like a linear component when used in some applications (e.g. an opamp connected as a voltage follower or an inverting amp) is full of highly non-linear devices and non-linearity does nasty things on signals. So the characteristics of the single noise sources are also modified heavily by the non-linear effects in BJTs, MOSFETs,diodes and other semiconductor devices inside an IC.

At the end of the day, in most applications where noise can be a problem you don't really care whether or not the actual noise distribution is Gaussian or not. Maybe it's just approximately so, but then? What's the advantage of that knowledge for the average design engineer. Most of the times the designer just cares about very basic statistical properties of the total excess noise the device is adding to your signal, because those properties are just what is needed for most noise-aware designs. So assuming the noise is Gaussian is just a convenient working hypothesis.

So peak or RMS noise values are used, because they can provide some guidance on the impact of noise in the time domain. For example peak value is more useful in applications where the peak of the signal is important, e.g. when you risk saturating something; on the other side, RMS is useful in applications when it's the energy content of the noise which is more problematic.

Another useful characteristic when dealing with systems with a well defined bandwidth is the voltage and current noise density, from which you can estimate the total noise of your system. And for components especially targeted at low noise applications, you can also see a plot of the noise densities vs. frequency, especially when it is not particularly flat.

For components used in noise-sensitive signal-processing/conditioning front-ends (e.g. low-noise receiving antenna amplifiers), there are also diagrams showing constant-noise-figure curves plotted against other meaningful design parameters (e.g. for an HF BJT they could be collector quiescent current or source impedance).

posted 9 months ago

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9mo ago

Lorenzo Donati‭

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