I have a white noise source (zener diode in full avalanche mode) and I've been amplifying it with an op amp. The op amp is a common device (TL081) with resistors set for a gain of 4. So there's an equation for the gain of -Rfeedback / Rin. Unremarkable, and this works for signals like a fixed frequency sine wave. I've tested this on a breadboard with a 1 kHz sine wave and the formula holds as expected.

But.

The -Rfeedback / Rin equation does not work for amplifying pure white noise. If you amplify a white noise signal, the gain is always a lot less than the formula predicts. In my case a 275mVp-p noise source is only amplified to 455mVp-p using the exact set up as for the sine wave. That's only a gain of 1.7. The standard gain equation fails badly. And the relationship get's worse for higher gains, so for a gain of 20, you might only raise the peak to peak level fourfold.

I encountered this a while ago. Then I theorised that it was due to the Arduino's ADC input impedance being incorrectly specified. I asked Is the ADC input impedance actually 3750 Ohms? and promptly was shown to be a fool

I have a better theory, but it's complicated. Please indulge me. White noise has a flat spectrum. So the signal spectrum covers the whole bandwidth of the op amp, 3 MHz in my case. The amplitude of white noise is proportional to the square root of the measurement bandwidth. More bandwidth means more amplitude. That's standard stuff. As you amplify white noise, you hit the gain bandwidth product of the op amp which attenuates the higher frequencies. So the signal amplitude drops. Yet you're still amplifying an ever more attenuated signal. Hence the traditional gain equation doesn't work as it makes no account of the op amps' bandwidth.

Does this make sense? I think I might be right this time, but I'm looking for educated opinions and perhaps a gain equation for noise only amplification...