Oscilloscopes give a Vrms noise and have a noise floor, but if I average for a long time to infinity, am I at the standard quantum limit and does SNR keep improving with the sqrt(N averages) forever?
Yes for random noise Vrms keeps reducing by N for sqrt N samples
- the SNR is then limited by drift, periodic noise and measurement accuracy
- However, if the measurement is a frequency which is nearly as stable ( high SNR) as the clock in the counter measuring it, then some phase noise must be added in order to obtain an accurate reading. Then the noise can be defined by std or pp deviation or Hz/rt(Hz)
- This can also be done by adding some AC voltage with a fixed trigger threshold.
- with sqrt(N) more samples you obtain N-times better resolution and stability on the result just as in DC.
- This AC exception example is due to harmonically-related synchronous effects with no jitter.
Remember that the averages are being computed by a digital processor using binary arithmetic. At some point the accuracy and resolution of the average will be limited by the resolution of the number format and the accuracy of the averaging algorithms. Algorithms that calculate the mean of a very large set of values run into numerical problems because the mean value may be very, very small compared to the sum of a few very large samples.
Your question suggests that you are averaging an extremely large set of values. I would be surprised if any commercial oscilloscope, performing these calculations in real time, would approach the accuracy of even a good desktop computer.
There is a couple of barriers that you will hit on the way there that have considerably more practical implications.
- 1/f noise. The oscilloscope amplifier bias points, oscillators, and ADC voltage reference assume DC conditions. You are basically doing an infinite integral of those DC values, 1/f noise is theoretically infinite at DC.
- ADC quantization. Regardless of the precision of the math and the oscilloscope ADC, the sqrt(N) gain in RMS noise only applies to noise that is uncorrelated to the signal, once your sqrt(N) gain drops your noise to a level below a single ADC step, your signal and your noise becomes the same. These become more and more correlated as you progress.
Interestingly, the way to bypass this correlation issue is to add larger and larger magnitudes of random noise to the signal. So the fundamental limit will end up being the dynamic range of the ADC. Which is the compromise achieved by sigma delta converters, as these use a 1-bit ADC.