Noise – Does ‘Signal Buried in Noise’ Mean Noise Amplitude Is Smaller?

noisephysicssignal processingsignal-to-noise

I heard that Lock-in amplifiers (LIAs) especially play to their strengths when the signals are weak compared to the noise level. But then I talked with someone about it, who understands the principles of lock-in amplification, and she said – which makes sense to me now – that of course the signal amplitude still needs to be larger than the level of noise. Otherwise we couldn't represent the signal V_s like this:
$$
V_{s} = R\cdot cos(\omega_{s} t + \phi)
$$

Is that correct? I find the formulation "buried in noise" a bit confusing then…

PS: I often get criticized for not explaining enough about the basics of the topic that I ask my question about. Since I don't want my question to get closed again, I would like to refer you to this page, which I used to learn about it: https://www.zhinst.com/others/en/resources/principles-of-lock-in-detection
Also, to forestall criticism that I just stipulate that "buried in noise" is an existent phrase in this context, I would refer you to this page, where you can see some examples of this phrase: https://preview.tinyurl.com/y64re9ln (secure URL: only preview of website, that would otherwise redirect to a Google domain)

Best Answer

What you're missing is the bandwidth, both of signal and noise.

If you look at, let's say, a 1 V rms sinewave signal, together with 10 V rms noise on an oscilloscope, you'll see only noise.

However, if the noise occupies a 1 MHz bandwidth, and is flat with frequency, and you pass the signal + noise through a 1 kHz bandwidth filter centred on the signal, then you will eliminate 99.9% of the noise power, dropping its amplitude to 0.3 V rms. The signal will then be clearly visible.

A lock-in-amplifier is a neat way to make a very narrow filter centred on the frequency you feed in as the reference.

You can use the same principle even without sine waves. Spread spectrum systems like CDMA and GPS use a pseudo-random square wave signal as the reference, and call the 'multiply and average' process convolution or correlation. As long as the reference is is the same as the underlying signal, and as long as the averaging process produces an effective bandwidth small enough to drop the noise power, the signal can be 'dug out of the noise'. A lock-in-amplifier is a special case of the more general 'correlation with a reference' that's used for CDMA.