The device, I‘m referencing here is called a lock-in amplifier.
When you try to measure an extremly noisy signal without all the noise, you can use one of these. If you‘re dealing with a DC-signal, you can chop it at the reference frequency.
But TLDR: After the reference signal is adjusted to have same frequency (and therefore constant phase difference), you get a signal that oscillates with ω_\text{in} - ω_\text{ref} and ω_\text{in} + ω_\text{ref}. Crucially, in the case, where ω_\text{in} = ω_\text{ref} the term becomes constant U(t) = U_0 |e^{i \theta}| while the other terms from other frequency components (Fourier-series) still oscillate. This is where the averaging comes in. An oscillating signal will average (roughly) 0 over a long enough duration. The output is then the amplitude of the desired signal without all the noise.
The device, I‘m referencing here is called a lock-in amplifier. When you try to measure an extremly noisy signal without all the noise, you can use one of these. If you‘re dealing with a DC-signal, you can chop it at the reference frequency.
Here‘s a great write up on the priciples of this technique: https://www.zhinst.com/sites/default/files/documents/2025-10/zi_whitepaper_principles_of_lock-in_detection.pdf
But TLDR: After the reference signal is adjusted to have same frequency (and therefore constant phase difference), you get a signal that oscillates with
ω_\text{in} - ω_\text{ref}andω_\text{in} + ω_\text{ref}. Crucially, in the case, whereω_\text{in} = ω_\text{ref}the term becomes constantU(t) = U_0 |e^{i \theta}|while the other terms from other frequency components (Fourier-series) still oscillate. This is where the averaging comes in. An oscillating signal will average (roughly) 0 over a long enough duration. The output is then the amplitude of the desired signal without all the noise.