The (effective) low-pass is only applied after the signals have been multiplied. This filters out the signal with the same frequency (and phase) as the reference. I have a more detailed comment on how this works in response to another person here in this comment section. This is useful for applications, where the signal can be drowned out by the noise, but you know what to look for, and where the phase information is important. (e. g. a DC signal that is chopped at a selected frequency or an AM-signal).

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} 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.