If you have about an hour to kill, you might want to check out [Shahriar’s] video about the Stanford Research SR530 lock in amplifier (see below). If you know what a lock in amplifier is, it is still a pretty interesting video and if you don’t know, then it really is a must see.
Most of the time, you think of an amplifier as just a circuit that makes a small signal bigger in some way — that is, increase the voltage or increase the current. But there are whole classes of amplifiers designed to reject noise and the lock in amplifier is one of them. [Shahriar’s] video discusses the math theory behind the amplifier, shows the guts, and demonstrates a few experiments (including measuring the speed of sound), as well.
The math behind the amplifier is mostly trig, although there is a little calculus involved. The idea is to mix a reference frequency together with the signal of interest. This will result in the sum and difference of the two signals. Integrating the signals will — over time — zero out.
That may not be intuitive, but consider this: thanks to Fourier analysis, we know that any signal can be decomposed into a bunch of sine waves. Since a sine wave has equal positive and negative excursions, the area under it is zero if you integrate over an entire period. If your calculus is rusty, integration is more or less adding up infinitely thin slices of the curve which gives the area under the curve. Since the sine wave has equal positive and negative areas, the area underneath is zero.
An amplifier that zeros out the input isn’t very useful. However, there is a catch. If the reference signal and the input signal are out of phase but equal in frequency, then the difference term will be a constant that doesn’t vary with time. When you integrate the entire signal, that constant will stick out like a sore thumb.
That’s the basic idea. If you want more details, the video does a nice job of explaining it as well as showing it in practice.
If you notice in the SR530’s block diagram, there are many PLLs, a topic we’ve covered before. You might wonder why you can’t just filter the frequency of interest. In theory, you could. But as [Shahriar] explains, to get the same performance out of a filter would require an impractically narrow filter.