Give the weak signal a known rhythm
Imagine a faint light made to brighten and dim periodically. A photodetector produces a small voltage at that rhythm. Ambient light and circuit noise also reach its output. Increasing the gain enlarges the desired variation and the noise together, so the small pattern can still be hard to see.
A separate reference cable tells the instrument when the light brightens. It supplies timing, not the unknown light strength. The measurement asks how much of the input follows that timing.
Sources: [2]
Multiplication moves the information to a slower signal
Consider two sinusoids with matching peaks. They are positive together and negative together, so their product has a positive average. Multiply by a very different rhythm and positive and negative contributions alternate. Much of that product cancels when averaged.
The multiplier performs this comparison. A low-pass filter then passes steady or slowly changing contributions and attenuates rapid variations. Even the matched pair produces a faster oscillation in addition to its average; the filter attenuates that part too. Information about the small input now appears as a slow voltage that is easier to read.
There is a catch. The peaks can arrive earlier or later within the cycle, a difference called phase. A signal displaced by exactly a quarter cycle gives equal positive and negative product areas. One averaging channel can read nearly zero despite a real signal.
A second reference catches the missing component
A second multiplier uses a reference shifted by a quarter cycle. It catches the component missed by the first. The outputs are called the in-phase and quadrature components, usually and . Think of them as the horizontal and vertical projections of an arrow. Turning the arrow changes both projections without changing its length.
For example, a three-millivolt RMS sinusoid leading the reference by thirty degrees gives about 2.60 millivolts in one channel and 1.50 in the other. Their combined magnitude is still three millivolts. RMS is a voltage scale: an equal DC voltage would produce the same average heating in the same resistor.
Combining the components removes the dependence on the chosen reference phase. After sufficient settling, a stable sinusoid at the reference frequency gives its RMS voltage as the magnitude.
A steadier reading takes longer to follow a change
The filter retains a memory of recent input. Its time constant describes how quickly it forgets an old value and approaches a new one. A longer time constant reduces brief fluctuations, but a real change in signal also takes longer to appear. A single RC stage completes about 99.3 percent of a step after five time constants.
Equivalent noise bandwidth describes the frequency band admitted in terms of noise power. A single RC stage with a tenth-second time constant has a noise bandwidth close to 2.5 hertz when sampling is fast enough. This bandwidth has a different meaning from the frequency where an oscillating signal falls to half power. Passing the same noisy capture through short and long filters reveals the exchange between output fluctuations and waiting time.
Sources: [2]
A slightly different rhythm leaves a beat
If the signal gains a little on the reference each second, their peaks gradually separate and meet again. Multiplication leaves their difference frequency, a beat. The two outputs rotate slowly, so their phase keeps changing. An angle at one instant describes that output direction; the filter also attenuates and delays the beat.
An unwanted component at exactly the target frequency, with a stable phase, survives too. The instrument measures the sum. Longer averaging cannot identify how much came from each source. Separating them requires another modulation, a changed experimental condition or another measurement.
Even noise alone produces a positive average magnitude because magnitude is never negative. Near the noise floor, inspect the signed components and their fluctuations together. The known rhythm and finite bandwidth make the measurement possible; matching that rhythm and allowing time to settle make the result useful.
Sources: [2]