In Part 1, we discussed ways to get better measurements by improving the *signal to noise ratio* (SNR), and saw that although it was often a win to measure more slowly and use lowpass filters, going too far actually makes things worse, because of the way noise concentrates at low frequency. Here we introduce a more sophisticated approach that generally works better: the *lock-in amplifier.*

We were considering a typical *baseband* signal, one that goes from near DC to some much higher frequency. Audio is a typical example, with a bandwidth usually quoted as 20 Hz to 20 kHz. To escape the low frequency noise, we need to move our signal up in frequency, out of baseband. In lock-in detection we make the signal periodic in time at some *carrier* frequency *f _{c}* chosen to be several times higher than the required bandwidth. This is generally pretty easy to do, as we'll see, and doing so ensures that none of the signal we care about remains near DC. Our noise rejection filter now needs to be a narrow bandpass centered at

An AC signal that passes through a narrowish filter can be looked at as a sine wave with some amplitude and phase: *g(t) = A* cos(2π *f t* + *φ*), where the signal information is contained in slowish variations of *A* and *φ*, the amplitude and phase (the modulation). This is familiar from broadcast radio: you can send music and speech program material over the air by encoding it as amplitude modulation (AM) or frequency modulation (FM). AM changes the heights of the peaks of the sinusoidal carrier wave in response to the audio signal (*A* varies), while FM changes the position of the peaks in time (*φ* varies). FM maps the baseband signal *s(t)* onto the instantaneous frequency, so d*φ*/d*t* is proportional to *s(t)*. In *phase modulation* (PM), which is less common in radios but more useful in measurements, the signal maps directly: *φ* is proportional to *s(t)*. The two are collectively known as *angle modulation*.

All types of modulation widen the carrier spectrum, forming *sidebands* above and below *f _{c}* that carry the signal information. It's generally preferable to talk about AM and PM, especially in discussions of noise, because in PM a flat baseband spectrum produces flat sidebands, whereas in FM it doesn't. That makes PM much easier to think about.

A carrier with both AM and PM can be written as *g*(*t*) = *A*(*t*) cos( 2*π* *f _{c} t* +

cos(*a+b*) = cos *a* cos *b* - sin *a* sin *b*, or in this case, cos( 2*π* *f _{c} t* +

Thus by measuring the amplitudes of the sine and cosine components of the signal, we can recover its phase. Rearranging the same trigonometric identity shows us how to do this:

cos* a* cos *b* = ( cos(*a-b*) + cos(*a+b*) ) / 2 and

sin *a* sin *b* = ( cos(*a-b*) - cos(*a+b*) ) / 2.

Thus if we multiply our signal by *local oscillator* (LO) signals sin(2π* f_{c} t *) and cos(2π

*I = A* cos *φ* cos(2*π f_{c} t*) cos(2

*Q = A* sin *φ* sin(2*πf _{c} t *) sin(2π

Lowpass filtering gets rid of the 2*f _{c}* components of

Thus the procedure of multiplying by the sine and cosine phases of the carrier converts the modulated carrier into a pair of baseband signals containing both the amplitude and phase information. Because of the lowpass filtering, the exact waveform of the modulated wave (sine, square, or something else) doesn't matter much--only sinusoidal components sufficiently close to *f _{c}* contribute. This property of sines and cosines is called

*A*= √( *I*^{ 2} + *Q*^{ 2} ) and *φ* = tan^{-1}(*Q / I *).

(One has to worry about a few other things when computing *φ*, such as which quadrant it's in, whether you're dividing by zero, and whether it needs unwrapping to avoid ambiguities of multiples of 2*π*.) The multiplications also of course produce the cross terms, proportional to

cos(2*πf _{c} t *) sin(2

but these have no baseband component and so get filtered out as well, showing that the sine and cosine components are orthogonal even though their frequencies are the same.

The sine and cosine LO signals can be derived from a reference frequency that you supply, or generated internally. Generally this reference is the same source used to generate the AC modulation of the measured signal, but it'll still work even if the two are different (the frequency error will show up as a ramp in *φ*(*t*), of course).

So that's the general principle of how lock-in amplifiers can improve our SNR by narrowing the measurement bandwidth while avoiding the low-frequency noise. In Part 3 we'll look at how that's done, in both analog and digital lock-in amplifiers.

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