= Solution
A <lock-in amplifier> multiplies the measured <signal> by a synchronized reference and <low-pass filters> the product. For $s(t)=A\cos(\omega t+\phi)+n(t)$ and reference $2\cos\omega t$, the <signal> product is $A\cos\phi+A\cos(2\omega t+\phi)$. Averaging away the second harmonic gives $X=A\cos\phi$ in this peak-amplitude convention. A second reference $-2\sin\omega t$ gives $Y=A\sin\phi$, so
$$
\boxed{A=\sqrt{X^2+Y^2},\qquad\phi=\operatorname{atan2}(Y,X).}
$$
A single quadrature could read zero for a substantial <signal> $90^\circ$ from its reference. Two quadratures avoid that blind spot; the phase is undefined at zero <amplitude>. Instrument RMS calibrations can include a different constant normalization.
The reference shifts the wanted <frequency> to zero, where a narrow low-pass <bandwidth> can select it. Uncorrelated broadband <noise> averages down, whereas a simple amplifier would enlarge <signal> and <noise> together. Longer averaging improves narrow-band sensitivity but slows the response to changing <amplitude>. The <signal> must remain phase coherent with the reference; <frequency> mismatch creates a beat instead of a constant output. <interference> at the same reference <frequency> survives, so shielding and independent background measurements remain necessary.<Model SR830 DSP Lock-In Amplifier manual by Stanford Research Systems>[printed pp.3-1–3-3, multiplication, dual-phase detection and RMS-versus-peak conventions]{ref}
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