= Lock-in harmonic rejection
Consider this original constructed input <electric potential difference>, with $t$ in seconds and $f_0=1\,\mathrm{Hz}$:
$$
s(t)=(0.4\,\mathrm V)\cos(2\pi f_0t+\pi/3)
+(0.9\,\mathrm V)\cos(6\pi f_0t+\pi/5).
$$
The unwanted component has three times the reference <frequency> and more than twice the wanted peak <amplitude>. Nevertheless, the <lock-in quadratures> over $T=4/f_0=4\,\mathrm s$ are exactly
$$
X=0.4\cos(\pi/3)\,\mathrm V=0.2\,\mathrm V,\qquad
Y=0.4\sin(\pi/3)\,\mathrm V=0.2\sqrt3\,\mathrm V\approx0.34641016\,\mathrm V,
$$
so $R=0.4\,\mathrm V$ and $\operatorname{atan2}(Y,X)=\pi/3=60^\circ$. This is a peak <amplitude> calculation, not an SR830 display prediction.
To see why the larger component vanishes, multiply $B\cos(3\omega t+\psi)$ by each reference:
$$
\begin{aligned}
2B\cos(3\omega t+\psi)\cos\omega t
&=B\cos(2\omega t+\psi)+B\cos(4\omega t+\psi),\\
-2B\cos(3\omega t+\psi)\sin\omega t
&=B\sin(2\omega t+\psi)-B\sin(4\omega t+\psi).
\end{aligned}
$$
Each <sinusoid> has zero <average value of a function> over a whole reference cycle. More generally, a component $B\cos(n\omega t+\psi)$ with <integer> $n\geq2$ gives products at $(n-1)\omega$ and $(n+1)\omega$, whose <integrals> vanish over any <integer> number of reference cycles. This is the <orthogonality> behind selecting a <Fourier coefficient> with sinusoidal references. A nonsinusoidal reference can contain additional <frequencies> and need not have the same rejection property.
\Image[/lock-in-multiplication-and-averaging.png]
{title=<Lock-in harmonic rejection>: multiplication reveals a constant component}
{description=Original calculated waveforms for the declared $0.4\,\mathrm V$ wanted component and $0.9\,\mathrm V$ interference; no measured data. Top: input and wanted <signal>. Middle: products with the two dimensionless references. Bottom: running <lock-in quadratures>; markers at complete reference cycles equal $X=0.2\,\mathrm V$, $Y=0.2\sqrt3\,\mathrm V$. Partial-cycle estimates need not equal those values.}
An unwanted component at the reference <frequency> survives. If $u(t)=B\cos(\omega t+\psi)$ is added to the input, its coordinate <vector> adds to the wanted one:
$$
Z=Ae^{\mathrm i\phi}+Be^{\mathrm i\psi}.
$$
For example, adding $B=0.1\,\mathrm V$ with $\psi=0$ to the wanted component gives $X=0.3\,\mathrm V$, $Y=0.2\sqrt3\,\mathrm V$, $R=\sqrt{0.21}\,\mathrm V\approx0.458258\,\mathrm V$, and <phase difference> $\operatorname{atan2}(0.2\sqrt3,0.3)\approx49.1066^\circ$. Longer averaging cannot separate these two synchronized contributions.
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