= Finite-window lock-in response
{title2=$F_T(\nu)=e^{\mathrm i\nu T/2}\operatorname{sinc}(\nu T/2)$}
An observation <interval> of finite duration is not an infinitely sharp <frequency> selector. Combine the <lock-in quadratures> into
$$
Z_T=\frac2T\int_0^T s(t)e^{-\mathrm i\omega t}\,dt.
$$
For unwanted <electric potential difference> $s(t)=B\cos((\omega+\delta)t+\psi)$, where $\delta$ is an <angular frequency> offset and $\omega+\delta>0$, expansion into <complex exponentials> gives the exact response
$$
\begin{aligned}
Z_T&=Be^{\mathrm i\psi}F_T(\delta)
+Be^{-\mathrm i\psi}F_T(-(2\omega+\delta)),\\
F_T(\nu)&=\frac1T\int_0^T e^{\mathrm i\nu t}\,dt
=e^{\mathrm i\nu T/2}\operatorname{sinc}(\nu T/2),\\
\operatorname{sinc}(x)&=\begin{cases}\sin x/x,&x\ne0,\\1,&x=0.\end{cases}
\end{aligned}
$$
The <sinc function> formula follows by integrating the <exponential function> and factoring $e^{\mathrm i\nu T}-1=2\mathrm i e^{\mathrm i\nu T/2}\sin(\nu T/2)$; its value at zero follows from a <limit>. The first term changes slowly after multiplication; the second oscillates at the sum <frequency>. It vanishes at exact matching when $T$ is an <integer> number of reference cycles, each lasting one <period of a function>, but cannot generally be discarded in a short-window calculation.
For $|\delta|\ll\omega$ and many reference cycles, the first term usually dominates: its <modulus> is $B|\operatorname{sinc}(\delta T/2)|$ and its <complex argument> includes $\delta T/2$. Thus $|\delta|T\ll1$ gives almost no rejection. A longer observation separates closer <frequencies>, with the first zeros of this term at $|\delta|=2\pi/T$. Those zeros are not a brick-wall cutoff: the <sinc function> has nonzero lobes beyond them. At $\delta=0$ a synchronized contribution remains however large $T$ becomes.
\Image[/lock-in-quadratures-and-window-response.png]
{title=<Lock-in quadratures> preserve <phase difference>; a finite <interval> limits rejection}
{description=Original calculated geometry and responses, not measurements. Left: $X=0.2\,\mathrm V$, $Y=0.2\sqrt3\,\mathrm V$, $R=0.4\,\mathrm V$, $\phi=60^\circ$; the dotted <vector> shows an added synchronized $0.1\,\mathrm V$ contribution. Right: exact $|Z_T|/B$ for $\psi=0$ and $T$ equal to four reference cycles, against dimensionless detuning $(f-f_0)T$. The dashed curve keeps only the difference-frequency term. Their difference shows why the sum-frequency term matters for a short observation.}
For a numerical check, use $f_0=1\,\mathrm{Hz}$, $T=4\,\mathrm s$, $\psi=0$ and an unwanted <frequency> $f=1.125\,\mathrm{Hz}$. Then $(f-f_0)T=1/2$ and the exact expression gives
$$
\frac{|Z_T|}{B}=\frac2\pi-\frac2{17\pi}=\frac{32}{17\pi}\approx0.599172,
$$
whereas the difference-frequency term alone gives $2/\pi\approx0.636620$. An interference <amplitude> $B=0.9\,\mathrm V$ therefore contributes $0.539254\,\mathrm V$ to $|Z_T|$ in this constructed example. A nearby <frequency> can survive strongly even though the three-times-frequency component in <lock-in harmonic rejection> cancels exactly.
A practical <lock-in amplifier>'s <low-pass filter> weights past products differently from this rectangular observation <interval>. Its <time constant> and roll-off determine near-reference rejection and <settling time>; narrowing detection slows the response.<Model SR830 DSP Lock-In Amplifier manual by Stanford Research Systems>[Model SR830 DSP Lock-In Amplifier manual, printed p. 3-2, “Narrow band detection”; pp. 3-11–3-12, time constants and filter roll-off]{ref}
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