= Input-time horizon of a delayed optical receiver record
{title2=$H=\max_j(T_j-d_j)$}
The \b[input-time horizon of a delayed optical receiver record] is the latest physical input time that could influence the recorded output, given the receiver's causal onset delay. If $h(t)=0$ for $t<d$ and the last output time is $T$, <convolution> depends only on input times
$$
u\leq T-d.
$$
Thus $T-d$ is a necessary timing limit on information in that record. Recoverability within the limit additionally depends on bandwidth, noise, sampling and the response's invertibility. <Causal indistinguishability of delayed optical inputs on a finite record> shows why later input changes cannot be inferred from the existing output samples.
For several channels with last recorded times $T_j$ and known onset delays $d_j$, the latest possible input horizon is
$$
H=\max_j(T_j-d_j).
$$
Any input perturbation starting strictly after $H$ is invisible in every stored channel. A channel's latest output timestamp alone is insufficient to compare its input horizon with another channel's.
For an original example, channel 1 records through $120\,\mathrm{ns}$ with delay $15\,\mathrm{ns}$, while channel 2 records through $100\,\mathrm{ns}$ with delay $2\,\mathrm{ns}$. Their input limits are $105$ and $98\,\mathrm{ns}$; the combined limit is $105\,\mathrm{ns}$. Knowing both response functions exactly still supplies no observation of an input pulse beginning later than that limit.
A <receiver-latency uncertainty in the inferred input-time horizon> also changes the timing claim. If channel 1's delay is known only to lie between $14$ and $16\,\mathrm{ns}$, its possible horizon ranges from $104$ to $106\,\mathrm{ns}$. An input beginning at $104.5\,\mathrm{ns}$ could affect the record for one allowed delay but arrive too late for another. Using the conservative endpoint $104\,\mathrm{ns}$ avoids claiming timing access that depends on an unresolved latency. It still does not guarantee successful reconstruction of every earlier input.
NIST characterized detector, cable and <oscilloscope> as a combined response chain using a short known optical pulse.<Calibration Service for Low-level Pulsed-Laser Radiometers at 1.06 µm: Pulse Energy and Peak Power by Rodney W. Leonhardt>[Appendix C.1–C.2, printed pp. 36–37, Figure C1 and response-measurement procedure]{ref} Such a calibration must relate output timing to the physical input time origin. The delays above are illustrative, not quoted apparatus values. De-embedding known latency aligns the record's time reference; it cannot extend observations into an unrecorded future interval.
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