= Instrument-response limit in sequential population inference
{title2=$S=K(R*p)+B$}
An \b[<instrument-response limit in sequential population inference>] is a practical limit on identifying fast transfer or decay rates from a temporally broadened measurement. It depends on the <instrument response function>, photon statistics, timing calibration and the assumed kinetic model. The response width is not a universal hard cutoff below which every rate becomes unknowable.
For an original sequential model, take normalized lower-state photon-arrival density
$$
p(t)=\frac{ab}{a-b}\left(e^{-bt}-e^{-at}\right),\qquad t\ge0,
$$
with $a,b>0$, $a\ne b$, and zero density before excitation. It corresponds to two successive independent exponential waiting times. A normalized instrument response $R$ produces modeled signal
$$
S(t)=K\int R(t-u)p(u)\,du+B(t),
$$
with total signal scale $K$ and background $B$. The measured curve is a <convolution>, not the molecular population curve alone. As $a$ grows with $b$ fixed, $p(t)$ approaches $be^{-bt}$: the fast-step signature becomes concentrated near the onset.
The arrival-time mean and variance are
$$
\mu_p=\frac1a+\frac1b,\qquad
\sigma_p^2=\frac1{a^2}+\frac1{b^2}.
$$
For a zero-mean response of variance $\sigma_R^2$, independent timing broadening adds that variance. With lower-state lifetime 100 ps, transfer times 2 and 10 ps give intrinsic means 102 and 110 ps. A response standard deviation of 25 ps broadens both traces and makes their rapid onsets less distinct. An uncertain time-zero shift can further mimic part of the mean difference.
\Image[/sequential-kinetics-response-broadening.png]
{title=Original sequential-arrival models before and after Gaussian instrument broadening; the response obscures but does not erase every difference}
A primary manufacturer technical note describes measuring an instrument response using a scattering sample and fitting by iterative reconvolution. It also explains why favorable photon counts can permit recovery of lifetimes shorter than the response width.<Time-Correlated Single Photon Counting (technical note v. 2.1) by Michael Wahl>[printed p. 3, instrument response, reconvolution and measurement resolution]{ref} Thus an unresolved-looking rise can support a model-dependent bound rather than a precise, independently identified transfer rate.
A useful experiment records the response under matching optical conditions, checks timing offsets and background, and compares response-convolved predictions across spectral channels. Confidence intervals or sensitivity to nuisance parameters reveal what the data actually constrain. A fine histogram bin width alone does not create temporal information absent from the measurement.
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