 Instrument response function

An instrument response function describes how a scientific instrument spreads or weights a specified ideal input. For a linear, time-invariant measurement system its temporal form $h(t)$ is the output produced by a unit impulse. An input signal $s(t)$ then giveswhere $b(t)$ is a background contribution. This convolution model requires linearity and time invariance; detector saturation or changing instrument settings can invalidate it. A spectral instrument similarly spreads a narrow input line across recorded wavelengths.

In time-correlated single-photon counting, a short laser pulse does not produce an infinitely narrow recorded event distribution. The pulse duration, photodetector timing and recording electronics all contribute to the measured instrument response function. Prompt scattered excitation can characterize the combined response under suitable measurement conditions. Extracting a fluorescence lifetime then often means fitting a decay after convolving it with the measured response, rather than treating the recorded histogram as the unblurred decay.^[ref]

An elementary example makes the distinction visible. If $h$ is a normalized Gaussian with standard deviation $\sigma$ and the normalized physical decay is $s(t)={\tau }^{-1}{e}^{-t/\tau }$ for $t\ge 0$, the convolved distribution has variance ${\sigma }^2+{\tau }^2$. Its width therefore combines instrumental broadening with the sample lifetime. A measured broad peak need not imply a slow physical process. Measuring the response, background and their uncertainties is part of learning the physical signal, not an optional cosmetic correction.