~~~~~~P_\mathrm{avg}(\tau)\simeq\frac{\Delta \langle I \rangle}{\Delta t}\bigg\vert_\tau\quad,$$

where $\langle I_m \rangle_{S_n}$ is the average integrated amplitude for time bin $m$ of data subsample $S_n$, $\tau=m\Delta t$ is the phase within that time bin, and $P_\mathrm{avg}(\tau)$ is the average value of $P(t)$ over $\Delta t$ at $t=\tau$. Equation \[eq:recopulse\] is valid to reconstruct the first 25 ns of the pulse shape, for which $m=0$. For each remaining 25 ns interval, bin $m$ is incremented and a second integral term appears in the equation that corresponds to the $\Delta t$-sized portion of the pulse “leaving” the time bin. In this manner, the whole pulse was reconstructed to approximately 0.8 ns resolution, as is shown in Fig. \[fig:recopulses\]. The functional form matched to these results is used in the simulation of the CMS detector response, and also to determine the second-order time reconstruction correction function (Fig. \[fig:corfun\]).

To measure both the channel-to-channel variations and the hadronic timing resolution limit of the HCAL, a 150 pion beam was directed at every cell in the unit under test. From these data the variation of the timing of signals as a function of $\eta$ between HCAL cells in the barrel was mapped. As the test setup mimicked the 