emission (in the case of @burgess14 [@preece2014], the sum of blackbody and synchrotron was used and the resulting temporal evolution of the characteristic energy was more consistent with a broken power law), we will restrict ourselves here to the evolution of the COMP function parameters.

A Model Pulse with Spectral Evolution
-------------------------------------

We start out with a simple simulation of a single pulse with HTS spectral evolution. Following @2005ApJ...627..324N and @HakkilaPreece, we adopt a parameterized pulse profile: $$p(t) = \exp\left(-\tau_1/(t-t_{\rm start}) - (t-t_{\rm start})/\tau_2\right),$$ where $\tau_1$ and $\tau_2$ are the rise and decay times of the pulse and the onset of the pulse is at time $t_{\rm start}$. In terms of these parameters, the duration of the pulse is approximately $$t_{\rm dur} = \tau_2 \sqrt{9 + 12 \sqrt{\tau_1 / \tau_2}}.$$ Next, we can specify a power-law temporal evolution for [$E_{\rm peak}$]{} from $E_{\rm hi}$ to $E_{\rm lo}$ over the pulse duration $t_{\rm dur}$ by the power-law index: $$i = \ln(E_{\rm hi}/E_{\rm lo})/\ln(t_{\rm dur}+ 1). \label{decay_index} $$ Finally, the power-law behavior of [$E_{\rm peak}$]{} is expressed as: $$E_{\rm peak}(t) = E_{\rm hi} \left(t -t_{\rm start} +  1\right)^{-i}.$$ All of these quantities are so far continuous; we must discretize the time $t$ 