Peaks

In the ACTest toolkit, the spectral function $A(\omega)$ is treated as a superposition of some peaks (i.e., features). That is to say:

\[\begin{equation} A(\omega) = \sum^{N_{p}}_{i = 1} p_i(\omega), \end{equation}\]

where $N_{p}$ is the number of peaks, $p(\omega)$ is the peak generation function. Now ACTest supports the following types of peaks:

• Gaussian peak

\[\begin{equation} p(\omega) = A\exp{\left[-\frac{(\omega - \epsilon)^2}{2\Gamma^2}\right]} \end{equation}\]

• Lorentzian peak

\[\begin{equation} p(\omega) = \frac{A}{\pi} \frac{\Gamma}{(\omega - \epsilon)^2 + \Gamma^2} \end{equation}\]

• $\delta$-like peak

\[\begin{equation} p(\omega) = A\exp{\left[-\frac{(\omega - \epsilon)^2}{2\gamma^2}\right]},~ \text{where}~\gamma = 0.01 \end{equation}\]

• Rectangular peak

\[\begin{equation} p(\omega) = \begin{cases} h, \quad \text{if}~\omega \in [c-w/2,c+w/2], \\ 0, \quad \text{else}. \\ \end{cases} \end{equation}\]

• Rise-And-Decay peak

\[\begin{equation} p(\omega) = h \exp{(-|\omega - c|^{\gamma})} \end{equation}\]

Here we just use a narrow Gaussian peak to mimic the $\delta$-like peak. In the above equations, $\mathcal{C} = \{A,~\Gamma,~\epsilon,~h,~c,~w,~\gamma\}$ is a collection for essential parameters. The ACTest toolkit will randomize $\mathcal{C}$ and use it to parameterize the peaks.