Click train detector#
In the following we try to develop a click train detector by means of a detection to click association algorithm
Consider Inter-click-interval (ICI)
(186)#\[\begin{equation}
(ICI)_n = t_n - t_{n-1}
\end{equation}\]
power variation
(187)#\[\begin{equation}
\Delta P_{n}=P_{n}-P_{n-1}
\end{equation}\]
and boundary reflection (multipath)
(188)#\[\begin{equation}
δτ_n=t_{r,n}-t_n
\end{equation}\]
While for each click train, there is only one \((ICI)\) and \(P\), there can be multiple boundary reflections
Assume variations in these quantities are stochastic processes that vary only randomly
(189)#\[\begin{equation}
\Delta (ICI)_n = \Delta (ICI)_{n-1} + N(0,\sigma_C)
\end{equation}\]
(190)#\[\begin{equation}
\Delta (\Delta P_n) = \Delta (\Delta P_{n-1}) + N(0,\sigma_P)
\end{equation}\]
(191)#\[\begin{equation}
\Delta δτ_n = \Delta \delta\tau_{n-1} + N(0,\sigma_\tau)
\end{equation}\]
That is
(192)#\[\begin{equation}
(ICI)_n = 2 (ICI)_{n-1} - (ICI)_{n-2} + N(0,\sigma_C)
\end{equation}\]
(193)#\[\begin{equation}
\Delta P_n = 2\Delta P_{n-1} - \Delta P_{n-2} + N(0,\sigma_P)
\end{equation}\]
(194)#\[\begin{equation}
δτ_n = 2\delta\tau_{n-1} -\delta\tau_{n-2}+ N(0,\sigma_\tau)
\end{equation}\]