\item$H(n)$ is the number of hits in the histogram at output $n$. A hit at output $n$ means that the signal propagated down to output $n$, without reaching output $n-1$.
\item$W(n)$ is the width of bin $n$.
\item$C =\displaystyle\sum\limits_{n=0}^{N-1} H(n)$ is the total number of hits in the histogram.
\item$R(n)$ is the timestamp of an event whose signal propagated down to output $n$ (without reaching output $n-1$), measured backwards from the clock tick.
\item$R(n)$ is the timestamp of an event whose signal propagated down to output $n$ (without reaching output $n-1$), measured backwards from the clock tick.
\item$f$ (respectively $f_{0}$) is the current (respectively reference) frequency of the online calibration ring oscillator.
\end{itemize}
\subsubsection{Offline calibration}
We take the first output of the delay line to be the origin of the time measurements, and we define:
\begin{equation}
W_{0}(N-1) = 0
\end{equation}
The width of other bins is proportional to their respective number of counts in the histogram. The widths sum up to a clock period. This leads to the following equation:
\begin{equation}
W_{0}(n) = \frac{H(n+1)}{C}\cdot T_{sys}
\end{equation}
The timestamp is the sum of the widths of the traversed bins:
In the TDC core, the unit is the clock period, and the output has $F$ base 2 digits after the radix points. The controller also chooses\footnote{This choice was made for design simplicity reasons, and the precision can be improved by increasing C.}$C=2^F$. Expressed in units of $2^{-F}$ clock periods (which is the weight of the least significant bit of the fixed-point output), we have:
\begin{equation}
\frac{T_{sys}}{C}=1
\end{equation}
\subsubsection{Online calibration}
Online calibration is performed with a simple linear interpolation of the delays relative to the ring oscillator frequencies:
\begin{equation}
R(n) = \frac{f_{0}}{f}\cdot R_{0}(n)
\end{equation}
Note that when $f < f_{0}$, some values can go above the maximum fractional part value of $1-2^{-F}$ and might not fit in the LUT anymore. To solve this problem, all LUT values contain a configurable amount of \textit{guard bits}, which are integer part bits used to contain the overflow when the situation appears.
