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Thesis: beacon: whitespace removal + parentheses around \Delta

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Eric Teunis de Boone 2023-03-30 23:57:32 +02:00
parent 04c8478f93
commit 7fc86a18dd
1 changed files with 17 additions and 17 deletions
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documents/thesis/chapters/beacon_discipline.tex
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@ -113,7 +113,7 @@ In that case, the differences between the true arrival times $(\tTrueArriv)_i$ a
\label{eq:interantenna_t0} \label{eq:interantenna_t0}
\phantom{.} \phantom{.}
\begin{aligned} \begin{aligned}
\Delta (\tTrueArriv)_{ij} (\Delta \tTrueArriv)_{ij}
&\equiv (\tTrueArriv)_i - (\tTrueArriv)_j \\ &\equiv (\tTrueArriv)_i - (\tTrueArriv)_j \\
&= \left[ \tTrueEmit + (\tProp)_i \right] - \left[ \tTrueEmit + (\tProp)_j \right] \\ &= \left[ \tTrueEmit + (\tProp)_i \right] - \left[ \tTrueEmit + (\tProp)_j \right] \\
%&= \left[ \tTrueEmit - \tTrueEmit \right] + \left[ (\tProp)_i - (\tProp)_j \right] \\ %&= \left[ \tTrueEmit - \tTrueEmit \right] + \left[ (\tProp)_i - (\tProp)_j \right] \\
@ -159,10 +159,10 @@ This can be resolved by knowledge on the $\tTrueEmit$ of the transmitter.
In general, we are interested in synchronising an array of antennas. In general, we are interested in synchronising an array of antennas.
As \eqref{eq:synchro_mismatch_clocks} applies for any two antennas in the array, all the antennas that record the signal can determine the synchronisation mismatches simultaneously. As \eqref{eq:synchro_mismatch_clocks} applies for any two antennas in the array, all the antennas that record the signal can determine the synchronisation mismatches simultaneously.
\\ \\
The mismatch terms for any two pairs of antennas sharing a single antenna $( (i,j), (j,k) )$ allows to find the closing mismatch term for $(i,k)$ since The mismatch terms for any two pairs of antennas sharing a single antenna $\{ (i,j), (j,k) \}$ allows to find the closing mismatch term for $(i,k)$ since
\begin{equation*} \begin{equation*}
\label{eq:synchro_closing} \label{eq:synchro_closing}
\Delta (\tClock)_{ij} + \Delta(\tClock)_{jk} + \Delta(\tClock)_{ki} = 0 (\Delta \tClock)_{ij} + (\Delta \tClock)_{jk} + (\Delta \tClock)_{ki} = 0
\end{equation*} \end{equation*}
Taking one antenna as the reference antenna with $(\tClock)_r = 0$, the mismatches across the array can be determined by applying \eqref{eq:synchro_mismatch_clocks} over consecutive pairs of antennas and thus all clock deviations $(\tClock)_i$. Taking one antenna as the reference antenna with $(\tClock)_r = 0$, the mismatches across the array can be determined by applying \eqref{eq:synchro_mismatch_clocks} over consecutive pairs of antennas and thus all clock deviations $(\tClock)_i$.
\\ \\