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Thesis: Beacon: wrong k in equation

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Eric Teunis de Boone 2023-08-08 16:02:45 +02:00
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documents/thesis/chapters/beacon_discipline.tex
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@ -369,7 +369,8 @@ changing the synchronisation mismatches in \eqref{eq:synchro_mismatch_clocks} to
(\Delta \tClock)_{ij} (\Delta \tClock)_{ij}
&\equiv (\tClock)_i - (\tClock)_j \\ &\equiv (\tClock)_i - (\tClock)_j \\
&= (\Delta \tMeasArriv)_{ij} - (\Delta \tTrueArriv)_{ij} \\ &= (\Delta \tMeasArriv)_{ij} - (\Delta \tTrueArriv)_{ij} \\
&= (\Delta \tMeasArriv)_{ij} - (\Delta \tProp)_{ij} - \Delta k_{ij} T\\ &= (\Delta \tMeasArriv)_{ij} - (\Delta \tProp)_{ij} \\
&= \left[ \frac{ (\Delta \pMeasArriv)_{ij}}{2\pi} - \Delta k'_{ij} \right] T - (\Delta \tProp)_{ij} \\
&= \left[ \frac{ (\Delta \pMeasArriv)_{ij}}{2\pi} - \frac{(\Delta \pProp)_{ij} }{2\pi} - \Delta k_{ij} \right] T\\ &= \left[ \frac{ (\Delta \pMeasArriv)_{ij}}{2\pi} - \frac{(\Delta \pProp)_{ij} }{2\pi} - \Delta k_{ij} \right] T\\
&\equiv \left[ \frac{ (\Delta \pClock)_{ij} }{2\pi} - \Delta k_{ij} \right] T &\equiv \left[ \frac{ (\Delta \pClock)_{ij} }{2\pi} - \Delta k_{ij} \right] T
.\\ .\\
@ -555,11 +556,11 @@ where $s$ is the amplitude of the beacon, $\sigma$ the noise amplitude and $\erf
For sake of brevity, it will be referred to as ``Random Phasor Sum''. For sake of brevity, it will be referred to as ``Random Phasor Sum''.
\\ \\
This Random Phasor Sum distribution collapses to a gaussian distribution when the beacon amplitude is (much) larger than the noise amplitude. This Random Phasor Sum distribution collapses to a gaussian distribution when the beacon amplitude is (much) larger than the noise amplitude.
This can be seen in Figure~\ref{fig:time_res_vs_snr} where both distributions are shown for a range of \glspl{SNR}. This can be seen in Figure~\ref{fig:sine:snr_time_resolution} where both distributions are shown for a range of \glspl{SNR}.
There, the phase residuals of the simulated waveforms closely follow the distribution. There, the phase residuals of the simulated waveforms closely follow the distribution.
\\ \\
From Figure~\ref{fig:time_res_vs_snr} we can conclude that depending on the \gls{SNR}, the timing accuracy of the beacon is below $1\ns$ for our beacon at $51.53\MHz$. From Figure~\ref{fig:sine:snr_time_resolution} we can conclude that depending on the \gls{SNR}, the timing accuracy of the beacon is below $1\ns$ for our beacon at $51.53\MHz$.
Since the time accuracy is derived from the phase accuracy, slightly lower frequencies could be used, but they would require a stronger signal to resolve to the same degree. Since the time accuracy is derived from the phase accuracy, slightly lower frequencies could be used, but they would require a stronger signal to resolve to the same degree.
Likewise, higher frequencies are an available method of linearly improving the time accuracy. Likewise, higher frequencies are an available method of linearly improving the time accuracy.
\\ \\