From 05ac28cb148fe97a3336b94b18b8a0f0d5858c70 Mon Sep 17 00:00:00 2001 From: Eric Teunis de Boone Date: Tue, 8 Aug 2023 16:02:45 +0200 Subject: [PATCH] Thesis: Beacon: wrong k in equation --- documents/thesis/chapters/beacon_discipline.tex | 7 ++++--- 1 file changed, 4 insertions(+), 3 deletions(-) diff --git a/documents/thesis/chapters/beacon_discipline.tex b/documents/thesis/chapters/beacon_discipline.tex index 9b72fe7..d26ee05 100644 --- a/documents/thesis/chapters/beacon_discipline.tex +++ b/documents/thesis/chapters/beacon_discipline.tex @@ -369,7 +369,8 @@ changing the synchronisation mismatches in \eqref{eq:synchro_mismatch_clocks} to (\Delta \tClock)_{ij} &\equiv (\tClock)_i - (\tClock)_j \\ &= (\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\\ &\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''. \\ 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. \\ -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. Likewise, higher frequencies are an available method of linearly improving the time accuracy. \\