dorsal/arxiv
View SchemaOn the measurement of frequency and of its sample variance with high-resolution counters
| Authors | Enrico Rubiola |
|---|---|
| Categories | |
| ArXiv ID | physics/0411227 |
| URL | https://arxiv.org/abs/physics/0411227 |
| DOI | 10.1063/1.1898203 |
| Journal | Review of Scientific Instruments vol. 76 no. 5 art. no. 054703, may 2005 |
Abstract
A frequency counter measures the input frequency $\bar{\nu}$ averaged over a suitable time $\tau$, versus the reference clock. High resolution is achieved by interpolating the clock signal. Further increased resolution is obtained by averaging multiple frequency measurements highly overlapped. In the presence of additive white noise or white phase noise, the square uncertainty improves from $\smash{\sigma^2_\nu\propto1/\tau^2}$ to $\smash{\sigma^2_\nu\propto1/\tau^3}$. Surprisingly, when a file of contiguous data is fed into the formula of the two-sample (Allan) variance $\smash{\sigma^2_y(\tau)=\mathbb{E}\{\frac12(\bar{y}_{k+1}-\bar{y}_k) ^2\}}$ of the fractional frequency fluctuation $y$, the result is the \emph{modified} Allan variance mod $\sigma^2_y(\tau)$. But if a sufficient number of contiguous measures are averaged in order to get a longer $\tau$ and the data are fed into the same formula, the results is the (non-modified) Allan variance. Of course interpretation mistakes are around the corner if the counter internal process is not well understood.
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"abstract": "A frequency counter measures the input frequency $\\bar{\\nu}$ averaged over a\nsuitable time $\\tau$, versus the reference clock. High resolution is achieved\nby interpolating the clock signal. Further increased resolution is obtained by\naveraging multiple frequency measurements highly overlapped. In the presence of\nadditive white noise or white phase noise, the square uncertainty improves from\n$\\smash{\\sigma^2_\\nu\\propto1/\\tau^2}$ to $\\smash{\\sigma^2_\\nu\\propto1/\\tau^3}$.\nSurprisingly, when a file of contiguous data is fed into the formula of the\ntwo-sample (Allan) variance\n$\\smash{\\sigma^2_y(\\tau)=\\mathbb{E}\\{\\frac12(\\bar{y}_{k+1}-\\bar{y}_k) ^2\\}}$ of\nthe fractional frequency fluctuation $y$, the result is the \\emph{modified}\nAllan variance mod $\\sigma^2_y(\\tau)$. But if a sufficient number of contiguous\nmeasures are averaged in order to get a longer $\\tau$ and the data are fed into\nthe same formula, the results is the (non-modified) Allan variance. Of course\ninterpretation mistakes are around the corner if the counter internal process\nis not well understood.",
"arxiv_id": "physics/0411227",
"authors": [
"Enrico Rubiola"
],
"categories": [
"physics.ins-det"
],
"doi": "10.1063/1.1898203",
"journal_ref": "Review of Scientific Instruments vol. 76 no. 5 art. no. 054703,\n may 2005",
"title": "On the measurement of frequency and of its sample variance with high-resolution counters",
"url": "https://arxiv.org/abs/physics/0411227"
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