On the measurement of frequency and of its sample variance with high-resolution counters

A frequency counter measures the input frequency nu averaged over a suitable time tau, versus the reference clock. Beside clock interpolation, modern counters improve the resolution by averaging multiple measurements highly overlapped. In the presence of white noise, the overlapping technique improves the square uncertainty from sigma
_nu

²
prop1/tau
²
to sigma
_nu

²
prop1/tau. This is important because the input trigger integrates white noise over the full instrument bandwidth, which is usually of at least 100 MHz. Due to insufficient technical information, the general user is inclined to make the implicit assumption that the counter takes the bare mean. After explaining the overlapped-average mechanism, we prove that feeding a file of contiguous data into the formula of the two-sample (Allan) variance sigma
_y

²
(tau) = E{frac12(ymacr
_k+1
-ymacr
_k
)
²
} gives the modified Allan variance mod sigma
_y

²
(tau). This conclusion is based on the mathematical reverse-engineering of the formulae found in technical specifications. More details are available on the web site arxiv.org, document arXiv:physics/0411227 (Rubiola, 2004), Our purpose is to warn the experimentalists against possible mistakes, and to encourage the manufacturers to explain what the instruments really do