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

A frequency counter measures the input frequency ν¯ averaged over a suitable time τ, 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 σ_ν²∝1/τ² to σ_ν²∝1/τ³. Surprisingly, when a file of contiguous data is fed into the formula of the two-sample (Allan) variance σ_y²(τ)=E{1/2(^{y¯k +1-y¯k)2}} of the fractional frequency fluctuation y, the result is the modified Allan variance mod σ_y²(τ). But if a sufficient number of contiguous measures are averaged in order to get a longer τ and the data are fed into the same formula, the results is the (nonmodified) Allan variance. Of course interpretation mistakes are around the corner if the counter internal process is not well understood. The typical domain of interest is the the short-term stability measurement of oscillators.