[Paper Review] On the measurement of frequency and of its sample variance with high-resolution counters

Enrico Rubiola|HAL (Le Centre pour la Communication Scientifique Directe)|Nov 25, 2004

Advanced Electrical Measurement Techniques4 citations

TL;DR

This paper analyzes the measurement of frequency and its sample variance using high-resolution counters, clarifying how different counter types (Π and Λ) affect Allan variance (AVAR) and modified Allan variance (MVAR) estimation. It demonstrates that overlapping averaging in Λ-type counters naturally yields MVAR for short-term stability, while long-term averaging converges to AVAR, resolving common misinterpretations in frequency stability analysis.

ABSTRACT

A frequency counter measures the input frequency $\barν$ 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 $\smash{σ^2_ν\propto1/τ^2}$ to $\smash{σ^2_ν\propto1/τ^3}$. Surprisingly, when a file of contiguous data is fed into the formula of the two-sample (Allan) variance $\smash{σ^2_y(τ)=\mathbb{E}\{\frac12(\bar{y}_{k+1}-\bar{y}_k) ^2\}}$ of the fractional frequency fluctuation $y$, the result is the \emph{modified} Allan variance mod $σ^2_y(τ)$. 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 (non-modified) Allan variance. Of course interpretation mistakes are around the corner if the counter internal process is not well understood.

Motivation & Objective

• To resolve widespread confusion in interpreting Allan variance (AVAR) and modified Allan variance (MVAR) when using high-resolution frequency counters.
• To clarify the impact of overlapping averaging and counter architecture (Π vs. Λ type) on the effective variance estimation.
• To provide a rigorous framework for correctly interpreting frequency counter outputs in terms of AVAR and MVAR for both short-term and long-term stability analysis.
• To address the critical issue of misinterpretation when feeding counter data into standard variance formulas without understanding the underlying averaging and windowing functions.

Proposed method

• Uses a mathematical model of frequency and phase fluctuations, defining fractional frequency fluctuation $ y(t) = \frac{\dot{\phi}(t)}{2\pi\nu_{00}} $, to analyze stability metrics.
• Analyzes the impulse response of frequency counters, showing that Π-type counters use rectangular averaging windows, while Λ-type counters use triangular (integrated) windows.
• Derives the effective weight functions for both counter types: $ w_{\Pi}(t) $ for Π-type and $ w_{\Lambda}(t) $ for Λ-type, which determine whether AVAR or MVAR is effectively measured.
• Demonstrates that overlapping measurements in Λ-type counters result in a triangular-shaped effective window, leading to MVAR when fed into the AVAR formula.
• Shows that averaging $ m $ contiguous Λ-measurements yields a trapezoidal window that asymptotically approaches a rectangular window, converging to AVAR for large $ m $.
• Uses wavelet-like interpretation of variance estimators, showing AVAR as a half-octave bandpass filter centered at $ 1/(2\tau) $.

Experimental results

Research questions

• RQ1 How do Π-type and Λ-type frequency counters differ in their effective averaging windows and resulting variance estimation?
• RQ2 Why does feeding Λ-counter data into the AVAR formula yield MVAR instead of AVAR?
• RQ3 Under what conditions does overlapping averaging in a Λ-counter converge to AVAR rather than MVAR?
• RQ4 What is the impact of window shape (rectangular vs. triangular) on the estimation of frequency stability variance?
• RQ5 Why is it impossible to convert Λ-counter data to AVAR without increasing the measurement time $ \tau $?

Key findings

• Λ-type counters with overlapping measurements produce a triangular-shaped effective averaging window, which results in the modified Allan variance (MVAR) when the standard AVAR formula is applied.
• For long-term stability analysis ($ m \gg 1 $), the averaging process in Λ-type counters converges to a rectangular window, making the AVAR formula yield the classical Allan variance (AVAR), not MVAR.
• Short-term stability measurements require weighted averaging with coefficients $ \{1,2,\ldots,\lceil m/2 \rceil,\ldots,2,1\} $ to preserve the triangular window and correctly estimate MVAR at $ \tau = m\tau_B $.
• The use of a picket-fence method—absolute timing of zero crossings—provides unambiguous data that can be used to compute AVAR, MVAR, and TotVar without interpretation errors.
• The paper resolves a key ambiguity: feeding raw Λ-counter data into the AVAR formula yields MVAR, not AVAR, due to the underlying triangular window function.
• For white phase noise, the uncertainty improves as $ \sigma^2_\nu \propto 1/\tau^3 $ when using overlapping averaging, surpassing the classical $ 1/\tau^2 $ scaling.