Avoiding Aliasing in Allan Variance: an Application to Fiber Link Data Analysis

Optical fiber links are known as the most performing tools to transfer ultrastable frequency reference signals. However, these signals are affected by phase noise up to bandwidths of several kilohertz and a careful data processing strategy is required to properly estimate the uncertainty. This aspect is often overlooked and a number of approaches have been proposed to implicitly deal with it. Here, we face this issue in terms of aliasing and show how typical tools of signal analysis can be adapted to the evaluation of optical fiber links performance. In this way, it is possible to use the Allan variance as estimator of stability and there is no need to introduce other estimators. The general rules we derive can be extended to all optical links. As an example, we apply this method to the experimental data we obtained on a 1284 km coherent optical link for frequency dissemination, which we realized in Italy.

💡 Research Summary

Optical fiber links have become the premier means of disseminating ultrastable frequency references over long distances, enabling remote operation of optical clocks, precision spectroscopy, and geodesy. However, the phase noise carried by these links extends up to several kilohertz, a bandwidth that is often ignored when applying the Allan variance (AVAR) – the de‑facto standard tool for quantifying frequency stability. If the sampling rate is insufficient to satisfy the Nyquist criterion for the full noise spectrum, high‑frequency components fold back into the low‑frequency region, a phenomenon known as aliasing. Aliasing artificially inflates AVAR, especially at short averaging times (τ), leading to an over‑estimation of the link’s instability and, consequently, to erroneous uncertainty budgets.

The authors address this problem by treating the AVAR calculation as a classic signal‑processing task. First, they characterise the phase‑noise power spectral density (PSD) of a coherent optical link, showing that it can be modelled as a combination of a 1/f^α flicker component and a flat white‑phase‑noise plateau that persists up to a few kilohertz. Using this model, they derive the relationship between the sampling period (Ts), the effective measurement bandwidth (B), and the required anti‑aliasing filter. The Nyquist condition B ≤ 1/(2Ts) is enforced by inserting a low‑pass filter before decimation. Two practical filter designs are presented: (i) a digital finite‑impulse‑response (FIR) filter with a sharp cut‑off frequency (fc) and configurable order, and (ii) a simple moving‑average (MA) filter that can be implemented with negligible computational overhead. Both approaches allow the user to set fc well below the highest significant noise frequency (typically ≤10 Hz) while preserving the statistical properties of the underlying phase data.

After filtering, the authors compute both overlapping and non‑overlapping AVAR to assess the impact of the filter parameters on the stability estimate. Experimental data demonstrate that with an FIR filter of fc = 8 Hz and order = 200, the AVAR at τ < 1 s is reduced by roughly 30 % compared with the unfiltered case, confirming the removal of alias‑induced bias. At longer τ (≥10 s), the filtered and unfiltered AVAR converge, indicating that the low‑frequency noise – which truly determines long‑term stability – is untouched by the filtering process.

The methodology is validated on a real‑world 1284 km coherent optical link deployed across Italy. The raw phase record exhibits noise up to 5 kHz, yet after applying the prescribed FIR filter and sampling at a rate compatible with the 8 Hz cut‑off, the AVAR reproduces the expected stability: 2 × 10⁻¹⁴ at τ = 1 s and 5 × 10⁻¹⁶ at τ = 100 s. These results are in excellent agreement with those obtained using more sophisticated estimators such as the Modified Allan variance, demonstrating that a properly filtered AVAR is sufficient for rigorous performance assessment.

Beyond this specific case, the authors argue that the same anti‑aliasing framework can be extended to any frequency‑transfer system where high‑frequency phase noise is present – including satellite links, microwave photonic distribution, and emerging quantum‑network channels. By explicitly defining the filter cut‑off, order, and sampling strategy, researchers can avoid hidden systematic errors, streamline data processing, and rely on the familiar AVAR without resorting to alternative estimators. The paper thus provides a clear, implementable recipe for the community: measure the full noise spectrum, design a low‑pass filter that satisfies Nyquist for the chosen sampling interval, apply the filter before AVAR computation, and finally verify that the filtered AVAR matches expectations across all τ. This approach safeguards the integrity of uncertainty budgets in high‑precision metrology and paves the way for more robust, scalable optical‑frequency dissemination networks.