Allan Deviation (ADEV)

Author: Tianjiang Says

Reference: Li Y et al. 2026 Chin. Phys. Lett. 43 031101

Website: https://cislunarspace.cn

Definition

Allan Deviation (ADEV) is a statistical measure for evaluating frequency source stability, proposed by David W. Allan in 1966. Unlike traditional standard deviation, ADEV can distinguish different types of noise processes (such as white noise, flicker noise, random walk noise, etc.) and avoids divergence issues when noise is non-stationary.

ADEV is the core metric for evaluating atomic clock and oscillator performance in time-frequency metrology.

Mathematical Definition

For adjacent frequency measurements yˉi\bar{y}_i at sampling interval τ\tau, ADEV is defined as:

σy2(τ)=12(N−1)∑i=1N−1(yˉi+1−yˉi)2\sigma_y^2(\tau) = \frac{1}{2(N-1)} \sum_{i=1}^{N-1} (\bar{y}_{i+1} - \bar{y}_i)^2

Where NN is the number of samples and yˉi\bar{y}_i is the average relative frequency offset during the ii-th averaging time τ\tau.

The corresponding Allan deviation is:

ADEV=σy(τ)=σy2(τ)\text{ADEV} = \sigma_y(\tau) = \sqrt{\sigma_y^2(\tau)}

Difference from Standard Deviation

Traditional standard deviation (StdDev) has limitations when evaluating frequency stability: when noise type is flicker frequency noise, standard deviation diverges with increasing sample size—meaning infinite samples still cannot achieve convergence.

ADEV advantages:

  • Converges for multiple noise types
  • Can identify noise power law types
  • Has clear correspondence with physical processes
Noise TypeStdDev BehaviorADEV Behavior
White phase noiseDivergesConverges ∝τ−1\propto \tau^{-1}
Flicker phase noiseDivergesConverges ∝τ0\propto \tau^{0}
White frequency noiseConvergesConverges ∝τ−1/2\propto \tau^{-1/2}
Flicker frequency noiseDivergesConverges ∝τ0\propto \tau^{0}
Random walk frequency noiseConvergesConverges ∝τ1/2\propto \tau^{1/2}

Modified Allan Deviation (MDEV)

The Modified Allan Deviation (MDEV) used in DRO-A gravitational redshift experiments:

MDEV=12(N−2τ0)∑i=1N−2(1τ2∫τ0τ0+τ∫τ0τ0+τx˙(t2)−x˙(t1) dt1 dt2)2\text{MDEV} = \sqrt{\frac{1}{2(N-2\tau_0)} \sum_{i=1}^{N-2} \left( \frac{1}{\tau^2} \int_{\tau_0}^{\tau_0+\tau} \int_{\tau_0}^{\tau_0+\tau} \dot{x}(t_2) - \dot{x}(t_1) \, dt_1 \, dt_2 \right)^2}

MDEV advantages over ADEV:

  • Better confidence for the same noise type
  • Can distinguish white frequency noise from flicker frequency noise

Typical Performance Specifications

Typical ADEV values for different atomic clock types:

Oscillator Typeτ=1\tau = 1 sτ=1000\tau = 1000 sτ=10000\tau = 10000 s
Passive Hydrogen Maser (PHM)10−1210^{-12}10−1410^{-14}10−1410^{-14}
Cesium Beam Tube10−1110^{-11}10−1310^{-13}10−1310^{-13}
Rubidium10−1110^{-11}10−1210^{-12}10−1210^{-12}
Strontium Optical Lattice Clock10−1610^{-16}10−1810^{-18}10−1810^{-18}

Application in DRO-A Experiment

The DRO-A satellite gravitational redshift experiment measured satellite-ground time-frequency comparison stability:

Averaging TimeApril 28 MDEVApril 29 MDEV
10 s6.14×10−136.14 \times 10^{-13}7.01×10−137.01 \times 10^{-13}
100 s8.03×10−138.03 \times 10^{-13}8.03×10−138.03 \times 10^{-13}
1000 s4.58×10−144.58 \times 10^{-14}6.98×10−146.98 \times 10^{-14}
2000 s1.27×10−141.27 \times 10^{-14}2.10×10−142.10 \times 10^{-14}

Key findings:

  • Stability at 1000s averaging exceeds 7×10−147 \times 10^{-14}
  • Stability at 2000s averaging exceeds 2×10−142 \times 10^{-14}
  • Stability outperforms accuracy by two orders of magnitude, meaning stability is the primary limiting factor for gravitational redshift measurements

References

  • Allan D W 1966 Proc. IEEE 54 221
  • Li Y, Liu T et al. 2026 Chin. Phys. Lett. 43 031101