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Allan deviation and related time/frequency statistics

Project description flake8 Status

A python library for calculating Allan deviation and related time & frequency statistics. LGPL v3+ license.

Input data should be evenly spaced observations of either fractional frequency, or phase in seconds. Deviations are calculated for given tau values in seconds.





Allan deviation

Classic - use only if required - relatively poor confidence.


Overlapping Allan deviation

General purpose - most widely used - first choice


Modified Allan deviation

Used to distinguish between White and Flicker Phase Modulation.


Time deviation

Based on modified Allan variance.


Hadamard deviation

Rejects frequency drift, and handles divergent noise.


Overlapping Hadamard deviation

Better confidence than normal Hadamard.


Parabolic deviation


Total deviation

Better confidence at long averages for Allan deviation.


Modified total deviation

Modified Total deviation. Better confidence at long averages for modified Allan


Time total deviation


Hadamard total deviation


Theo1 deviation

Theo1 is a two-sample variance with improved confidence and extended averaging factor range.


Maximum Time Interval Error


Time Interval Error RMS


Gap resistant overlapping Allan deviation


Groslambert Covariance

Noise generators for creating synthetic datasets are also included:

  • violet noise with f^2 PSD

  • white noise with f^0 PSD

  • pink noise with f^-1 PSD

  • Brownian or random walk noise with f^-2 PSD

More details on available statistics and noise generators : full list of available functions

see /tests for tests that compare allantools output to other (e.g. Stable32) programs. More test data, benchmarks, ipython notebooks, and comparisons to known-good algorithms are welcome!


Install from pypi:

pip install allantools

Latest version + examples, tests, test data, iPython notebooks : clone from github, then install

python install

(see python –help install for install options)

These commands should be run as root for system-wide installation, or you can use the –user option to install for your account only. Exact command names may vary depending on your OS / package manager / target python version.

Basic usage

Minimal example, phase data

We can call allantools with only one parameter - an array of phase data. This is suitable for time-interval measurements at 1 Hz, for example from a time-interval-counter measuring the 1PPS output of two clocks.

>>> import allantools
>>> x = allantools.noise.white(10000)        # Generate some phase data, in seconds.
>>> (taus, adevs, errors, ns) = allantools.oadev(x)

when only one input parameter is given, phase data in seconds is assumed when no rate parameter is given, rate=1.0 is the default when no taus parameter is given, taus=’octave’ is the default

Frequency data example

Note that allantools assumes non-dimensional frequency data input. Normalization, by e.g. dividing all data points with the average frequency, is left to the user.

>>> import allantools
>>> import pylab as plt
>>> import numpy as np
>>> t = np.logspace(0, 3, 50)  # tau values from 1 to 1000
>>> y = allantools.noise.white(10000)  # Generate some frequency data
>>> r = 12.3  # sample rate in Hz of the input data
>>> (t2, ad, ade, adn) = allantools.oadev(y, rate=r, data_type="freq", taus=t)  # Compute the overlapping ADEV
>>> fig = plt.loglog(t2, ad) # Plot the results
>>> #

New in 2016.11 : simple top-level API, using dedicated classes for data handling and plotting.

import allantools #
import numpy as np

# Compute a deviation using the Dataset class
a = allantools.Dataset(data=np.random.rand(1000))

# New in 2019.7 : write results to file

# Plot it using the Plot class
b = allantools.Plot()
# New in 2019.7 : additional keyword arguments are passed to
# matplotlib.pyplot.plot()
b.plot(a, errorbars=True, grid=True)
# You can override defaults before "show" if needed"Tau (s)")

Jupyter notebooks with examples

Jupyter notebooks are interactive python scripts, embedded in a browser, allowing you to manipulate data and display plots like easily. For guidance on installing jupyter, please refer to

See /examples for some examples in notebook format.

github formats the notebooks into nice web-pages, for example


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