Implemented statistics functions

Low-level access to the statistics functions

The deviation functions are generally of the form:

(tau_out, adev, adeverr, n) = allantools.adev(data, rate=1.0, data_type="phase", taus=None)

Inputs:

  • data = list of phase measurements in seconds, or list of fractional frequency measurements (nondimensional)

  • rate = sample rate of data in Hz , i.e. interval between phase measurements is 1/rate seconds.

  • data_type= = either “phase” or “freq”

  • taus = list of tau-values for ADEV computation. The keywords “all”, “octave”, or “decade” can also be used.

Outputs

  • tau_out = list of tau-values for which deviations were computed

  • adev = list of ADEV (or another statistic) deviations

  • adeverr = list of estimated errors of allan deviations. some functions instead return a confidence interval (err_l, err_h)

  • n = list of number of pairs in allan computation. standard error is adeverr = adev/sqrt(n)

Statistics

allantools.adev(data, rate=1.0, data_type='phase', taus=None)
Allan deviation.

Classic - use only if required - relatively poor confidence.

\[\sigma^2_{ADEV}(\tau) = { 1 \over 2 \tau^2 } \langle ( {x}_{n+2} - 2x_{n+1} + x_{n} )^2 \rangle = { 1 \over 2 (N-2) \tau^2 } \sum_{n=1}^{N-2} ( {x}_{n+2} - 2x_{n+1} + x_{n} )^2\]

where \(x_n\) is the time-series of phase observations, spaced by the measurement interval \(\tau\), and with length \(N\).

Or alternatively calculated from a time-series of fractional frequency:

\[\sigma^{2}_{ADEV}(\tau) = { 1 \over 2 } \langle ( \bar{y}_{n+1} - \bar{y}_n )^2 \rangle\]

where \(\bar{y}_n\) is the time-series of fractional frequency at averaging time \(\tau\)

Parameters

data: np.array

Input data. Provide either phase or frequency (fractional, adimensional).

rate: float

The sampling rate for data, in Hz. Defaults to 1.0

data_type: {‘phase’, ‘freq’}

Data type, i.e. phase or frequency. Defaults to “phase”.

taus: np.array

Array of tau values, in seconds, for which to compute statistic. Optionally set taus=[“all”|”octave”|”decade”] for automatic tau-list generation.

Returns

(taus2, ad, ade, ns): tuple

Tuple of values

taus2: np.array

Tau values for which td computed

ad: np.array

Computed adev for each tau value

ade: np.array

adev errors

ns: np.array

Values of N used in each adev calculation

References

allantools.oadev(data, rate=1.0, data_type='phase', taus=None)

Overlapping Allan deviation. General purpose - most widely used - first choice.

\[\sigma^2_{OADEV}(m\tau_0) = { 1 \over 2 (m \tau_0 )^2 (N-2m) } \sum_{n=1}^{N-2m} ( {x}_{n+2m} - 2x_{n+1m} + x_{n} )^2\]

where \(\sigma_{OADEV}(m\tau_0)\) is the overlapping Allan deviation at an averaging time of \(\tau=m\tau_0\), and \(x_n\) is the time-series of phase observations, spaced by the measurement interval \(\tau_0\), with length \(N\).

Parameters

data: np.array

Input data. Provide either phase or frequency (fractional, adimensional).

rate: float

The sampling rate for data, in Hz. Defaults to 1.0

data_type: {‘phase’, ‘freq’}

Data type, i.e. phase or frequency. Defaults to “phase”.

taus: np.array

Array of tau values, in seconds, for which to compute statistic. Optionally set taus=[“all”|”octave”|”decade”] for automatic tau-list generation.

Returns

(taus2, ad, ade, ns): tuple

Tuple of values

taus2: np.array

Tau values for which td computed

ad: np.array

Computed oadev for each tau value

ade: np.array

oadev errors

ns: np.array

Values of N used in each oadev calculation

References

allantools.mdev(data, rate=1.0, data_type='phase', taus=None)

Modified Allan deviation. Used to distinguish between White and Flicker Phase Modulation.

\[\sigma^2_{MDEV}(m\tau_0) = { 1 \over 2 (m \tau_0 )^2 (N-3m+1) } \sum_{j=1}^{N-3m+1} \left[ \sum_{i=j}^{j+m-1} {x}_{i+2m} - 2x_{i+m} + x_{i} \right]^2\]

Parameters

data: np.array

Input data. Provide either phase or frequency (fractional, adimensional).

rate: float

The sampling rate for data, in Hz. Defaults to 1.0

data_type: {‘phase’, ‘freq’}

Data type, i.e. phase or frequency. Defaults to “phase”.

taus: np.array

Array of tau values, in seconds, for which to compute statistic. Optionally set taus=[“all”|”octave”|”decade”] for automatic tau-list generation.

Returns

(taus2, md, mde, ns): tuple

Tuple of values

taus2: np.array

Tau values for which td computed

md: np.array

Computed mdev for each tau value

mde: np.array

mdev errors

ns: np.array

Values of N used in each mdev calculation

References

allantools.pdev(data, rate=1.0, data_type='phase', taus=None)

Parabolic deviation.

Use for evaluating uncertainty of omega-average of frequency.

\[\sigma^2_{PDEV}(m\tau_0) = { 72 \over (N-2m) (m \tau_0 )^2 } \sum_{i=0}^{N-2m-1} \left[ \sum_{k=0}^{m-1} \left( { m-1 \over 2} - k \right) {x}_{i+k} - x_{i+k+m} \right]^2\]

for \(m>1\) and for an averaging-factor of \(m=1\) PDEV equals ADEV/MDEV: \(\sigma_{PDEV}(\tau_0)=\sigma_{ADEV}(\tau_0)\).

Parameters

data: np.array

Input data. Provide either phase or frequency (fractional, adimensional).

rate: float

The sampling rate for data, in Hz. Defaults to 1.0

data_type: {‘phase’, ‘freq’}

Data type, i.e. phase or frequency. Defaults to “phase”.

taus: np.array

Array of tau values, in seconds, for which to compute statistic. Optionally set taus=[“all”|”octave”|”decade”] for automatic tau-list generation.

Returns

(taus2, ad, ade, ns): tuple

Tuple of values

taus2: np.array

Tau values for which td computed

ad: np.array

Computed adev for each tau value

ade: np.array

adev errors

ns: np.array

Values of N used in each adev calculation

References

allantools.hdev(data, rate=1.0, data_type='phase', taus=None)
Hadamard deviation.

Rejects frequency drift, and handles divergent noise.

\[\sigma^2_{HDEV}( \tau ) = { 1 \over 6 \tau^2 (N-3) } \sum_{i=1}^{N-3} ( {x}_{i+3} - 3x_{i+2} + 3x_{i+1} - x_{i} )^2\]

where \(x_i\) is the time-series of phase observations, spaced by the measurement interval \(\tau\), and with length \(N\).

Parameters

datanp.array

Input data. Provide either phase or frequency (fractional, adimensional).

ratefloat

The sampling rate for data, in Hz. Defaults to 1.0

data_typestring, {‘phase’, ‘freq’}

Data type, i.e. phase or frequency. Defaults to “phase”.

tausnp.array

Array of tau values, in seconds, for which to compute statistic. Optionally set taus=[“all”|”octave”|”decade”] for automatic tau-list generation.

References

  • NIST [SP1065] eqn (17) and (18), page 20

allantools.ohdev(data, rate=1.0, data_type='phase', taus=None)
Overlapping Hadamard deviation.

Better confidence than normal Hadamard.

\[\sigma^2_{OHDEV}(m\tau_0) = { 1 \over 6 (m \tau_0 )^2 (N-3m) } \sum_{i=1}^{N-3m} ( {x}_{i+3m} - 3x_{i+2m} + 3x_{i+m} - x_{i} )^2\]

where \(x_i\) is the time-series of phase observations, spaced by the measurement interval \(\tau_0\), and with length \(N\).

Parameters

data: np.array

Input data. Provide either phase or frequency (fractional, adimensional).

rate: float

The sampling rate for data, in Hz. Defaults to 1.0

data_type: {‘phase’, ‘freq’}

Data type, i.e. phase or frequency. Defaults to “phase”.

taus: np.array

Array of tau values, in seconds, for which to compute statistic. Optionally set taus=[“all”|”octave”|”decade”] for automatic tau-list generation.

Returns

(taus2, hd, hde, ns): tuple

Tuple of values

taus2: np.array

Tau values for which td computed

hd: np.array

Computed hdev for each tau value

hde: np.array

hdev errors

ns: np.array

Values of N used in each hdev calculation

References

allantools.tdev(data, rate=1.0, data_type='phase', taus=None)

Time deviation.

Based on modified Allan variance.

\[\sigma^2_{TDEV}( \tau ) = { \tau^2 \over 3 } \sigma^2_{MDEV}( \tau )\]

Note that TDEV has a unit of seconds.

Parameters

data: np.array

Input data. Provide either phase or frequency (fractional, adimensional).

rate: float

The sampling rate for data, in Hz. Defaults to 1.0

data_type: {‘phase’, ‘freq’}

Data type, i.e. phase or frequency. Defaults to “phase”.

taus: np.array

Array of tau values, in seconds, for which to compute statistic. Optionally set taus=[“all”|”octave”|”decade”] for automatic tau-list generation.

Returns

(taus, tdev, tdev_error, ns): tuple

Tuple of values

taus: np.array

Tau values for which td computed

tdev: np.array

Computed time deviations (in seconds) for each tau value

tdev_errors: np.array

Time deviation errors

ns: np.array

Values of N used in mdev_phase()

References

allantools.totdev(data, rate=1.0, data_type='phase', taus=None)
Total deviation.

Better confidence at long averages for Allan deviation.

\[\sigma^2_{TOTDEV}( m\tau_0 ) = { 1 \over 2 (m\tau_0)^2 (N-2) } \sum_{i=2}^{N-1} ( {x}^*_{i-m} - 2x^*_{i} + x^*_{i+m} )^2\]

Where \(x^*_i\) is a new time-series of length \(3N-4\) derived from the original phase time-series \(x_n\) of length \(N\) by reflection at both ends. The original data \(x_n\) is in the center of \(x^*\):

\[ \begin{align}\begin{aligned}x^*_{1-j} = 2x_1 - x_{1+j} \quad \text{for} j=1..N-2\\x^*_i = x_i \quad \text{for} i=1..N\\x^*_{N+j} = 2x_N - x_{N-j} \quad \text{for} j=1..N-2\end{aligned}\end{align} \]

FIXME: bias correction http://www.wriley.com/CI2.pdf page 5

Parameters

phase: np.array

Phase data in seconds. Provide either phase or frequency.

frequency: np.array

Fractional frequency data (nondimensional). Provide either frequency or phase.

rate: float

The sampling rate for phase or frequency, in Hz

taus: np.array

Array of tau values for which to compute measurement

References

allantools.mtotdev(data, rate=1.0, data_type='phase', taus=None)

Modified Total deviation.

Better confidence at long averages for modified Allan

FIXME: bias-correction http://www.wriley.com/CI2.pdf page 6

The variance is scaled up (divided by this number) based on the noise-type identified.

noise type

bias correction

WPM

0.94

FPM

0.83

WFM

0.73

FFM

0.70

RWFM

0.69

Parameters

data: np.array

Input data. Provide either phase or frequency (fractional, adimensional).

rate: float

The sampling rate for data, in Hz. Defaults to 1.0

data_type: {‘phase’, ‘freq’}

Data type, i.e. phase or frequency. Defaults to “phase”.

taus: np.array

Array of tau values, in seconds, for which to compute statistic. Optionally set taus=[“all”|”octave”|”decade”] for automatic tau-list generation.

References

allantools.ttotdev(data, rate=1.0, data_type='phase', taus=None)

Time Total Deviation

Modified total variance scaled by \(\tau^2 / 3\)

\[\sigma^2_{TTOTDEV}( \tau ) = { \tau^2 \over 3 } \sigma^2_{MTOTDEV}( \tau )\]

Note that [SP1065] erroneously has tau-cubed here (!).

References

allantools.htotdev(data, rate=1.0, data_type='phase', taus=None)

Hadamard Total deviation.

Better confidence at long averages for Hadamard deviation

PRELIMINARY - REQUIRES FURTHER TESTING.

Computed for N fractional frequency points y_i with sampling period tau0, analyzed at tau = m*tau0 1. remove linear trend by averaging first and last half, and dividing by interval 2. extend sequence by uninverted even reflection 3. compute Hadamard for extended, length 9m, sequence.

FIXME: bias corrections from http://www.wriley.com/CI2.pdf W FM 0.995 alpha= 0 F FM 0.851 alpha=-1 RW FM 0.771 alpha=-2 FW FM 0.717 alpha=-3 RR FM 0.679 alpha=-4

Parameters

data: np.array

Input data. Provide either phase or frequency (fractional, adimensional).

rate: float

The sampling rate for data, in Hz. Defaults to 1.0

data_type: {‘phase’, ‘freq’}

Data type, i.e. phase or frequency. Defaults to “phase”.

taus: np.array

Array of tau values, in seconds, for which to compute statistic. Optionally set taus=[“all”|”octave”|”decade”] for automatic tau-list generation.

allantools.theo1(data, rate=1.0, data_type='phase', taus=None)

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

PRELIMINARY - REQUIRES FURTHER TESTING.

\[\sigma^2_{THEO1}(m\tau_0) = { 1 \over (m \tau_0 )^2 (N-m) } \sum_{i=1}^{N-m} \sum_{\delta=0}^{m/2-1} {1\over m/2-\delta}\lbrace ({x}_{i} - x_{i-\delta +m/2}) + (x_{i+m}- x_{i+\delta +m/2}) \rbrace^2\]

Where \(10<=m<=N-1\) is even.

FIXME: bias correction

Parameters

data: np.array

Input data. Provide either phase or frequency (fractional, adimensional).

rate: float

The sampling rate for data, in Hz. Defaults to 1.0

data_type: {‘phase’, ‘freq’}

Data type, i.e. phase or frequency. Defaults to “phase”.

taus: np.array

Array of tau values, in seconds, for which to compute statistic. Optionally set taus=[“all”|”octave”|”decade”] for automatic tau-list generation.

References

allantools.mtie(data, rate=1.0, data_type='phase', taus=None)

Maximum Time Interval Error.

Parameters

data: np.array

Input data. Provide either phase or frequency (fractional, adimensional).

rate: float

The sampling rate for data, in Hz. Defaults to 1.0

data_type: {‘phase’, ‘freq’}

Data type, i.e. phase or frequency. Defaults to “phase”.

taus: np.array

Array of tau values, in seconds, for which to compute statistic. Optionally set taus=[“all”|”octave”|”decade”] for automatic tau-list generation.

Notes

this seems to correspond to Stable32 setting “Fast(u)” Stable32 also has “Decade” and “Octave” modes where the dataset is extended somehow?

allantools.tierms(data, rate=1.0, data_type='phase', taus=None)

Time Interval Error RMS.

Parameters

data: np.array

Input data. Provide either phase or frequency (fractional, adimensional).

rate: float

The sampling rate for data, in Hz. Defaults to 1.0

data_type: {‘phase’, ‘freq’}

Data type, i.e. phase or frequency. Defaults to “phase”.

taus: np.array

Array of tau values, in seconds, for which to compute statistic. Optionally set taus=[“all”|”octave”|”decade”] for automatic tau-list generation.

allantools.gcodev(data_1, data_2, rate=1.0, data_type='phase', taus=None)

Groslambert codeviation a.k.a. Allan Covariance

Similarly to the three-cornered hat method, we consider three uncorrelated oscillators A, B, C. The Groslambert codeviation estimates the noise of one oscillator (e.g. B), given two synchronous measurements AB and BC. Unlike three-cordenred hat, Gcodev is not affected by the (uncorrelated) noise of the measurement devices (time-interval or frequency counter) used for the measurements AB and BC.

Parameters

data_1: np.array

Oscillator 1 input data. Provide either phase or frequency

data_2: np.array

Oscillator 2 input data. Provide either phase or frequency

rate: float

The sampling rate for data, in Hz. Defaults to 1.0

data_type: {‘phase’, ‘freq’}

Data type, i.e. phase or frequency. Defaults to “phase”.

taus: np.array

Array of tau values, in seconds, for which to compute statistic. Optionally set taus=[“all”|”octave”|”decade”] for automatic tau-list generation.

Returns

(taus, gd): tuple

Tuple of values

taus: np.array

Tau values for which gcodev computed

gd: np.array

Computed gcodev for each tau value

References

Real-Time Statistics

class allantools.oadev_realtime(afs=[1], tau0=1.0, auto_afs=False, pts_per_decade=4)

Overlapping Allan deviation in real-time from a stream of phase/frequency samples.

Reference [Dobrogowski2007]

add_phase(xnew)

add new phase point, in units of seconds

update_S(idx)

update S, sum-of-squares

class allantools.tdev_realtime(afs=[1], tau0=1.0, auto_afs=False, pts_per_decade=4)

Time deviation and Modified Allan deviation in real-time from a stream of phase/frequency samples.

Reference [Dobrogowski2007]

add_phase(xnew)

add new phase point

mdev()

scale tdev to output mdev

update_S(idx)

update S, sum-of-squares

update_S3n(idx)

eqn (13) of paper

class allantools.ohdev_realtime(afs=[1], tau0=1.0, auto_afs=False, pts_per_decade=4)

Overlapping Hadamard deviation in real-time from a stream of phase/frequency samples.

Reference [Dobrogowski2007]

add_phase(xnew)

add new phase point

update_S(idx)

update S, sum-of-squares

Noise Generation

class allantools.noise_kasdin.Noise(nr=2, qd=1, b=0)

Generate discrete colored noise

Python / Numpy implementation of [Kasdin1992] Kasdin, N.J., Walter, T., “Discrete simulation of power law noise [for oscillator stability evaluation],” Frequency Control Symposium, 1992. 46th., Proceedings of the 1992 IEEE, pp.274,283, 27-29 May 1992 http://dx.doi.org/10.1109/FREQ.1992.270003

Parameters

nr: integer

length of generated time-series must be power of two

qd: float

discrete variance

b: float

noise type

b

noise type

0

White Phase Modulation (WPM)

-1

Flicker Phase Modulation (FPM)

-2

White Frequency Modulation (WFM)

-3

Flicker Frequency Modulation (FFM)

-4

Random Walk Frequency Modulation (RWFM)

Returns

Noise()

A Noise() instance

Example:
import numpy as np
noise = allantools.Noise(nr=2*8, qd=1.0e-20, b=-1)
noise.generateNoise()
print noise.time_series
adev(tau0, tau)

return predicted ADEV of noise-type at given tau

adev_from_qd(tau0=1.0, tau=1.0)

prefactor for Allan deviation for noise type defined by (qd, b, tau0)

Colored noise generated with (qd, b, tau0) parameters will show an Allan variance of:

\[AVAR = prefactor \cdot h_a \cdot \tau^c\]

where \(a = b + 2\) is the slope of the frequency PSD. and \(h_a\) is the frequency PSD prefactor \(S_y(f) = h_a f^a\)

The relation between a, b, c is:

a

b

c(AVAR)

c(MVAR)

-2

-4

1

1

-1

-3

0

0

0

-2

-1

-1

+1

-1

-2

-2

+2

0

-2

-3

Coefficients from [Dawkins2007].

Vernotte2015 Table I

c_avar()

return tau exponent “c” for noise type. AVAR = prefactor * h_a * tau^c

c_mvar()

return tau exponent “c” for noise type. MVAR = prefactor * h_a * tau^c

frequency_psd_from_qd(tau0=1.0)

return frequency power spectral density coefficient \(h_a\) for the noise type defined by (qd, b, tau0)

Colored noise generated with (qd, b, tau0) parameters will show a frequency power spectral density of

\[S_y(f) = Frequency_{PSD}(f) = h_a f^a\]

where the slope \(a\) comes from the phase PSD slope \(b\): \(a = b + 2\)

[Kasdin1992] eqn (39)

generateNoise()

Generate noise time series based on input parameters

Returns

time_series: np.array

Time series with colored noise. len(time_series) == nr

mdev(tau0, tau)

return predicted MDEV of noise-type at given tau

mdev_from_qd(tau0=1.0, tau=1.0)

prefactor for Modified Allan deviation for noise type defined by (qd, b, tau0)

Colored noise generated with (qd, b, tau0) parameters will show an Modified Allan variance of:

\[MVAR = prefactor \cdot h_a \cdot \tau^c\]

where \(a = b + 2\) is the slope of the frequency PSD. and \(h_a\) is the frequency PSD prefactor \(S_y(f) = h_a f^a\)

pdev_from_qd(tau0=1.0, tau=1.0)

prefactor for Parabolic Allan deviation for noise type defined by (qd, b, tau0)

Colored noise generated with (qd, b, tau0) parameters will show an Parabolic Allan variance of:

\[PVAR = prefactor \cdot h_a \cdot \tau^c\]

where \(a = b + 2\) is the slope of the frequency PSD. and \(h_a\) is the frequency PSD prefactor \(S_y(f) = h_a f^a\)

phase_psd_from_qd(tau0=1.0)

return phase power spectral density coefficient \(g_b\) for noise-type defined by (qd, b, tau0) where tau0 is the interval between data points

Colored noise generated with (qd, b, tau0) parameters will show a phase power spectral density of \(S_x(f) = Phase_{PSD}(f) = g_b f^b\)

[Kasdin1992] eqn (39)

set_input(nr=2, qd=1, b=0)

Set inputs after initialization

Parameters

nr: integer

length of generated time-series number must be power of two

qd: float

discrete variance

b: float

noise type

b

noise type

0

White Phase Modulation (WPM)

-1

Flicker Phase Modulation (FPM)

-2

White Frequency Modulation (WFM)

-3

Flicker Frequency Modulation (FFM)

-4

Random Walk Frequency Modulation (RWFM)
allantools.noise.white(num_points=1024, b0=1.0, fs=1.0)

White noise generator

Generate time series with white noise that has constant PSD = b0, up to the nyquist frequency fs/2.

The PSD is at ‘height’ b0 and extends from 0 Hz up to the nyquist frequency fs/2 (prefactor math.sqrt(b0*fs/2.0))

Parameters

num_points: int, optional

number of samples

b0: float, optional

desired power-spectral density in [X^2/Hz] where X is the unit of x

fs: float, optional

sampling frequency, i.e. 1/fs is the time-interval between datapoints

Returns

White noise sample: numpy.array

allantools.noise.brown(num_points=1024, b_minus2=1.0, fs=1.0)

Brownian or random walk (diffusion) noise with 1/f^2 PSD

Not really a color… rather Brownian or random-walk. Obtained by integrating white-noise.

Parameters

num_points: int, optional

number of samples

b_minus2: float, optional

desired power-spectral density is b2*f^-2

fs: float, optional

sampling frequency, i.e. 1/fs is the time-interval between datapoints

Returns

Random walk sample: numpy.array

allantools.noise.violet(num_points=1024, b2=1, fs=1)

Violet noise with f^2 PSD

Obtained by differentiating white noise

Parameters

num_points: int, optional

number of samples

b2: float, optional

desired power-spectral density is b2*f^2

fs: float, optional

sampling frequency, i.e. 1/fs is the time-interval between datapoints

Returns

Violet noise sample: numpy.array

allantools.noise.pink(num_points=1024, depth=80)

Pink noise (approximation) with 1/f PSD

Fills a sample with results from a pink noise generator from http://pydoc.net/Python/lmj.sound/0.1.1/lmj.sound.noise/, based on the Voss-McCartney algorithm, discussion and code examples at http://www.firstpr.com.au/dsp/pink-noise/

Parameters

num_points: int, optional

number of samples

depth: int, optional

number of iteration for each point. High numbers are slower but generates a more correct spectrum on low-frequencies end.

Returns

Pink noise sample: numpy.array

Utilities

allantools.frequency2phase(freqdata, rate)

integrate fractional frequency data and output phase data

Parameters

freqdata: np.array

Data array of fractional frequency measurements (nondimensional)

rate: float

The sampling rate for phase or frequency, in Hz

Returns

phasedata: np.array

Time integral of fractional frequency data, i.e. phase (time) data in units of seconds. For phase in units of radians, see phase2radians()

allantools.phase2frequency(phase, rate)

Convert phase in seconds to fractional frequency

Parameters

phase: np.array

Data array of phase in seconds, length N

rate: float

The sampling rate for phase, in Hz

Returns

y: np.array

Data array of fractional frequency, length N-1

allantools.phase2radians(phasedata, v0)

Convert phase in seconds to phase in radians

Parameters

phasedata: np.array

Data array of phase in seconds

v0: float

Nominal oscillator frequency in Hz

Returns

fi: np.array

phase data in radians

allantools.psd2allan(S_y, f=1.0, kind='adev', base=2)
Convert a given (one-sided) power spectral density \(S_y(f)\) to Allan

deviation or modified Allan deviation

For ergodic noise, the Allan variance or modified Allan variance is related to the power spectral density \(S_y\) of the fractional frequency deviation:

\[\sigma^2_y(\tau) = 2 \int_0^\infty S_y(f) \left| \sin(\pi f \tau)^{(k+1)} \over (\pi f \tau)^k \right|^2 df,\]

where \(f\) is the Fourier frequency and \(\tau\) the averaging time. The exponent \(k\) is 1 for the Allan variance and 2 for the modified Allan variance.

psd2allan() implements the integral by discrete numerical integration via a sum.

Parameters

S_y: np.array

Single-sided power spectral density (PSD) of fractional frequency deviation S_y in 1/Hz^2. First element is S_y(f=0).

f: np.array or scalar numeric (float or int)

if np.array: Spectral frequency vector in Hz if numeric scalar: Spectral frequency step in Hz default: Spectral frequency step 1 Hz

kind: {‘adev’, ‘mdev’}

Which kind of Allan deviation to compute. Defaults to ‘adev’

base: float

Base for logarithmic spacing of tau values. E.g. base= 10: decade, base= 2: octave, base <= 1: all

Returns

(taus_used, ad): tuple

Tuple of 2 values

taus_used: np.array

tau values for which ad computed

ad: np.array

Computed Allan deviation of requested kind for each tau value

References

allantools.tau_generator(data, rate, taus=None, v=False, even=False, maximum_m=-1)

pre-processing of the tau-list given by the user (Helper function)

Does sanity checks, sorts data, removes duplicates and invalid values. Generates a tau-list based on keywords ‘all’, ‘decade’, ‘octave’. Uses ‘octave’ by default if no taus= argument is given.

Parameters

data: np.array

data array

rate: float

Sample rate of data in Hz. Time interval between measurements is 1/rate seconds.

taus: np.array

Array of tau values for which to compute measurement. Alternatively one of the keywords: “all”, “octave”, “decade”. Defaults to “octave” if omitted.

keyword

averaging-factors

“all”

1, 2, 3, 4, …, len(data)

“octave”

1, 2, 4, 8, 16, 32, …

“decade”

1, 2, 4, 10, 20, 40, 100, …

“log10”

approx. 10 points per decade
v: bool

verbose output if True

even: bool

require even m, where tau=m*tau0, for Theo1 statistic

maximum_m: int

limit m, where tau=m*tau0, to this value. used by mtotdev() and htotdev() to limit maximum tau.

Returns

(data, m, taus): tuple

List of computed values

data: np.array

Data

m: np.array

Tau in units of data points

taus: np.array

Cleaned up list of tau values

allantools.edf_simple(N, m, alpha)

Equivalent degrees of freedom. Simple approximate formulae.

Parameters

Nint

the number of phase samples

mint

averaging factor, tau = m * tau0

alpha: int

exponent of f for the frequency PSD: ‘wp’ returns white phase noise. alpha=+2 ‘wf’ returns white frequency noise. alpha= 0 ‘fp’ returns flicker phase noise. alpha=+1 ‘ff’ returns flicker frequency noise. alpha=-1 ‘rf’ returns random walk frequency noise. alpha=-2 If the input is not recognized, it defaults to idealized, uncorrelated noise with (N-1) degrees of freedom.

Notes

Returns

edffloat

Equivalent degrees of freedom

allantools.edf_greenhall(alpha, d, m, N, overlapping=False, modified=False, verbose=False)

returns Equivalent degrees of freedom

Parameters

alpha: int

noise type, +2…-4

d: int

1 first-difference variance 2 Allan variance 3 Hadamard variance require alpha+2*d>1

m: int

averaging factor tau = m*tau0 = m*(1/rate)

N: int

number of phase observations (length of time-series)

overlapping: bool

True for oadev, ohdev

modified: bool

True for mdev, tdev

Returns

edf: float

Equivalent degrees of freedom

Notes

Reference [Greenhall2004].

Used for the following deviations (see [Riley_CI] page 8) adev() oadev() mdev() tdev() hdev() ohdev()

allantools.edf_totdev(N, m, alpha)

Equivalent degrees of freedom for Total Deviation FIXME: what is the right behavior for alpha outside 0,-1,-2?

NIST [SP1065] page 41, Table 7

allantools.edf_mtotdev(N, m, alpha)

Equivalent degrees of freedom for Modified Total Deviation

NIST [SP1065] page 41, Table 8

allantools.three_cornered_hat_phase(phasedata_ab, phasedata_bc, phasedata_ca, rate, taus, function)

Three Cornered Hat Method

Given three clocks A, B, C, we seek to find their variances \(\sigma^2_A\), \(\sigma^2_B\), \(\sigma^2_C\). We measure three phase differences, assuming no correlation between the clocks, the measurements have variances:

\[ \begin{align}\begin{aligned}\sigma^2_{AB} = \sigma^2_{A} + \sigma^2_{B}\\\sigma^2_{BC} = \sigma^2_{B} + \sigma^2_{C}\\\sigma^2_{CA} = \sigma^2_{C} + \sigma^2_{A}\end{aligned}\end{align} \]

Which allows solving for the variance of one clock as:

\[\sigma^2_{A} = {1 \over 2} ( \sigma^2_{AB} + \sigma^2_{CA} - \sigma^2_{BC} )\]

and similarly cyclic permutations for \(\sigma^2_B\) and \(\sigma^2_C\)

Parameters

phasedata_ab: np.array

phase measurements between clock A and B, in seconds

phasedata_bc: np.array

phase measurements between clock B and C, in seconds

phasedata_ca: np.array

phase measurements between clock C and A, in seconds

rate: float

The sampling rate for phase, in Hz

taus: np.array

The tau values for deviations, in seconds

function: allantools deviation function

The type of statistic to compute, e.g. allantools.oadev

Returns

tau_ab: np.array

Tau values corresponding to output deviations

dev_a: np.array

List of computed values for clock A

References