Responses and Degrees of Freedom of PVAR for a Continuous Power-Law PSD

05/27/2020 ∙ by François Vernotte, et al. ∙ 0

This paper is devoted to the use of the Parabolic Variance (PVAR) to characterize continuous power-law noises. First, from a theoretical calculation, we give the responses of PVAR extended to continuous power-law noises. Second, from an empirical study, we provide an approximated but relevant expression of the uncertainties of PVAR estimates. This simple expression easily applies to continuous power-law noises.

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I Introduction

We generally assume that the FM noises affecting an oscillator are five and can be modeled by a PSD following 5 integer power-laws, from to . It is sometimes necessary to consider fractional or even real exponents, whether due to the process itself or to the needs of the estimation. In our case, the millisecond pulsar timing analysis, we are faced to both cases: on the one hand, we are searching for a red noise which is expected to follow a FM slope [1, 2]

; and on the other hand, we aim to estimate the exponent of this red noise by a Bayesian statistics analysis of the PVAR estimates. We have then to consider this exponent as a real number around which we have to place a confidence interval. For these reasons, we need an expression of the PVAR response for continuous power-law noises as well as its uncertainty, i.e. the number of degrees of freedom of a PVAR estimate versus the power-law exponent.

Ii Continuous PVAR response

The response of a variance XVAR (X may be A for AVAR, M for MVAR, H for HVAR, P for PVAR, etc.) for a noise is given by the following relationship

(1)

where is the transfer function associated to XVAR and the noise level.

Following the pioneer work of Walter [3], nothing forces to be an integer and we can calculate the response of a variance for continuous power-law noise. Thus, using Eq. (1) leads to the continuous response of AVAR:

(2)

which is equal to [3, Eq. (14)] since we don’t consider a high cut-off frequency in (1).

Fig. 1: Continuous response of AVAR and PVAR compared to the known responses for .

Similarly, by using the transfer function of PVAR [4, Eq. (17)], we found

(3)

Since PVAR converges from to FM noise, we assume that this expression is valid for .

Fig. 1 shows that this relationship gives results which are in perfect accordance with the PVAR responses for integer power-law noises [4, Table I].

Iii Degrees of Freedom of PVAR estimates

First, we have to find a simplified expression of the number of degrees of freedom (dof) of PVAR estimates for integer power-law noises. Since this equation was solved for a white PM noise (see Eq. (24) in [4]), we assume that the following expression should have the same form:

(4)

with , i.e. the integration time divided by the sampling time , is the number of summations in the computation of the variance estimates, i.e. ( is the total number of samples in the data-run) for PVAR and are coefficients to be determined for each exponent. Thanks to [4, Eq. (24)], we already know that and .

We determined these coefficients by using massive Monte-Carlo simulations and verified the results by comparing them to the dof computed for continuous power-law noises.

Iii-a Determination of the coefficients from Monte-Carlo simulations

Fig. 2: Above: comparison of the empirical dof (crosses) and the approximations (lines) given by Eq. (4) and (6) for all types of noise. Below: relative difference (in %) between the empirical dof and the approximations.

The Monte-Carlo simulation was performed by computing 10 000 sequences of frequency deviations for each and for each data-run length , i.e. 150 000 simulated sequences. For given , and , we deduced the dof from the averages and the variances of the PVAR for the corresponding set of sequences by using the following well-known property of distributions [4]:

(5)

where and stands respectively for the mathematical expectation and the variance of the quantity between the brackets. By using a least square fit, we obtained the following results:

  • , , , ,

  • for all .

We have then modeled by the following order polynomial and assumed that is constant:

(6)

Thanks to Eq. (4) and (6), we are now able to assess the dof of all PVAR estimates whatever the integration time or the number of samples.

The upper plot of Fig. 2 compares the dof obtained by the Monte-Carlo simulations and by Eq. (4) and (6) for all integer types of noise. The pretty good agreement is confirmed by the lower plot which shows that the discrepancies are within except for the very first -values ().

Therefore, the approximated dof assessment by using Eq. (4) and (6) can then be applied to all integer power-law noises. In order to check if it remains valid for non-integer power-law noises when let us compute the continuous dof of PVAR.

Iii-B Verification for continuous power-law noises

The dof may be computed from Eq. (5). The mathematical expectation is the response of PVAR given above in (3) and the variance can be computed from (21) and (22) of [4]:

(7)

where is the autocorrelation function of the phase-time samples , i.e. . We used the following continuous expression of versus the power-law exponent (see [5, 3]):

However, since this expression involves -functions with arguments of the order of , we have limited these computations to samples (!). Thanks to this equation, we have computed the theoretical variance of PVAR versus continuous and deduced the dof from (5).

Let us call . From Eq. (4), we see that . The top plot of Fig. 3 shows computed from (7) (crosses) and approximated from (6) (solid lines) versus the noise power-law for (we prefer to plot than for reasons of readability of the graph). The agreement is quite good for and , but there is a notable difference for and . The lower plot of Fig. 3 shows that this discrepancy is around 20% at worst but generally remains within % in most cases (all for and all for ). This agreement is satisfactory to get an acceptable assessment of the PVAR uncertainties since the relative uncertainties are proportional to : they are therefore always below 10% and mostly within .

Fig. 3: Above: comparison of computed from (7) () and approximated by (4) (solid lines) for data. The blue squares and the green circles are respectively the values obtained for and from the Monte-Carlo simulations. Below: Error (in %) between the approximated values of and the computed values.

Iii-C Case of the largest integration times

The approximation given by Eq. (4) and (6) remains enough close to the empirical dof if . Moreover, we know that for . On the other hand, we can note that the approximation diverges beyond (see dashed lines in the upper plot of Fig. 4). However, it is of importance to assess the uncertainties within this interval, particularly if is not a power of 2.

In order to fill this gap, we decided to interpolate the dof within

, i.e. between and , by using the following semi-logarithmic fit

For , the dof are set to 1.

Fig. 4 (top) is an enlargement of the highest 2 decades of , i.e. fo data, to focus on the result of the semi-logarithmic fit. The bottom plot shows the error between the fit and the dof computed from the Monte-Carlo simulations. We can see than most of these errors are within except for the case of the white FM which ranges from to and even reaches -24% for . However, this fit is sufficient to ensure an estimation of the PVAR uncertainty for the highest value within at worst.

Fig. 4: Above: comparison of the empirical dof (crosses) and the semi-logarithmic fits (solid lines) for and for random walk FM, white FM and white PM. The dashed lines represent the approximations given by Eq. (4) and (6). In this example samples and the logarithmic increment of the -values is within . Below: Error (in %) between the empirical dof and the semi-logarithmic fits for all types of noise.

Iv Conclusion

From a theoretical calculation, we have determined the response of PVAR for continuous power-law noises. From Monte-Carlo simulations, we have got a simplified expression providing the dof of the the PVAR estimates within 10 %. We have proved that this expression remains valid for non-integer power-law noises. Finally, we have shown that a simple interpolation is efficient to fit the dof for the highest octave of integration times. Thanks to these results, we will be able to use PVAR to analyze millisecond pulsar timings and to estimate the non-integer exponent of a red noise if it is detected. On the other hand, this paper make it possible to generalize the use of PVAR to process any signal with non-integer power-law noise.

Acknowledgement

This work was partially funded by the ANR Programmes d’Investissement d’Avenir (PIA) Oscillator IMP (Project 11-EQPX-0033) and FIRST-TF (Project 10-LABX-0048).

References

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  • [4] F. Vernotte, M. Lenczner, P.-Y. Bourgeois, and E. Rubiola, “The parabolic variance (PVAR), a wavelet variance based on least-square fit,” IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control, vol. 63, no. 4, pp. 611–623, 2016.
  • [5] N. J. Kasdin, “Discrete simulation of colored noise and stochastic processes and power law noise generation,” Proceedings of the IEEE, vol. 83, no. 5, pp. 802–827, May 1995.