A method for evaluating the expectation value of a power spectrum using the probability density function of phases

TL;DR

This paper introduces a statistical method to evaluate the expected power spectrum of a time series by analyzing the probability density function of phases, validated through simulations and applied to pulsar data with uncertain parameters.

Contribution

The paper presents a novel statistical approach for calculating the expectation value of the power spectrum, accounting for phase distributions and parameter uncertainties, including in blind searches.

Findings

Method accurately reproduces known power spectrum properties.

Effectively handles uncertainties in pulsar ephemeris parameters.

Provides analytical solutions involving Fresnel's integrals for complex cases.

Abstract

Here, we present a new method to evaluate the expectation value of the power spectrum of a time series. A statistical approach is adopted to define the method. After its demonstration, it is validated showing that it leads to the known properties of the power spectrum when the time series contains a periodic signal. The approach is also validated in general with numerical simulations. The method puts into evidence the importance that is played by the probability density function of the phases associated to each time stamp for a given frequency, and how this distribution can be perturbed by the uncertainties of the parameters in the pulsar ephemeris. We applied this method to solve the power spectrum in the case the first derivative of the pulsar frequency is unknown and not negligible. We also undertook the study of the most general case of a blind search, in which both the frequency and its first derivative are uncertain. We found the analytical solutions of the above cases invoking the sum of Fresnel's integrals squared.