The Statistics of the Cross-Spectrum and the Spectrum Average: Generalization to Multiple Instruments

TL;DR

This paper compares the statistical properties of the spectrum average and cross-spectrum estimators for red noise processes, extending the analysis to multiple instruments and applying it to radio astronomy observations.

Contribution

It generalizes the cross-spectrum estimator to multiple instruments and analyzes its statistical distribution, providing new insights for low-frequency power spectrum measurement.

Findings

Cross-spectrum estimator follows a Variance-Gamma distribution with two instruments.

Generalization to q devices is derived.

Application to millisecond pulsar observations with 5 radio telescopes.

Abstract

This article addresses the measurement of the power spectrum of red noise processes at the lowest frequencies, where the minimum acquisition time is so long that it is impossible to average on a sequence of data record. Therefore, averaging is possible only on simultaneous observation of multiple instruments. This is the case of radio astronomy, which we take as the paradigm, but examples may be found in other fields such as climatology and geodesy. We compare the Bayesian confidence interval of the red-noise parameter using two estimators, the spectrum average and the cross-spectrum. While the spectrum average is widely used, the cross-spectrum using multiple instruments is rather uncommon. With two instruments, the cross-spectrum estimator leads to the Variance-Gamma distribution. A generalization to $q$ devices is provided, with the example of the observation of millisecond pulsars with 5 radio telescopes.