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

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 based on the Fourier transform of characteristic functions is provided, with the example of the observation of millisecond pulsars with five radio telescopes (RTs). The simulations show that the spectrum average is by a small amount more efficient than the cross-spectrum, chiefly when the background exceeds the signal. However, some notable differences between their upper limit indicate that it should be wiser to compute both estimators.

Cross-Spectrum Measurement Statistics: Uncertainties and Detection Limit.

The cross-spectrum method consists in measuring a signal c(t) simultaneously with two independent instruments. Each of these instruments contributes to the global noise by its intrinsic (white) noise, whereas the signal c(t) that we want to characterize could be a (red) noise. We first define the real part of the cross spectrum as a relevant estimator. Then, we characterize the probability density function (pdf) of this estimator knowing the noise level (direct problem) as a Variance-gamma (VG) distribution. Next, we solve the "inverse problem" due to Bayes' theorem to obtain an upper limit of the noise level knowing the estimate. Checked by massive Monte Carlo simulations, VG proves to be perfectly reliable for any number of degrees of freedom (DOFs). Finally, we compare this method with another method using the Karhunen-Loève transform (KLT). We find an upper limit of the signal level slightly different as the one of VG since KLT better considers the available information.