Statistics of Correlated Noise

Distribution of the product of complex numbers drawn from covariant Gaussian distributions, with covariance 0.8.

Noiselike Signals: Nearly all signals from astrophysical sources are noise. More precisely, one sense of polarization of the electric field from an astrophysical source at any instant is drawn from a complex Gaussian distribution. (The real and imaginary parts represent the instantaneous phase of the signal.) All the interesting parameters of the source -- spectrum, size, polarization, and so on -- can be expressed as the variance of that Gaussian distribution, or its mean square; and as the covariances among the noises measured at different times, places, polarizations, and so on. These variances and covariances completely characterize the signal.

Correlated Noise: In radio astronomy, the statistics of the Gaussian distribution are often measured by correlation. Specialized correlators multiply electric fields together and average their product, to determine their covariance. For reasons similar to the success of digital cell phones, music, and video, radio-astronomical signals are usually digitized: digitized signals can be amplified and transmitted much more easily. Of course, the process of digitization reduces the amount of information in the signal. A fundamental understanding of the consequences of digitization and multiplication on Gaussian noise is important to understanding the statistics of astrophysical measurements.

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Last Modified: 24 Sept 2003