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.

- The product of correlated elements drawn from
Gaussian distributions and multiplied
together has an interesting probability distribution,
as shown above.

Gwinn 2001, ApJ, 554, 1197

- Description of continuous signals as a set of discrete levels (a
part of
the digitization process known as quantization) destroys some of the
information in the signals. Quantization changes both the deterministic
and random parts of the correlation
(that is, the signal and the noise)
when the signals are correlated, in a way that can be
calculated mathematically. Perhaps surprisingly, correlation of
quantized signals can yield higher signal-to-noise ratio than would
correlation of the original, continuous signals!

"Correlation Statistics of Quantized Noise", C.R. Gwinn, PASP, 116, 84 (2004)

- The spectrum of a source is the Fourier transform of the
autocorrelation function. Radiotelescopes often apply this fact to
measure spectra, from autocorrelation of a signal using an
autocorrelator; or from cross-correlation of two spectra, to find the
cross-power spectrum. Quantization affects the signal and noise of
the result, for both cross- and autocorrelation.

"Correlation Statistics of Spectrally-Varying Quantized Noise", C.R. Gwinn, PASP, 118, 461 (2006)

- Propagation through a scattering, dispersing or absorbing medium
is well approximated by a convolution. Propagation of a signal
from a noiselike source with intermittent emission, such as a pulsar,
leaves the average spectrum, average correlation function, and
distribution of noise in the spectrum unchanged; however, intermittency
changes the distribution of noise in the correlation function, and
introduces correlations among noise in different channels of the
spectrum.

"Noise and Signal for Spectra of Intermittent Noiselike Emission", C.R. Gwinn and M.D. Johnson, ApJ in press (2011).

- Inventory of noise for the pulsar B0834+06, as observed with VLBI
and single-dish spectrometers, matches theoretical expectation, as long
as effects of intermittency of the pulsar on short, intermediate, and
long timescales are included in the analysis.

"Effects of Intermittent Emission: Noise Inventory for Scintillating Pulsar B0834+06", C.R. Gwinn, M.D. Johnson, T.V. Smirnova, and D.R. Stinebring, ApJ in press (2011).

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Carl Gwinn

Last Modified: 24 Sept 2003