Correlation Statistics of Spectrally-Varying Quantized Noise

I calculate the noise in the measured correlation functions and spectra of digitized, noiselike signals. In the spectral domain, the signals are drawn from a Gaussian distribution with variance that depends on frequency. Nearly all astrophysical signals have noiselike statistics of this type, many with important spectral variations. Observation and analysis of such signals at millimeter and longer wavelengths typically involves sampling in the time domain, and digitizing the sampled signal. (Quantum-mechanical effects, not discussed here, are important at infrared and shorter wavelengths.) The digitized noise is then correlated to form a measured correlation function, which is then Fourier transformed to produce a measured spectrum. When averaged over many samples, the elements of the correlation function and of the spectrum, follow Gaussian distributions. For each element, the mean of that distribution is the deterministic part of the measurement. The standard deviation of the Gaussian is the noise. Here I calculate that noise, as a function of the parameters of digitization. The noise of the correlation function is related to the underlying spectrum, by constants that depend on the digitization parameters. Noise affects variances of elements of the correlation function and covariances between them. In the spectral domain, noise also produces variances and covariances. I show that noise is correlated between spectral channels, for digitized spectra, and calculate the correlation. These statistics of noise are important for understanding of signals sampled with very high signal-to-noise ratio, or signals with rapidly-changing levels such as pulsars. Subject headings: methods: data analysis – techniques