Van Vleck correction generalization for complex correlators with multilevel quantization
Remote sensing with phased antenna arrays is based on measurement of the cross correlations between the signals from each antenna pair. Digital correlators have systematic errors due to the quantization losses. The correlation errors allow substantial abatement based on the assumption that the analog signals are stochastic processes sampled from a statistical distribution (usually the Gaussian) The correlation correction technique is named after Van Vleck who was the first to apply it to two level clipping quantizers. The correction is especially important for high correlation levels, e.g. in studies of solar radio emissions. We offer a generalized method that for every antenna pair inputs the quantized signals' covariance and standard deviations, and outputs high precision estimates of the analog correlation. Although correlation correction methods have been extensively investigated in the past, there are several problems that, as far as we know, have not been published yet. We consider a very general quantization scheme with arbitrary set of transition thresholds and output levels, and our correction method is designed for correlations obtained from signals with generally unequal standard deviations. We also provide a method for estimation of the analog standard deviation from the quantized one for subsequent use in the correlation correction. We apply the correction to the the complex valued analytic signals, overwhelmingly used in modern remote sensing systems with arrays of antennas. The approach is valid not only for analytic signals with the imaginary part being the Hilbert transform of the real one, but also for more general, circularly symmetric complex processes whose real and imaginary parts may have arbitrary relationships to each other. This work was motivated by the need for greater precision in analysis of data from the Murchison Widefield Array (MWA)