Low-rank approximation of improper complex random vectors

In reduced-rank signal processing for radar, sonar, and digital communications, we seek the right tradeoff between model bias and model variance for reconstructing signals from noisy data. Here, we extend the classical theory by considering the low-rank approximation of complex random vectors, which may or may not be proper. We show that, in general, widely linear approximation is superior to strictly linear approximation, unless the vector to be approximated is proper, in which case the optimum procedure is strictly linear. We analyze the case where the approximated random vector becomes proper in its internal coordinate system. This class of random vector, which we call generalized proper, possesses qualities similar to proper random vectors.