Untangling the SVD's of Random Matrix Sample Paths

Abstract

Singular Value Decomposition (SVD) is a powerful tool for multivariate analysis. However, independent computation of the SVD for each sample taken from a bandlimited matrix random process will result in singular value sample paths whose tangled evolution is not consistent with the structure of the underlying random process. The solution to this problem is developed as follows: (i) a SVD with relaxed identification conditions is proposed, (ii) an approach is formulated for computing the SVD's of two adjacent matrices in the sample path with the objective of maximizing the correlation between corresponding singular vectors of the two matrices, and (iii) an efficient algorithm is given for untangling the singular value sample paths. The algorithm gives a unique solution conditioned on the seed matrix's SVD. Its effectiveness is demonstrated on bandlimited Gaussian random-matrix sample paths. Results are shown to be consistent with those predicted by random-matrix theory. A primary application of the algorithm is in multiple-antenna radio systems. The benefit promised by using SVD untangling in these systems is that the fading rate of the channel's SVD factors is greatly reduced so that the performance of channel estimation, channel feedback and channel prediction can be increased.