Novel Matrix Singular Value Inequalities and Their Applications to Uncertain MIMO Channels

• Published in: IEEE transactions on information theory Vol. 61; no. 12; pp. 6623 - 6634
• Main Authors: Loyka, Sergey, Charalambous, Charalambos D.
• Format: Journal Article
• Language: English
• Published: New York IEEE 01.12.2015; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Additives; Channel estimation; Compounds; Information theory; Linear matrix inequalities; Matrix; MIMO; Probability distribution; Signal to noise ratio; Uncertainty; MMSE; singular value inequality; Compound channel; channel capacity; optimal precoding; channel uncertainty; multiple-input multiple-output (MIMO)
• ISSN: 0018-9448; 1557-9654
• DOI: 10.1109/TIT.2015.2485986

Novel matrix singular value inequalities are established for a sum/product of three matrices. Their application to the uncertain (compound) multiple-input multiple-output (MIMO) channel subject to normed additive uncertainty establishes the saddle-point property for a wide range of performance metrics monotonic in the channel singular values, including, among others, the mutual information, MMSE, error exponent, and pairwise error probability. This, in turn, implies that the transmission on the eigenmodes of the nominal (or worst case) channel is also optimal for the whole set of channels under a general power constraint and hence achieves the compound channel capacity. The worst case channel turns out to be antiparallel of the nominal one for all these performance metrics. An application of these results to beamforming over compound MIMO channels is discussed. An optimal robust precoder for the uncertain MIMO channel is obtained in a closed-form under the sum-MSE criterion and the total power constraint. The saddle-point property is shown to hold and the optimal strategy is to diagonalize the nominal (or worst case) channel.