Quantization Bounds on Grassmann Manifolds and Applications to MIMO Communications

Abstract

The Grassmann manifold G <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n,p</sub> (L) is the set of all p-dimensional planes (through the origin) in the n-dimensional Euclidean space L <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> , where L is either R or C. This paper considers the quantization problem in which a source in G <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n,p</sub> (L) is quantized through a code in G <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n,q</sub> (L), with p and q not necessarily the same. The analysis is based on the volume of a metric ball in G <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n,p</sub> (L) with center in G <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n,q</sub> (L), and our chief result is a closed-form expression for the volume of a metric ball of radius at most one. This volume formula holds for arbitrary n, p, q, and L, while previous results pertained only to some special cases. Based on this volume formula, several bounds are derived for the rate-distortion tradeoff assuming that the quantization rate is sufficiently high. The lower and upper bounds on the distortion rate function are asymptotically identical, and therefore precisely quantify the asymptotic rate-distortion tradeoff. We also show that random codes are asymptotically optimal in the sense that they achieve the minimum possible distortion in probability as n and the code rate approach infinity linearly. Finally, as an application of the derived results to communication theory, we quantify the effect of beamforming matrix selection in multiple-antenna communication systems with finite rate channel state feedback.