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The Grassmann manifold Gn,p (L) is the set of all p-dimensional planes (through the origin) in the n-dimensional Euclidean space Ln, where L is either R or C. This paper considers the quantization problem in which a source in Gn,p (L) is quantized through a code in Gn,q (L), with p and q not necessarily the same. The analysis is based on the volume of a metric ball in Gn,p (L) with center in Gn,q (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.
Date of Publication: March 2008