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This paper considers the quantization problem on the Grassmann manifold withdimension n and p. The unique contribution is the derivation of a closed-formformula for the volume of a metric ball in the Grassmann manifold when theradius is sufficiently small. This volume formula holds for Grassmann manifoldswith arbitrary dimension n and p, while previous results are only valid foreither p=1 or a fixed p with asymptotically large n. Based on the volumeformula, the Gilbert-Varshamov and Hamming bounds for sphere packings areobtained. Assuming a uniformly distributed source and a distortion metric basedon the squared chordal distance, tight lower and upper bounds are establishedfor the distortion rate tradeoff. Simulation results match the derived results.As an application of the derived quantization bounds, the information rate of aMultiple-Input Multiple-Output (MIMO) system with finite-rate channel-statefeedback is accurately quantified for arbitrary finite number of antennas,while previous results are only valid for either Multiple-Input Single-Output(MISO) systems or those with asymptotically large number of transmit antennasbut fixed number of receive antennas.

Quantization Bounds on Grassmann Manifolds of Arbitrary Dimensions and MIMO Communications with Feedback