Volume of Metric Balls in High-Dimensional Complex Grassmann Manifolds

Volume of metric balls relates to rate-distortion theory and packing bounds on codes. In this paper, the volume of balls in complex Grassmann manifolds is evaluated for an arbitrary radius. The ball is defined as a set of hyperplanes of a fixed dimension with reference to a center of possibly different dimension, and a generalized chordal distance for unequal dimensional subspaces is used. First, the volume is reduced to one-dimensional integral representation. The overall problem boils down to evaluating a determinant of a matrix of the same size as the subspace dimensionality. Interpreting this determinant as a characteristic function of the Jacobi ensemble, an asymptotic analysis is carried out. The obtained asymptotic volume is moreover refined using moment-matching techniques to provide a tighter approximation in finite-size regimes. Lastly, the pertinence of the derived results is shown by rate-distortion analysis of source coding on Grassmann manifolds.Comment: 26 page

Modulation-Specific Multiuser Transmit Precoding and User Selection for BPSK Signalling

Motivated by challenges to existing multiuser transmission methods in a low signal to noise ratio (SNR) regime, and emergence of massive numbers of low data rate ehealth and internet of things (IoT) devices, in this paper we show that it is beneficial to incorporate knowledge of modulation type into multiuser transmit precoder design. Particularly, we propose a transmit precoding (beamforming) specific to BPSK modulation, which has maximum power efficiency and capacity in poor channel conditions. To be more specific, in a multiuser scenario, an objective function is formulated based on the weighted sum of error probabilities of BPSK modulated users. Convex optimization is used to transform and solve this ill-behaved non-convex minimum probability of error (MPE) precoding problem. Numerical results confirm significant performance improvement. We then develop a low-complexity user selection algorithm for MPE precoding. Based on line packing principles in Grassmannian manifolds, the number of supported users is able to exceed the number of transmit antennas, and hence the proposed approach is able to support more simultaneous users compared with existing multiuser transmit precoding methods

Frequency Selective Hybrid Precoding for Limited Feedback Millimeter Wave Systems

Hybrid analog/digital precoding offers a compromise between hardware complexity and system performance in millimeter wave (mmWave) systems. This type of precoding allows mmWave systems to leverage large antenna array gains that are necessary for sufficient link margin, while permitting low cost and power consumption hardware. Most prior work has focused on hybrid precoding for narrowband mmWave systems, with perfect or estimated channel knowledge at the transmitter. MmWave systems, however, will likely operate on wideband channels with frequency selectivity. Therefore, this paper considers wideband mmWave systems with a limited feedback channel between the transmitter and receiver. First, the optimal hybrid precoding design for a given RF codebook is derived. This provides a benchmark for any other heuristic algorithm and gives useful insights into codebook designs. Second, efficient hybrid analog/digital codebooks are developed for spatial multiplexing in wideband mmWave systems. Finally, a low-complexity yet near-optimal greedy frequency selective hybrid precoding algorithm is proposed based on Gram-Schmidt orthogonalization. Simulation results show that the developed hybrid codebooks and precoder designs achieve very good performance compared with the unconstrained solutions while requiring much less complexity.Comment: 42 pages, 8 figures, IEEE Transactions on Communications (Invited Paper) [Some typos are fixed in this version