Density of Spherically-Embedded Stiefel and Grassmann Codes

The density of a code is the fraction of the coding space covered by packing balls centered around the codewords. This paper investigates the density of codes in the complex Stiefel and Grassmann manifolds equipped with the chordal distance. The choice of distance enables the treatment of the manifolds as subspaces of Euclidean hyperspheres. In this geometry, the densest packings are not necessarily equivalent to maximum-minimum-distance codes. Computing a code's density follows from computing: i) the normalized volume of a metric ball and ii) the kissing radius, the radius of the largest balls one can pack around the codewords without overlapping. First, the normalized volume of a metric ball is evaluated by asymptotic approximations. The volume of a small ball can be well-approximated by the volume of a locally-equivalent tangential ball. In order to properly normalize this approximation, the precise volumes of the manifolds induced by their spherical embedding are computed. For larger balls, a hyperspherical cap approximation is used, which is justified by a volume comparison theorem showing that the normalized volume of a ball in the Stiefel or Grassmann manifold is asymptotically equal to the normalized volume of a ball in its embedding sphere as the dimension grows to infinity. Then, bounds on the kissing radius are derived alongside corresponding bounds on the density. Unlike spherical codes or codes in flat spaces, the kissing radius of Grassmann or Stiefel codes cannot be exactly determined from its minimum distance. It is nonetheless possible to derive bounds on density as functions of the minimum distance. Stiefel and Grassmann codes have larger density than their image spherical codes when dimensions tend to infinity. Finally, the bounds on density lead to refinements of the standard Hamming bounds for Stiefel and Grassmann codes.Comment: Two-column version (24 pages, 6 figures, 4 tables). To appear in IEEE Transactions on Information Theor

Coding on Flag Manifolds for Limited Feedback MIMO Systems

The efficiency of the physical layer in modern communication systems using multi-input multi-output (MIMO) techniques is largely based on the availability of channel state information (CSI) at the transmitter. In many practical systems, CSI needs to be quantized at the receiver side before transmission through a limited rate feedback channel. This is typically done using a codebook-based precoding transmission, where the receiver transmits the index of a codeword from a pre-designed codebook shared with the transmitter. To construct such codes one has to discretize complex flag manifolds. For single-user MIMO with a maximum likelihood receiver, the spaces of interest are Grassmann manifolds. With a linear receiver and network MIMO, the codebook design is related to discretization of Stiefel manifolds and more general flag manifolds. In this thesis, coding in flag manifolds is studied. In a first part, flag manifolds are defined as metric spaces corresponding to subsurfaces of hyperspheres. The choice of distance defines the geometry of the space and impacts clustering and averaging (centroid computation) in vector quantization, as well as coding theoretical packing bounds and optimum constructions. For two transmitter antenna systems, the problem reduces to designing spherical codes. A simple isomorphism enables to analytically derive closed-form codebooks with inherent low-implementation complexity. For more antennas, the concept of orbits of symmetry groups is investigated. Optimum codebooks, having desirable implementation properties as described in industry standardization, can be obtained using orbits of specific groups. For large antenna systems and base station cooperation, a product codebook strategy is also considered. Such a design requires to jointly discretize the Grassmann and Stiefel manifolds. A vector quantization algorithm for joint Grassmann-Stiefel quantization is proposed. Finally, the pertinence of flag codebook design is illustrated for a MIMO system with linear receiver

From Random Matrix Theory to Coding Theory : Volume of a Metric Ball in Unitary Group

Volume estimates of metric balls in manifolds find diverse applications in information and coding theory. In this paper, new results for the volume of a metric ball in unitary group are derived via tools from random matrix theory. The first result is an integral representation of the exact volume, which involves a Toeplitz determinant of Bessel functions. A simple but accurate limiting volume formula is then obtained by invoking Szego's strong limit theorem for large Toeplitz matrices. The derived asymptotic volume formula enables analytical evaluation of some coding-theoretic bounds of unitary codes. In particular, the Gilbert-Varshamov lower bound and the Hamming upper bound on the cardinality as well as the resulting bounds on code rate and minimum distance are derived. Moreover, bounds on the scaling law of code rate are found. Finally, a closed-form bound on the diversity sum relevant to unitary space-time codes is obtained, which was only computed numerically in the literature.Peer reviewe

Convergence of Gradient Descent for Low-Rank Matrix Approximation

This paper provides a proof of global convergence of gradient search for low-rank matrix approximation. Such approximations have recently been of interest for large scale problems, as well as for dictionary learning for sparse signal representations and matrix completion. The proof is based on the interpretation of the problem as an optimization on the Grassmann manifold and Fubiny-Study distance on this space

Fastening the Initial Access in 5G NR Sidelink for 6G V2X Networks

The ever-increasing demand for intelligent, automated, and connected mobility solutions pushes for the development of an innovative sixth Generation (6G) of cellular networks. A radical transformation on the physical layer of vehicular communications is planned, with a paradigm shift towards beam-based millimeter Waves or sub-Terahertz communications, which require precise beam pointing for guaranteeing the communication link, especially in high mobility. A key design aspect is a fast and proactive Initial Access (IA) algorithm to select the optimal beam to be used. In this work, we investigate alternative IA techniques to fasten the current fifth-generation (5G) standard, targeting an efficient 6G design. First, we discuss cooperative position-based schemes that rely on the position information. Then, motivated by the intuition of a non-uniform distribution of the communication directions due to road topology constraints, we design two Probabilistic Codebook (PCB) techniques of prioritized beams. In the first one, the PCBs are built leveraging past collected traffic information, while in the second one, we use the Hough Transform over the digital map to extract dominant road directions. We also show that the information coming from the angular probability distribution allows designing non-uniform codebook quantization, reducing the degradation of the performances compared to uniform one. Numerical simulation on realistic scenarios shows that PCBs-based beam selection outperforms the 5G standard in terms of the number of IA trials, with a performance comparable to position-based methods, without requiring the signaling of sensitive information

Relay Deployment in Single Frequency Network

Future wireless communication systems would have to support very high data rates. As this vision is not feasible with the conventional cellular architecture without increasing the density of the base stations, in order to increase the capacity, different transmit diversity schemes have been heavily investigated in the past years. Relaying has been one of those areas investigated and has been proposed as a costly advantageous method to increase the radio coverage area. The scope of this thesis is to build a Matlab simulator for Digital Video Broadcasting- Handheld/Terrestrial (DVB-H/T) and Multimedia Broadcast Multicast Service (MBMS) in 3G systems and evaluate the performance of different relaying concepts. The simulated network is a single-frequency network (SFN) using orthogonal frequency-division multiplexing (OFDM) and the program is a physical layer simulator. The performance metric is the cumulative distribution function of the signal-to-interference-plus-noise-ratio (SINR) for different user positions in the network. Different simulations have been run to assess the impact of inter-site distance, asynchronity, and other factors such as distance from the border of two SFNs and effec