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

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

Quantized Multimode Precoding in Spatially Correlated Multi-Antenna Channels

Multimode precoding, where the number of independent data-streams is adapted optimally, can be used to maximize the achievable throughput in multi-antenna communication systems. Motivated by standardization efforts embraced by the industry, the focus of this work is on systematic precoder design with realistic assumptions on the spatial correlation, channel state information (CSI) at the transmitter and the receiver, and implementation complexity. For spatial correlation of the channel matrix, we assume a general channel model, based on physical principles, that has been verified by many recent measurement campaigns. We also assume a coherent receiver and knowledge of the spatial statistics at the transmitter along with the presence of an ideal, low-rate feedback link from the receiver to the transmitter. The reverse link is used for codebook-index feedback and the goal of this work is to construct precoder codebooks, adaptable in response to the statistical information, such that the achievable throughput is significantly enhanced over that of a fixed, non-adaptive, i.i.d. codebook design. We illustrate how a codebook of semiunitary precoder matrices localized around some fixed center on the Grassmann manifold can be skewed in response to the spatial correlation via low-complexity maps that can rotate and scale submanifolds on the Grassmann manifold. The skewed codebook in combination with a lowcomplexity statistical power allocation scheme is then shown to bridge the gap in performance between a perfect CSI benchmark and an i.i.d. codebook design.Comment: 30 pages, 4 figures, Preprint to be submitted to IEEE Transactions on Signal Processin

A Geometric Framework for Analyzing the Performance of Multiple-Antenna Systems under Finite-Rate Feedback

We study the performance of multiple-antenna systems under finite-rate feedback of some function of the current channel realization from a channel-aware receiver to the transmitter. Our analysis is based on a novel geometric paradigm whereby the feedback information is modeled as a source distributed over a Riemannian manifold. While the right singular vectors of the channel matrix and the subspace spanned by them are located on the traditional Stiefel and Grassmann surfaces, the optimal input covariance matrix is located on a new manifold of positive semi-definite matrices - specified by rank and trace constraints - called the Pn manifold. The geometry of these three manifolds is studied in detail; in particular, the precise series expansion for the volume of geodesic balls over the Grassmann and Stiefel manifolds is obtained. Using these geometric results, the distortion incurred in quantizing sources using either a sphere-packing or a random code over an arbitrary manifold is quantified. Perturbative expansions are used to evaluate the susceptibility of the ergodic information rate to the quality of feedback information, and thereby to obtain the tradeoff of the achievable rate with the number of feedback bits employed. For a given system strategy, the gap between the achievable rates in the infinite and finite-rate feedback cases is shown to be $O(2^{-\frac{2N_f}{N}})$ for Grassmann feedback and $O(2^{-\frac{N_f}{N}})$ for other cases, where $N$ is the dimension of the manifold used for quantization and $N_f$ is the number of bits used by the receiver per block for feedback. The geometric framework developed enables the results to hold for arbitrary distributions of the channel matrix and extends to all covariance computation strategies including, waterfilling in the short-term/long-term power constraint case, antenna selection and other rank-limited scenarios that could not be analyzed using previous probabilistic approaches

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

Limited feedback MIMO techniques for temporally correlated channels and linear receivers

Advanced mobile wireless networks will make extensive use of multiantenna (MIMO) transceivers to comply with high requirements of data rates and reliability. The use of feedback channels is of paramount importance to achieve this goal in systems employing frequency division duplexing (FDD). The design of the feedback mechanism is challenging due to the severe constraints imposed by computational complexity and feedback bandwidth restrictions. This thesis addresses the design of transmission strategies in both single-user and multi-user MIMO systems, based on compact feedback messages. First, recursive feedback mechanisms for single-user transmission scenarios are proposed, including stochastic gradient techniques, deterministic updates based on Givens rotations and low computational complexity schemes based on partial update filtering concepts. Thereafter, channel feedback algorithms are proposed, and a convergence analysis for static channels is presented. These algorithms can be used to provide channel side information to any multi-user MIMO solution. A limited-feedback decentralized multi-user MIMO solution is proposed, which avoids the need for explicit channel feedback. A feed-forward technique is proposed, which allows our methods to operate in presence of feedback errors. The performance of all the proposed algorithms is illustrated via link-level simulations, where the effect of different parameter values is assessed. Our results show that the proposed methods outperform existing limited-feedback counterparts over a range of low to medium mobile speeds, for moderate antenna array sizes that are deemed practical for commercial deployment. The computational complexity reduction of some of the proposed algorithms is also shown to be considerable, when compared to existing techniques

Limited Feedback Techniques in Multiple Antenna Wireless Communication Systems

Multiple antenna systems provide spatial multiplexing and diversity benefits.These systems also offer beamforming and interference mitigation capabilities in single-user (SU) and multi-user (MU) scenarios, respectively. Although diversity can be achieved without channel state information (CSI) at the transmitter using space-time codes, the knowledge of instantaneous CSI at the transmitter is essential to the above mentioned gains. In frequency division duplexing (FDD) systems, limited feedback techniques are employed to obtain CSI at the transmitter from the receiver using a low-rate link. As a consequence, CSI acquired by the transmitter in such manner have errors due to channel estimation and codebook quantization at the receiver, resulting in performance degradation of multi-antenna systems. In this thesis, we examine CSI inaccuracies due to codebook quantization errors and investigate several other aspects of limited feedback in SU, MU and multicell wireless communication systems with various channel models. For SU multiple-input multiple-output (MIMO) systems, we examine the capacity loss using standard codebooks. In particular, we consider single-stream and two-stream MIMO transmissions and derive capacity loss expressions in terms of minimum squared chordal distance for various MIMO receivers. Through simulations, we investigate the impact of codebook quantization errors on the capacity performance in uncorrelated Rayleigh, spatially correlated Rayleigh and standardized MIMO channels. This work motivates the need of effective codebook design to reduce the codebook quantization errors in correlated channels. Subsequently, we explore the improvements in the design of codebooks in temporally and spatially correlated channels for MU multiple-input single-output (MISO) systems, by employing scaling and rotation techniques. These codebooks quantize instantaneous channel direction information (CDI) and are referred as differential codebooks in the thesis. We also propose various adaptive scaling techniques for differential codebooks where packing density of codewords in the differential codebook are altered according to the channel condition, in order to reduce the quantization errors. The proposed differential codebooks improve the spectral efficiency of the system by minimizing the codebook quantization errors in spatially and temporally correlated channels. Later, we broaden the scope to massive MISO systems and propose trellis coded quantization (TCQ) schemes to quantize CDI. Unlike conventional codebook approach, the TCQ scheme does not require exhaustive search to select an appropriate codeword, thus reducing computational complexity and memory requirement at the receiver. The proposed TCQ schemes yield significant performance improvements compared to the existing TCQ based limited feedback schemes in both temporally and spatially correlated channels. Finally, we investigate interference coordination for multicell MU MISO systems using regularized zero-forcing (RZF) precoding. We consider random vector quantization (RVQ) codebooks and uncorrelated Rayleigh channels. We derive expected SINR approximations for perfect CDI and RVQ codebook-based CDI. We also propose an adaptive bit allocation scheme which aims to minimize the network interference and moreover, improves the spectral efficiency compared to equal bit allocation and coordinated zero-forcing (ZF) based adaptive bit allocation schemes