Z-Opt: A Near-Optimal Reduced-Complexity Two-Dimensional Grassmannian Constellation

Grassmannian constellations are known to achieve the capacity of noncoherent communications over Rayleigh fading channels in the high-SNR regime, yet their efficient construction remains challenging. In this paper, we propose two construction methods for Grassmannian constellations of one-dimensional subspaces in a two-dimensional space, termed S-Opt and Z-Opt, along with two low-complexity detectors. Both the construction and detection procedures are performed on the unit sphere, known as the Bloch sphere in quantum computing. We show that the chordal distance on the Grassmann manifold is proportional to the Euclidean distance on the Bloch sphere and derive a corresponding theoretical upper bound based on the Fejes--Tóth bound on the minimum chordal distance. The S-Opt constellation is constructed from sphere-packing solutions and attains the derived upper bound for the optimal Bloch-sphere packings considered. The S-Opt detector can be applied to arbitrary Grassmannian constellations on $\mathcal{G}(2,1)$, and its time complexity scales linearly with the number of receive antennas and logarithmically with the constellation size, while yielding the same detection performance as the GLRT detector. Furthermore, based on the insight obtained through the S-Opt construction, the Z-Opt constellation is constructed by stacking regular polygons on the Bloch sphere, and its minimum chordal distance approaches the derived upper bound over the evaluated constellation sizes. The Z-Opt detector's time complexity scales linearly with the number of receive antennas, while yielding the same detection performance as the GLRT detector for Z-Opt.

Sparsification of Precoding Codebooks for PAPR Reduction via Grassmannian Representations

In this letter, we propose a sparsification method for precoding codebooks that reduces the peak-to-average power ratio (PAPR) while preserving the achievable rate. By exploiting the fact that precoder matrices lie on the Grassmann manifold, we formulate a codebook design problem that enables sparsification without modifying the existing feedback mechanism. We develop two sparsification approaches, namely exact sparsification via unitary transformation and approximate sparsification via sparse principal component analysis, and integrate them into a unified design algorithm. The proposed sparsified codebooks incur negligible performance loss while reducing PAPR by more than 1 dB in uplink scenarios.

Low-Complexity and Power-Efficient Precoding Codebook Design on Sparse Grassmannian

We propose a sparse Grassmannian design for precoding codebooks. Due to their sparse structure, our proposed codebooks achieve low peak-to-average power ratio (PAPR), low complexity of precoder multiplication, and low storage cost, while demonstrating performance comparable to the optimal codebook. Specifically, we introduce a method for constructing codebooks based on Schubert cell decomposition on the Grassmann manifold. Designing an optimal Grassmannian precoding codebook generally requires high computational complexity. In the proposed approach, by exploiting its sparsity, the objective function can be simplified, and the search space can also be significantly reduced compared to state-of-the-art codebooks. Numerical simulations in uplink systems demonstrate that the proposed sparse codebook asymptotically approaches the optimal codebook and outperforms the codebook currently adopted in 5G NR, in terms of achievable rate under uncorrelated Rayleigh fading channels, while maintaining substantially lower PAPR than conventional dense designs. These results confirm that the proposed sparse codebook can be a practical and power-efficient alternative to conventional codebooks for a wide range of uplink transmission scenarios.

Sparse Grassmannian Design for Noncoherent Codes via Schubert Cell Decomposition

In this paper, we propose a method for designing sparse Grassmannian codes for noncoherent multiple-input multiple-output systems. Conventional pairwise error probability formulations under uncorrelated Rayleigh fading channels fail to account for rank deficiency induced by sparse configurations. We revise these formulations to handle such cases in a unified manner. Furthermore, we derive a closed-form metric that effectively maximizes the noncoherent average mutual information (AMI) at a given signal-to-noise ratio. We focus on the fact that the Schubert cell decomposition of the Grassmann manifold provides a mathematically sparse property, and establish design criteria for sparse noncoherent codes based on our analyses. In numerical results, the proposed sparse noncoherent codes outperform conventional methods in terms of both symbol error rate and AMI, and asymptotically approach the performance of the optimal Grassmannian constellations in the high-signal-to-noise ratio regime. Moreover, they reduce the time and space complexity, which does not scale with the number of transmit antennas.

Maximizing Spectrum Efficiency of Data-Carrying Reference Signals via Bayesian Optimization

Data-carrying reference signals are a type of reference signal (RS) constructed on the Grassmann manifold, which allows for simultaneous data transmission and channel estimation to achieve boosted spectral efficiency at high signal-to-noise ratios (SNRs). However, they do not improve spectral efficiency at low to middle SNRs compared with conventional RSs. To address this problem, we propose a numerical optimization-based Grassmann constellation design on the Grassmann manifold that accounts for both data transmission and channel estimation. In our numerical optimization, we derive an upper bound on the normalized mean squared error (NMSE) of estimated channel matrices and a lower bound on the noncoherent average mutual information (AMI), and these bounds are optimized simultaneously by using a Bayesian optimization technique. The proposed objective function outperforms conventional design metrics in obtaining Pareto-optimal constellations for NMSE and AMI. The constellation obtained by our method achieves an NMSE comparable to conventional non-data-carrying RSs while enabling data transmission, resulting in superior AMI performance and improved spectral efficiency even at middle SNRs.

Superimposed Pilot-Based OTFS -- Will it Work?

Orthogonal time frequency space (OTFS) modulation is a promising solution to handle doubly-selective fading, but its channel estimation is a nontrivial task in terms of maximizing spectral efficiency. Conventional pilot assignment approaches face challenges: the standard embedded pilot-based scheme suffers from low transmission rates, and the single superimposed pilot (SP)-based scheme experiences inevitable data-pilot interference, leading to coarse channel estimation. To cope with this issue, focusing on the SP-based OTFS system in channel coded scenarios, we propose a novel pilot assignment scheme and an iterative algorithm. The proposed scheme allocates multiple SPs per frame to estimate channel coefficients accurately. Furthermore, the proposed algorithm performs refined interference cancellation, utilizing a replica of data symbols generated from soft-decision outputs provided by a decoder. Assuming fair and unified conditions, we evaluate each pilot assignment scheme in terms of reliability, channel estimation accuracy, effective throughput, and computational complexity. Our numerical simulations demonstrate that the multiple SP-based scheme, which balances the transmission rate and the interference cancellation performance, has the best throughput at the expense of slightly increased complexity. In addition, we confirm that the multiple SP-based scheme achieves further improved throughput due to the proposed interference cancellation algorithm.

Covert Communications Without Pre-Sharing of Side Information and Channel Estimation Over Quasi-Static Fading Channels

We propose a new covert communication scheme that operates without pre-sharing side information and channel estimation, utilizing a Gaussian-distributed Grassmann constellation for noncoherent detection. By designing constant-amplitude symbols on the Grassmann manifold and multiplying them by random variables, we generate signals that follow an arbitrary probability distribution, such as Gaussian or skew-normal distributions. The mathematical property of the manifold enables the transmitter's random variables to remain unshared with the receiver, and the elimination of pilot symbols that could compromise covertness. The proposed scheme achieved higher covertness and achievable rates compared to conventional coherent Gaussian signaling schemes, without any penalty in terms of complexity.

Joint Channel and Data Estimation for Multiuser Extremely Large-Scale MIMO Systems

This paper proposes a joint channel and data estimation (JCDE) algorithm for uplink multiuser extremely large-scale multiple-input-multiple-output (XL-MIMO) systems. The initial channel estimation is formulated as a sparse reconstruction problem based on the angle and distance sparsity under the near-field propagation condition. This problem is solved using non-orthogonal pilots through an efficient low complexity two-stage compressed sensing algorithm. Furthermore, the initial channel estimates are refined by employing a JCDE framework driven by both non-orthogonal pilots and estimated data. The JCDE problem is solved by sequential expectation propagation (EP) algorithms, where the channel and data are alternately updated in an iterative manner. In the channel estimation phase, integrating Bayesian inference with a model-based deterministic approach provides precise estimations to effectively exploit the near-field characteristics in the beam-domain. In the data estimation phase, a linear minimum mean square error (LMMSE)-based filter is designed at each sub-array to address the correlation due to energy leakage in the beam-domain arising from the near-field effects. Numerical simulations reveal that the proposed initial channel estimation and JCDE algorithm outperforms the state-ofthe-art approaches in terms of channel estimation, data detection, and computational complexity.

Boosting Spectral Efficiency with Data-Carrying Reference Signals on the Grassmann Manifold

In wireless networks, frequent reference signal transmission for accurate channel reconstruction may reduce spectral efficiency. To address this issue, we consider to use a data-carrying reference signal (DC-RS) that can simultaneously estimate channel coefficients and transmit data symbols. Here, symbols on the Grassmann manifold are exploited to carry additional data and to assist in channel estimation. Unlike conventional studies, we analyze the channel estimation errors induced by DC-RS and propose an optimization method that improves the channel estimation accuracy without performance penalty. Then, we derive the achievable rate of noncoherent Grassmann constellation assuming discrete inputs in multi-antenna scenarios, as well as that of coherent signaling assuming channel estimation errors modeled by the Gauss-Markov uncertainty. These derivations enable performance evaluation when introducing DC-RS, and suggest excellent potential for boosting spectral efficiency, where interesting crossings with the non-data carrying RS occurred at intermediate signal-to-noise ratios.