Rigid Body Localization via Gaussian Belief Propagation with Quadratic Angle Approximation

Gaussian belief propagation (GaBP) is a technique that relies on linearized error and input-output models to yield low-complexity solutions to complex estimation problems, which has been recently shown to be effective in the design of range-based GaBP schemes for stationary and moving rigid body localization (RBL) in three-dimensional (3D) space, as long as an accurate prior on the orientation of the target rigid body is available. In this article we present a novel range-based RBL scheme via GaBP that removes the latter limitation. To this end, the proposed method incorporates a quadratic angle approximation to linearize the relative orientation between the prior and the target rigid body, enabling high precision estimates of corresponding rotation angles even for large deviations. Leveraging the resulting linearized model, we derive the corresponding message-passing (MP) rules to obtain estimates of the translation vector and rotation matrix of the target rigid body, relative to a prior reference frame. Numerical results corroborate the good performance of the proposed angle approximation itself, as well as the consequent RBL performance in terms of root mean square errors (RMSEs) in comparison to the state-of-the-art (SotA), while maintaining a low computational complexity

Fundamental Limits of Rigid Body Localization

We consider a novel approach to formulate the Cram\'er-Rao Lower Bound (CRLB) for the rigid body localization (RBL) problem, which allows us to assess the fundamental accuracy limits on the estimation of the translation and rotation of a rigid body with respect to a known reference. To that end, we adopt an information-centric construction of the Fisher information matrix (FIM), which allows to capture the contribution of each measurement towards the FIM, both in terms of input measurement types, as well as of their error distributions. Taking advantage of this approach, we derive a generic framework for the CRLB formulation, which is applicable to any type of rigid body localization scenario, extending the conventional FIM formulation suitable for point targets to the case of a rigid body whose location include both translation vector and the rotation matrix (or alternative the rotation angles), with respect to a reference. Closed-form expressions for all CRLBs are given, including the bound incorporating an orthonormality constraint onto the rotation matrix. Numerical results illustrate that the derived expression correctly lower-bounds the errors of estimated localization parameters obtained via various related state-of-the-art (SotA) estimators, revealing their accuracies and suggesting that SotA RBL algorithms can still be improved.

Robust Egoistic Rigid Body Localization

We consider a robust and self-reliant (or "egoistic") variation of the rigid body localization (RBL) problem, in which a primary rigid body seeks to estimate the pose (i.e., location and orientation) of another rigid body (or "target"), relative to its own, without the assistance of external infrastructure, without prior knowledge of the shape of the target, and taking into account the possibility that the available observations are incomplete. Three complementary contributions are then offered for such a scenario. The first is a method to estimate the translation vector between the center point of both rigid bodies, which unlike existing techniques does not require that both objects have the same shape or even the same number of landmark points. This technique is shown to significantly outperform the state-of-the-art (SotA) under complete information, but to be sensitive to data erasures, even when enhanced by matrix completion methods. The second contribution, designed to offer improved performance in the presence of incomplete information, offers a robust alternative to the latter, at the expense of a slight relative loss under complete information. Finally, the third contribution is a scheme for the estimation of the rotation matrix describing the relative orientation of the target rigid body with respect to the primary. Comparisons of the proposed schemes and SotA techniques demonstrate the advantage of the contributed methods in terms of root mean square error (RMSE) performance under fully complete information and incomplete conditions.

Enabling Next-Generation V2X Perception: Wireless Rigid Body Localization and Tracking

Vehicle-to-everything (V2X) perception describes a suite of technologies used to enable vehicles to perceive their surroundings and communicate with various entities, such as other road users, infrastructure, or the network/cloud. With the development of autonomous driving, V2X perception is becoming increasingly relevant, as can be seen by the tremendous attention recently given to integrated sensing and communication (ISAC) technologies. In this context, rigid body localization (RBL) also emerges as one important technology which enables the estimation of not only target's positions, but also their shape and orientation. This article discusses the need for RBL, its benefits and opportunities, challenges and research directions, as well as its role in the standardization of the sixth-generation (6G) and beyond fifth generation (B5G) applications.

Massive Index Modulation with Combinatorics-Free Detection for Integrated Sensing and Communications

Integrated sensing and communications (ISAC) and index modulation (IM) are promising technologies for beyond fifth generation (B5G) and sixth generation (6G) systems. While ISAC enables new applications, IM is attractive for its inherent energy and spectral efficiencies. In this article we propose massive IM as an enabler of ISAC, by considering transmit signals with information conveyed through the indexation of the resources utilized in their transmission, and pilot symbols exploited for sensing. In order to overcome the complexity hurdle arising from the large sizes of IM codebooks, we propose a novel message passing (MP) decoder designed under the Gaussian belief propagation (GaBP) framework exploiting a novel unit vector decomposition (UVD) of IM signals with purpose-derived novel probability distributions. The proposed method enjoys a low decoding complexity that is independent of combinatorial factors, while still approaching the performance of unfeasible state-of-the-art (SotA) search-based methods. The effectiveness of the proposed approach is demonstrated via complexity analysis and numerical results for piloted generalized quadrature spatial modulation (GQSM) systems of large sizes (up to 96 antennas).

Discrete Aware Matrix Completion via Convexized $\ell_0$-Norm Approximation

We consider a novel algorithm, for the completion of partially observed low-rank matrices in a structured setting where each entry can be chosen from a finite discrete alphabet set, such as in common recommender systems. The proposed low-rank matrix completion (MC) method is an improved variation of state-of-the-art (SotA) discrete aware matrix completion method which we previously proposed, in which discreteness is enforced by an $\ell_0$-norm regularizer, not by replaced with the $\ell_1$-norm, but instead approximated by a continuous and differentiable function normalized via fractional programming (FP) under a proximal gradient (PG) framework. Simulation results demonstrate the superior performance of the new method compared to the SotA techniques as well as the earlier $\ell_1$-norm-based discrete-aware matrix completion approach.

From OTFS to AFDM: A Comparative Study of Next-Generation Waveforms for ISAC in Doubly-Dispersive Channels

Next-generation wireless systems will offer integrated sensing and communications (ISAC) functionalities not only in order to enable new applications, but also as a means to mitigate challenges such as doubly-dispersive channels, which arise in high mobility scenarios and/or at millimeter-wave (mmWave) and Terahertz (THz) bands. An emerging approach to accomplish these goals is the design of new waveforms, which draw from the inherent relationship between the doubly-dispersive nature of time-variant (TV) channels and the environmental features of scatterers manifested in the form of multi-path delays and Doppler shifts. Examples of such waveforms are the delay-Doppler domain orthogonal time frequency space (OTFS) and the recently proposed chirp domain affine frequency division multiplexing (AFDM), both of which seek to simultaneously combat the detrimental effects of double selectivity and exploit them for the estimation (or sensing) of environmental information. This article aims to provide a consolidated and comprehensive overview of the signal processing techniques required to support reliable ISAC over doubly-dispersive channels in beyond fifth generation (B5G)/sixth generation (6G) systems, with an emphasis on OTFS and AFDM waveforms, as those, together with the traditional orthogonal frequency division multiplexing (OFDM) waveform, suffice to elaborate on the most relevant properties of the trend. The analysis shows that OTFS and AFDM indeed enable significantly improved robustness against inter-carrier interference (ICI) arising from Doppler shifts compared to OFDM. In addition, the inherent delay-Doppler domain orthogonality of the OTFS and AFDM effective channels is found to provide significant advantages for the design and the performance of integrated sensing functionalities.

Enabling Energy-Efficiency in Massive-MIMO: A Scalable Low-Complexity Decoder for Generalized Quadrature Spatial Modulation

Generalized quadrature spatial modulation (GQSM) schemes are known to achieve high energy- and spectral- efficiencies by modulating information both in transmitted symbols and in coded combinatorial activations of subsets of multiple transmit antennas. A challenge of the approach is, however, the decoding complexity which scales with the efficiency of the scheme. In order to circumvent this bottleneck and enable high-performance and feasible GQSM in massive multiple-input multiple-output (mMIMO) scenarios, we propose a novel decoding algorithm which enjoys a complexity order that is independent of the combinatorial factor. This remarkable feature of the proposed decoder is a consequence of a novel vectorized Gaussian belief propagation (GaBP) algorithm, here contributed, whose message passing (MP) rules leverage both pilot symbols and the unit vector decomposition (UVD) of the GQSM signal structure. The effectiveness of the proposed UVD-GaBP method is illustrated via computer simulations including numerical results for systems of a size never before reported in related literature (up to 32 transmit antennas), which demonstrates the potential of the approach in paving the way towards high energy and spectral efficiency for wireless systems in a truly mMIMO setting.

Soft-connected Rigid Body Localization: State-of-the-Art and Research Directions for 6G

This white paper describes a proposed article that will aim to provide a thorough study of the evolution of the typical paradigm of wireless localization (WL), which is based on a single point model of each target, towards wireless rigid body localization (W-RBL). We also look beyond the concept of RBL itself, whereby each target is modeled as an independent multi-point three-dimensional (3D), with shape enforced via a set of conformation constraints, as a step towards a more general approach we refer to as soft-connected RBL, whereby an ensemble of several objects embedded in a given environment, is modeled as a set of soft-connected 3D objects, with rigid and soft conformation constraints enforced within each object and among them, respectively. A first intended contribution of the full version of this article is a compact but comprehensive survey on mechanisms to evolve WL algorithms in W-RBL schemes, considering their peculiarities in terms of the type of information, mathematical approach, and features the build on or offer. A subsequent contribution is a discussion of mechanisms to extend W-RBL techniques to soft-connected rigid body localization (SCW-RBL) algorithms.

AFDM vs OTFS: A Comparative Study of Promising Waveforms for ISAC in Doubly-Dispersive Channels

This white paper aims to briefly describe a proposed article that will provide a thorough comparative study of waveforms designed to exploit the features of doubly-dispersive channels arising in heterogeneous high-mobility scenarios as expected in the beyond fifth generation (B5G) and sixth generation (6G), in relation to their suitability to integrated sensing and communications (ISAC) systems. In particular, the full article will compare the well-established delay-Doppler domain-based orthognal time frequency space (OTFS) and the recently proposed chirp domain-based affine frequency division multiplexing (AFDM) waveforms. Both these waveforms are designed based on a full delay- Doppler representation of the time variant (TV) multipath channel, yielding not only robustness and orthogonality of information symbols in high-mobility scenarios, but also a beneficial implication for environment target detection through the inherent capability of estimating the path delay and Doppler shifts, which are standard radar parameters. These modulation schemes are distinct candidates for ISAC in B5G/6G systems, such that a thorough study of their advantages, shortcomings, implications to signal processing, and performance of communication and sensing functions are well in order. In light of the above, a sample of the intended contribution (Special Issue paper) is provided below.