International audience

Integrated Sensing and Communication (ISAC) systems combine sensing and communication functionalities within a unified framework, enhancing spectral efficiency and reducing costs by utilizing shared hardware components. This paper investigates multipath component power delay profile (MPCPDP)-based joint range and Doppler estimation for Affine Frequency Division Multiplexing (AFDM)-ISAC systems. The path resolvability of the equivalent channel in the AFDM system allows the recognition of Line-of-Sight (LoS) and Non-Line-of-Sight (NLoS) paths within a single pilot symbol in fast time-varying channels. We develop a joint estimation model that leverages multipath Doppler shifts and delays information under the AFDM waveform. Utilizing the MPCPDP, we propose a novel ranging method that exploits the range-dependent magnitude of the MPCPDP across its delay spread by constructing a Nakagami-m statistical fading model for MPC channel fading and correlating the distribution parameters with propagation distance in AFDM systems. without requiring additional ranging-specific time synchronization beyond standard receiver coarse synchronization, since the method exploits relative delays and MPCPDP statistics. We also transform the nonlinear Doppler estimation problem into a bilinear estimation problem using a First-order Taylor expansion. Moreover, we introduce the Expectation Maximization algorithm to estimate the hyperparameters and leverage the Expectation Consistent algorithm to cope with high-dimensional integration challenges. Extensive numerical simulations demonstrate the effectiveness of our MPCPDP-based joint range and Doppler estimation in ISAC systems.

Grid-based methods in sparse signal reconstruction (SSR) are well-regarded for their efficacy in direction-of-arrival (DoA) estimation. This paper presents the EP (Expectation Propagation)-SURE (Stein's Unbiased Risk Estimate)-SBL (Sparse Bayesian Learning) algorithm, designed for single snapshot DoA estimation. The algorithm divides DoA estimation into two parts: grid-on estimation and off-grid error estimation, employing first-order and second-order Taylor expansions. In grid-on estimation, sparse Bayesian learning is employed for sparse modeling. To tackle hyperparameter estimation challenges within sparse Bayesian learning, the algorithm adopts SURE estimator instead of the commonly-used expectation-maximization (EM) approach. For off-grid error estimation, the algorithm utilizes the EP technique to handle high-dimensional, non-tractable integration in posterior mean calculations. The feasibility and effectiveness of the proposed algorithm are validated through extensive simulations.

When derived from the Bethe Free Energy (BFE) of the Generalized Linear Model (GLM), Approximate Message Passing (AMP) algorithms combine two asymptotic Large System Limit (LSL) simplifications which are asymptotic Gaussianity of extrinsics and large random matrix theory based asymptotic variance computations. In the provably convergent AMBGAMP algorithm, a LSL version of the BFE is derived. In Expectation Propagation (EP) style minimization, the LSL BFE cost function is augmented with Lagrangian terms for mean and variance consistency constraints, augmented with a quadratic version of the mean constraints as in the Method of Multipliers (MM). The mean Lagrange multipliers then get updated ADMM-style (Alternating Direction of MM). In this approach, the weights of the MM terms need to be carefully chosen, which is not part of the MM philosophy, and the Lagrange multipliers have no particular meaning. On the other hand, AMP can be derived by directly introducing LSL simplifications in the Belief Propagtion (BP) algorithm that minimizes the original GLM BFE. This allows to relate extrinsic messages to posterior pdfs by first-order Taylor series expansion based perturbations. We also apply LSL approximations to the variances of the various Gaussians involved, which in fact leads to a rederivation of a fundamental LSL theorem describing the deterministic limit of posterior variances. We show that this LSL version of BP leads to BFE modifications that correspond to the augmented Lagrangian of the LSL BFE, explaining its weights and Lagrange Multipliers. These insights should facilitate the extension of AMP to more complex settings such as bilinear models.

In the context of the expectation-maximization (EM) algorithm, which often faces challenges due to intractable posterior distributions, this study explores an innovative approach by integrating the EM algorithm with expectation consistent (EC) approximate inference. Our method involves the incorporation of the EC algorithm into the M-step of the EM algorithm, resulting in the EM-EC algorithm. We demonstrate that the fixed points of the proposed EM-EC algorithm correspond to stationary points of a specific constrained auxiliary function, thereby providing a variational interpretation of the algorithm. Through simulations, we showcase the effectiveness and robustness of this novel approach, highlighting its potential for advancing the field of Bayesian network estimation.

Many signal processing problems involve a Generalized Linear Model (GLM), which is a type of linear model where the unknowns may be non-identically independently distributed (n.i.i.d.). Vector Approximate Message Passing for Generalized Linear Models (GVAMP) is a computationally efficient belief propagation technique used for Bayesian inference. However, the posterior variances obtained from GVAMP with limited complexity are only exact under the assumption of an independent and identically distributed (i.i.d.) prior, owing to the averaging operations involved. In numerous problems, it is beneficial not just to estimate the unknowns but also to obtain accurate posterior distributions. While VAMP, and especially AMP, are applicable to high-dimensional problems, many applications involve dimensions that are not excessively high, allowing for more complex operations. Furthermore, in finite dimensions, the asymptotic regime that leads to correct variances under certain measurement matrix model assumptions is not applicable. To overcome these challenges, we propose a revised version of GVAMP, named reGVAMP. This method provides a multivariate Gaussian posterior approximation, which includes inter-parameter correlations, and yields accurate posterior marginals requiring only the extrinsic distributions to become Gaussian.

Sparse signal reconstruction (SSR) involves tackling large underdetermined systems of linear equations while incorporating constraints or regularizers. Expectation propagation (EP) emerges as a robust method for SSR, converting these constraints into prior information. However, the cubic complexity of matrix inversion per EP cycle hinders its implementation in large systems without approximation. In various applications like direction of arrival estimation (DoA), radar imaging etc., the signal to be recovered exhibits sparsity in the Fourier dictionary. To address this, we present a fast EP algorithm based on the Gohberg-Semencul (G-S) formula and Levinson-Durbin (L-D) type algorithm, boasting only quadratic complexity. Notably, no approximation operations or random measurement matrices are required for matrix inversion compared to approximate message passing (AMP) and other message passing based algorithms. Furthermore, it is compatible with non-identically and independently distributed (n.i.i.d.) priors. Numerical simulations conclusively demonstrate the efficacy of fast EP.

The Channel State Information (CSI) of the orthogonal frequency division multiplexing (OFDM) comprises data pertaining to the attenuation of multipath propagation. In this paper, we assume that the amplitude fading of both line-of-sight (LoS) and non-line-of-sight (NLoS) paths conforms to the Nakagami-m distribution. Via establishing a relationship between the distribution parameters and the propagation distance, we propose a CSI-based ranging method utilizing the Expectation Maximization (EM)-Revisited Approximate Message Passing (reVAMP) algorithm. This algorithm is not only applicable to the CSI-based ranging estimation but can also be extended to other parameter estimation scenarios. It effectively tackles challenges associated with generalized linear models (GLMs) that involve hidden random variables and the intractability of posterior distributions during the EM iterations.