Bayesian 3D User Localization and Channel Reconstruction with Planar Extremely Large-Scale Antenna Array

Abstract

An extremely large-scale antenna array (ELAA) can potentially provide significantly increased spatial multiplexing and beamforming gains, as well as enhanced localization capability. While presenting new potential, its near-field propagation and spatial non-stationary properties also impose a great challenge on the receiver design. This paper focuses on the receiver design in an uplink orthogonal frequency-division multiplexing system with a planar ELAA deployed at the base station. To solve the challenging problem of 3D user localization and channel estimation with the planar ELAA, a space-frequency user localization and channel reconstruction (SF-ULCR) receiver is proposed. Under the Bayesian framework, an extended probability model is first established, to capture the channel structural information comprehensively, based on which an iterative receiver consisting of three modules is derived: element-wise line spectrum estimation (ELSE), distance parameter estimation (DPE), and near-field localization (NFL). In particular, the ELSE module handles the line spectrum relationships among multiple subcarriers in the frequency domain, the DPE module extracts and integrates the distance information from the line spectrum parameters, and the NFL module utilizes the messages of distances for user localization based on near-field spatial characteristics. Our numerical results demonstrate that the proposed SF-ULCR algorithm outperforms existing baselines in terms of channel estimation and localization performance, and that it approaches the Cramèr–Rao bound.

1. Introduction

The extremely large-scale antenna array (ELAA) is a promising candidate technology for the next generation of wireless communications, due to its significantly increased spatial multiplexing and beamforming gains, as well as its enhanced localization capability [1,2]. However, the channel under an ELAA exhibits different characteristics from the traditional ones, due to its exceptionally large array size. One main difference is the near-field spherical wavefronts propagation. Due to its significantly increased array size, the Rayleigh distance [3] for an ELAA can be quite large. In this case, the conventional far-field planar wavefront propagation assumption will not hold, and the more accurate near-field spherical wavefront modeling is required [2,4]. Another difference is spatial non-stationarity [5,6,7]. Due to its extremely large array size, different parts of an ELAA can have different statistics. In particular, different parts of the array are possibly visible to totally different users, or the visibility of the same user may vary across different parts of the array, which can be described by the visibility region (VR) [5,6,7,8,9,10,11,12]. The above near-field propagation and spatial non-stationarity characteristics impose a great challenge on the receiver design of ELAA systems, such as channel estimation and user localization.

Many works have been proposed to tackle the challenges imposed by near-field propagation on channel estimation and user localization in ELAA systems. Depending on whether the configuration of the ELAA is linear (configured as uniform linear array) or planar (configured as uniform planar array), 2D or 3D user localization can be involved. For a linear ELAA, a polar-coordinate discretization scheme for the 2D user position was proposed in [13,14], based on which the channel estimation and user localization were formulated as a compressed sensing problem that could be efficiently solved by the orthogonal matching pursuit (OMP) algorithm. With the grid-based estimation as the initial point, a gridless user localization algorithm was further derived, using the gradient descent method [14]. For a planar ELAA, the complexity of discretization for the 3D user position becomes much higher. By exploiting the structural properties of the channel autocorrelation function after the Fresnel approximation, [15] proved that searching over the angle domain and distance domain can be decoupled, based on which an efficient searching method was derived for 3D user localization.

However, as demonstrated in [16,17], for user localization from the near-field channels of ELAA systems the global optima of the maximum likelihood (ML) estimation has a very sharp peak and there are also many local optima. In this case, grid-based search algorithms and gradient-based refining algorithms will require a prohibitively high complexity to approach the optimal solution, especially in 3D cases. To solve the problem, [17] proposed a sub-array-based method of processing and angle fusion. Specifically, the whole large array is divided into small sub-arrays, such that the far-field assumption holds within each sub-array and the near-field assumption applies between sub-arrays. Then, the spatial angles within each sub-array are estimated and fused for the user localization. Nevertheless, for ELAA systems with spatial non-stationarity, the active antenna elements that are visible for the user in sub-arrays can be sparse, imposing challenges on spatial angles estimation, preventing the application of [17] in this case.

A few works have studied the problem of channel estimation and user localization in ELAA systems with both near-field propagation and spatial non-stationarity [18,19,20]. For the linear ELAA setup,  refs. [18,19] assumed the antenna’s VR was composed of several contiguous sub-arrays of equal size. By discretizing the user’s spatial location into a coordinate grid and formulating it as a compressed sensing problem, the OMP algorithm was then leveraged to jointly estimate the user’s position and identify the VR. In [20], a large-scale linear reconfigurable intelligent surface (RIS)–assisted communication system was considered. Assuming that the RIS had only one continuous VR, the extent of the VR was determined based on variations in the channel energy, and then a search method in the polar coordinate domain was derived for user localization. Nevertheless, all the above studies focused on 2D user localization and channel estimation problems with a linear ELAA. For 3D user localization and channel estimation with a planar ELAA, these grid/search-based methods would involve much higher complexity to approach the optima, preventing their direct application.

For wideband communications, the frequency domain provides an extra dimension for user localization. In [14], near-field channel estimation with multiple carriers was studied. While the parameter estimation in the polar coordinate domain was focused, the received signals from multiple carriers provided multiple observations for such parameter estimation, and so enhanced performance.

In this paper, we focus on the challenging problem of 3D user localization and channel estimation with a planar ELAA. To tackle the issues incurred by both near-field propagation and spatial non-stationarity, we utilize the frequency-domain and the spatial-domain channel characteristics jointly under a Bayesian framework. The main contributions of the paper are outlined as follows:

• An uplink orthogonal frequency-division multiplexing (OFDM) communication scenario with the base station (BS) equipped with a planar ELAA was considered, and the corresponding channel and system models were derived. Then, the user localization and channel reconstruction problem was formulated from a Bayesian perspective.
• Under the Bayesian framework, an extended probability model was induced, based on which a space-frequency user localization and channel reconstruction (SF-ULCR) algorithm is proposed. The frequency-domain and spatial-domain characteristics of the channels are utilized iteratively, to achieve high-precision user localization and channel reconstruction.
• The Cramér–Rao bounds (CRBs) for channel estimation and user localization are derived, providing useful insights for system design.
• Our numerical results demonstrate that the proposed SF-ULCR algorithm outperforms existing alternatives, and its performance approaches the CRB, especially for systems with large bandwidth or/and high signal-to-noise ratio (SNR) values.

The rest of this paper is organized as follows: Section 2 outlines the channel and system models and formulates the corresponding user localization and channel estimation problem. Section 3 develops the proposed SF-ULCR algorithm. The CRBs for channel estimation and user localization are derived in Section 4. Our numerical results are provided in Section 5, followed by the conclusions.

Notation: The scalars are denoted by letters in normal fonts. The vectors and the matrices are denoted by bold lowercase letters and bold capital letters, respectively. We use superscripts ${\left(·\right)}^T$, ${\left(·\right)}^{*}$, and ${\left(·\right)}^H$ to denote the operations of transpose, conjugate, and conjugate transpose, respectively. We use $\mathrm{tr}(·)$ to denote the trace operator, $\Re \left(·\right)$ to denote the real part operator, ${\left[·\right]}_{i,j}$ to denote the element in the i-th row, j-th column of a matrix. We denote by $\mathcal{A}$ a set, and by $\left|\mathcal{A}\right|$ the cardinality of $\mathcal{A}$. $\mathcal{N}(x;\mu ,\tau )$ denotes the probability distribution function (pdf) for a variable x following a Gaussian distribution with mean $\mu$ and variance $\tau$; $\mathcal{C}\mathcal{N}(x;\mu ,\tau )$ denotes the pdf for a variable x following a complex Gaussian distribution with mean $\mu$ and variance $\tau$; $\mathcal{V}\mathcal{M}\left(x;{\kappa \mathrm{e}}^{\mathrm{j}\mu }\right)$ denotes the pdf for a variable x following a von Mises (VM) distribution with mean direction $\mu$ and concentration parameter $\kappa$. We denote by $\mathrm{E}\left[·\right]$ and $\mathrm{Var}\left[·\right]$ the expectation and variance operators. We denote by $\left|·\right|$ and $∥·∥$ the absolute value of a scalar and the modulus of a vector/matrix, respectively. We denote by $\angle \left(·\right)$ the angle of a complex number.

2. System Model and Problem Formulation

2.1. System Model

Consider an uplink MIMO OFDM communication system where the BS is equipped with an $N_b$-element ELAA and each user is equipped with a single antenna. As the transmissions of different users are orthogonal, we will focus on one certain user. As shown in Figure 1, the ELAA is in a UPA form composed of $N_b=N_x\times N_y$ elements, where $N_x$ and $N_y$ are the number of elements in the horizontal and vertical directions, respectively. The uniform antenna spacings of the UPA along the horizontal direction and vertical direction are denoted by $L_x$ and $L_y$, respectively. We number the $(I_x,I_y)$ element of the UPA as $n=I_x{N}_y+I_y$, where $I_x\in \left\{1,\dots ,N_x\right\}$ and $I_y\in \left\{1,\dots ,N_y\right\}$ are the element indexes along the horizontal and vertical directions, respectively.

We further establish a 3D Cartesian coordinate system, as illustrated in Figure 1, with the center of the first element of the UPA as the origin. The x-axis and the y-axis are parallel to the horizontal and vertical directions of the UPA plane, respectively, while the z-axis is perpendicular to the UPA plane. Let ${\mathbf{p}}_{\mathrm{U}}={\left[x_{\mathrm{U}},y_{\mathrm{U}},z_{\mathrm{U}}\right]}^T$ denote the location of the user, and let ${\mathbf{p}}_{\mathrm{BS},n}={\left[x_n,y_n,z_n\right]}^T$ denote the location of the n-th ($\forall n$) antenna element.

To characterize the channel between the user and the BS with an ELAA, we need to consider both near-field spherical wavefront propagation and spatial non-stationarity.

(1) Near-Field Spherical Wavefront Propagation: The wireless signal has a spherical wavefront. Following [4], the near-field line of sight (LoS) channel seen by the n-th BS antenna element at the m-th subcarrier satisfies

$$h_{n,m}\propto {\displaystyle \frac{1}{d_n}}{e}^{-\mathrm{j}2\pi {\displaystyle \frac{(m-1)\Delta f+f_c}{c}}{d}_n},$$

(1)

where $f_c$ denotes the carrier frequency, $\Delta f$ denotes the subcarrier spacing, and $d_n$ denotes the distance between the user and the n-th antenna element of the BS given by

$$\begin{array}{cc}\hfill d_n& =∥{\mathbf{p}}_{\mathrm{U}}-{\mathbf{p}}_{\mathrm{BS},n}∥\hfill \\ \hfill & =\sqrt{{\left(x_{\mathrm{U}}-x_n\right)}^2+{\left(y_{\mathrm{U}}-y_n\right)}^2+{\left(z_{\mathrm{U}}-z_n\right)}^2}.\hfill \end{array}$$

(2)

We denote by $\beta$ the unknown complex ratio in (1), the channel can be further expressed as

$$h_{n,m}=\beta {\displaystyle \frac{1}{d_n}}{e}^{-\mathrm{j}2\pi {\displaystyle \frac{(m-1)\Delta f+f_c}{c}}{d}_n}.$$

(3)

(2) Non-Stationarity: For the spatial non-stationarity, we consider a special case that only LoS paths exist in the channel. The non-visibility of the user is primarily caused by obstacles such as trees and cars, as depicted in Figure 1. We assume that the user is only visible to a part of the BS array and that only the antenna elements within the VR of the BS array can receive the user’s signals [5,7]. We define a subset $\mathcal{A}\subseteq \left\{1,\dots ,N_x{N}_y\right\}$ to characterize the VR of the BS array, and we denote by $N_a=\left|\mathcal{A}\right|$ the number of visible antenna elements. Let $u_n\in \{0,1\}$ be an indicator variable for the “visibility” of the antenna elements, namely, $u_n=1$ if the user is visible to the n-th antenna element, i.e., $n\in \mathcal{A}$, and $u_n=0$ otherwise (We assume $u_n=1$ with probability $\alpha$ for any n). Thus, we can model the channel as

$$\begin{array}{cc}\hfill {\tilde{h}}_{n,m}& =u_n{h}_{n,m}\hfill \\ \hfill & =u_n·\beta {\displaystyle \frac{1}{d_n}}{e}^{-\mathrm{j}2\pi {\displaystyle \frac{\left(m-1\right)\Delta f+f_c}{c}}{d}_n}.\hfill \end{array}$$

(4)

We denote the user pilot by $\mathit{s}\in {\mathbb{C}}^M$, configured on M subcarriers of a single OFDM symbol. Without loss of generality, let $s_m=1(\forall 1\le m\le M)$. When the number of subcarriers M is large, it is relatively straightforward to determine whether a BS antenna element is visible. In this paper, we focus on the issues of the discontinuity (non-stationarity) of the ELAA antenna elements receiving signals and the near-field wavefront propagation. Thus, we assume that the VR $\mathcal{A}$ is known at the BS. Then, we focus on the received signal from each visible antenna $n\in \mathcal{A}$, which is given by

$$\begin{array}{cc}\hfill y_{n,m}& =s_m{\tilde{h}}_{n,m}+w_{n,m}\hfill \\ \hfill & ={\displaystyle \frac{\beta }{d_n}}{e}^{-\mathrm{j}2\pi {\displaystyle \frac{(m-1)\Delta f}{c}}{d}_n}{e}^{-\mathrm{j}2\pi {\displaystyle \frac{f_c}{c}}{d}_n}+w_{n,m},\hfill \end{array}$$

(5)

where $w_{n,m}$ is the normalized additive white Gaussian noise (AWGN) with zero mean and variance ${\sigma }^2$.

2.2. Problem Formulation

In this paper, we focus on 3D user localization and channel estimation with an ELAA. From a Bayesian perspective, the estimation of $\left\{h_{n,m}\right\}$ and ${\mathbf{p}}_{\mathrm{U}}$ can be formulated as a maximum a posteriori (MAP) estimation problem:

$$\begin{array}{cc}\hfill \underset{\left\{h_{n,m}\right\},{\mathbf{p}}_{\mathrm{U}}}{argmax}& p\left(\left\{h_{n,m}\right\},{\mathbf{p}}_{\mathrm{U}}\mid \left\{y_{n,m}\right\}\right)\hfill \\ \hfill \mathrm{s}.\mathrm{t}.& h_{n,m}={\displaystyle \frac{\beta }{d_n}}{e}^{-\mathrm{j}2\pi {\displaystyle \frac{(m-1)\Delta f+f_c}{c}}{d}_n},\hfill \\ \hfill & n\in \mathcal{A},m=1,\dots ,M.\hfill \\ \hfill & d_n=∥{\mathbf{p}}_{\mathrm{U}}-{\mathbf{p}}_{\mathrm{BS},n}∥,n\in \mathcal{A},\hfill \end{array}$$

(6)

where $\left\{y_{n,m}\right\}$ denotes the signals received by the BS array.

However, solving (6) is challenging, due to the complicated coupling of the related variables $\left\{d_n\right\}$, ${\mathbf{p}}_{\mathrm{U}}$, and $\left\{h_{n,m}\right\}$. Due to the sharpness of the global optimal peak and the presence of numerous local optima in the ML estimation of the user position [17], gradient-based methods and grid-based search methods tend to converge to local optima, while an exhaustive search over the 3D position is very costly. Additionally, the array partitioning-based location estimation method proposed in [17] cannot be applied directly, due to the discontinuity of the receiving array. In this paper, we propose a Bayesian iterative algorithm to solve (6) by progressively utilizing and integrating both the frequency-domain and the spatial-domain structural information of channels.

3. Proposed Algorithm

To facilitate the Bayesian receiver design, we first construct an extended probability model of the considered problem by introducing appropriate auxiliary variables. We then construct the factor graph accordingly, upon which the SF-ULCR algorithm is proposed and detailed.

3.1. Extended Probability Model and Receiver Structure

For each BS antenna $n\in \mathcal{A}$, we denote the phase gap between the adjacent subcarriers in (5) by

$$\Delta {\theta }_n=-{\displaystyle \frac{2\pi \Delta f}{c}}{d}_n,$$

(7)

and the corresponding complex gain by

$$g_n={\displaystyle \frac{\beta }{d_n}}{e}^{-\mathrm{j}2\pi {\displaystyle \frac{f_c}{c}}{d}_n}.$$

(8)

Then, $h_{n,m}$ in (3) can be rewritten as a line spectrum estimation (LSE) model with a single path given below

$$h_{n,m}=g_n{e}^{\mathrm{j}(m-1)\Delta {\theta }_n},m=1,\cdots ,M.$$

(9)

For convenience, we will denote by ${\mathit{h}}_n={[h_{n,1},\cdots ,h_{n,M}]}^T$.

Based on (7)–(9), the corresponding extended probabilistic model can be written as   

$$\begin{array}{cc}\hfill & p\left(\left\{h_{n,m}\right\},\left\{g_n\right\},\left\{\Delta {\theta }_n\right\},\left\{d_n\right\},\beta ,{\mathbf{p}}_{\mathrm{U}}\left|\left\{y_{n,m}\right\}\right.\right)\hfill \\ \hfill \stackrel{\left(a\right)}{\propto }& p\left(\left\{y_{n,m}\right\}\left|\left\{h_{n,m}\right\},\left\{g_n\right\},\left\{\Delta {\theta }_n\right\},\left\{d_n\right\},\beta ,{\mathbf{p}}_{\mathrm{U}}\right.\right)\hfill \\ \hfill ·& p\left(\left\{h_{n,m}\right\}\left|\left\{g_n\right\},\left\{\Delta {\theta }_n\right\},\left\{d_n\right\},\beta ,{\mathbf{p}}_{\mathrm{U}}\right.\right)\hfill \\ \hfill ·& p\left(\left\{g_n\right\}\left|\left\{\Delta {\theta }_n\right\},\left\{d_n\right\},\beta ,{\mathbf{p}}_{\mathrm{U}}\right.\right)p\left(\left\{\Delta {\theta }_n\right\}\left|\left\{d_n\right\},\beta ,{\mathbf{p}}_{\mathrm{U}}\right.\right)\hfill \\ \hfill ·& p\left(\left\{d_n\right\}\left|\beta ,{\mathbf{p}}_{\mathrm{U}}\right.\right)p\left(\beta \left|{\mathbf{p}}_{\mathrm{U}}\right.\right)p\left({\mathbf{p}}_{\mathrm{U}}\right)\hfill \\ \hfill \stackrel{\left(b\right)}{=}& p\left(\left\{y_{n,m}\right\}\left|\left\{h_{n,m}\right\}\right.\right)p\left(\left\{h_{n,m}\right\}\left|\left\{g_n\right\},\left\{\Delta {\theta }_n\right\}\right.\right)\hfill \\ \hfill ·& p\left(\left\{g_n\right\}\left|\left\{d_n\right\},\beta \right.\right)p\left(\beta \right)p\left(\left.\left\{\Delta {\theta }_n\right\}\right|\left\{d_n\right\}\right)\hfill \\ \hfill ·& p\left(\left.\left\{d_n\right\}\right|{\mathbf{p}}_{\mathrm{U}}\right)p\left({\mathbf{p}}_{\mathrm{U}}\right)\hfill \\ \hfill \stackrel{\left(c\right)}{=}& \prod _{n,m}p\left(y_{n,m}\left|h_{n,m}\right.\right)·\prod _{n}p\left({\mathit{h}}_n\left|g_n,\Delta {\theta }_n\right.\right)\hfill \\ \hfill ·& \prod _{n}p\left(g_n\left|d_n,\beta \right.\right)·p\left(\beta \right)·\prod _{n}p\left(\Delta {\theta }_n\left|d_n\right.\right)\hfill \\ \hfill ·& p\left(\left.\left\{d_n\right\}\right|{\mathbf{p}}_{\mathrm{U}}\right)·p\left({\mathbf{p}}_{\mathrm{U}}\right)\hfill \end{array}$$

(10)

where (a) is due to the Bayesian formula; (b) is due to the Markov property of the involved variables; and (c) is due to the independence or conditional independence of the involved variables.

Based on (10), the factor graph of the proposed Bayesian receiver is shown in Figure 2, involving two types of nodes:

• Variable nodes $\left\{y_{n,m}\right\}$, $\left\{h_{n,m}\right\}$, $\left\{g_n\right\}$, $\left\{{\Delta \theta }_n\right\}$, $\left\{d_n\right\}$, $\beta$, and ${\mathbf{p}}_{\mathrm{U}}$ depicted as red circles in Figure 2, corresponding to the variables in (10);
• Check nodes $\left\{f_{y_{n,m}}\right\}$, $\left\{f_{h_n}\right\}$, $\left\{f_{g_n}\right\}$, $\left\{f_{\Delta {\theta }_n}\right\}$, $f_L$, $p\left(\beta \right)$, and $p\left({\mathbf{p}}_{\mathrm{U}}\right)$ depicted as blue boxes in Figure 2, corresponding to the conditional distributions $\left\{p\left(y_{n,m}\left|h_{n,m}\right.\right)\right\}$, $\left\{p\left({\mathit{h}}_n\left|g_n,\Delta {\theta }_n\right.\right)\right\}$, $\left\{p\left(g_n\left|d_n,\beta \right.\right)\right\}$, $\left\{p\left(\left.\Delta {\theta }_n\right|d_n\right)\right\}$, and $p\left(\left.\left\{d_n\right\}\right|{\mathbf{p}}_{\mathrm{U}}\right)$, the a prior distribution of $\beta$, the a prior distribution of ${\mathbf{p}}_{\mathrm{U}}$, respectively.

• where a variable node is connected to a check node when the variable is involved in the check constraint.

As shown in Figure 2, we divide the whole receiver into three modules, i.e., the element-wise line spectrum estimation (ELSE) module, the distance parameter estimation (DPE) module, and near-field localization (NFL) module. The main functions of these modules are outlined below:

• ELSE Module: This module focuses on the conditional probability $p\left(y_{n,m}\left|h_{n,m}\right.\right)$ and $p\left({\mathit{h}}_n\right|$$g_n,\Delta {\theta }_n)$ in (10). Based on the observations $\left\{y_{n,m}\right\}$ and the messages of $\left\{g_n\right\}$ and $\left\{\Delta {\theta }_n\right\}$ fed back from the DPE module, the ELSE module calculates the posterior messages of $\left\{g_n\right\}$ and $\left\{\Delta {\theta }_n\right\}$, reconstructs the channels $\left\{h_{n,m}\right\}$, and passes the extrinsic messages of $\left\{g_n\right\}$ and $\left\{\Delta {\theta }_n\right\}$ to the DPE module.
• DPE Module: This module focuses on the conditional probability $p\left(g_n\right|d_n,\beta )$ and $p(\Delta {\theta }_n|d_n)$, to extract and fuse the distance information of $d_n$ from the $d_n$-dependent complex gain $g_n$ and the $d_n$-dependent phase gap $\Delta {\theta }_n$ between the adjacent subcarriers. With the messages of $\left\{d_n\right\}$ fed back from the NFL module and the messages of $\left\{g_n\right\}$ and $\left\{\Delta {\theta }_n\right\}$ from the ELSE module, the estimate of $\left\{d_n\right\}$ can be refined and forwarded to the NFL module, while the messages of $\left\{g_n\right\}$ and $\left\{\Delta {\theta }_n\right\}$ are updated and fed back to the ELSE module.
• NFL Module: This module focuses on the conditional probability $p\left(\left.\left\{d_n\right\}\right|{\mathbf{p}}_{\mathrm{U}}\right)$. Using the messages of $\left\{d_n\right\}$ from the DPE module and constraint (2), the user’s location ${\mathbf{p}}_{\mathrm{U}}$ is estimated, and the refined estimate of $\left\{d_n\right\}$ is fed back to the DPE module.

3.2. Element-Wise Line Spectrum Estimation Module

The ELSE module utilizes the observation $\left\{y_{n,m}\right\}$ and the messages $\left\{m_{g_n}^{\left(\mathrm{DPE}\to \mathrm{ELSE}\right)}\left(g_n\right)\right\}$ and $\left\{m_{{\theta }_n}^{\left(\mathrm{DPE}\to \mathrm{ELSE}\right)}\left({\theta }_n\right)\right\}$ from the DPE module to update the estimates of $\left\{g_n\right\}$ and $\left\{\Delta {\theta }_n\right\}$, while reconstructing the channels $\left\{h_{n,m}\right\}$. Then, it calculates and passes the external messages $\left\{m_{g_n}^{(\mathrm{ELSE}\to \mathrm{DPE})}\left(g_n\right)\right\}$ and $\left\{m_{\Delta {\theta }_n}^{(\mathrm{ELSE}\to \mathrm{DPE})}\left(\Delta {\theta }_n\right)\right\}$ to the DPE module.

For each visible antenna element $n\in \mathcal{A}$, the observed equivalent signal ${\mathit{y}}_n$ from all subcarriers can be expressed as

$$\begin{array}{cc}\hfill {\mathit{y}}_n& ={\mathit{h}}_n+{\mathit{w}}_n\hfill \\ \hfill & =g_n{\left[1,e^{\mathrm{j}\Delta {\theta }_n},\dots ,e^{\mathrm{j}(M-1)\Delta {\theta }_n}\right]}^T+{\mathit{w}}_n.\hfill \end{array}$$

(11)

Note that it is indeed a single-path LSE problem to exact the information of the parameters $g_n$ and $\Delta {\theta }_n$ in (11). Compared to traditional LSE problems, the number of spectral components in (11) is fixed to be 1. Note that the variational Bayesian line spectral estimation (VALSE) algorithm [21,22] achieves state-of-the-art performance in solving LSE problems, and its soft output is well-suited for our Bayesian framework. Here, we adopt a VALSE-based algorithm to solve the single-path LSE problem in (11).

For each visible antenna element $n\in \mathcal{A}$, the modified VALSE method is employed to calculate the posterior messages of $\left\{g_n\right\}$ and $\left\{\Delta {\theta }_n\right\}$ that are denoted by [21,22]:

$$\begin{array}{cc}\hfill p\left(g_n\mid {\mathit{y}}_n\right)& \approx \mathcal{C}\mathcal{N}\left(g_n;g_n^{\mathrm{post}},C_{g_n}^{\mathrm{post}}\right),\hfill \end{array}$$

(12a)

$$\begin{array}{cc}\hfill p\left(\Delta {\theta }_n\mid {\mathit{y}}_n\right)& \approx \mathcal{V}\mathcal{M}\left(\Delta {\theta }_n;{\eta }_{\Delta {\theta }_n}^{\mathrm{post}}\right),\hfill \end{array}$$

(12b)

where ${\eta }_{\Delta {\theta }_n}^{\mathrm{post}}={\kappa }_{\Delta {\theta }_n}^{\mathrm{post}}{e}^{\mathrm{j}\Delta {\theta }_n^{\mathrm{post}}}$, $\Delta {\theta }_n^{\mathrm{post}}$, and ${\kappa }_{\Delta {\theta }_n}^{\mathrm{post}}$ are the mean direction and concentration parameter, respectively.

With the above posterior messages of $g_n$ and $\Delta {\theta }_n$, we can reconstruct the channel $h_{n,m}$ as [21]

$${\widehat{h}}_{n,m}=g_n^{\mathrm{post}}{\displaystyle \frac{I_{m-1}\left({\kappa }_{\Delta {\theta }_n}^{\mathrm{post}}\right)}{I_0\left({\kappa }_{\Delta {\theta }_n}^{\mathrm{post}}\right)}}{e}^{\mathrm{j}\left(m-1\right)\Delta {\theta }_n^{\mathrm{post}}},$$

(13)

where $I_p\left(·\right)$ is the modified Bessel function of the first kind and order p.

Then, the extrinsic messages of $g_n$ and $\Delta {\theta }_n$ passed to the DPE module can be calculated, respectively, as

$$\begin{array}{cc}\hfill m_{g_n}^{(\mathrm{ELSE}\to \mathrm{DPE})}\left(g_n\right)& \propto {\displaystyle \frac{p\left(g_n\mid {\mathit{y}}_n\right)}{m_{g_n}^{(\mathrm{DPE}\to \mathrm{ELSE})}\left(g_n\right)}}\hfill \\ \hfill & \propto \mathcal{C}\mathcal{N}\left(g_n;g_n^{\mathrm{ext}},C_{g_n}^{\mathrm{ext}}\right)\hfill \end{array}$$

(14)

$$\begin{array}{cc}\hfill m_{\Delta {\theta }_n}^{(\mathrm{ELSE}\to \mathrm{DPE})}\left(\Delta {\theta }_n\right)& \propto {\displaystyle \frac{p\left(\Delta {\theta }_n\mid {\mathit{y}}_n\right)}{m_{\Delta {\theta }_n}^{(\mathrm{DPE}\to \mathrm{ELSE})}\left(\Delta {\theta }_n\right)}}\hfill \\ \hfill & \propto \mathcal{V}\mathcal{M}\left(\Delta {\theta }_n;{\eta }_{\Delta {\theta }_n}^{\mathrm{ext}}\right)\hfill \end{array}$$

(15)

where ${\eta }_{\Delta {\theta }_n}^{\mathrm{ext}}={\kappa }_{\Delta {\theta }_n}^{\mathrm{ext}}{e}^{\mathrm{j}\Delta {\theta }_n^{\mathrm{ext}}}$ and satisfies ${\eta }_{\Delta {\theta }_n}^{\mathrm{ext}}={\eta }_{\Delta {\theta }_n}^{\mathrm{post}}-{\eta }_{\Delta {\theta }_n}^{\mathrm{pri}}$.

3.3. Distance Parameter Estimation Module

This module utilizes the messages $\left\{m_{\Delta {\theta }_n}^{(\mathrm{ELSE}\to \mathrm{DPE})}\left(\Delta {\theta }_n\right)\right\}$ and $\left\{m_{g_n}^{(\mathrm{ELSE}\to \mathrm{DPE})}\left(g_n\right)\right\}$ passed from the ELSE module, as well as the feedback messages $\left\{m_{d_n}^{(\mathrm{NFL}\to \mathrm{DPE})}\left(d_n\right)\right\}$ from the NFL module, to estimate the distance parameters. Then, it updates and passes the messages $\left\{m_{d_n}^{(\mathrm{DPE}\to \mathrm{NFL})}\left(d_n\right)\right\}$ to the NFL module and the messages $\left\{m_{\Delta {\theta }_n}^{(\mathrm{DPE}\to \mathrm{ELSE})}\left(\Delta {\theta }_n\right)\right\}$ and $\left\{m_{g_n}^{(\mathrm{DPE}\to \mathrm{ELSE})}\left(g_n\right)\right\}$ to the ELSE module.

The distance parameter $d_n$ is primarily related to the complex gain $g_n$ and the phase gap $\Delta {\theta }_n$ between the adjacent subcarriers. While extracting distance information from $\Delta {\theta }_n$ is relatively straightforward, the extraction of distance information from $g_n$ is more challenging, due to the complex coupling relationship between $g_n$ and $d_n$ in (8). To facilitate the discussion, we rewrite $g_n$ as

$$\begin{array}{cc}\hfill g_n& ={\displaystyle \frac{\beta }{d_n}}{e}^{-\mathrm{j}2\pi {\displaystyle \frac{f_c}{c}}{d}_n}\hfill \\ \hfill & =b_n{e}^{\mathrm{j}\phi }{e}^{\mathrm{j}{\theta }_n},\hfill \end{array}$$

(16)

where $b_n=\left|g_n\right|={\displaystyle \frac{\left|\beta \right|}{d_n}}$ denotes the amplitude of $g_n$, $\phi =\angle \left(\beta \right)$ denotes the phase of $\beta$, and ${\theta }_n=-2\pi {\displaystyle \frac{f_c}{c}}{d}_n$ denotes the part of the $g_n$ phase that depends on $d_n$.

We observe numerically that compared to ${\theta }_n$, $b_n$ provides much less information for distance $d_n$. In this case, to simplify the calculation, we ignore the dependence of $b_n$ on $d_n$. Then, the simplified probabilistic model related to the DPE module is given by

$$\begin{array}{c}\hfill p\left(\left.{\theta }_n,d_n,\phi \right|g_n\right)\propto p\left(\left.g_n\right|{\theta }_n,d_n,\phi \right)p\left(\left.{\theta }_n\right|d_n\right)p\left(d_n\right)p\left(\phi \right)\end{array}.$$

(17)

Based on the probabilistic model (17), the factor graph corresponding to the DPE module can be drawn as Figure 3. The message passing involved in the DPE module is outlined as below:

3.3.1. Check Nodes$\left\{f_{g_n}\right\}$

According to (16), the check node $f_{g_n}$ presenting $f_{g_n}=\delta \left(g_n-b_n{e}^{\mathrm{j}\phi }{e}^{\mathrm{j}{\theta }_n}\right)$. As illustrated by Figure 3, there are three types of messages updated at $f_{g_n}$.

Messages passed from $\left\{f_{g_n}\right\}$ to $\phi$: According to the sum-product rule, we have

$$\begin{array}{c}\hfill m_{f_{g_n}\to \phi }\left(\phi \right)={\int }_{g_n,{\theta }_n}\delta \left(g_n-b_n{e}^{\mathrm{j}\phi }{e}^{\mathrm{j}{\theta }_n}\right)\mathcal{C}\mathcal{N}\left(g_n;g_n^{\mathrm{ext}},C_{g_n}^{\mathrm{ext}}\right)\mathcal{V}\mathcal{M}\left({\theta }_n;{\eta }_n^{\mathrm{pri}}\right).\end{array}$$

(18)

Leveraging the relationship between the von Mises distribution and the complex Gaussian distribution, and applying appropriate approximations, we can obtain

$$\begin{array}{c}\hfill m_{f_{g_n}\to \phi }\left(\phi \right)\propto \mathcal{V}\mathcal{M}\left(\phi ;{\eta }_{{\phi }_n}^{\mathrm{ext}}\right).\end{array}$$

(19)

The detailed derivation and calculations of (19) are given in Appendix A.

Messages passed from $\left\{f_{g_n}\right\}$ to $\left\{{\theta }_n\right\}$: Similar to (19), we have

$$m_{f_{g_n\to {\theta }_n}}\left({\theta }_n\right)\propto \mathcal{V}\mathcal{M}\left({\theta }_n;{\eta }_{{\theta }_n}^{\mathrm{ext}}\right),$$

(20)

where ${\eta }_{{\theta }_n}^{\mathrm{ext}}$ satisfies ${\eta }_{{\theta }_n}^{\mathrm{ext}}={\displaystyle \frac{2g_n^{\mathrm{ext}}{e}^{-\mathrm{j}{\phi }_n^{\mathrm{pri}}}}{\frac{C_{g_n}^{\mathrm{ext}}}{b_n}+\frac{2b_n}{{\kappa }_{{\phi }_n}^{\mathrm{pri}}}}}={\kappa }_{{\theta }_n}^{\mathrm{ext}}{e}^{\mathrm{j}{\theta }_n^{\mathrm{ext}}}$.

Messages passed from $\left\{f_{g_n}\right\}$ to $\left\{g_n\right\}$: According to the sum-product rule, we have

$$\begin{array}{cc}\hfill m_{f_{g_n}\to g_n}\left(g_n\right)& ={\int }_{{\theta }_n,\phi }\delta \left(g_n-b_n{e}^{\mathrm{j}\phi }{e}^{\mathrm{j}{\theta }_n}\right)\mathcal{V}\mathcal{M}\left({\theta }_n;{\eta }_n^{\mathrm{pri}}\right)\mathcal{V}\mathcal{M}\left(\phi ;{\eta }_{{\phi }_n}^{\mathrm{pri}}\right)\hfill \\ \hfill & \propto \mathcal{C}\mathcal{N}\left(g_n;g_n^{\mathrm{pri}},C_{g_n}^{\mathrm{pri}}\right).\hfill \end{array}$$

(21)

The detailed derivation and calculations are given in Appendix B.

3.3.2. Check Nodes$\left\{f_{{\theta }_n}\right\}$

Check node $f_{{\theta }_n}$ corresponds to the function

$$f_{{\theta }_n}=\delta \left[{\theta }_n-\left(-{\displaystyle \frac{2\pi f_c{d}_n}{c}}+2k_n\pi \right)\right],k_n\in \mathbb{N},$$

(22)

where $k_n$ is an unknown positive integer that ensures $(-{\displaystyle \frac{2\pi f_c{d}_n}{c}}+2k_n\pi )$ is in the range $\left[-\pi ,\pi \right]$, which can be determined using the distance estimate obtained from the previous iteration.

As illustrated by Figure 3, there are two types of messages updated at $\left\{f_{{\theta }_n}\right\}$, as detailed below.

Messages passed from $\left\{f_{{\theta }_n}\right\}$ to $\left\{d_n\right\}$: According to the sum-product rule, we have

$$\begin{array}{cc}\hfill m_{f_{{\theta }_n}\to d_n}\left(d_n\right)=& \mathcal{V}\mathcal{M}\left(-{\displaystyle \frac{2\pi f_c{d}_n}{c}}+2k_n\pi ;{\eta }_{{\theta }_n}^{\mathrm{ext}}\right)\hfill \\ \hfill \stackrel{\left(a\right)}{\approx }& \mathcal{N}\left(-{\displaystyle \frac{2\pi f_c{d}_n}{c}}+2k_n\pi ;{\theta }_n^{\mathrm{ext}},{\displaystyle \frac{2}{{\kappa }_{{\theta }_n}^{\mathrm{ext}}}}\right)\hfill \\ \hfill \propto & \mathcal{N}\left(d_n;{\displaystyle \frac{\left(2\pi k_n-{\theta }_n^{\mathrm{ext}}\right)c}{2\pi f_c}},{\displaystyle \frac{2}{{\kappa }_{{\theta }_n}^{\mathrm{ext}}}}{\left({\displaystyle \frac{c}{2\pi f_c}}\right)}^2\right)\hfill \\ \hfill =& \mathcal{N}\left(d_n;d_n^{\theta \mathrm{ext}},C_{d_n}^{\theta \mathrm{ext}}\right),\hfill \end{array}$$

(23)

where (a) is the approximation of the von Mises distribution to a Gaussian distribution, which becomes accurate when the concentration of the von Mises distribution is sufficiently large.

Messages passed from $\left\{f_{{\theta }_n}\right\}$ to $\left\{{\theta }_n\right\}$: According to the sum-product rule, we have

$$\begin{array}{cc}\hfill m_{f_{{\theta }_n}\to {\theta }_n}\left({\theta }_n\right)=& \mathcal{N}\left({\displaystyle \frac{\left(2\pi k_n-{\theta }_n\right)c}{2\pi f_c}};d_n^{\theta \mathrm{pri}},C_{d_n}^{\theta \mathrm{pri}}\right)\hfill \\ \hfill \propto & \mathcal{N}\left({\theta }_n;2\pi k_n-{\displaystyle \frac{2\pi f_c{d}_n^{\theta \mathrm{pri}}}{c}},C_{d_n}^{\theta \mathrm{pri}}{\left({\displaystyle \frac{2\pi f_c}{c}}\right)}^2\right).\hfill \end{array}$$

(24)

Note that the phase is generally periodic, and a Gaussian distribution can be well approximated by a von Mises distribution when its variance is small. We periodize the Gaussian distribution in (24) to a von Mises distribution, and we obtain

$$m_{f_{{\theta }_n}\to {\theta }_n}\left({\theta }_n\right)\propto \mathcal{V}\mathcal{M}\left({\theta }_n;{\eta }_{\theta n}^{\mathrm{pri}}\right)$$

(25)

where ${\eta }_{{\theta }_n}^{\mathrm{pri}}={\kappa }_{{\theta }_n}^{\mathrm{pri}}{e}^{\mathrm{j}{\theta }_n^{\mathrm{pri}}}$ with

$$\begin{array}{cc}\hfill {\theta }_n^{\mathrm{pri}}& =2\pi k_n-{\displaystyle \frac{2\pi f_c{d}_n^{\theta \mathrm{pri}}}{c}},\hfill \end{array}$$

(26a)

$$\begin{array}{cc}\hfill {\kappa }_{{\theta }_n}^{\mathrm{pri}}& ={\displaystyle \frac{1}{C_{d_n}^{\theta \mathrm{pri}}}}{\left({\displaystyle \frac{c}{2\pi f_c}}\right)}^2.\hfill \end{array}$$

(26b)

3.3.3. Check Nodes$\left\{f_{\Delta {\theta }_n}\right\}$

Similar to the processing of $\left\{f_{{\theta }_n}\right\}$, two types of messages updated at $\left\{f_{\Delta {\theta }_n}\right\}$ can be computed as

$$\begin{array}{cc}\hfill m_{f_{\Delta {\theta }_n}\to d_n}\left(d_n\right)& \propto \mathcal{N}\left(d_n;d_n^{\Delta \theta \mathrm{ext}},C_{d_n}^{\Delta \theta \mathrm{ext}}\right),\hfill \end{array}$$

(27a)

$$\begin{array}{cc}\hfill m_{f_{\Delta {\theta }_n}\to \Delta {\theta }_n}\left(\Delta {\theta }_n\right)& \propto \mathcal{V}\mathcal{M}\left(\Delta {\theta }_n;{\eta }_{\Delta {\theta }_n}^{\mathrm{pri}}\right).\hfill \end{array}$$

(27b)

3.3.4. Variable Nodes $\left\{d_n\right\}$

There are three types of messages updated at $\left\{d_n\right\}$, passed to the check nodes $\left\{f_{{\theta }_n}\right\}$, $\left\{f_{\Delta {\theta }_n}\right\}$, and $f_L$:

Messages passed from $\left\{d_n\right\}$ to $\left\{f_{{\theta }_n}\right\}$:

$$\begin{array}{cc}\hfill m_{d_n\to f_{{\theta }_n}}\left({\theta }_n\right)\propto & \mathcal{N}\left(d_n;d_n^{\Delta \theta \mathrm{ext}},C_{d_n}^{\Delta \theta \mathrm{ext}}\right)\mathcal{N}\left(d_n;d_n^{\mathrm{pri}},C_{d_n}^{\mathrm{pri}}\right)\hfill \\ \hfill \propto & \mathcal{N}\left(d_n;d_n^{\theta \mathrm{pri}},C_{d_n}^{\theta \mathrm{pri}}\right).\hfill \end{array}$$

(28)

Messages passed from $\left\{d_n\right\}$ to $\left\{f_{\Delta {\theta }_n}\right\}$:

$$\begin{array}{cc}\hfill m_{d_n\to f_{{\Delta }_n}}\left(\Delta {\theta }_n\right)\propto & \mathcal{N}\left(d_n;d_n^{\theta \mathrm{ext}},C_{d_n}^{\theta \mathrm{ext}}\right)\mathcal{N}\left(d_n;d_n^{\mathrm{pri}},C_{d_n}^{\mathrm{pri}}\right)\hfill \\ \hfill \propto & \mathcal{N}\left(d_n;d_n^{\Delta \theta \mathrm{pri}},C_{d_n}^{\Delta \theta \mathrm{pri}}\right).\hfill \end{array}$$

(29)

Messages passed from $\left\{d_n\right\}$ to $\left\{f_L\right\}$:

$$\begin{array}{cc}\hfill m_{d_n}^{(\mathrm{DPE}\to \mathrm{NFL})}\left(d_n\right)\propto & \mathcal{N}\left(d_n;d_n^{\Delta \theta \mathrm{ext}},C_{d_n}^{\Delta \theta \mathrm{ext}}\right)\mathcal{N}\left(d_n;d_n^{\theta \mathrm{ext}},C_{d_n}^{\theta \mathrm{ext}}\right)\hfill \\ \hfill \propto & \mathcal{N}\left(d_n;d_n^{\mathrm{ext}},C_{d_n}^{\mathrm{ext}}\right).\hfill \end{array}$$

(30)

3.4. Near-Field Localization Module

Using the messages $\left\{m_{d_n}^{(\mathrm{DPE}\to \mathrm{NFL})}\left(d_n\right)\right\}$ passed from the DPE module and the prior message of ${\mathbf{p}}_{\mathrm{U}}$, the NFL module focuses on constraint (2) to estimate the user position ${\mathbf{p}}_{\mathrm{U}}$. Simultaneously, it calculates and feeds back the distance messages $\left\{m_{d_n}^{(\mathrm{NFL}\to \mathrm{DPE})}\left(d_n\right)\right\}$ to the DPE module.

3.4.1. Near-Field Localization

With the distance messages $\left\{\mathcal{N}\left(d_n;d_n^{\mathrm{ext}},C_{d_n}^{\mathrm{ext}}\right)\right\}$ and constraint (2), we model

$$d_n^{\mathrm{ext}}=\sqrt{{\left(x_{\mathrm{U}}-x_n\right)}^2+{\left(y_{\mathrm{U}}-y_n\right)}^2+{\left(z_{\mathrm{U}}-z_n\right)}^2}+w_{d_n},n\in \mathcal{A}$$

(31)

where $w_{d_n}\sim \mathcal{N}\left(w_{d_n};0,C_{d_n}^{\mathrm{ext}}\right)$. The above equation can be further expressed as

$${\left(d_n^{\mathrm{ext}}\right)}^2={\left(x_{\mathrm{U}}-x_n\right)}^2+{\left(y_{\mathrm{U}}-y_n\right)}^2+{z_{\mathrm{U}}}^2+2d_n{w}_{d_n}+w_{d_n}^2.$$

(32)

To rewrite (32) in a compact matrix form, we define

$$\begin{array}{cc}\hfill \mathit{q}& =\left[\begin{array}{c}⋮\hfill \\ {\left(d_n^{\mathrm{ext}}\right)}^2-x_n^2-y_n^2\hfill \\ ⋮\hfill \end{array}\right]\in {\mathbb{N}}^{N_a\times 1},n\in \mathcal{A}\hfill \end{array}$$

(33a)

$$\begin{array}{cc}\hfill \mathit{A}& =\left[\begin{array}{ccc}⋮& ⋮& ⋮\\ 1& -2x_n& -2y_n\\ ⋮& ⋮& ⋮\end{array}\right]\in {\mathbb{N}}^{N_a\times 3},n\in \mathcal{A}\hfill \end{array}$$

(33b)

$$\begin{array}{cc}\hfill \mathit{s}& =\left[\begin{array}{c}{\left(x_{\mathrm{U}}\right)}^2+{\left(y_{\mathrm{U}}\right)}^2+{\left(z_{\mathrm{U}}\right)}^2\\ x_{\mathrm{U}}\\ y_{\mathrm{U}}\end{array}\right]\in {\mathbb{N}}^{3\times 1}.\hfill \end{array}$$

(33c)

Then, we have

$$\mathit{q}=\mathit{A}\mathit{s}+{\mathit{w}}_d.$$

(34)

Note that (34) is in a linear-model form. We adopt the linear minimum mean square error (LMMSE) estimator for posterior inference of the location parameter $\mathit{s}$, yielding

$$\begin{array}{cc}\hfill \widehat{\mathit{s}}& ={\mu }_{\mathit{s}}+{\mathit{C}}_{\mathit{s}q}{\mathit{C}}_q^{-1}\left(\mathit{q}-\mathit{A}{\mu }_{\mathit{s}}\right),\hfill \end{array}$$

(35a)

$$\begin{array}{cc}\hfill {\mathit{v}}_s& ={\mathit{C}}_s-{\mathit{C}}_{sq}{\mathit{C}}_q^{-1}\mathit{A}{\mathit{C}}_s,\hfill \end{array}$$

(35b)

where ${\mu }_{\mathit{s}}$ denotes the a priori mean of $\mathit{s}$, ${\mathit{C}}_{\mathit{s}}$ denotes the a priori covariance matrix of $\mathit{s}$, ${\mathit{C}}_{\mathit{sq}}$ denotes the covariance matrix between $\mathit{s}$ and $\mathit{q}$, and ${\mathit{C}}_{\mathit{q}}$ denotes the auto-covariance matrix of $\mathit{q}$.

After obtaining a posteriori estimate of the location parameter $\mathit{s}$, a posteriori estimate of the user location can be obtained:

$$\begin{array}{cc}\hfill {\widehat{x}}_{\mathrm{U}}& =\widehat{\mathit{s}}\left(2\right),\hfill \end{array}$$

(36a)

$$\begin{array}{cc}\hfill {\widehat{y}}_{\mathrm{U}}& =\widehat{\mathit{s}}\left(3\right),\hfill \end{array}$$

(36b)

$$\begin{array}{cc}\hfill {\widehat{z}}_{\mathrm{U}}& =\sqrt{\widehat{\mathit{s}}\left(1\right)-{\left[\widehat{\mathit{s}}\left(2\right)\right]}^2-{\left[\widehat{\mathit{s}}\left(3\right)\right]}^2}.\hfill \end{array}$$

(36c)

3.4.2. Calculation of Feedback Messages

Based on the posterior estimate of $\mathit{s}$ and the relationship between $\mathit{s}$ and $\left\{d_n\right\}$, the posterior estimate of $\left\{d_n\right\}$ can be approximated using the following Gaussian distribution:

$$p\left(d_n\mid \widehat{\mathit{s}},{\mathit{v}}_s\right)\sim \mathcal{N}\left(d_n;d_n^{\mathrm{post}},C_{d_n}^{\mathrm{post}}\right),$$

(37)

where $d_n^{\mathrm{post}}$ and $C_{d_n}^{\mathrm{post}}$ satisfy

$$\begin{array}{cc}\hfill d_n^{\mathrm{post}}& =\mathrm{E}\left[d_n\mid \widehat{\mathit{s}},{\mathit{v}}_s\right],\hfill \end{array}$$

(38a)

$$\begin{array}{cc}\hfill C_{d_n}^{\mathrm{post}}& =\mathrm{Var}\left[d_n\mid \widehat{\mathit{s}},{\mathit{v}}_s\right].\hfill \end{array}$$

(38b)

Thus, the extrinsic messages passed to the DPE module can be calculated as   

$$\begin{array}{cc}\hfill m_{d_n}^{(\mathrm{NFL}\to \mathrm{DPE})}\left(d_n\right)& \propto {\displaystyle \frac{\mathcal{N}\left(d_n;d_n^{\mathrm{post}},C_{d_n}^{\mathrm{post}}\right)}{\mathcal{N}\left(d_n;d_n^{\mathrm{ext}},C_{d_n}^{\mathrm{ext}}\right)}}\hfill \\ \hfill & \propto \mathcal{N}\left(d_n;d_n^{\mathrm{pri}},C_{d_n}^{\mathrm{pri}}\right).\hfill \end{array}$$

(39)

3.5. Overall Algorithm

With the procedures described in Section 3.1, Section 3.2, Section 3.3 and Section 3.4, the proposed SF-ULCR algorithm is outlined in Algorithm 1. The operations of the ELSE module, the DPE module, and the NFL module iterate until convergence or the number of iterations reaches T. Finally, the channel estimate obtained from the ELSE module and the user position estimate obtained from the NFL module during the last iteration are output.

Algorithm 1 SF-ULCR Algorithm

Input: equivalent received signal$\left\{{\mathit{y}}_n\right\}$ repeat    // ELSE Module    for $n\in \mathcal{A}$ do       Compute $p\left(g_n\mid {\mathit{y}}_n\right)$ and $p\left(\Delta {\theta }_n\mid {\mathit{y}}_n\right)$ by the modified VALSE alogrithm.       Reconstruct $\left\{h_{n,m}\right\}$ by (13).       Update $m_{g_n}^{(\mathrm{ELSE}\to \mathrm{DPE})}\left(g_n\right)$ and $m_{\Delta {\theta }_n}^{(\mathrm{ELSE}\to \mathrm{DPE})}\left(\Delta {\theta }_n\right)$ by (14) and (15).    end    // DPE Module    for $n\in \mathcal{A}$       Update $m_{f_{{\theta }_n}\to {\theta }_n}\left({\theta }_n\right)$ by (24)–(26b) and (28).       Update $m_{f_{\Delta {\theta }_n}\to \Delta {\theta }_n}\left(\Delta {\theta }_n\right)$ by (27b) and (29).       Update messages passed from $\left\{f_{g_n}\right\}$ by (19)–(21).       Update $m_{d_n}^{(\mathrm{DPE}\to \mathrm{NFL})}\left(d_n\right)$ by (23), (27a), and (30).    end    // NFL Module    Update $\widehat{\mathit{s}}$ and ${\mathit{v}}_s$ according to (35a) and (35b).    Estimate ${\mathbf{p}}_{\mathbf{U}}$ by (36a)–(36c).    for $n\in \mathcal{A}$       Update $m_{d_n}^{(\mathrm{NFL}\to \mathrm{DPE})}\left(d_n\right)$ by (37)–(39).    end until the maximum number of iterations T is reached. Output: channel estimate$\left\{{\widehat{h}}_{n,m}\right\}$; user location estimate ${\widehat{\mathbf{p}}}_{\mathrm{U}}={[{\widehat{x}}_{\mathrm{U}},{\widehat{y}}_{\mathrm{U}},{\widehat{z}}_{\mathrm{U}}]}^T$.

The complexity of the SF-ULCR algorithm is outlined below. The complexity of the ELSE module is primarily determined by the complexity of the VALSE algorithm, which is $\mathcal{O}\left(N_i{N}_{a}MT\right)$ for the single path case in this paper where $N_i$ is the number of iterations in the VALSE. The complexity of both the DPE module and the NFL module is $\mathcal{O}\left(N_{a}T\right)$. Thus, the total complexity of the algorithm is $\mathcal{O}\left(N_i{N}_{a}MT\right)$, which is modest for practical communication systems.

4. Cramér–Rao Bound

In this section, we derive the CRB for the considered channel estimation and localization problem, serving as a performance bound for the proposed method. We define the parameter vector for the considered problem as $\gamma ={[x_{\mathrm{U}},y_{\mathrm{U}},z_{\mathrm{U}},b,\phi ]}^T\in {\mathbb{R}}^5$, where $b=\left|\beta \right|$ and $\phi =\angle \left(\beta \right)$. The CRB of $\gamma$ is equal to the inverse of the Fisher information matrix (FIM) $\mathit{F}\left(\gamma \right)\in {\mathbb{R}}^{5\times 5}$. Each element of the $\mathit{F}\left(\gamma \right)$ can be computed as [23]

$$\mathit{F}{\left(\gamma \right)}_{i,j}={\displaystyle \frac{2}{{\sigma }^2}}\mathfrak{R}\left(\sum _{m=1}^M\sum _{n\in V}\left({\displaystyle \frac{\partial h_{n,m}^{*}}{\partial {\gamma }_i}}\right)\left({\displaystyle \frac{\partial h_{n,m}}{\partial {\gamma }_j}}\right)\right),$$

(40)

where $i,j\in \left\{1,2,3,4,5\right\}$ and the computation of the corresponding partial derivatives satisfies

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial h_{n,m}}{\partial x_{\mathrm{U}}}}& ={\displaystyle \frac{\partial h_{n,m}}{\partial d_n}}{\displaystyle \frac{\partial d_n}{\partial x_{\mathrm{U}}}},\hfill \end{array}$$

(41a)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial h_{n,m}}{\partial y_{\mathrm{U}}}}& ={\displaystyle \frac{\partial h_{n,m}}{\partial d_n}}{\displaystyle \frac{\partial d_n}{\partial y_{\mathrm{U}}}},\hfill \end{array}$$

(41b)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial h_{n,m}}{\partial z_{\mathrm{U}}}}& ={\displaystyle \frac{\partial h_{n,m}}{\partial d_n}}{\displaystyle \frac{\partial d_n}{\partial z_{\mathrm{U}}}},\hfill \end{array}$$

(41c)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial h_{n,m}}{\partial b}}& ={\displaystyle \frac{1}{d_n}}{e}^{\mathrm{j}\phi }{e}^{-\mathrm{j}2\pi {\displaystyle \frac{\left[f_c+(m-1)\Delta f\right]}{c}}{d}_n},\hfill \end{array}$$

(41d)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial h_{n,m}}{\partial \phi }}& =\mathrm{j}{h}_{n,m},\hfill \end{array}$$

(41e)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial h_{n,m}}{\partial d_n}}& =-{\displaystyle \frac{h_{n,m}}{d_n}}-\mathrm{j}2\pi {\displaystyle \frac{\left[f_c+(m-1)\Delta f\right]}{c}}{h}_{n,m},\hfill \end{array}$$

(41f)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial d_n}{\partial x_{\mathrm{U}}}}& ={\displaystyle \frac{x_{\mathrm{U}}-x_n}{d_n}},{\displaystyle \frac{\partial d_n}{\partial y_{\mathrm{U}}}}={\displaystyle \frac{y_{\mathrm{U}}-y_n}{d_n}},{\displaystyle \frac{\partial d_n}{\partial z_{\mathrm{U}}}}={\displaystyle \frac{z_{\mathrm{U}}-z_n}{d_n}}.\hfill \end{array}$$

(41g)

Then, the CRB for the user’s position in each direction can be calculated as ${\mathrm{CRB}}_x={\left[{\mathit{F}}^{-1}\left(\gamma \right)\right]}_{1,1}$, ${\mathrm{CRB}}_y={\left[{\mathit{F}}^{-1}\left(\gamma \right)\right]}_{2,2}$, and ${\mathrm{CRB}}_z={\left[{\mathit{F}}^{-1}\left(\gamma \right)\right]}_{3,3}$.

We can also derive the CRB for the estimation error of the user’s position as

$$\mathrm{PEB}=\sqrt{tr\left\{{\left[{\mathit{F}}^{-1}\left(\gamma \right)\right]}_{1:3,1:3}\right\}},$$

(42)

where PEB denotes the position error bound (PEB), which denotes the performance limit of the user localization error.

Based on the FIM, we can further derive the CRB for the channel estimation error. First, we define the entire channel state information as follows:

$$\overline{\mathit{h}}=vector\left({\left[h_{n,m}\right]}_{n\in \mathcal{A},m\in \{1,\dots ,M\}}\right)\in {\mathbb{C}}^{N_{a}M\times 1}.$$

(43)

We denote the channel estimate by $\widehat{\mathit{h}}$. Then, the channel estimation error satisfies [23]:

$$\mathrm{E}\left(\parallel \widehat{\mathit{h}}-\overline{\mathit{h}}{\parallel }^2\right)\ge {\mathrm{CRB}}_{\mathit{h}}=\mathrm{t}\mathrm{r}\left({\displaystyle \frac{\partial \overline{\mathit{h}}}{\partial \gamma }}{\mathit{F}}^{-1}\left(\gamma \right){\displaystyle \frac{\partial {\overline{\mathit{h}}}^{\mathrm{H}}}{\partial \gamma }}\right).$$

(44)

5. Numerical Results

In this section, we provide numerical results to verify the effectiveness of the proposed method. The uplink communication scenario as illustrated in Figure 1 was considered. The carrier frequency was set to be $f_c=30$ GHz with subcarrier spacing $\Delta f=12.5$ MHz. A strip-shaped UPA array with $N_x=128$ and $N_y=24$ was adopted, with half-wavelength antenna spacing in both horizontal and vertical directions (i.e., $L_x=L_y=0.5\lambda$). Due to the spatial non-stationarity, only a proportion $\alpha =0.1$ of the antenna elements could receive signals. The users were randomly distributed within a 3D space, as shown below:

$$\Omega =\{(x,y,z)\mid X_{min}\le x\le X_{max},Y_{min}\le y\le Y_{max},Z_{min}\le z\le Z_{max}\},$$

(45)

where $X_{min}=-9$ m, $X_{max}=9$ m, $Y_{min}=1$ m, $Y_{max}=3$ m, $Z_{min}=1$ m, $Z_{max}=9$ m.

5.1. Channel Reconstruction Performance

We first considered the performance of the channel reconstruction, which was measured using the normalized mean square error (NMSE) of the channel estimate. We used the CRB derived in Section 4 as the theoretical lower bound. For comparison, we also included the performance of the least squares (LS) method, the orthogonal matching pursuit (OMP) method, and the VALSE method [21], as described below:

• LS: The classic LS estimation method ignores the structural characteristics of the channel, treating the channels between different antenna elements and subcarriers as independent unknown variables.
• OMP: To utilize the near-field characteristics of channels, the user location is grided in the 3D Cartesian coordinate domain; then, the OMP-based method is adopted for user localization and channel reconstruction. The grid resolutions in the x, y, z directions are $0.05$ m, $0.2$ m, and $0.1$ m, respectively.
• VALSE: To utilize the frequency-domain characteristics of the channels, the state-of-the art VALSE method [21] is adopted for the parameter estimation of the LSE model and the channel reconstruction.

Figure 4 shows the NMSE performance of channel reconstruction under different SNR conditions. It can be seen that the OMP method worked well in the low SNR range, due to its utilization of the spatial-domain characteristics of the channels. But its performance in the high SNR range was limited by the grid resolution. A higher-resolution grid may result in a better NMSE performance. However, for an acceptable complexity, more sophisticated grid techniques are required, which is beyond the scope of this paper. The VALSE method exhibited better performance than the LS method and the OMP method. This was possibly due to the efficient utilization of the frequency-domain characteristic of the channels. The proposed SF-ULCR algorithm outperformed both the OMP and VALSE methods, and approximated the CRB in the high SNR region. This was expected, since the proposed method utilizes both the frequency-domain and near-field characteristics of the channels.

Figure 5 shows the NMSE performance of channel reconstruction under different number of subcarriers M, with the carrier spacing kept constant. It can be observed that the NMSE performance of the LS and OMP methods did not vary significantly with the number of carriers. This was possibly because these two methods do not utilize the frequency-domain structural information of the channel efficiently. On the other hand, by characterizing the channel frequency-domain information with the LSE model explicitly, the performance of both the VALSE method and the proposed SF-ULCR method improved as the number of carriers increased. Similar to the observation as in Figure 4, the proposed SF-ULCR algorithm outperformed all the other alternatives and approached the CRB as M became large, providing an attractive solution for wide-band near-field communications.

Comprehensive utilization of channel structure information can achieve excellent channel reconstruction performance. However, the challenge lies in how to fully utilize the available channel structure information. The proposed SF-ULCR method is based on a Bayesian framework, achieving comprehensive utilization of channel information by gradually leveraging and integrating both frequency-domain and spatial-domain information. Specifically, the ELSE module utilizes partial frequency-domain information by using the line spectrum relationships across multiple subcarriers. The DPE module utilizes additional frequency-domain information by using the relationship between line spectrum parameters and distance parameters. The NFL module utilizes spatial-domain information by using the relationship between distance parameters and user location in the near-field. The gradual utilization and fusion of the information is achieved through the iterative interaction between the modules.

5.2. User Localization Performance

We next considered the performance of the user localization, measured by the root mean square error (RMSE) of the user position estimate $\mathrm{RMSE}=\sqrt{\mathbb{E}[\parallel {\mathbf{p}}_{\mathrm{U}}-{\widehat{\mathbf{p}}}_{\mathrm{U}}{\parallel }^2]}$. We used the CRB derived in Section 4 as the theoretical lower bound. For comparison, we also included the performance of the OMP method and the VALSE-TOA method. The OMP method is consistent with the approach described in Section 5.1. The VALSE-TOA method refers to estimating the delays between the user and visible antenna elements through VALSE, and then obtaining the user’s position estimate through multi-point positioning based on time of arrival (TOA).

Figure 6 and Figure 7, respectively, show the RMSE performance of user localization under different SNR conditions and numbers of subcarriers. Similar observations as in Figure 4 and Figure 5 can be made. The OMP method worked well in the low SNR range or small number of subcarrriers, but its performance was limited by the related grid resolution, and increasing the SNR value or the number of subcarriers M resulted in negligible performance improvement. On the other hand, both the VALSE-TOA method and the proposed SF-ULCR method can benefit from increasing the SNR values or the signal bandwidth, due to the better channel parameter estimates. Compared to the VALSE-TOA method, the proposed SF-ULCR method provided a comprehensive utilization of channel structural information under a Bayesian framework, including the frequency-domain linear spectrum property, the spatial near-field propagation, and the non-stationary characteristics. Consequently, it achieved both better channel reconstruction performance and higher-precision user localization, and it approached the CRB as the SNR value or the bandwidth of the signal became moderately large.

Figure 8 illustrates the simulated 3D user localization performance of the proposed method. The true user positions are denoted by the blue circular markers, while the estimated user positions are represented by the red star symbols. As can be observed from the figure, the proposed method is capable of achieving high-precision 3D user localization.

6. Conclusions

In this paper, we focused on the challenging receiver design problem in planar ELAA systems with both near-field propagation and spatial non-stationarity. We proposed an SF-ULCR algorithm, to achieve 3D user localization and channel reconstruction by utilizing the frequency-domain and spatial-domain channel characteristics iteratively under the Bayesian framework. We also derived the CRBs for channel reconstruction and user localization. Our numerical results demonstrate that the proposed algorithm achieves excellent performance in both user localization and channel reconstruction, and that it approaches the CRBs, especially when the system bandwidth or/and the SNR value is relatively large.