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 , , and to denote the operations of transpose, conjugate, and conjugate transpose, respectively. We use to denote the trace operator, to denote the real part operator, to denote the element in the i-th row, j-th column of a matrix. We denote by a set, and by the cardinality of . denotes the probability distribution function (pdf) for a variable x following a Gaussian distribution with mean and variance ; denotes the pdf for a variable x following a complex Gaussian distribution with mean and variance ; denotes the pdf for a variable x following a von Mises (VM) distribution with mean direction and concentration parameter . We denote by and the expectation and variance operators. We denote by and the absolute value of a scalar and the modulus of a vector/matrix, respectively. We denote by the angle of a complex number.
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
, we denote the phase gap between the adjacent subcarriers in (
5) by
and the corresponding complex gain by
Then,
in (
3) can be rewritten as a line spectrum estimation (LSE) model with a single path given below
For convenience, we will denote by .
Based on (
7)–(
9), the corresponding extended probabilistic model can be written as
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
,
,
,
,
,
, and
depicted as red circles in
Figure 2, corresponding to the variables in (
10);
Check nodes
,
,
,
,
,
, and
depicted as blue boxes in
Figure 2, corresponding to the conditional distributions
,
,
,
, and
, the a prior distribution of
, the a prior distribution of
, respectively.
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
and
in (
10). Based on the observations
and the messages of
and
fed back from the DPE module, the ELSE module calculates the posterior messages of
and
, reconstructs the channels
, and passes the extrinsic messages of
and
to the DPE module.
DPE Module: This module focuses on the conditional probability and , to extract and fuse the distance information of from the -dependent complex gain and the -dependent phase gap between the adjacent subcarriers. With the messages of fed back from the NFL module and the messages of and from the ELSE module, the estimate of can be refined and forwarded to the NFL module, while the messages of and are updated and fed back to the ELSE module.
NFL Module: This module focuses on the conditional probability
. Using the messages of
from the DPE module and constraint (
2), the user’s location
is estimated, and the refined estimate of
is fed back to the DPE module.
3.2. Element-Wise Line Spectrum Estimation Module
The ELSE module utilizes the observation and the messages and from the DPE module to update the estimates of and , while reconstructing the channels . Then, it calculates and passes the external messages and to the DPE module.
For each visible antenna element
, the observed equivalent signal
from all subcarriers can be expressed as
Note that it is indeed a single-path LSE problem to exact the information of the parameters
and
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
, the modified VALSE method is employed to calculate the posterior messages of
and
that are denoted by [
21,
22]:
where
,
, and
are the mean direction and concentration parameter, respectively.
With the above posterior messages of
and
, we can reconstruct the channel
as [
21]
where
is the modified Bessel function of the first kind and order
p.
Then, the extrinsic messages of
and
passed to the DPE module can be calculated, respectively, as
where
and satisfies
.
3.3. Distance Parameter Estimation Module
This module utilizes the messages and passed from the ELSE module, as well as the feedback messages from the NFL module, to estimate the distance parameters. Then, it updates and passes the messages to the NFL module and the messages and to the ELSE module.
The distance parameter
is primarily related to the complex gain
and the phase gap
between the adjacent subcarriers. While extracting distance information from
is relatively straightforward, the extraction of distance information from
is more challenging, due to the complex coupling relationship between
and
in (
8). To facilitate the discussion, we rewrite
as
where
denotes the amplitude of
,
denotes the phase of
, and
denotes the part of the
phase that depends on
.
We observe numerically that compared to
,
provides much less information for distance
. In this case, to simplify the calculation, we ignore the dependence of
on
. Then, the simplified probabilistic model related to the DPE module is given by
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
According to (
16), the check node
presenting
. As illustrated by
Figure 3, there are three types of messages updated at
.
Messages passed from
to
: According to the sum-product rule, we have
Leveraging the relationship between the von Mises distribution and the complex Gaussian distribution, and applying appropriate approximations, we can obtain
The detailed derivation and calculations of (
19) are given in
Appendix A.
Messages passed from
to
: Similar to (
19), we have
where
satisfies
.
Messages passed from
to
: According to the sum-product rule, we have
The detailed derivation and calculations are given in
Appendix B.
3.3.2. Check Nodes
Check node
corresponds to the function
where
is an unknown positive integer that ensures
is in the range
, 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
, as detailed below.
Messages passed from
to
: According to the sum-product rule, we have
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
to
: According to the sum-product rule, we have
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
where
with
3.3.3. Check Nodes
Similar to the processing of
, two types of messages updated at
can be computed as
3.3.4. Variable Nodes
There are three types of messages updated at , passed to the check nodes , , and :
Messages passed from
to
:
Messages passed from
to
:
Messages passed from
to
:
3.4. Near-Field Localization Module
Using the messages
passed from the DPE module and the prior message of
, the NFL module focuses on constraint (
2) to estimate the user position
. Simultaneously, it calculates and feeds back the distance messages
to the DPE module.
3.4.1. Near-Field Localization
With the distance messages
and constraint (
2), we model
where
. The above equation can be further expressed as
To rewrite (
32) in a compact matrix form, we define
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
, yielding
where
denotes the a priori mean of
,
denotes the a priori covariance matrix of
,
denotes the covariance matrix between
and
, and
denotes the auto-covariance matrix of
.
After obtaining a posteriori estimate of the location parameter
, a posteriori estimate of the user location can be obtained:
3.4.2. Calculation of Feedback Messages
Based on the posterior estimate of
and the relationship between
and
, the posterior estimate of
can be approximated using the following Gaussian distribution:
where
and
satisfy
Thus, the extrinsic messages passed to the DPE module can be calculated as
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 repeat // ELSE Module for do Compute and by the modified VALSE alogrithm. Reconstruct by ( 13). Update and by ( 14) and ( 15). end // DPE Module for Update by ( 24)–( 26b) and ( 28). Update by ( 27b) and ( 29). Update messages passed from by ( 19)–( 21). Update by ( 23), ( 27a), and ( 30). end // NFL Module Update and according to ( 35a) and ( 35b). for Update by ( 37)–( 39). end until the maximum number of iterations T is reached. Output: channel estimate; user location estimate . |
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 for the single path case in this paper where is the number of iterations in the VALSE. The complexity of both the DPE module and the NFL module is . Thus, the total complexity of the algorithm is , 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
, where
and
. The CRB of
is equal to the inverse of the Fisher information matrix (FIM)
. Each element of the
can be computed as [
23]
where
and the computation of the corresponding partial derivatives satisfies
Then, the CRB for the user’s position in each direction can be calculated as , , and .
We can also derive the CRB for the estimation error of the user’s position as
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:
We denote the channel estimate by
. Then, the channel estimation error satisfies [
23]: