Near-Field Channel Estimation for Extremely Large-Scale Reconfigurable Intelligent Surface (XL-RIS)-Aided Wideband mmWave Systems

The research on near-field XL-RIS is still in the preliminary stage. The authors of [19, 20] provided complementary perspectives on the use of large intelligent surfaces and near-field beamforming for improving wireless communications in the near-field region. They both highlight the potential of these techniques to enhance the performance and efficiency of wireless communication systems, and provide insights into the design and optimization of these systems for different communication modes and scenarios. Since the focus of near-field RIS is on the spatial beamspace, several related topics have been studied, including beam training [21, 22, 23], localization [24, 25, 26, 27, 28], and channel estimation [26, 27, 29]. In the near-field region, angular beam training using the planar steering vector is no longer effective, making it necessary to redesign the codebook for beam training with the spherical steering vector. In the Cartesian coordinate system, two near-field codebooks have been designed, namely the near-field uniform codebook [21] and the near-field hierarchical codebook [21, 22]. In [21], it was shown that the near-field hierarchical codebook incurs significantly less training overhead than the near-field uniform codebook. Alternatively, the authors of [23] utilized the polar-domain codebook [30] to perform near-field beam training in the polar domain. On the other hand, near-field localization is a popular topic due to the advantageous spherical wavefront for distance sensing. In this context, various studies have investigated near-field localization with the XL-RIS system. Notably, due to the similarity between compressive channel estimation and localization, references [26] and [27] investigated near-field localization while also studying near-field channel estimation. In [29], the authors investigated near-field channel estimation for the hybrid beamforming system with the aid of the XL-RIS.

However, the above references just considered narrowband systems. In practical systems such as orthogonal frequency division multiplex (OFDM), wideband signals are commonly used, which will cause a well-known spatial effect called beam squint in phased array antennas. Beam squint occurs due to the change in the angle of incidence of the incoming wave with frequency, and it can have a significant impact on the performance of the wideband system. To mitigate the effects of beam squint, researchers have proposed effective methods for wideband channel estimation and hybrid beamforming, as demonstrated in [31, 32, 33, 34]. However, dealing with beam squint can be even more challenging in near-field XL-RIS systems. Using the polar-domain dictionary proposed in [30], researchers in [35] adopted the Kronecker compressive sensing (KCS) framework to estimate the wideband channel by exploiting common sparsity on each subcarrier. Moreover, the near-field beam squint effect was explored in [36], which unveiled two important findings in the polar domain: 1) the mathematical correlation between the beam of the central subcarrier and the beam of other subcarriers, and 2) the variation of beam trajectory with respect to frequency. To address beam squint issues in RIS systems at the hardware level, [37] proposed a delay adjustable metasurface (DAM) architecture, inspired by the true-time delay unit design of phased array antennas. The DAM architecture was subsequently used by [38] and [36] to mitigate the near-field beam squint effect. However, as the number of time delay units increases, the hardware cost and energy consumption also increase.

Previous research on near-field XL-RIS with beam squint considered uniform linear arrays (ULAs) for channel estimation or localization. However, in the near-field region, the beam squint effect of uniform planar arrays (UPAs) is reflected in the spherical domain, which presents particular challenges for signal processing such as spherical-domain dictionary design [39]. Last but not least, the XL-RIS system is sensitive to the algorithm complexity due to the numerous elements. In order to fully exploit the potential of XL-RIS technology for future communication applications, it is essential to improve channel estimation and other signal processing techniques.

Taking the aforementioned context into account, this paper focuses on the efficient estimation of near-field wideband channels in XL-RIS systems, capitalizing on both the channel’s sparsity in the spherical domain and the common sparsity support across all subcarriers. Our approach involves a thorough analysis of the near-field beam squint effect, the development of a wideband spherical-domain dictionary, the proposal of an efficient CS framework, and the derivation of a lower bound. The details of these endeavors are listed below.

• •

  To begin with, we employ the Fresnel approximation [7] to approximate the spherical array response in terms of the elevation/azimuth angle and distance. By separately analyzing the linear and quadratic phase terms, we establish a mathematical relationship between the desired beam at the central frequency and the corresponding beam at other frequencies, captured by the error function ${\rm erf}(\cdot)$. Our exploration of the near-field beam squint effect in the 3D space enables us to produce a comprehensive visualization of the beam trajectory across the frequency spectrum in the spherical domain. Furthermore, drawing on the error function, we formulate a wideband spherical-domain dictionary that minimizes the coherence between any two arbitrary dictionary atoms.

• •

  We propose the multi-frequency parallelizable subspace recovery (MMPSR) framework for solving the wideband channel estimation problem, using the designed wideband spherical-domain dictionary. This framework converts the 2D-CS problem into multiple sparse vector recovery problems, with multi-frequency joint processing. It is particularly suitable for recovering multi-parameter and large-scale dictionary problems, outperforming commonly used recovery frameworks such as Kronecker CS. To achieve optimal evaluation of the phase relation between the measured signal and the sensing atom, we propose correlation coefficient (CC)-based atom matching, as opposed to inner product (IN)-based atom matching used in some greedy algorithms like orthogonal matching pursuit (OMP).

• •

  Finally, we propose a 2D-OLS estimator to serve as a reliable benchmark that any estimation method cannot surpass. Furthermore, we derive its lower bound with the evaluation index of the normalized mean squared error (NMSE). Significantly, this lower bound accounts for multi-frequency channels, highlighting the influence of both system hyperparameters (such as the number of antennas, subcarriers, and channel paths) and the sensing matrix of each subcarrier on estimation performance.

The rest of this paper is organized as follows: Section II describes the wideband near-field channels and the uplink training model. Section III discusses the near-field beam squint effect and designs the wideband spherical-domain dictionary. Section IV proposes the CS framework to process the 2D recovery problem, and gives an efficient solution with CC-based atom matching. Section V derives the lower bound for the benchmark of any estimation method. Section VI conducts several experiments to demonstrate our proposed methods’ effectiveness. Finally, Section VII concludes this paper.

Notations: ${\left(\cdot\right)}^{*}$, ${\left(\cdot\right)}^{T}$, ${\left(\cdot\right)}^{H}$, and $\left(\cdot\right)^{-1}$ denote conjugate, transpose, conjugate transpose, and inverse, respectively. $\|\cdot\|_{2}$ represent $\ell_{0}$ norm and $\ell_{2}$ norm, respectively. $\|\mathbf{A}\|_{F}$ denotes the Frobenius norm of matrix $\mathbf{A}$. ${\rm Tr}\{\mathbf{A}\}$ denotes the trace of matrix $\mathbf{A}$. $|\cdot|$ denotes the modulus. Furthermore, $\otimes$ is the Kronecker product. $[\mathbf{a}]_{i}$ and $[\mathbf{A}]_{i,j}$ denote the $i$-th element of vector $\mathbf{a}$, the $(i,j)$-th element of matrix $\mathbf{A}$, respectively. $\rm{vec}(\cdot)$ represents the vectorization operation. $\mathbb{E}\{\cdot\}$ and $\mathbb{V}\{\cdot\}$ denote the expectation and variance operations , respectively. $\mathbf{I}_{M}$ denotes the $M$-by-$M$ identity matrix. Moreover, $\rm{diag}(\mathbf{a})$ is a square diagonal matrix with entries of $\mathbf{a}$ on its diagonal. Finally, $\mathcal{CN}(\mathbf{a},\mathbf{A})$ is the complex Gaussian distribution with mean $\mathbf{a}$ and covariance matrix $\mathbf{A}$.

Considering that the size of the RIS is much larger than that of the user, we utilize a one-side near-field channel model for the RIS-user channel. In the context, the RIS-user channel on the $k$-th subcarrier can be expressed with the Saleh-Valenzuela model:

where we make a far-field assumption at the user end. $P$ is the number of channel paths, $\{\tau_{p}\}_{p=1}^{P}$ and$\{\beta_{p}\}_{p=1}^{P}$ are the path delays and complex path gains, respectively. $r_{R,p}$ denotes the distance from the RIS to the $p$-th object (the scatter or the user). $\widetilde{\vartheta}_{U,p}\triangleq\cos(\vartheta_{U,p})$ and $\widetilde{\varphi}_{U,p}\triangleq\sin(\vartheta_{U,p})\sin(\varphi_{U,p})$ represent the virtual angles, where $\vartheta_{U,p}$ and $\varphi_{U,p}$ denote the $p$-th path’s elevation and azimuth angles from the user to the RIS, respectively. $f_{k}\triangleq f_{c}+\frac{f_{s}}{K}(k-1-\frac{K-1}{2})$, $k\in\{1,\cdots,K\}$, with $f_{c}$ being the central frequency and $f_{s}$ being the bandwidth, denotes the frequency of subcarrier $k$. Moreover, $\mathbf{b}_{\rm R}$ and $\mathbf{a}_{\rm U}$ are the spherical array response and planar array response, respectively. The spherical array response on subcarrier $k$ is given by

where $M^{z/y}_{R}\triangleq\frac{N^{z/y}_{R}-1}{2}$. As shown in Fig. 1, the distance between the $(m^{y}_{R},m^{z}_{R})$-th element with vector coordinate $(0,m^{y}_{R}d,m^{z}_{R}d)$ and the object with $(r\sin(\theta_{R})\cos(\phi_{R}),r\sin(\theta_{R})\sin(\phi_{R}),r\cos(\theta_{R}))$ is denoted by

where $(a)$ holds due to $\sqrt{1+x+y}\approx 1+\frac{x+y}{2}-\frac{(x+y)^{2}}{8}$.

With the far-field assumption, the user’s planar array response on subcarrier $k$ can be written as

where $M^{z/y}_{U}\triangleq\frac{N^{z/y}_{U}-1}{2}$.

Due to the long distance between the BS and the RIS, the far-field assumption can be used for simple approximation. The BS-RIS channel on the $k$-th subcarrier is modeled with a LoS path [26]:

where $\alpha$ is the complex path gain and $\tau_{0}$ is the path delay. $\widetilde{\theta}_{B}\triangleq\cos(\theta_{B})$ and $\widetilde{\phi}_{B}\triangleq\sin(\theta_{B})\sin(\phi_{B})$, in which $\theta_{B}$ and $\phi_{B}$ are the elevation and azimuth angles from the BS to the RIS, respectively. $\widetilde{\vartheta}_{R}$ and $\widetilde{\varphi}_{R}$ are defined similarly. Moreover, $\mathbf{a}_{\rm B/R}$ is the planar array response, which has a similar form of Eqn. (4).

In the TDD system, uplink channel estimation is performed through continuous transmission of pilots by the user and continuous adjustment of the phase matrix by the RIS. Assuming that $Q$ subframes are utilized for uplink training, where each subframe corresponds to one RIS phase matrix, the RIS fixes the phase matrix in each subframe, while the user transmits $N_{X}$ pilot sequences. As a result, $QN_{X}$ symbol durations are required for channel estimation of a single user. The $n$-th training beam on subcarrier $k$ is denoted as $\mathbf{f}_{n}[k]$ with $n=1,\cdots,N_{X}$, and the $q$-th RIS phase matrix on all subcarriers is represented as $\mathbf{V}_{q}\triangleq{\rm diag}(\mathbf{v}_{q})$, where $q=1,\cdots,Q$. The training model is described as

where $y_{q,n}[k]$ is the received training signal on subcarrier $k$, $\mathbf{f}_{n}[k]\triangleq f_{{\rm BB},n}[k]\mathbf{f}_{{\rm RF},n}\in\mathbb{C}^{N_{T}\times 1}$ is the hybrid precoder vector on subcarrier $k$ such that $\mathbf{f}^{H}_{n}[k]\mathbf{f}_{n}[k]=\sigma_{p}^{2}$ with $\sigma_{p}^{2}$ denoting the transmit power. $\mathbf{w}[k]\triangleq w_{\rm BB}[k]\mathbf{w}_{\rm RF}\in\mathbb{C}^{N_{R}\times 1}$ is the hybrid combiner vector on subcarrier $k$, and $\mathbf{n}_{q,n}[k]\in\mathbb{C}^{N_{R}\times 1}$ is the noise following $\mathcal{CN}(0,\sigma_{n}^{2}\mathbf{I}_{N_{R}})$. $f_{{\rm BB},n}[k]$ and $w_{\rm BB}[k]$ are the baseband coefficients. Note that $\mathbf{f}_{{\rm RF},n}$ and $\mathbf{w}_{\rm RF}$, the analog vectors produced by the UPA, are fixed at different subcarriers due to analog hardware limitations. With an appropriate location deployment of the BS and the RIS, the BS-RIS channel can be approximated as a LoS channel. In this way, we just need to estimate the RIS-user channel. Denoting the effective channel $\widetilde{\mathbf{h}}_{\rm B}[k]\triangleq\mathbf{w}^{H}[k]\mathbf{H}_{\rm B}[k]$ and the effective noise $\widetilde{n}_{q,n}[k]\triangleq\mathbf{w}^{H}[k]\mathbf{n}_{q,n}[k]$, where $\mathbf{w}[k]$ can be set towards the angle-of-arrival of the BS. Then, we have

By collecting the training signals in $Q$ subframes on subcarrier $k$, yielding

where $\mathbf{Y}[k]\in\mathbb{C}^{Q\times N_{X}}$, $\widetilde{\mathbf{V}}[k]\triangleq[\mathbf{v}_{1}^{T},\cdots,\mathbf{v}_{Q}^{T}]^{T}{\rm diag}(\widetilde{\mathbf{h}}_{\rm B}[k])\in\mathbb{C}^{Q\times L}$, $\mathbf{F}[k]\triangleq\left[\mathbf{f}^{T}_{1}[k],\cdots,\mathbf{f}^{T}_{N_{X}}[k]\right]^{T}\in\mathbb{C}^{N_{T}\times N_{X}}$, and $\widetilde{\mathbf{N}}[k]\in\mathbb{C}^{Q\times N_{X}}$.

Let the index of the central frequency $f_{c}$ be $\bar{k}\triangleq\frac{K+1}{2}$ such that $f_{\bar{k}}=f_{c}$. Given a desired beam on frequency $f_{c}$, the near-field effect indicates that the steering beams of other frequencies will have a spatial shift, as shown in Fig. 2. Next, we will derive the mathematical expression to describe the spatial shift.

Reference [7] showed that the Fresnel approximation, with neglection of terms greater than quadratic, was sufficient to characterize near-field communications. According to this, we can simplify the near-field array response. Defining the phase term $\delta_{k,(m^{y}_{R},m^{z}_{R})}\triangleq\frac{f_{k}}{c}(-m^{y}_{R}d\widetilde{\phi}-m^{z}_{R}d\widetilde{\theta}+\frac{(m^{z}_{R}d)^{2}}{2r}(1-\widetilde{\theta}^{2})+\frac{(m^{y}_{R}d)^{2}}{2r}(1-\widetilde{\phi}^{2})-\frac{m^{y}_{R}m^{z}_{R}d^{2}\widetilde{\theta}\widetilde{\phi}}{r})$ by simplifying the phase term of Eqn. (2). Then, the spherical array response is defined as

Suppose that the $\bar{k}$-th subcarrier’s phase with a desired beam $(\widetilde{\theta}_{\bar{k}},\widetilde{\phi}_{\bar{k}},{r}_{\bar{k}})$ is defined by $\delta_{\bar{k},(m^{y}_{R},m^{z}_{R})}\triangleq\frac{f_{\bar{k}}}{c}(-m^{y}_{R}d\widetilde{\phi}_{\bar{k}}-m^{z}_{R}d\widetilde{\theta}_{\bar{k}}+\frac{(m^{z}_{R}d)^{2}}{2r}(1-\widetilde{\theta}_{\bar{k}}^{2})+\frac{(m^{y}_{R}d)^{2}}{2r}(1-\widetilde{\phi}_{\bar{k}}^{2})-\frac{m^{y}_{R}m^{z}_{R}d^{2}\widetilde{\theta}_{\bar{k}}\widetilde{\phi}_{\bar{k}}}{r})$. The pattern function pointing to the beam $(\widetilde{\theta}_{\bar{k}},\widetilde{\phi}_{\bar{k}},{r}_{\bar{k}})$ on subcarrier $k$, denoted by $\bm{\mathcal{S}}(\widetilde{\theta},\widetilde{\phi},r)[k]$ with $\{\widetilde{\theta},\widetilde{\phi}\}\in[-1,1]$ and $r\in(0,+\infty)$, is expressed as

where $I_{k,(m_{R}^{y},m_{R}^{z})}$ and $\bm{\mathcal{E}}_{(m_{R}^{y},m_{R}^{z})}[k]$ are the excitation amplitude and the element pattern of the $(m_{R}^{y},m_{R}^{z})$-th antenna on subcarrier $k$, respectively. As the passive RIS structure is considered, we have $I_{k,(m_{R}^{y},m_{R}^{z})}=1$. Moreover, we assume each antenna element on each subcarrier is isotropic such that $\bm{\mathcal{E}}_{(m_{R}^{y},m_{R}^{z})}(\widetilde{\theta},\widetilde{\phi},r)[k]=1$.

To make the pattern independent of the subcarrier index, we have $e^{-j{2\pi}\left(\delta_{k_{1},(m^{y}_{R},m^{z}_{R})}-\delta_{\bar{k},(m^{y}_{R},m^{z}_{R})}\right)}=e^{-j{2\pi}\left(\delta_{k_{2},(m^{y}_{R},m^{z}_{R})}-\delta_{\bar{k},(m^{y}_{R},m^{z}_{R})}\right)}$, where $k_{1},k_{2}\in\{1,\cdots,K\}$, $k_{1}\neq k_{2}$. This equation is equivalent to

where $Z$ is any integer. By assuming $Z=0$, $k_{1}=k$, and $k_{2}=\bar{k}$, we can calculate $\bar{\delta}_{k}\triangleq\delta_{k,(m^{y}_{R},m^{z}_{R})}-\delta_{\bar{k},(m^{y}_{R},m^{z}_{R})}$ as

Now, the problem is equivalent to maximizing $\left|\bm{\mathcal{S}}(\widetilde{\theta},\widetilde{\phi},r)[k]\right|$ when assigning zero to $\bar{\delta}_{k}$. In this way, the method in [36] used for ULAs is extended to calculate the $k$-the subcarrier’s $(\widetilde{\theta},\widetilde{\phi},r)$ that is independent on the antenna index. By assigning zero to $\bar{\delta}_{k}$, the following system of equations is derived:

Nevertheless, the system of equations under consideration fails to yield a unique solution due to the interdependent nature of the elevation and azimuth angles on a shared distance parameter, leading to a spatial mismatch. To resolve this issue, we conduct a sequential analysis of each component of the near-field phase.

For channel estimation using the CS technique, the virtual channel representation with dictionaries is a critical component. One commonly used method for designing an effective dictionary $\mathbf{A}\in\mathbb{C}^{N\times G}$ is to minimize the total coherence $\mu=\sum_{g_{1}}^{G}\sum_{g_{2},g_{2}\neq g_{1}}^{G}\left|[\mathbf{A}]_{:,g_{1}}^{H}[\mathbf{A}]_{:,g_{2}}\right|$, which is equivalent to minimizing $\left|[\mathbf{A}]_{:,g_{1}}^{H}[\mathbf{A}]_{:,g_{2}}\right|$ for $\forall g_{1},g_{2}$. To design the dictionaries $\mathbf{A}_{\rm U}[k]$ and $\mathbf{B}_{\rm R}[k]$, we first derive them with a narrowband system at frequency $f_{c}$ and then obtain their wideband form.

It is well established that the 2D discrete Fourier transform (2D-DFT) matrix can be adopted to design the 2D angle sampling for dictionary $\mathbf{A}_{\rm U}[\bar{k}]$, which is given by

where $G_{U}^{y}G_{U}^{z}=G_{U}$ is the total number of atoms.

We aim to orthogonalize two arbitrary atoms (with indices $g_{1}$ and $g_{2}$, where $g_{1}\neq g_{2}$) in $\mathbf{B}_{\rm R}[\bar{k}]$ by minimizing the coherence, $\mu_{g_{1},g_{2}}$, approximating it to $0$, where

This method is similar to the one described in Eqn. (15), with the difference being that Section III-A focuses on exploring the maximal pattern gain on different subcarriers, while Section III-B discusses the minimal coherence of arbitrary atoms in a dictionary independent of the subcarrier index.

As the angle and distance parameters are coupled in $\mu_{g_{1},g_{2}}$, we separate the sampling of angle and distance, as done in [30]. By considering only the linear term of $\mu_{g_{1},g_{2}}$ and assigning $d=\frac{c}{2f_{c}}$, it can be expressed as

In this case, the method for sampling angles is the same as that described in Eqn. (21), which is designed using the 2D-DFT matrix.

After sampling the angle space, we focus on sampling the distance of each angle point. Since the angle is fixed ($\widetilde{\theta}_{g_{1}}=\widetilde{\theta}_{g_{2}}=\widetilde{\theta}$ and $\widetilde{\phi}_{g_{1}}=\widetilde{\phi}_{g_{2}}=\widetilde{\phi}$), the distance sampling becomes solvable. The coherence, $\mu_{g_{1},g_{2}}$, can now be re-written as