RIS-aided Near-Field MIMO Communications: Codebook and Beam Training Design

Signals in the far field of an array lead to planar wavefronts that produce linear phase variations and negligible amplitude differences across the array. However, the situation is different for near-field signals. In order to achieve better performance through reasonable resource allocation, accurate channel models are essential for NFC. Hybrid-field channel modeling schemes were studied in [21] and [22], where the far-field and near-field components are separately estimated. A polar-domain representation for near-field channel models was proposed in [23], accounting for the information in both the angular and distance domains. Considering the different multi-path characteristics observed by antennas at different distances in large-scale MIMO NFC systems, the authors in [24] proposed a non-stationary distance modeling framework.

The beam training process is implemented by searching for the optimal beam from multiple predefined directional beams (codewords). For beam training, testing all codewords in the codebook during the training process is the most straightforward approach. However, the codebook and the resulting training overhead can be extremely large for large-scale massive MIMO systems, especially when considering both angular and distance information in NFC. The authors in [15] proposed a new hierarchical codebook to achieve uniform beam alignment performance with low overhead for mmWave communication systems. For mmWave and sub-THz systems with multiple users, [4] and [16] proposed hierarchical beamforming training strategies to carry out simultaneous training for multiple users. Aiming at overcoming the resolution limit of directional angle estimation, [25] proposed a codebook-based beam search approach for mmWave massive MIMO transmissions. However, these codebooks are designed for the far-field case. Considering the spherical wavefront propagation model for NFC, [17, 26] proposed a two-phase beam training method to sequentially perform the beam sweeping in the angular and distance domains.

To implement beam training for RIS-based systems, one must first construct an appropriate RIS codebook based on information about the cascaded channel. Then, based on the information obtained during the training phase, the RIS selects the optimal codeword from the predesigned codebook to perform passive beamforming. For RISs with massive numbers of elements, highly efficient beam training schemes with good performance are required to reduce the training overhead. The authors in [19] first proposed a hierarchical codebook-generating method using pattern synthesis, then provided a hierarchical beam training method using two multi-mainlobe codewords in each layer. To train multiple users at the same time, a multi-lobe beam training mechanism was proposed in [27] based on the BS-RIS joint codebook, thus achieving reduced overhead. A multi-beam training method was proposed for a RIS-assisted multiuser system in [28], by dividing the RIS into multiple sub-surfaces and designing multi-beam directions over time. Although [20, 29, 30] proposed beam training schemes for RIS-assisted NFC systems, only limited scenarios with single-antenna users were considered.

When the far-field channel model is considered, the channel between the BS and RIS is characterized by the Saleh-Valenzuela channel model [32], given by

where $L_{\rm{B}}$ is the number of paths between the BS and RIS, ${\beta_{0}}$ represents the complex gain of the line-of-sight (LoS) component, ${\beta_{l}}\left(1\leq l\leq L_{\rm{B}}-1\right)$ denotes the complex gain of the $l$-th non-line-of-sight (NLoS) path [33], ${\alpha_{l}^{\rm{A}}}$ and $\varphi_{l}^{\rm{A}}$ denote the azimuth and elevation angle of arrival (AoA) associated with the RIS, ${\alpha_{l}^{\rm{D}}}$ denotes the angle of departure (AoD) associated with the BS, and ${\bf{a}}_{\rm{R}}\left({\alpha_{l}^{\rm{A}}},{\varphi_{l}^{\rm{A}}}\right)$ and ${{\bf{a}}_{\rm{B}}}\left({\alpha_{l}^{\rm{D}}}\right)$ are the array response vectors associated with the RIS and the BS, respectively.

For a UPA with $M_{x}$ and $M_{y}$ elements (${M_{x}}{M_{y}}=M$) on the horizontal and the vertical axes respectively, the RIS array response vector is given by

where $0\leq x\leq(M_{x}-1)$ and $0\leq y\leq(M_{y}-1)$, and $\otimes$ is the Kronecker product. For the ULA with $N_{\rm{B}}$ elements at the BS, the array response vector is expressed as [34]

where $0\leq n_{b}\leq(N_{\rm{B}}-1)$. Since the response vector of a ULA is constant in the elevation domain, we omit the parameter ${\varphi}$ in the response vector of the BS in (3).

Similarly, the channel between the RIS and user in the far-field region is given by

where $L_{\rm{U}}$ is the number of paths between the RIS and user, ${\alpha_{l}^{{\rm{A}}}}$ denotes the azimuth AoA associated with the user, ${\alpha_{l}^{\rm{D}}}$ and ${\varphi_{l}^{\rm{D}}}$ denote the azimuth and elevation AoD associated with the RIS respectively, and the array response vectors ${\bf{a}}_{\rm{R}}\left({\alpha_{l}^{\rm{D}}},{\varphi_{l}^{\rm{D}}}\right)$ and ${\bf{a}}_{\rm{U}}\left({\alpha_{l}^{{\rm{A}}}}\right)$ can be generated similarly to (2) and (3).

For communications in the near-field domain, we consider a three-dimensional topology. In FFC, the signal propagation paths from transmitters to receivers can be regarded as parallel to each other if they share the same AoA. However, in NFC, the angle distribution is also related to the distance between specific points of transmitters and receivers. Assume that all the BS antennas are located on the $x$-axis, where the coordinate of the midpoint of the BS array is denoted as $\left(0,0,0\right)$. Thus, the coordinate of the $n_{b}$-th antenna at the BS can be denoted as

where ${{\tilde{n}}_{b}}=n_{b}-\frac{N_{\rm{B}}-1}{2}$.

Similarly, assume that all of the user antennas are parallel to the $x$-axis, where the coordinate of the midpoint of the user array is $\left(x_{\rm{U}},y_{\rm{U}},z_{\rm{U}}\right)$. Thus, the location of $n_{u}$-th antenna at the user can be denoted as

where ${{\tilde{n}}_{u}}=n_{u}-\frac{N_{\rm{U}}-1}{2}$.

Assume that all elements of the RIS are parallel to the XY-plane. Denote the coordinate of the midpoint of the RIS as $\left(x_{\rm{R}},y_{\rm{R}},z_{\rm{R}}\right)$. Thus, the coordinate of the $\left(m_{x},m_{y}\right)$-th element of the RIS can be expressed as

where ${{\tilde{m}}_{x}}=m_{x}-\frac{M_{x}-1}{2}$ and ${{\tilde{m}}_{y}}=m_{y}-\frac{M_{y}-1}{2}$. Further, assume that the location and topology of the RIS are known.

Based on the above geometry, the LoS link for the near-field channel between BS and RIS can be modeled as [36]

where ${\bf{g}}_{{\rm{B}},{\rm{R}}}^{m_{xy}}=\Big{[}\kappa_{m_{xy},1}e^{-j\frac{{2\pi}}{\lambda_{c}}d_{m_{xy},1}^{\rm{B}}},\cdots,$ $\kappa_{m_{xy},N_{\rm{B}}}e^{-j\frac{{2\pi}}{\lambda_{c}}d_{m_{xy},N_{\rm{B}}}^{\rm{B}}}\Big{]}^{T}$ and $\kappa_{m_{xy},n_{b}}\propto\frac{1}{d_{m_{xy},n_{b}}^{\rm{B}}}$ denotes the free-space large-scale path loss between the $n_{b}$-th antenna of the BS and the $\left(m_{x},m_{y}\right)$-th element of the RIS. The distance between the $n_{b}$-th antenna of the BS and the $\left(m_{x},m_{y}\right)$-th element of the RIS is given by

which can be expanded as

with $r_{m_{xy}}^{\rm{B}}$ denoting the distance from $\left(0,0,0\right)$ and the $\left(m_{x},m_{y}\right)$-th element of the RIS, $\alpha_{m_{xy}}^{\rm{B}}$, $\varphi_{m_{xy}}^{\rm{B}}$ denoting the corresponding azimuth and elevation angle, respectively, and ${\left(a\right)}$ is obtained based on the first-order Taylor expansion $\sqrt{1+x}=1+\frac{1}{2}x-\frac{1}{8}x^{2}+{\cal{O}}\left(x^{3}\right)$.

Similarly, the LoS near-field channel between the RIS and user can be modeled as

where ${\bf{g}}_{{\rm{R}},{\rm{U}}}^{n_{u}}=\left[\kappa_{n_{u},1}e^{-j\frac{{2\pi}}{\lambda_{c}}d_{n_{u},1}^{\rm{U}}},\cdots,\kappa_{n_{u},M}e^{-j\frac{{2\pi}}{\lambda_{c}}d_{n_{u},M}^{\rm{U}}}\right]^{T}$ and $\kappa_{n_{u},m_{xy}}\propto\frac{1}{d_{n_{u},m_{xy}}^{\rm{U}}}$ denote the free-space large-scale path loss between the $n_{u}$-th antenna of the user and the $\left(m_{x},m_{y}\right)$-th element of the RIS. The distance between the $n_{u}$-th antenna of the user and the $\left(m_{x},m_{y}\right)$-th element of the RIS is given by

which can be approximated as

with $r_{m_{xy}}^{\rm{U}}$ denoting the distance from the midpoint of the user array and the $\left(m_{x},m_{y}\right)$-th element of the RIS, $\alpha_{m_{xy}}^{\rm{U}}$, $\varphi_{m_{xy}}^{\rm{U}}$ denoting the corresponding azimuth and elevation angle, respectively.

Denote the phase-shift matrix of the RIS as ${\bf{\Theta}}={\rm diag}\big{(}\phi_{1},\cdots,$ $\phi_{m},\cdots,\phi_{M}\big{)}$, with $\phi_{m}=e^{-j\theta_{m}},\theta_{m}\in\left[0,2\pi\right)$ denoting the phase-shift coefficient of the $m$-th RIS element. Therefore, the cascaded channels of the four scenarios mentioned above can be expressed as

To align the phase on the FFC side, the full angular range will be covered in the predesigned codebook. Therefore, the first component of the RIS codebook for hybrid cascaded channel models is given by

where $\beta_{m_{x}}=\frac{2{m_{x}}-{M_{x}}-1}{M_{x}}$ with ${m_{x}}=1,\cdots,{M_{x}}$, $\delta_{m_{y}}=\frac{2{m_{y}}-{M_{y}}-1}{M_{y}}$ with ${m_{y}}=1,\cdots,{M_{y}}$.

For each given angle pair $\beta$ and $\delta$, all possible sampling points in the feasible range should be included to align the phase on the NFC side. Therefore, the second component of the RIS codebook for hybrid cascaded channel models is given by

where $x_{s_{x}}^{k}$ and $y_{s_{x}}^{k}$, $k\in\left\{{\rm{B}},{\rm{U}}\right\}$, are defined as in (26). In particular, $k={\rm{B}}$ for the NF case, and $k={\rm{U}}$ for the FN case.

Accordingly, the codebook at the RIS for hybrid cascaded channel models can be designed as

where the definition of $\star$ is given in (24).

The most direct method to perform beam training for this case is sweeping through all possible angles if the resulting training overhead is acceptable. The size of the FF codebook in (20) depends on the number of RIS reflecting elements. Based on the given angle pair $\left\langle{\beta_{m_{x}},\delta_{m_{y}}}\right\rangle$, the corresponding codeword is ${{\bf{b}}}\left({\beta_{m_{x}},\delta_{m_{y}}}\right)$, and the achievable rate can be calculated as

with given beamforming matrix ${\bf{W}}$. The optimal codeword is selected as the one that yields the maximum rate after sweeping through all candidate codewords in (20).

From (25) and (26), we see that the NN codebook size is determined by the number of sampled points on the $x$-axis and $y$-axis. The beam training overhead can be reduced by reducing the codebook size (i.e., increasing the step size between the codebook samples), but this will reduce the performance of the beam training. In order to solve this problem, we design a hierarchical beam training approach that consists of several different levels of sub-codebooks. These sub-codebook levels are determined using different sampling ranges and sampling step sizes.

As illustrated in Fig. 2, we divide the sampling range parallel to the XY-plane into four sub-ranges in each layer, and find the range corresponding to the optimal codeword as the sampling range of the next layer. More particularly, in the $l$-th layer, the rectangular sampling range defined by the coordinate pairs $\left(x^{k}_{\min},y^{k}_{\min}\right)$ and $\left(x^{k}_{\max},y^{k}_{\max}\right)$, $k\in\left\{{\rm{B}},{\rm{U}}\right\}$, is divided into the following four sub-ranges:

where $x_{\Delta}=\frac{1}{2}\left[x^{k}_{\max}-x^{k}_{\min}\right]$ and $y_{\Delta}=\frac{1}{2}\left[y^{k}_{\max}-y^{k}_{\min}\right]$. Therefore, the sub-codebook for sub-range $i$ of ${\rm{B}}$ and sub-range $j$ of ${\rm{U}}$ in the $l$-th layer, denoted by ${\cal{F}}_{\rm{NN}}\left(l,i,j\right),i,j=1,2,3,4$, can be generated based on (25). Each column in ${\cal{F}}_{\rm{NN}}\left(l,i,j\right)$ corresponds to a codeword ${\bm{\phi}}_{l,i,j,s}$, with $s$ denoting the column index in sub-codebook ${\cal{F}}_{\rm{NN}}\left(l,i,j\right)$. Based on the given beamforming matrix ${\bf{W}}$ and codeword ${\bm{\phi}}_{l,i,j,s}$, the achievable rate can be calculated as

By adopting the ES method, the optimal codeword can be obtained for the first layer, as well as the optimal sampling sub-range pair $\left\langle i,j\right\rangle$, which will be the sampling range in the next layer. This process is repeated until the maximum number of layers $L$ is reached.

The hierarchical beam training procedure is summarized in Algorithm 1. We define the size of the codebook as the number of candidate codewords, i.e., the number of column vectors. There are 16 sub-codebooks in each layer, and the size of each sub-codebook is $\left(S_{x}S_{y}\right)^{2}$. Therefore, with a given maximum number of training layers $L$, the training overhead can be calculated as $16L\left(S_{x}S_{y}\right)^{2}$. For the same sampling grid resolution and hence beam training accuracy, the training overhead of ES will be $4^{L+1}\left(S_{x}S_{y}\right)^{2}$.

The size of codebooks ${\cal{F}}_{\rm{NF}}$ and ${\cal{F}}_{\rm{FN}}$ is determined by the product of the number of RIS elements and the number of sample points, which can be extremely large especially for large RISs. Thus, while ES is a straightforward way to find the optimal codeword from the predefined codebooks, the training overhead and consumed time can be unacceptable. We therefore propose a two-stage beam training scheme for the FN and NF channel models. Specifically, all candidate angles are swept through in stage one, and the distance is determined in stage two based on the selected direction. The details are given as follows.

Stage 1 Angular-domain beam sweeping: In the first stage, we aim to find the optimal beamforming angle based on the far-field codebook given in (20), with the objective of maximizing the achievable rate in (17). All sampled angles will be swept through in this stage, and the one yielding the maximum achievable rate will be selected. Therefore, the selected indices in the angular-domain sweeping stage can be expressed mathematically as

where ${\bf{\Theta}}_{{m_{x},m_{y}}}={\rm diag}\left({{\bf{b}}}\left({\beta_{m_{x}},\delta_{m_{y}}}\right)\right)$. Denote the optimal index pair as $\left\langle m^{\prime}_{x},m^{\prime}_{y}\right\rangle$.

Stage 2 Distance-domain sampling: In NFC, the beam can be designed to point at a particular location, rather than just a specific angle. Thus, in the second stage, we aim to determine the optimal focusing point for the optimal direction. With a given angle pair $\left\langle m^{\prime}_{x},m^{\prime}_{y}\right\rangle$, the selected sampling point in the distance domain can be determined by

where ${\cal{D}}_{k}$ lies in the sampling range for $k$, $k\in\left\{{\rm{B}},{\rm{U}}\right\}$, and ${\bf{\Theta}}_{x_{s_{x}}^{k},y_{s_{y}}^{k}}={\rm diag}\left({{\bf{c}}}\left(x_{s_{x}}^{k},y_{s_{y}}^{k},\beta_{m^{\prime}_{x}},\delta_{m^{\prime}_{y}}\right)\right)$.

Similar to the analysis in Section IV-A2, we adopt a hierarchical beam training scheme in this stage to strike a balance between the training overhead and accuracy. Thus, the two-stage beam training procedure is summarized in Algorithm 2. The overhead of the proposed two-stage beam training scheme is $M+4L{S_{x}}{S_{y}}$, while that of the ES method is $M\times 4L{S_{x}}{S_{y}}$.