Error Propagation and Overhead Reduced Channel Estimation for RIS-Aided Multi-User mmWave Systems

Estimating the AoA steering matrix of the common RIS-BS channel, i.e., $\mathbf{A}_{N}$, from (7) is a classical directional of arrival (DOA) estimation problem in array processing, and can be solved by many mature signal processing techniques [4]. Due to the large scale antenna arrays employed at the BS with typically $L\ll N$, the DFT-based method in [8] can be adopted to obtain the estimate of the common AoA steering matrix.

Denote the estimate of $\mathbf{A}_{N}$ as $\mathbf{\widehat{A}}_{N}=[\mathbf{a}_{N}(\widehat{\psi}_{1}),\ldots,\mathbf{a}_{N}(\widehat{\psi}_{L})]\in\mathbb{C}^{N\times L}$. Due to the property that $\mathrm{rank}(\mathbf{\widehat{A}}_{N})=L$ [9], by substituting $\mathbf{A}_{N}=\mathbf{\widehat{A}}_{N}+\Delta\mathbf{A}_{N}$, we can obtain the equivalent received matrix $\mathbf{\widehat{A}}_{N}^{\mathrm{\dagger}}\mathbf{Y}$ as

where $\Delta\mathbf{A}_{N}\triangleq\mathbf{A}_{N}-\mathbf{\widehat{A}}_{N}$ is treated as the estimation error between the common AoA and its estimate, and $\bar{\mathbf{N}}\triangleq\mathbf{N}+\Delta\mathbf{A}_{N}\boldsymbol{\Lambda}\mathbf{A}_{M}^{\mathrm{H}}\mathrm{Diag}\{\mathbf{h}\}\mathbf{E}$ represents the equivalent noise matrix.

With the estimated common AoA, i.e., $\mathbf{\widehat{A}}_{N}$, a correlation relationship between different paths in the common RIS-BS channel will be revealed, which helps us to construct the ambiguous common RIS-BS channel. Specifically, calculating the $l$-th row and the $r$-th row of $\mathbf{\widehat{A}}_{N}^{\mathrm{\dagger}}\mathbf{Y}$ in (8), and then taking their conjugate transpose, we have

where $\mathbf{\tilde{n}}_{l}\triangleq[(\mathbf{\widehat{A}}_{N}^{\mathrm{\dagger}}\bar{\mathbf{N}})_{l,:}]^{\mathrm{H}}$ and $\mathbf{\tilde{n}}_{r}\triangleq[(\mathbf{\widehat{A}}_{N}^{\mathrm{\dagger}}\bar{\mathbf{N}})_{r,:}]^{\mathrm{H}}$. Apparently, the dominant terms of $[(\mathbf{\widehat{A}}_{N}^{\mathrm{\dagger}}\mathbf{Y})_{l,:}]^{\mathrm{H}}$ and $[(\mathbf{\widehat{A}}_{N}^{\mathrm{\dagger}}\mathbf{Y})_{r,:}]^{\mathrm{H}}$, i.e., $\mathbf{E}^{\mathrm{H}}\mathrm{Diag}\{\mathbf{h}^{*}\}\mathbf{a}_{M}(\omega_{l})\alpha_{l}^{*}$ and $\mathbf{E}^{\mathrm{H}}\mathrm{Diag}\{\mathbf{h}^{*}\}\mathbf{a}_{M}(\omega_{r})\alpha_{r}^{*}$, contain the whole information regarding the $l$-th path and the $r$-th path, respectively, and there exists a relationship between these two terms, reflecting the correlation between the corresponding two paths.

To illustrate this correlation relationship, define a matching matrix $\mathbf{A_{\mathit{l}}}$ for the $l$-th path with respect to the $r$-th path as

where $\mathbf{A}(\varpi_{l})\triangleq[\mathbf{U}_{V}\odot(\mathbf{a}_{V}(\varpi_{l})\mathbf{1_{\mathit{V}}^{\mathrm{T}}})]^{\mathrm{H}}\frac{1}{V}\mathbf{U}_{V}\in\mathbb{C}^{V\times V}$ is a complex nonlinear function of $\varpi_{l}$. $\mathbf{U}_{V}$ is a $V\times V$ DFT matrix with the $(n,m)$-th entry given by $[\mathbf{U}_{V}]_{n,m}=e^{-\mathrm{i}\frac{2\pi}{V}(n-1)(m-1)}$. $\mathbf{a}_{V}(\varpi_{l})$ can be regarded as the array manifold with dimension $V$ and $\mathbf{1}_{\mathit{V}}$ is an all-one vector of size $V$. Here, the $r$-th path is treated as the reference path.¹¹1The reference index $r$ can be chosen based on the maximum received power criterion, i.e., $r=\mathrm{arg}\max_{p\in[1,L]}||[(\mathbf{\widehat{A}}_{N}^{\mathrm{\dagger}}\mathbf{Y})_{p,:}]^{\mathrm{H}}||^{2}$. $\varpi_{l}$ and $x_{l}$ are the rotation factor and scaling factor for the $l$-th path with respect to the $r$-th path, respectively, which are given by

Clearly, $\varpi_{l}\in[-2\frac{d_{\mathrm{RIS}}}{\lambda_{c}},2\frac{d_{\mathrm{RIS}}}{\lambda_{c}}]$. During this stage, the RIS phase shift matrix $\mathbf{E}$ in (7) needs to be designed carefully, which should satisfy the following structure

Where $\mathbf{U}_{V}\in\mathbb{C}^{V\times V}$ is a DFT matrix defined in (11), satisfying the constant modulus constraint. To show how the matching matrix $\mathbf{A_{\mathit{l}}}$ works intuitively, the noise terms of (9) and (10) are momentarily omitted. Then, we have

where the third equality is obtained using $(\mathbf{yx^{\mathrm{H}}\mathrm{)}\odot A=\mathrm{Diag}\{y\}A\mathrm{Diag}^{*}\{x\}}$. From Eq. (14), the term $\mathbf{E}^{\mathrm{H}}\mathrm{Diag}\{\mathbf{h}^{*}\}\mathbf{a}_{M}(\omega_{l})\alpha_{l}^{*}$ can be expressed as the result of the term $\mathbf{E}^{\mathrm{H}}\mathrm{Diag}\{\mathbf{h}^{*}\}\mathbf{a}_{M}(\omega_{r})\alpha_{r}^{*}$ via the linear transformation $\mathbf{A_{\mathit{l}}}$. Thus the correlation between the $l$-th path and the $r$-th path is depicted by the two variables of matching matrix $\mathbf{A_{\mathit{l}}}$, i.e., $\varpi_{l}$ and $x_{l}$. By analogy, we conclude that the correlation relationship between any two paths in the common RIS-BS channel can be described by their rotation factors and scaling factors. Now we will show how to estimate these two kinds of factors, i.e., rotation factors and scaling factors, from the equivalent received signal $\mathbf{\widehat{A}}_{N}^{\mathrm{\dagger}}\mathbf{Y}$.

We still take the $l$-th path and the $r$-th path, i.e., the reference path, as an example to illustrate the method for estimating the rotation factors and scaling factors. Similar to Eq. (14), we process the received signal $[(\mathbf{\widehat{A}}_{N}^{\mathrm{\dagger}}\mathbf{Y})_{r,:}]^{\mathrm{H}}$ in (10) via the linear transformation $\mathbf{A_{\mathit{l}}}$ as

where $\Delta\mathbf{\tilde{n}}_{r}\triangleq-\mathbf{A_{\mathit{l}}}\mathbf{\tilde{n}}_{r}$. By replacing $\mathbf{E}^{\mathrm{H}}\mathrm{Diag}\{\mathbf{h}^{*}\}\mathbf{a}_{M}(\omega_{l})\alpha_{l}^{*}=\mathbf{A_{\mathit{l}}}[(\mathbf{\widehat{A}}_{N}^{\mathrm{\dagger}}\mathbf{Y})_{r,:}]^{\mathrm{H}}+\Delta\mathbf{\tilde{n}}_{r}$ and $\mathbf{A_{\mathit{l}}}=\mathbf{A}(\varpi_{l})x_{l}$, (9) is re-expressed as

Here, $\mathbf{n}_{\mathrm{noise}}\triangleq\Delta\mathbf{\tilde{n}}_{r}+\mathbf{\tilde{n}}_{l}$ represents the corresponding noise vector. Moreover, denote $[(\mathbf{\widehat{A}}_{N}^{\mathrm{\dagger}}\mathbf{Y})_{l,:}]^{\mathrm{H}}$ as $\mathbf{\tilde{y}}_{l}$ and $\mathbf{A}(\varpi_{l})[(\mathbf{\widehat{A}}_{N}^{\mathrm{\dagger}}\mathbf{Y})_{r,:}]^{\mathrm{H}}$ as $\mathbf{B}(\varpi_{l})\in\mathbb{C}^{V}$, (16) can be written as

So our aim is to estimate the rotation factor $\varpi_{l}$ and scaling factor $x_{l}$ with the received $\mathbf{\tilde{y}}_{l}$, which can be achieved by solving the problem shown below as

where $\mathbf{B}(\varpi)$ is an extremely complex nonlinear function with respect to $\varpi$. This problem is known as a separable nonlinear least squares (SNL-LS) problem, in which the optimal $\varpi_{l}$ and $x_{l}$ can be found via the following steps.

First, when the $\varpi$ is fixed, the optimal $x$ is given by the least square (LS) solution, i.e., $x_{\mathrm{opt}}=\mathbf{B}^{\dagger}(\varpi)\mathbf{\tilde{y}}_{l}$. Then, by substituting $x_{\mathrm{opt}}$ into Problem (18), the optimal $\varpi$ can be acquired by solving the new problem as

where the objective function of Problem (19), i.e., $||\mathbf{\tilde{y}}_{l}-\mathbf{B}(\varpi)\mathbf{B}^{\dagger}(\varpi)\mathbf{\tilde{y}}_{l}||^{2}$, can be simplified as

where $\phi(\varpi)\triangleq[\mathbf{B}^{\mathrm{H}}(\varpi)\mathbf{B}(\varpi)]^{-1}$. With Eq. (20), Problem (19) is equivalent to the optimization problem as

For Problem (21), one-dimension search method can be adopted to obtain the optimal $\varpi$, i.e., $\varpi_{l}$, whose performance depends on the number of search grids. Once the rotation factor $\varpi_{l}$ is obtained by solving Problem (21), the scaling factor $x_{l}$ is given by

Finally, the rotation factors and the scaling factors for any paths in the common RIS-BS channel with respect to the reference path, i.e., the $r$-th path, can be estimated. Specifically, for $\forall l\neq r$, the rotation factors and the scaling factors between the $l$-th path and the $r$-th path, i.e., $\varpi_{l}$ and $x_{l}$, are obtained via the solution to Problem (21) and Eq. (22). While for $l=r$, according to the definitions in (12), the corresponding rotation factor and the scaling factor are given by $\varpi_{r}=0$ and $x_{r}=1$.

In previous subsections, we have shown that the rotation factors and scaling factors, i.e., $\varpi_{l}$ and $x_{l}$ for $\forall l=\{1,2,...,L\}$, can be estimated effectively by all users jointly with the designed $\mathbf{E}$. Based on the obtained parameters, an ambiguous complex gain matrix of the common RIS-BS channel, i.e., denoted by $\boldsymbol{\Lambda}_{\mathrm{s}}$, and an ambiguous AoD steering matrix of the common RIS-BS channel, i.e., denoted by $\mathbf{A}_{\mathrm{s}}$, are constructed as

It is observed that the relationship between $\boldsymbol{\Lambda}_{\mathrm{s}}$ and $\boldsymbol{\Lambda}$, and the relationship between $\mathbf{A}_{\mathrm{s}}$ and $\mathbf{A}_{M}$ can be described as

With the obtained $\mathbf{A}_{N}$, $\boldsymbol{\Lambda}_{\mathrm{s}}$ and $\mathbf{A}_{\mathrm{s}}$, the corresponding ambiguous common RIS-BS channel, denoted by $\mathbf{H}_{\mathrm{s}}$, is naturally constructed as

Then, substituting $\boldsymbol{\Lambda}=\alpha_{r}\boldsymbol{\Lambda}_{\mathrm{s}}$ and $\mathbf{A}_{M}=\mathrm{Diag}\{\mathbf{a}_{M}(\omega_{r})\}\mathbf{A}_{\mathrm{s}}$ into $\mathbf{G}_{k}$ in (5), we have

where $\mathbf{A}_{\mathrm{s},k}\triangleq\mathrm{Diag}\{\mathbf{a}_{M}(-\omega_{r})\}\mathbf{A}_{M,k}=[\mathbf{a}_{M}(\varphi_{k,1}-\omega_{r}),\ldots,\mathbf{a}_{M}(\varphi_{k,J_{k}}-\omega_{r})]\in\mathbb{C}^{M\times J_{k}}$ and $\boldsymbol{\beta}_{\mathrm{s},k}\triangleq\alpha_{r}\boldsymbol{\beta}_{k}=[\alpha_{r}\beta_{k,1},\ldots,\alpha_{r}\beta_{k,J_{k}}]^{\mathrm{T}}\in\mathbb{C}^{J_{k}\times 1}$ are the corresponding ambiguous AoA steering matrix and ambiguous complex gain vector of the specific user-RIS channel for user $k$, respectively. Accordingly, $\mathbf{h}_{\mathrm{s},k}\triangleq\mathbf{A}_{\mathrm{s},k}\boldsymbol{\beta}_{\mathrm{s},k}=\alpha_{r}\mathrm{Diag}\{\mathbf{a}_{M}(-\omega_{r})\}\mathbf{h}_{k}$ is the corresponding ambiguous specific user-RIS channel for user $k$, that must still be found. In next subsection we will show how to estimate $\mathbf{h}_{\mathrm{s},k}$ for $\forall k\in\mathcal{K}$ with the constructed $\mathbf{H}_{\mathrm{s}}$, leading to a significant reduction in the pilot overhead.