On the Sum Secrecy Rate of Multi-User Holographic MIMO Networks

We employ the electromagnetic compliant channel model for HMIMO communications from [6], which accurately approximates the HMIMO electromagnetic multi-path propagation through an asymptotic Fourier transform-based Karhunen-Loeve channel expansion. More specifically, the wireless channel between Alice and the $u$-th user, for $u\in\{\mathcal{B},\text{E}\}$, i.e., valid for all the Bobs and Eve, can be given by

where $\bm{\Phi}_{u}\in\mathbb{C}^{N_{u}\times n_{u}}$ and $\bm{\Phi}_{\text{A}}\in\mathbb{C}^{N_{\text{A}}\times n_{\text{A}}}$ are semi-unitary matrices, i.e., $\bm{\Phi}^{\mathsf{H}}_{u}\bm{\Phi}_{u}={\mathbf{I}}_{n_{u}}$ and $\bm{\Phi}^{\mathsf{H}}_{\text{A}}\bm{\Phi}_{\text{A}}={\mathbf{I}}_{n_{\text{A}}}$, comprising the array response vectors $\bm{\theta}(l_{u,x},l_{u,y},{\mathbf{C}}_{u})\in\mathbb{C}^{N_{u}}$ and $\bm{\theta}(l_{\text{A},x},l_{\text{A},y},{\mathbf{C}}_{\text{A}})\in\mathbb{C}^{N_{\text{A}}}$ of the $u$-th user and Alice, respectively, in which the $n$-th entry of $\bm{\theta}(l_{i,x},l_{i,y},{\mathbf{C}}_{i})$, for $n=1,\cdots,N_{i}$, and $i\in\{\text{A},\mathcal{B},\text{E}\}$, can be computed by

where $\gamma(l_{i,x},l_{i,y})=\sqrt{\kappa^{2}-\left(\frac{2\pi}{L_{i,x}}l_{i,x}\right)^{2}-\left(\frac{2\pi}{L_{i,y}}l_{i,y}\right)^{2}}$ with $\kappa=\frac{2\pi}{\lambda}$ denoting the wavenumber of the system, and $l_{i,x}$ and $l_{i,y}$ are the sampling points in the wavenumber domain, which lead to non-zero angular responses only when the points are within the lattice ellipse [6] $\mathcal{E}_{i}=\left\{(l_{i,x},l_{i,y})\in\mathbb{Z}^{2}:\left(\frac{\lambda}{L_{i,x}}l_{i,x}\right)^{2}+\left(\frac{\lambda}{L_{i,y}}l_{i,y}\right)^{2}\leq 1\right\}$. In particular, with a uniform sampling, these points can be obtained through $l_{i,x}\in\mathcal{E}_{i,x}=\left\{\left\lceil\frac{-L_{i,x}+(q_{i,x}-1)\lambda}{\lambda}\right\rceil\right\}$, for $q_{i,x}=1,\cdots,\left\lceil\frac{L_{i,x}}{\lambda}\right\rceil$, and $l_{i,y}\in\mathcal{E}_{i,y}=\left\{\left\lceil\frac{-L_{i,y}+(q_{i,y}-1)\lambda}{\lambda}\right\rceil\right\}$, for $q_{i,y}=1,\cdots,\left\lceil\frac{L_{i,y}}{\lambda}\right\rceil$, which results in $n_{i}=4\left\lceil\frac{L_{i,x}L_{i,y}}{\lambda^{2}}\right\rceil$, for $u\in\{\text{A},\mathcal{B},\text{E}\}$ [7]. Moreover, the matrix $\tilde{{\mathbf{H}}}_{u}$ collects the small-scale fading coefficients in the angular domain, which can be structured as $\tilde{{\mathbf{H}}}_{u}=\bm{\Sigma}_{u}\odot{\mathbf{G}}_{u}\in\mathbb{C}^{n_{u}\times n_{\text{A}}}$, where ${\mathbf{G}}_{u}\in\mathbb{C}^{n_{u}\times n_{\text{A}}}$ is a random matrix with entries following the complex Gaussian distribution with zero mean and unity variance, and $\bm{\Sigma}_{u}\in\mathbb{R}^{n_{u}\times n_{\text{A}}}$ is a matrix that collects $n_{u}\times n_{\text{A}}$ scaled standard deviations $\{\sqrt{N_{\text{A}}N_{u}}\sigma(l_{u,x},l_{u,y},l_{\text{A},x},l_{\text{A},y})\}$ of the channel, where the variances $\sigma^{2}(\cdot)$’s describe the power transferred from Alice to the $u$-th receiver in the corresponding wavenumber sampling points, with $u\in\{\mathcal{B},\text{E}\}$. Under the assumption of isotropic scattering, the variances observed at Alice and receivers can be decoupled, i.e., $\sigma^{2}(l_{u,x},l_{u,y},l_{\text{A},x},l_{\text{A},y})\approx\sigma^{2}(l_{u,x},l_{u,y})\sigma^{2}(l_{\text{A},x},l_{\text{A},y})$. Thus, $\sigma^{2}(l_{i,x},l_{i,y})$, for $i\in\{\text{A},\mathcal{B},\text{E}\}$, can be calculated as follows [8]

for $l_{i,x}\in\mathcal{E}_{i,x}$ and $l_{i,y}\in\mathcal{E}_{i,y}$, where $\mathcal{D}=\{(x,y)\in\mathbb{R}^{2}:x^{2}+y^{2}\leq 1\}$ is a disk of radius $1$ centered at the origin. The closed-form expression of $\sigma^{2}(l_{i,x},l_{i,y})$ is derived in [8, Appendix IV.C]. As a result, the matrix of standard deviations can be obtained as

where $\bm{\sigma}_{i}\in\mathbb{R}^{n_{i}}$ is the vector that collects the standard deviations $\{\sqrt{N_{i}}\sigma(l_{i,x},l_{i,y})\},\forall l_{i,x}\in\mathcal{E}_{i,x},\forall l_{i,y}\in\mathcal{E}_{i,y}$, for $i\in\{\text{A},\mathcal{B},\text{E}\}$.

Recall that $\bm{\Phi}_{u}$ and $\bm{\Phi}_{\text{A}}$ are deterministic, depending only on the structure of the antenna arrays. Also, the entries of $\bm{\Sigma}_{u}$ change slowly compared to the coherence interval of the fast-fading channel coefficients. Given these facts, we assume that $\bm{\Phi}_{u}$, $\bm{\Phi}_{A}$, and $\bm{\Sigma}_{u}$ are perfectly known in the system. However, we introduce imperfectness on ${\mathbf{G}}_{u}$, which is modeled by a first-order Gauss-Markov process

where ${\mathbf{E}}_{u}$ is a complex standard Gaussian distributed error matrix, and $\xi^{2}$ represents the variance of the channel estimation error. The effect of imperfect ${\mathbf{G}}_{u}$ on secrecy performance will be evaluated comprehensively in Section IV.

Under the above channel model, Alice transmits an information symbol $s_{b}$ to the $b$-th Bob, $\forall b\in\mathcal{B}$. We assume that Alice does not have any knowledge of the channel or location information of Eve. As a result, it becomes challenging to avoid information leakage to Eve through beamforming only. To mitigate this security threat, Alice superimposes a random AN $w_{b}\in\mathbb{C}$ onto the information symbol of each Bob, satisfying $\mathrm{E}\{|w_{b}|^{2}\}=1$. More specifically, Alice transmits the following beamformed data stream

where ${\mathbf{f}}_{b}\in\mathbb{C}^{N_{\text{A}}}$ is the beamforming vector for the $b$-th Bob, such that $\|{\mathbf{f}}_{b}\|_{2}^{2}=1$, $\alpha_{b}$ and $\beta_{b}$ are the PA coefficients for the information symbol and AN, respectively, with a total transmit power constraint $P_{T}=\sum_{b=1}^{B}\alpha_{b}+\beta_{b}$. Furthermore, the information symbols $s_{b}$’s are assumed to have zero mean and unity variance, i.e., $\mathrm{E}\{|s_{b}|^{2}\}=1$. With these assumptions, the signals received by the $b$-th user and Eve can be written, respectively, as

where $\zeta_{b}=d_{b}^{-\eta}\Lambda$ and $\zeta_{\text{E}}=d_{\text{E}}^{-\eta}\Lambda$ model the large-scale fading coefficients, in which $d_{b}$ and $d_{\text{E}}$ denote the distances from Alice to the $b$-th Bob and Eve, respectively, $\eta$ represents the path-loss exponent, and $\Lambda$ is the array gain parameter. Moreover, ${\mathbf{z}}_{b}$ and ${\mathbf{z}}_{\text{E}}$ are the corresponding additive noise vectors, whose entries follow the complex Gaussian distribution with zero mean and variance $\sigma^{2}_{z}$.

In this subsection, we focus on the design of ${\mathbf{f}}_{b}\in\mathbb{C}^{N_{\text{A}}}$, $\forall b\in\mathcal{B}$. Specifically, we wish to avoid information leakage to non-intended Bobs. Before introducing the beamforming design, we expand the HMIMO channel model in Eq. (1) as follows

where $\bm{\Delta}_{b}\triangleq\mathrm{diag}(\bm{\sigma}_{b})$ and $\bm{\Delta}_{\text{A}}\triangleq\mathrm{diag}(\bm{\sigma}_{\text{A}})$. Given the expansion in Eq. (II-C) and the aforementioned property $\bm{\Phi}_{\text{A}}^{\mathsf{H}}\bm{\Phi}_{\text{A}}={\mathbf{I}}_{n_{\text{A}}}$, we can design the desired beamforming vector with the following structure ${\mathbf{f}}_{b}=\bm{\Phi}_{\text{A}}{\mathbf{p}}_{b}$, where ${\mathbf{p}}_{b}\in\mathbb{C}^{n_{\text{A}}}$ is an inner beamformer computed based on the null space spanned by the reduced-dimension effective matrices of unintended users given by $\bm{\Phi}_{\text{A}}^{\mathsf{H}}{\mathbf{H}}_{b^{\prime}}^{\mathsf{H}}=\bm{\Delta}_{\text{A}}^{\mathsf{H}}{\mathbf{G}}_{b^{\prime}}^{\mathsf{H}}\bm{\Delta}_{b^{\prime}}^{\mathsf{H}}\bm{\Phi}_{b^{\prime}}^{\mathsf{H}}\in\mathbb{C}^{n_{\text{A}}\times N_{\text{B}}}$, with rank denoted by $r_{b^{\prime}}$, $\forall b^{\prime}\neq b$. More specifically, we collect all the reduced-dimension effective matrices of unintended users and stack them in a column-wise fashion as

for $b\in\mathcal{B}$, with a rank $\bar{r}_{b}=\sum\limits_{b^{\prime}\in\mathcal{B},b^{\prime}\neq b}r_{b^{\prime}}$. Then, given that $\bar{r}_{b}<BN_{\text{B}},\forall b\in\mathcal{B}$ due to the correlated entries of $\bm{\Phi}_{\text{A}}^{\mathsf{H}}{\mathbf{H}}_{b}^{\mathsf{H}}$, the beamformer ${\mathbf{p}}_{b}\in\mathbb{C}^{n_{\text{A}}}$ can be obtained from the orthonormal basis of the nontrivial null space of $\bm{\Xi}_{b}$, which we can choose from the left singular vectors of $\bm{\Xi}_{b}$ that are associated with zero singular values. To this end, we perform singular value decomposition (SVD) and write

where $\bm{\Omega}^{(1)}_{b}$ and $\bm{\Omega}^{(0)}_{b}$ are diagonal matrices that comprise the nonzero and zero singular values of $\bm{\Xi}_{b}$, respectively, ${\mathbf{U}}^{(1)}_{b}$ and ${\mathbf{U}}^{(0)}_{b}$ are semi-unitary matrices that comprise the corresponding left singular vectors, and ${\mathbf{V}}_{b}$ comprises the right singular vectors of $\bm{\Xi}_{b}$. More specifically, given that the matrix ${\mathbf{U}}^{(0)}_{b}\in\mathbb{C}^{n_{\text{A}}\times(n_{\text{A}}-\bar{r}_{b})}$ comprises $n_{\text{A}}-\bar{r}_{b}$ orthonormal basis vectors of the null space of $\bm{\Xi}_{b}$, the desired inner beamformer can be

which satisfies $\|{\mathbf{p}}_{b}\|_{2}^{2}=1$ and ${\mathbf{H}}_{b^{\prime}}\bm{\Phi}_{\text{A}}{\mathbf{p}}_{b}=\bm{0},\forall b\neq b^{\prime}\in\mathcal{B}$, as long as the rank $\bar{r}_{b}<BN_{\text{B}}$ and the constraints $n_{\text{A}}>\bar{r}_{b}$ and $n_{\text{A}}-\bar{r}_{b}\geq 1$ are met.

With the beamformer design presented in the previous subsection, all inter-user interference among the Bobs can be eliminated. This fact allows the $b$-th Bob to exploit its effective channel ${\mathbf{H}}_{b}{\mathbf{f}}_{b}=\bm{\Phi}_{b}\bm{\Delta}_{b}{\mathbf{G}}_{b}\bm{\Delta}_{\text{A}}{\mathbf{p}}_{b}\in\mathbb{C}^{n_{\text{A}}}$ for computing its reception combining vector, as follows

which is a matched filter vector, satisfying $\|{\mathbf{q}}_{b}\|_{2}^{2}=1$, constructed based on the effective channel matrix observed only by the $b$-th Bob. It has a reduced dimension that is determined by the number of receive antennas $N_{\text{B}}$, where we adopt $N_{\text{B}}\ll N_{\text{A}}$ in this work. This property makes the proposed approach much less demanding than relying on the full channel matrix ${\mathbf{H}}_{b}\in\mathbb{C}^{N_{\text{B}}\times N_{\text{A}}}$, which has a much higher dimension. By employing the reception combining vector in Eq. (12), the $b$-th legitimate user will have the post-processed signal as

On the other hand, we assume that Eve infiltrates into the system and gets access to the effective channels ${\mathbf{H}}_{\text{E}}{\mathbf{f}}_{b},\forall b\in\mathcal{B}$ of the legitimate users. Note, however, that because ${\mathbf{f}}_{b},\forall b\in\mathcal{B}$, is computed based on the channels of legitimate users only, the signals intended for other Bobs, i.e., $\forall b^{\star}\in\mathcal{B},b^{\star}\neq b$, will cause interference to Eve when Eve eavesdrops on the $b$-th Bob. In addition, Eve is not aware that Alice is transmitting AN. Under these assumptions, Eve computes its reception vector ${\mathbf{q}}_{\text{E}}$ following the same approach as in Eq. (12) but based on Eve’s effective channel matrix ${\mathbf{H}}_{\text{E}}{\mathbf{f}}_{b}\in\mathbb{C}^{n_{\text{A}}}$ associated with the target user $b$. More specifically, Eve’s receive combining vector is obtained as

Then, after filtering the eavesdropped signal of user $b$ through ${\mathbf{q}}_{\text{E}}$, Eve has the post-processed signal as

The corresponding signal-to-interference-plus-noise ratios (SINRs) as well as the secrecy capacity experienced in the system are investigated in the sequel.

Alice informs all legitimate users of the exploitation of AN. Therefore, we assume that $w_{b}$ can be successfully subtracted from the signal in Eq. (13) with the aid of the successive interference cancellation (SIC) technique. As a result, the SINR observed by the $b$-th Bob when recovering its information symbol, for $\forall b\in\mathcal{B}$, can be given by

In contrast to Bobs, Eve cannot decode the AN $w_{b}$ and, thus, it will be able to eavesdrop only on a noisy version of the transmitted information symbol, which is also corrupted by inter-user interference. To be specific, when detecting the symbol of the target user $b\in\mathcal{B}$, Eve observes the following SINR

where the numerator $|{\mathbf{q}}_{\text{E}}^{\mathsf{H}}{\mathbf{H}}_{\text{E}}{\mathbf{f}}_{b}|^{2}\zeta_{\text{E}}\alpha_{b}$ represents the received power of the signal of interest. The denominator, on the other hand, represents the total interference and noise power observed by Eve. It consists of four components: (i) the power of the AN intended for Bob $b$, $|{\mathbf{q}}_{\text{E}}^{\mathsf{H}}{\mathbf{H}}_{\text{E}}{\mathbf{f}}_{b}|^{2}\zeta_{\text{E}}\beta_{b}$, (ii) the sum powers of the signals intended for all the other legitimate users, $\sum_{b^{\star}\in\mathcal{B},b^{\star}\neq b}|{\mathbf{q}}_{\text{E}}^{\mathsf{H}}{\mathbf{H}}_{\text{E}}{\mathbf{f}}_{b^{\star}}|^{2}\zeta_{\text{E}}\alpha_{b^{\star}}$, (iii) the sum powers of AN intended for all the other legitimate users, $\sum_{b^{\star}\in\mathcal{B},b^{\star}\neq b}|{\mathbf{q}}_{\text{E}}^{\mathsf{H}}{\mathbf{H}}_{\text{E}}{\mathbf{f}}_{b^{\star}}|^{2}\zeta_{\text{E}}\beta_{b^{\star}}$, and (iv) the noise power $\sigma_{z}^{2}$.

With the above derivations of SINRs, the rates achieved by the $b$-th Bob and Eve are given by $R_{b}=\log_{2}\big{(}1+\gamma_{b}\big{)},$ and $R^{b}_{\text{E}}=\log_{2}\big{(}1+\gamma^{b}_{\text{E}}\big{)},$ respectively. As a result, the secrecy capacity in bits per channel use (bpcu) observed for the legitimate user $b\in\mathcal{B}$ can be computed by

where $[a]^{+}=\max\{a,0\}$.

By following the MMF tradition, we aim to maximize the minimum of the secrecy rates of Bobs. The associated optimization problem can be formulated as follows:

under the constraint of sum transmit power in (18b). We conduct PA based on the instantaneous CSI, either perfect or imperfect. It is noted that the MMF form in the objective function makes the problem $\mathcal{P}_{1}$ intractable. To address this issue, we reformulate the optimization problem $\mathcal{P}_{1}$ as

by introducing the auxiliary variable $\tau$. However, the non-convexity of constraint (19b) remains an obstacle for solving problem $\mathcal{P}_{2}$. To address this issue, we introduce an auxiliary variable $C_{E}^{b}$ to transform (19b) into the following two constraints: i.e., $R_{b}-C_{E}^{b}\geq\tau$ and $C_{E}^{b}-R_{E}^{b}\geq 0$. The latter is non-convex, and can be further transformed into $2^{C_{E}^{b}}-1\geq|\bm{q}_{E}^{\mathsf{H}}\bm{H}_{E}\bm{f}_{b}|^{2}\zeta_{E}\frac{\alpha_{b}}{I_{E}^{b}}$ and $|\bm{q}_{E}^{\mathsf{H}}\bm{H}_{E}\bm{f}_{b}|^{2}\zeta_{E}\beta_{b}+\sum\limits_{b^{\star}=1,b^{\star}\neq b}^{B}|\bm{q}_{E}^{\mathsf{H}}\bm{H}_{E}\bm{f}_{b^{\star}}|^{2}\zeta_{E}\alpha_{b^{\star}}+\sum\limits_{b^{\star}=1,b^{\star}\neq b}^{B}|\bm{q}_{E}^{\mathsf{H}}\bm{H}_{E}\bm{f}_{b^{\star}}|^{2}\zeta_{E}\beta_{b^{\star}}+\sigma_{z}^{2}\geq I_{E}^{b}$, where $I_{E}^{b}$ is another newly introduced auxiliary variable. The tight coupling of optimization variables in the constraint $2^{C_{E}^{b}}-1\geq|\bm{q}_{E}^{\mathsf{H}}\bm{H}_{E}\bm{f}_{b}|^{2}\zeta_{E}\frac{\alpha_{b}}{I_{E}^{b}}$ introduces further challenges in solving the optimization problem $\mathcal{P}_{2}$. By introducing a series of auxiliary variables, i.e., $X_{b}$, $Y_{b}$, and $Z_{b}$, for $b\in\mathcal{B}$, we can further transform it into four constraints, i.e., $\exp(Z)\geq|\bm{q}_{E}^{\mathsf{H}}\bm{H}_{E}\bm{f}_{b}|^{2}\zeta_{E}{\exp(X_{b}-Y_{b})}$, $\alpha_{b}\leq\exp(X_{b})$, $I_{E}^{b}\geq\exp(Y_{b})$, and $2^{C_{E}^{b}}-1\geq\exp(Z_{b})$. Summarizing the above steps, the optimization problem $\mathcal{P}2$ becomes

Although the problem $\mathcal{P}_{3}$ becomes more tractable than the original problem $\mathcal{P}_{1}$, constraints (20e) and (20g) remain non-convex. To address this, we employ the successive convex approximation (SCA) method with first-order Taylor expansion to tackle them. In particular, the problem $\mathcal{P}_{3}$ can be finally rewritten as

The right-hand side of  (21b) and the left-hand side of (21c) are the first-order approximations of $\exp(X_{b})$ and $2^{C_{E}^{b}}-1$ at points $\bar{X}_{b}[n]$ and $\bar{C}_{E}^{b}[n]$, respectively, which are the solutions of $X_{b}$ and $C_{E}^{b}$ from the $n$-th iteration. Obviously, $\mathcal{P}_{4}$ is a convex problem that can be solved by using the well-known Matlab CVX toolbox [12].