Robust Beamforming Design for Covert Communications

From Willie’s perspective, Alice’s transmitted signal is given by

where ${{\bf{w}}_{\rm{c,0}}}$ and ${{\bf{w}}_{\rm{c,1}}}$ denote the transmit beamformer vectors for ${{x_{{\rm{c}}}}}$ in hypothesis ${{\cal H}_{0}}$ and hypothesis ${{\cal H}_{1}}$, respectively; ${{\bf{w}}_{\rm{b}}}$ denotes the transmit beamformer vector for ${{x_{{\rm{b}}}}}$. Let ${P_{{\rm{total}}}}$ denote the maximum transmit power of Alice. Therefore, the beamformer vectors satisfy: ${\left\|{{{\bf{w}}_{{\rm{c,0}}}}}\right\|^{2}}\leq{P_{{\rm{total}}}}$ under ${{\cal H}_{0}}$ and ${\left\|{{{\bf{w}}_{{\rm{c,1}}}}}\right\|^{2}}+{\left\|{{{\bf{w}}_{\rm{b}}}}\right\|^{2}}\leq{P_{{\rm{total}}}}$ under ${{\cal H}_{1}}$.

For Carol, the received signal is given by

where ${z_{\rm{c}}}\sim{\cal CN}\left({{\rm{0}},\sigma_{\rm{c}}^{2}}\right)$ is the received noise at Carol²²2 Here, the inter cell interference is modeled as white Gaussian noise..

For Bob, the received signal is given by

where ${z_{\rm{b}}}\sim{\cal CN}\left({{\rm{0}},\sigma_{\rm{b}}^{2}}\right)$ is the received noise at Bob.

In addition, the signals received by Willie can be written as

where ${z_{\rm{w}}}\sim{\cal CN}\left({{\rm{0}},\sigma_{\rm{w}}^{2}}\right)$ is the received noise at Willie.

According to (7), we assume that the instantaneous rates at Carol are expressed as ${R_{{\rm{c}},0}}\left({{{\bf{w}}_{{\rm{c}},0}}}\right)$ and ${R_{{\rm{c}},1}}\left({{{\bf{w}}_{{\rm{c}},1}}},{{{\bf{w}}_{{\rm{b}}}}}\right)$ under ${{\cal H}_{0}}$ and ${{\cal H}_{1}}$, respectively, and can be written as

Similarly, based on (10), we assume that ${R_{\rm{b}}}\left({{{\bf{w}}_{{\rm{c}},1}}},{{{\bf{w}}_{{\rm{b}}}}}\right)$ is the instantaneous rate at Bob under hypothesis ${{\cal H}_{1}}$, which is given by

Since Willie needs to distinguish between the two hypotheses from its received signal ${y_{{\rm{w}}}}$, we further characterize the probability of ${y_{{\rm{w}}}}$. Let ${p_{0}}\left({{y_{\rm{w}}}}\right)$ and ${p_{1}}\left({{y_{\rm{w}}}}\right)$ denote the likelihood functions of the received signals of Willie under ${{{\cal H}_{\rm{0}}}}$ and ${{{\cal H}_{\rm{1}}}}$, respectively. Based on (13), ${p_{0}}\left({{y_{\rm{w}}}}\right)$ and ${p_{1}}\left({{y_{\rm{w}}}}\right)$ are given as

where ${\lambda_{0}}\buildrel\Delta\over{=}{\left|{{\bf{h}}_{\rm{w}}^{H}{{\bf{w}}_{{\rm{c}},0}}}\right|^{2}}+\sigma_{\rm{w}}^{2}$ and ${\lambda_{1}}\buildrel\Delta\over{=}{\left|{{\bf{h}}_{\rm{w}}^{H}{{\bf{w}}_{{\rm{c}},1}}}\right|^{2}}+{\left|{{\bf{h}}_{\rm{w}}^{H}{{\bf{w}}_{\rm{b}}}}\right|^{2}}+\sigma_{\rm{w}}^{2}$.

Recall from the previous section that Willie wants to minimize the detection error probability ${\xi}$ (1) by applying an optimal detector. To take $\xi$ into our problem formulation, we next specify conditions on the likelihood functions such that covert communication can be achieved with the given $\varepsilon$. First, we let

where ${V_{T}}\left({{p_{0}},{p_{1}}}\right)$ is the total variation between ${p_{0}}\left({{y_{\rm{w}}}}\right)$ and ${p_{1}}\left({{y_{\rm{w}}}}\right)$. In general, computing ${V_{T}}\left({{p_{0}},{p_{1}}}\right)$ analytically is intractable. Thus, we adopt Pinsker’s inequality [33], and can obtain

where $D\left({{p_{0}}\left\|{{p_{1}}}\right.}\right)$ denotes the KL divergence from $p_{0}(y_{\rm{w}})$ to $p_{1}(y_{\rm{w}})$, and $D\left({{p_{1}}\left\|{{p_{0}}}\right.}\right)$ is the KL divergence from $p_{1}(y_{\rm{w}})$ to $p_{0}(y_{\rm{w}})$. Furthermore, $D\left({{p_{0}}\left\|{{p_{1}}}\right.}\right)$ and $D\left({{p_{1}}\left\|{{p_{0}}}\right.}\right)$ are respectively given as

Therefore, to achieve covert communication with the given $\varepsilon$, i.e., $\xi\geq 1-\varepsilon$, the KL divergences of the likelihood functions should satisfy one of the following constraints:

In this subsection, we assume that Alice can accurately estimate the CSI of Bob and Carol. In most cases, such CSI can be learned at both the receiver side and the transmitter side by training and feedback. However, the WCSI may not be always accessible to Alice because of the potential limited cooperation between Alice and Willie. As a result, we consider the following two scenarios³³3 When Willie is totally passive, the covert communications scheme design may turn to exploit the channel distribution information of Willie [34, 31, 32, 35] :

1) Scenario 1. Perfect WCSI: We first consider a scenario that often arises in practice, where Willie is a legitimate user and is only hostile to Bob. In this case, Alice knows the full CSI of the channel ${{\bf{h}}_{\rm{w}}}$, and uses it to help Bob to hide from Willie [5, 22, 28].

2) Scenario 2. Imperfect WCSI: We consider a more practical scenario where Willie is a regular user with only limited cooperation to Alice. In this case, Alice has imperfect CSI knowledge due to the passive warden and channel estimation errors [16, 36]. Here, the imperfect WCSI is modeled as

where ${{{\bf{\hat{h}}}}_{\rm{w}}}$ denotes the estimated CSI vector between Alice and Willie, and $\Delta{{\bf{h}}_{\rm{w}}}$ denotes corresponding CSI error vector. Moreover, the CSI error vector $\Delta{{\bf{h}}_{\rm{w}}}$ is characterized by an ellipsoidal region, i.e.,

where ${{\bf{C}}_{\rm{w}}}={{\bf{C}}_{\rm{w}}^{H}}\underline{\succ}{\bf{0}}$ controls the axes of the ellipsoid, and ${v_{\rm{w}}}>0$ determines the volume of the ellipsoid [37, 32].

To simplify the derivation, define $\tau_{1}\buildrel\Delta\over{=}{\left|{{\mathbf{h}}_{{\rm{c}}}^{H}{{\mathbf{w}}_{{\rm{c}},0}}}\right|^{2}}$ and $\tau_{2}\buildrel\Delta\over{=}{\left|{{\mathbf{h}}_{{\rm{w}}}^{H}{{\mathbf{w}}_{{\rm{c}},0}}}\right|^{2}}$, and introduce an auxiliary variable ${r_{b}}$. Then, problem (23) can be reformulated as the following equivalent form:

Next, we apply the SDR technique [38] to relax problem (24). Towards this end, by using the following conditions

and ignoring the rank-one constraints, we can obtain a relaxed version of problem (24) as

Note that for any fixed ${r_{\rm{b}}}\geq 0$, problem (26) is a SDP. Therefore, problem (26) is quasi-concave, and its optimal solution can be found by checking its feasibility under any given ${r_{\rm{b}}}$.

After that, it can be checked that the problem of maximizing (26b) is concave with respect to $r_{\rm{b}}$. To be more specific, let $\mathcal{W}=\{{{\bf{W}}_{\rm{b}}},{{\bf{W}}_{{\rm{c}},1}}|\eqref{C_15b}-\eqref{C_15e}\}$, $\phi({\bf{W}}_{\rm{b}}):={\rm{Tr}}\left({{\bf{h}}_{\rm{b}}^{H}{{\bf{W}}_{\rm{b}}}{{\bf{h}}_{\rm{b}}}}\right)$, and $\theta({\bf{W}}_{{\rm{c}},1}):={{\rm{Tr}}\left({{\bf{h}}_{\rm{b}}^{H}{{\bf{W}}_{{\rm{c}},1}}{{\bf{h}}_{\rm{b}}}}\right)+\sigma_{\rm{b}}^{2}}$. Then, we have the following result.

Lemma 1: Function

is concave for $r_{b}\geq 0$.

Thus, we first transform problem (26) into a series of convex subproblems with a given ${r_{\rm{b}}}\geq 0$, which can be optimally solved by standard convex optimization solvers such as CVX. Next, we adopt a bisection search method to find the proposed covert beamformers ${{\bf{W}}_{\rm{b}}}$ and ${{\bf{W}}_{{\rm{c}},1}}$. The details of the bisection search method are summarized as Algorithm 1 in Table I, which outputs the optimal solutions ${\bf{W}}_{{\rm{c}},1}^{*}$ and ${\bf{W}}_{\rm{b}}^{*}$. The computational complexity of Algorithm 1 is ${\cal O}\left({\max{{\left\{{4,2N}\right\}}^{4}}\sqrt{2N}\log\left({{1\mathord{\left/{\vphantom{1{{\xi_{1}}}}}\right.\kern-1.2pt}{{\xi_{1}}}}}\right)\log\left({{1\mathord{\left/{\vphantom{1\zeta_{1}}}\right.\kern-1.2pt}\zeta_{1}}}\right)}\right)$, where ${\xi_{1}}>0$ is the pre-defined accuracy of problem (26) [38, 39, 40].

Finally, we can reconstruct the beamformers ${{\bf{w}}_{{\rm{c}},1}}$ and ${{\bf{w}}_{\rm{b}}}$ based on the solutions given by Algorithm 1. Note that due to relaxation of SDR, the ranks of the optimal solutions ${\bf{W}}_{{\rm{c}},1}^{*}$,${\bf{W}}_{\rm{b}}^{*}$ may not be the optimal solutions of problem (23) or, equivalently, (24). In particular, if ${\rm{rank}}\left({{\bf{W}}_{{\rm{c}},1}^{*}}\right)=1$ and ${\rm{rank}}\left({{\bf{W}}_{\rm{b}}^{*}}\right)=1$, then ${\bf{W}}_{{\rm{c}},1}^{*}$,${\bf{W}}_{\rm{b}}^{*}$ are also the optimal solutions of problem (23), and the optimal beamformers ${{\bf{w}}_{{\rm{c}},1}}$ and ${{\bf{w}}_{\rm{b}}}$ can be obtained using the singular value decomposition (SVD), i.e.,${\bf{W}}_{{\rm{c}},1}^{*}={{\bf{w}}_{{\rm{c}},1}}{\bf{w}}_{{\rm{c}},1}^{H}$ and ${\bf{W}}_{\rm{b}}^{*}={{\bf{w}}_{\rm{b}}}{\bf{w}}_{\rm{b}}^{H}$. However, if ${\rm{rank}}\left({{\bf{W}}_{{\rm{c}},1}^{*}}\right)>1$ or ${\rm{rank}}\left({{\bf{W}}_{\rm{b}}^{*}}\right)>1$, we can adopt the Gaussian randomization procedure [38] to produce a high-quality rank-one solution to problem (23).

It is worth mentioning that the above SDR based beamformers design approach requires solving a series of feasibility subproblems. Therefore, this approach leads to high computational complexity, which motivates us to further develop an alternative approach with less intensive computational complexity.

In this subsection, we propose a ZF beamformers design with iterative processing, which is able to achieve a desirable tradeoff between complexity and performance. In particular, the interference signals ${\bf{h}}_{\rm{w}}^{H}{{\bf{w}}_{\rm{b}}}{s_{\rm{b}}}$ and ${\bf{h}}_{\rm{c}}^{H}{{\bf{w}}_{\rm{b}}}{s_{\rm{b}}}$ are eliminated by designing ${{\bf{w}}_{\rm{b}}}$ such that ${\bf{h}}_{\rm{w}}^{H}{{\bf{w}}_{\rm{b}}}=0$ and ${\bf{h}}_{\rm{c}}^{H}{{\bf{w}}_{\rm{b}}}=0$. Meanwhile, the interference signal ${\bf{h}}_{\rm{b}}^{H}{{\bf{w}}_{{\rm{c}},1}}{s_{{\rm{c}},1}}$ can be removed by designing ${{\bf{w}}_{{\rm{c}},1}}$ such that ${\bf{h}}_{\rm{b}}^{H}{{\bf{w}}_{{\rm{c}},1}}=0$. Note that ${{\bf{w}}_{\rm{b}}}$ has to be orthogonal to ${{\bf{h}}_{\rm{w}}}$ and ${{\bf{h}}_{\rm{c}}}$; and ${{\bf{w}}_{\rm{c,1}}}$ has to have non-zero projections on ${{\bf{h}}_{\rm{w}}}$ and ${{\bf{h}}_{\rm{c}}}$, which means it has to be orthogonal to ${{\bf{h}}_{\rm{b}}}$. Since the probability that ${{\bf{h}}_{\rm{b}}}$ falls in the space spanned by ${{\bf{h}}_{\rm{w}}}$ and ${{\bf{h}}_{\rm{c}}}$ is zero, the number of antennas at Alice is no less than three [41, 42].

Mathematically, applying the ZF beamformers design principle, problem (24) can be reformulated as

To address the joint ZF beamformers design problem (29), we first optimize the beamformer ${\bf{w}}_{\rm{c},1}$ by minimizing the transmission power ${\left\|{{{\bf{w}}_{{\rm{c}},{\rm{1}}}}}\right\|^{2}}$ under constraints (29d), (29e) and (29f). This is because the objective function (29a) does not depend on ${{\bf{w}}_{\rm{c},1}}$, but it increases with the power of beamformer ${{\bf{w}}_{\rm{b}}}$. The total transmission power constraint (29g) includes both ${{\bf{w}}_{\rm{b}}}$ and ${{\bf{w}}_{\rm{c},1}}$. Therefore, in order to maximize the objective function (29a), we need to design the beamformer ${\bf{w}}_{\rm{c},1}$ with the minimum transmission power. Therefore, the ZF beamformer ${\bf{w}}_{\rm{c},1}$ design problem can be formulated as

which is also non-convex.

To handle the non-convexity issue, we relax problem (30) to a convex form by applying SDR as well, which is similar to the approach we followed in the previous section. Specifically, by relaxing ${{\bf{W}}_{{\rm{c}},1}}={{\bf{w}}_{{\rm{c}},1}}{\bf{w}}_{{\rm{c}},1}^{H}$ to ${{\bf{W}}_{{\rm{c}},1}}\succeq{\bf{0}}$, problem (30) can be reformulated as

which is a convex SDP.

Let ${\bf{W}}_{{\rm{c}},1}^{{\rm{opt}}}$ denote the optimal solution of problem (31). Due to relaxation, the rank of ${\bf{W}}_{{\rm{c}},1}^{{\rm{opt}}}$ may not equal to one. Therefore if ${\rm{rank}}\left({{\bf{W}}_{{\rm{c}},1}^{{\rm{opt}}}}\right)=1$, then ${\bf{W}}_{{\rm{c}},1}^{{\rm{opt}}}$ is the optimal solution of problem (29), and the optimal beamformer ${{\bf{w}}_{{\rm{c}},1}}$ can be obtained by SVD, i.e.,${\bf{W}}_{{\rm{c}},1}^{{\rm{opt}}}={{\bf{w}}_{{\rm{c}},1}}{\bf{w}}_{{\rm{c}},1}^{H}$. Otherwise, if ${\rm{rank}}\left({{\bf{W}}_{{\rm{c}},1}^{{\rm{opt}}}}\right)>1$, we can adopt the Gaussian randomization procedure [38] to produce a high-quality rank-one solution to problem (30).

Next, we consider the design of ${{\bf{w}}_{{\rm{b}}}}$. Let ${{\bf{w}}_{{\rm{c}},{\rm{1}}}^{{\rm{opt}}}}$ denote the beamformer of problem (31). Then, let ${P_{c}}={\left\|{{\bf{w}}_{{\rm{c}},{\rm{1}}}^{{\rm{opt}}}}\right\|^{2}}$ denote the transmission power of ${{\bf{w}}_{{\rm{c}},{\rm{1}}}^{{\rm{opt}}}}$. With the notations just defined, problem (29) can be formulated as

which is equivalent to

Problem (33) is a SOCP that can be optimally solved by standard convex optimization solvers such as CVX [39]. Therefore, the ZF transmit beamformers of problem (29) are finally obtained.

Furthermore, we analyze the multiplexing gains of the covert communication system based on ZF beamformer design [42]. Specifically, based on the definitions ${{\bf{H}}_{{\rm{w,c}}}}\buildrel\Delta\over{=}\left[{{{\bf{h}}_{\rm{w}}},{{\bf{h}}_{\rm{c}}}}\right]$ and $\prod_{{\rm{w,c}}}^{\bot}\buildrel\Delta\over{=}{\bf{I}}-\frac{{{{\bf{H}}_{{\rm{w,c}}}}{\bf{H}}_{{\rm{w,c}}}^{H}}}{{{{\left\|{{{\bf{H}}_{{\rm{w,c}}}}}\right\|}^{2}}}}$, the ZF beamformer ${{\bf{w}}_{\rm{b}}}$ of problem (29c) is given as

where $\alpha\geq 0$ is a non-negative real-valued scalar. Then, by substituting ${{\bf{w}}_{\rm{b}}}$ into problem (29c), the optimal ZF beamformer of Bob is given by

Thus, the multiplexing gain covert communication system is given as

where ${\rm{SNR}}\buildrel\Delta\over{=}\frac{{{{\left\|{{{\bf{w}}_{\rm{b}}}}\right\|}^{2}}}}{{\sigma_{\rm{b}}^{2}}}$.

With imperfect WCSI, we aim to maximize ${R_{b}}$ via the joint design of the beamformers ${{\bf{w}}_{{\rm{c}},1}}$ and ${{\bf{w}}_{\rm{b}}}$ under the QoS of Carol, the covertness constraint and the total power constraint. Mathematically, the robust covert rate maximization problem is formulated as

Recall in Section II.B that the CSIs of Bob and Carol, i.e., ${{\bf{h}}_{\rm{b}}}$ and ${{\bf{h}}_{\rm{c}}}$, are perfectly known.

It is clear that problem (37) is not convex, and thereby it is difficult to obtain the optimal solution directly. To deal with this issue, we first reformulate the covertness constraint (37d), by exploiting the property of the function $f\left(x\right)=\ln x+\frac{1}{x}-1$ for $x>0$. More specifically, the covertness constraint $D\left({{p_{0}}\left\|{{p_{1}}}\right.}\right)=\ln\frac{{{\lambda_{1}}}}{{{\lambda_{0}}}}+\frac{{{\lambda_{0}}}}{{{\lambda_{1}}}}-1\leq 2{\varepsilon^{2}}$ can be equivalently transformed as

where $\bar{a}$ and $\bar{b}$ are the two roots of the equation $\ln\frac{{{\lambda_{1}}}}{{{\lambda_{0}}}}+\frac{{{\lambda_{0}}}}{{{\lambda_{1}}}}-1=2{\varepsilon^{2}}$. Therefore, constraint (37c) can be equivalently reformulated as

Here, due to $\Delta{{\bf{h}}_{\rm{w}}}\in{{\cal{E}}_{\rm{w}}}$, there are infinite choices for $\Delta{{\bf{h}}_{\rm{w}}}$, in constraint (37e), which makes problem (37) non-convex and intractable. To overcome this challenge, we propose a relaxation and restriction approach. Specifically, in the relaxation step, the nonconvex robust design problem is transformed into a convex SDP; while in the restriction step, infinite number of complicated constraints are reformulated into a finite number of linear matrix inequalities (LMIs).

For mathematical convenience, let us define ${{\bf{W}}_{\rm{b}}}={{\bf{w}}_{\rm{b}}}{{\bf{w}}_{\rm{b}}}^{H}$, ${{\bf{W}}_{{\rm{c}},1}}={{\bf{w}}_{{\rm{c}},1}}{\bf{w}}_{{\rm{c}},1}^{H}$, $\widehat{\bf{W}}_{1}\buildrel\Delta\over{=}{{\bf{W}}_{b}}+{{\bf{W}}_{c,1}}-\bar{a}{{\bf{w}}_{{\rm{c}},0}}{\bf{w}}_{{\rm{c}},0}^{H}$, and $\widetilde{\bf{W}}_{1}\buildrel\Delta\over{=}{{\bf{W}}_{{\rm{c}},1}}+{{\bf{W}}_{\rm{b}}}-\bar{b}{{\bf{w}}_{{\rm{c}},0}}{\bf{w}}_{{\rm{c}},0}^{H}$. Then, constraint (39) can be equivalently re-expressed as

By applying SDR, we ignore the rank-one constraints of ${{\bf{W}}_{{\rm{c}},1}}$ and ${{\bf{W}}_{\rm{b}}}$, which is similar to the approach used in (25) and (26). Then, problem (37) can be relaxed as follows

where ${{\tilde{r}}_{\rm{b}}}\geq 0$ is a slack variable.

Note that the SDR problem (41) is quasi-concave, since the objective function and constraints are linear in ${{\bf{W}}_{{\rm{c}},1}}$ and ${{\bf{W}}_{\rm{b}}}$. However, problem (41) is still computationally intractable because it involves an infinite number of constraints due to $\Delta{{\bf{h}}_{\rm{w}}}\in{{\cal E}_{\rm{w}}}$.

Next, we employ the S-Procedure to recast the infinitely many constraints as a certain set of LMIs, which is a tractable approximation.

Lemma 2 (S-Procedure[43]): Let a function ${f_{m}}\left(x\right),m\in\left\{{1,2}\right\},x\in{{\mathbb{C}}^{N\times 1}}$, be defined as

where ${{\bf{A}}_{m}}\in{{\mathbb{C}}^{N}}$ is a complex Hermitian matrix, ${{\bf{b}}_{m}}\in{{\mathbb{C}}^{N\times 1}}$ and ${c_{m}}\in{{\mathbb{R}}^{1\times 1}}$. Then, the implication relation ${f_{1}}\left(x\right)\leq 0\Rightarrow{f_{2}}\left(x\right)\leq 0$ holds if and only if there exists a variable $\eta\geq 0$ such that

Consequently, by using the S-Procedure of Lemma 2, constraints (40a) and (40b) can be respectively recast as a finite number of LMIs: