Beamforming Design for Intelligent Reflecting Surface-Enhanced Symbiotic Radio Systems

Recently, intelligent reflecting surface (IRS)-assisted symbiotic radio (SR) systems [1] have been proposed as one of the promising technologies to achieve spectrally and energy-efficient transmission towards the sixth-generation (6G) communications. By exploiting the IRS, SR systems not only enhance the quality of the primary transmission from the access point (AP) to its primary users (PUs), but also allow the IRS to be served as a secondary transmitter to convey its information to the desired secondary users (SUs). Conventionally, an IRS consists of a large number of low-cost passive reflection elements (REs) and the phase shift of each element can reflect/redirect the incident signals to the desired users in a nearly-passive manner. Thus, the introduction of an IRS can further enhance the quality of primary transmission by providing a controllable additional signal propagation path for dedicated energy-focusing and energy-nulling [2]. On the other hand, in an SR system, an environment sensor serving as an information source can be connected to the IRS [3] for collecting the environmental information such as light intensity, temperature, and humidity. Thus, the IRS has its need to transmit the sensed information to a low data-rate SU. In these scenarios, the IRS can embed its information to the reflected radio frequency signals originating from the AP. As such, a mutualistic SR system can be established by intelligent synergistic resources exchanges. Specifically, the SU shares the same frequency spectrum, energy, and infrastructure with PUs, which results in more spectrally and energy-efficient communications compared with conventional networks [4, 5].

To realize practical IRS-assisted SR systems, various schemes have been proposed. For instance, in [6, 7], the IRS is able to transmit information to the SU by adopting binary phase shift keying modulation to modulate its information over the incident signals from the AP. Yet, having the off state of all IRS elements concurrently deflects the purpose of deploying an IRS as it leads to a low spectral efficiency in end-to-end information transmission in the primary system. To further improve the spectral efficiency of IRS-assisted systems, [3] and [8] adopted a higher order IRS modulation by exploiting spatial modulation over IRS elements. However, the problem formulations in [3] and [8] did not take into account the quality-of-service (QoS) requirement of decoding IRS symbols at the SU. As such, the performance of the SU cannot be guaranteed. Moreover, all the aforementioned papers, i.e., [6, 3, 7, 8], ideally assumed that the SU is capable to perform sophisticated multiuser detection or successive interference cancellation for decoding the information symbols of the AP and the IRS jointly, which is generally impossible for a low-cost SU. Besides, without knowing the symbols transmitted by the AP, the effective channel state information (CSI) is generally unknown at the desired SU as it is a product of instantaneous CSI and AP symbols. As a result, it is challenging, if not impossible, for the implementation of the coherent detection proposed in [6, 7]. Thus, a more practical resource allocation design for non-coherent detection in SR systems is desired.

In this paper, we consider an IRS-assisted multiuser multi-input single-output (MISO) downlink multiuser SR system, where the IRS can transmit its information to the SU while assisting the primary transmission between the AP and PUs. In particular, the IRS can modulate its information by applying an on/off multi-level amplitude modulation to the index of the IRS elements. In contrast to existing methods, e.g. [6, 3, 7, 8], we investigate a practical SR system which is able to acquire modulated IRS symbols at the SU without decoding AP symbols. In particular, non-coherent detection is adopted at the SU for decoding the IRS symbols. To quantify the decoding performance of the SU, we first derive an upper bound of the average symbol error rate (SER). Furthermore, by jointly designing the precoding vector at the AP and the phase shift matrix at the IRS, the average sum-rate of the primary system is maximized subject to an SER-based constraint for the SU. Due to the coupling among the optimization variables and the explicit expression of the derived SER upper bound, the formulated problem is non-convex such that obtaining an optimal solution in polynomial time is generally intractable. As a compromise, we exploit the Schur complement and adopt successive convex approximation (SCA) to obtain a suboptimal solution of the beamforming design problem. Simulation results show that the proposed scheme not only guarantees the decoding performance of the secondary system, but also significantly improves the average sum-rate of the primary system compared with various benchmarks.

Notations: Scalars, vectors, and matrices are represented by lowercase letter $x$, boldface lowercase letter $\mathbf{x}$, and boldface uppercase letter $\mathbf{X}$, respectively. $\mathbb{B}^{N\times M}$ and $\mathbb{C}^{N\times M}$ denote the spaces of $N\times M$ matrices with binary and complex entries, respectively. $\mathbf{X}(n,m)$ denotes the element at the $n$-th row and the $m$-column of the matrix. The Euclidean norm and Frobenius norm of a vector/matrix are denoted by $\|\cdot\|$ and $\|\cdot\|_{\mathrm{F}}$, respectively. The absolute value of a complex-valued scalar is denoted by $|\cdot|$. The occurrence probability of an event is denoted by $\mathrm{Pr}\{\cdot\}$. The conditional probability density function of $x$ on event $\mathcal{H}$ is denoted by $p(x|\mathcal{H})$. The transpose, conjugate transpose, conjugate, expectation, and trace of a matrix/vector are denoted by $(\cdot)^{\mathrm{T}}$, $(\cdot)^{\mathrm{H}}$, $(\cdot)^{*}$, $\mathbb{E}[\cdot]$, and $\mathrm{Tr(\cdot)}$, respectively. $\mathbf{X}\succeq\mathbf{0}$ and $\mathbf{X}\preceq\mathbf{0}$ mean that matrix $\mathbf{X}$ is positive semi-definite and negative semi-definite, respectively. $\mathrm{diag(\mathbf{x})}$ denotes a diagonal matrix with its diagonal elements given by vector $\mathbf{x}$. $j$ denotes the imaginary unit. The distribution of a circularly symmetric complex Gaussian (CSCG) random variable with mean $\mu$ and variance $\sigma^{2}$ is denoted by $\mathcal{CN}(\mu,\sigma^{2})$ and $\sim$ stands for “distributed as”. The distribution of Erlang distribution is denoted by $\mathrm{Erlang}(\alpha,n)$ with scale parameter $\alpha$ and shape parameter $n$. $\mathbf{I}_{N}$ denotes an $N\times N$ identity matrix.

As shown in Fig. 1, we consider an IRS-assisted SR system, which includes an AP equipped with $N_{\mathrm{t}}>1$ antennas, an IRS with $M>1$ elements, $K>1$ single-antenna PUs, and a single-antenna SU¹¹1The extension to the case of multiple SUs will be considered in our future work.. In particular, the AP transmits $K$ independent data streams to $K$ PUs simultaneously with the assistance from the IRS. The precoding vector for the $k$-th PU ($\mathrm{PU}$_k) adopted at the AP is defined as $\mathbf{w}_{k}\in\mathbb{C}^{N_{\mathrm{t}}\times 1},\forall k\in\mathcal{K}=\{1,\ldots,K\}$. Meanwhile, the IRS passively transmits the sensed environmental information of the connected sensor to the SU by altering the IRS reflection patterns via index modulation, which will be detailed in Sections II-B and II-C. The IRS reflection matrix is defined as $\mathbf{\Phi}=\mathrm{diag}($$e^{j\theta_{1}},\ldots,e^{j\theta_{m}}$$,\ldots,e^{j\theta_{M}})$$\in\mathbb{C}^{M\times M}$ with $\theta_{m}\in[0,2\pi),\forall m\in\mathcal{M}=\{1,\ldots,M\}$, denoting the phase shift at the $m$-th IRS element. For the ease of practical implementation of the IRS, the amplitude coefficients of all the elements are fixed to be unity in this paper. Besides, as depicted in Fig. 1, the direct links of AP-to-PUs and AP-to-SU are blocked due to heavy shadowing. The aforementioned assumptions are commonly adopted in the literature [2, 4, 5]. On the other hand, this paper considers a quasi-static flat fading channel model. We assume that handshaking has been performed between the AP, PUs, and the SU at the beginning of transmission. As such, the channel coefficients of the AP-to-IRS link, the IRS-to-PU_k link, and the IRS-to-SU link can be acquired by exploiting some existing advanced channel estimation methods, e.g. [9], which are denoted by $\mathbf{G}\!\in\!\mathbb{C}^{M\!\times\!N_{\mathrm{t}}}$, $\mathbf{h}_{\mathrm{P},k}\!\in\!\mathbb{C}^{M\!\times\!1}$, and $\mathbf{h}_{\mathrm{S}}\!\in\!\mathbb{C}^{M\!\times\!1}$, respectively.

We aim to maximize the average achievable sum-rate of the primary system while guaranteeing the power budget at the AP and the QoS requirements of the SU by jointly designing the precoding vectors $\mathbf{w}_{k},\forall k,$ and the phase shifts $\theta_{m},\forall m$. The joint design can be formulated as the following:

Constraint C1 ensures that the transmit power consumption of the precoder at the AP is less than the maximum available power budget $P_{\max}$. Constraint C2 is imposed to restrict the upper bound of the SER for decoding the modulated IRS symbols at the SU to be less than a constant $P_{\mathrm{e}}^{\max}$ defined by the target application. Constraint C3 specifies that $\theta_{m}$ can only vary from $0$ to $2\pi$. The formulated problem is non-convex due to the non-convexities in both the upper and lower incomplete Gamma functions in constraint C2 and the couplings among optimization variables $\mathbf{w}_{k}$ and $\theta_{m}$ in both the objective function and constraint C2. In general, the application of a brute-force search is required for obtaining the globally optimal solution of (11), which is computationally prohibited even for a moderate system size. As an alternative, a computationally efficient suboptimal algorithm is proposed in the next section.

To address the proposed optimization problem in (11), we first transform the objective function and constraint C2 into their equivalent forms, such that they are convex with respect to (w.r.t.) $\mathbf{\Phi}\mathbf{H}_{\mathrm{P},k}\mathbf{G}\mathbf{w}_{j},\forall k,j,$ and $\mathbf{\Phi}\mathbf{H}_{\mathrm{S}}\mathbf{G}\mathbf{w}_{k},\forall k$, respectively. Then, we decouple the coupling between optimization variables $\mathbf{w}_{k}$ and $\theta_{m}$ by utilizing the Schur complement. Finally, SCA is applied to address the non-convex constraints in the transformed optimization problem.

As shown in (8) and (11), constraint C2 in (11) is non-convex due to the highly coupled variables. We first introducing auxiliary optimization variables $\gamma_{\hat{q}}$, $\Gamma_{\tilde{q}},\xi_{\hat{q}}^{\gamma},\epsilon_{\hat{q}}^{\gamma 1},\epsilon_{\hat{q}}^{\gamma 2}$ $\xi_{\tilde{q}}^{\Gamma},\epsilon_{\tilde{q}}^{\Gamma 1}$, and $\epsilon_{\tilde{q}}^{\Gamma 2},\forall\hat{q},\tilde{q},$ for decoupling. Then, by exploiting the properties of logarithm and quadratic equation, constraint C2 can be equivalently transformed as:

In particular, $\gamma_{\hat{q}}$, $\Gamma_{\tilde{q}},\xi_{\hat{q}}^{\gamma}$, and $\xi_{\tilde{q}}^{\Gamma}$ replace $\gamma(L,L\lambda_{\hat{q}}d_{\hat{q}-1})$, $\Gamma(L,L\lambda_{\tilde{q}}d_{\tilde{q}})$, $\sum_{l=0}^{L-1}\frac{(L\lambda_{\hat{q}}d_{{\hat{q}}-1})^{l}}{l!}$, and $\sum_{l=0}^{L-1}\frac{(L\lambda_{\tilde{q}}d_{{\tilde{q}}})^{l}}{l!}$ in constraint C2, respectively. Besides, $\epsilon_{\hat{q}}^{\gamma 1}$ and $\epsilon_{\tilde{q}}^{\Gamma 1}$ replace $\lambda_{\hat{q}}d_{{\hat{q}}-1}$ and $\lambda_{\tilde{q}}d_{{\tilde{q}}}$ in $e^{-L\lambda_{\hat{q}}d_{{\hat{q}}-1}}$ and $e^{-L\lambda_{\tilde{q}}d_{{\tilde{q}}}}$ of constraint C2, respectively. Also, $\epsilon_{\hat{q}}^{\gamma 2}$ and $\epsilon_{\tilde{q}}^{\Gamma 2}$ replace $\lambda_{\hat{q}}d_{{\hat{q}}-1}$ and $\lambda_{\tilde{q}}d_{{\tilde{q}}}$ in $\sum_{l=0}^{L-1}\frac{(L\lambda_{\hat{q}}d_{{\hat{q}}-1})^{l}}{l!}$ and $\sum_{l=0}^{L-1}\frac{(L\lambda_{\tilde{q}}d_{{\tilde{q}}})^{l}}{l!}$ of constraint C2, respectively. It can be verified that constraints C2b3, C2b4, C2c3, and C2c4 are convex w.r.t. $\mathbf{\Phi}\mathbf{H}_{\mathrm{S}}\mathbf{G}\mathbf{w}_{k}$ that paves the way for further simplification in the sequel.

On the other hand, since $\mathbf{s}_{q}$ has a finite number of choices, i.e., $Q+1$, the average achievable capacity in (4) can be directly expressed as $\overline{R}_{k}^{\mathrm{PU}}=\sum_{q=1}^{Q+1}R_{q,k}^{\mathrm{PU}}\mathrm{Pr}\{\mathcal{H}_{q}\}$. Thus, the objective function in (11) can be equivalently transformed to a new objective function as

where $P_{q,k,j}^{\mathrm{PU}}=\mathrm{Tr}(\mathbf{s}_{q}^{\mathrm{T}}\mathbf{\Phi}\mathbf{H}_{\mathrm{P},k}\mathbf{G}\mathbf{w}_{j}\mathbf{w}_{j}^{\mathrm{H}}\mathbf{G}^{\mathrm{H}}\mathbf{H}_{\mathrm{P},k}^{\mathrm{H}}\mathbf{\Phi}^{\mathrm{H}}\mathbf{s}_{q}),\forall k,j,q$.

Next, we decouple the coupled variables $\{\mathbf{w}_{k},\bm{\Phi}\}$ in the objective function in (13) via utilizing the Schur complement. We first introduce auxiliary optimization variables $\mathbf{f}_{k,j}^{\mathrm{PU}}\!\in\!\mathbb{C}^{M\!\times\!1}$, $\!\mathbf{U}_{k,j}^{\mathrm{PU}}\!\in\!\mathbb{C}^{M\!\times\!M}\!$, $\!{c}_{k,j}^{\mathrm{PU}},\forall k,j$, and $\mathbf{A}\!\in\!\mathbb{C}^{M\!\times\!M}$. Then, by substituting $\mathbf{\Phi}\mathbf{H}_{\mathrm{P},k}\mathbf{G}\mathbf{w}_{j}\mathbf{w}_{j}^{\mathrm{H}}\mathbf{G}^{\mathrm{H}}\mathbf{H}_{\mathrm{P},k}^{\mathrm{H}}\mathbf{\Phi}^{\mathrm{H}}=\mathbf{U}_{k,j}^{\mathrm{PU}}$ into (13), the objective function in (13) and constraint C3 in (11) can be equivalently transformed as

respectively, where $\mathbf{D}_{k,j}^{\mathrm{PU}}=\begin{bmatrix}\mathbf{A}&\mathbf{f}_{k,j}^{\mathrm{PU}}\\[1.42262pt] (\mathbf{f}_{k,j}^{\mathrm{PU}})^{\mathrm{H}}&{c}_{k,j}^{\mathrm{PU}}\end{bmatrix}$ and $\mathbf{E}_{k,j}^{\mathrm{PU}}=\begin{bmatrix}\bm{\Phi}\\ (\mathbf{H}_{\mathrm{P},k}\mathbf{G}\mathbf{w}_{k})^{\mathrm{H}}\end{bmatrix}$.

According to the Schur complement[14, Th. 1.12], constraints C4a and C4b ensure that $\mathbf{U}_{k,j}=\mathbf{f}_{k,j}^{\mathrm{PU}}(\mathbf{f}_{k,j}^{\mathrm{PU}})^{\mathrm{H}}$ holds. Similarly, by combining constraints C3b, C4c, and C4d, we have $\mathbf{f}_{k,j}=\mathbf{\Phi}\mathbf{H}_{\mathrm{P},k}\mathbf{G}\mathbf{w}_{j}$ and $\mathbf{A}=\mathbf{\Phi}\mathbf{\Phi}^{\mathrm{H}}$, such that $\mathbf{U}_{k,j}^{\mathrm{PU}}$ is equivalent to $\mathbf{\Phi}\mathbf{H}_{\mathrm{P},k}\mathbf{G}\mathbf{w}_{j}\mathbf{w}_{j}^{\mathrm{H}}\mathbf{G}^{\mathrm{H}}\mathbf{H}_{\mathrm{P},k}^{\mathrm{H}}\mathbf{\Phi}^{\mathrm{H}}$ in (13).

Likewise, by introducing auxiliary optimization variables $\mathbf{f}_{k}^{\mathrm{SU}}\in\mathbb{C}^{M\times 1}$, $\mathbf{U}_{k}^{\mathrm{SU}}\in\mathbb{C}^{M\times M}$, and ${c}_{k}^{\mathrm{SU}}$, constraints C2b3, C2b4, C2c3, and C2c4 can be equivalently transformed as (15) via substituting ${P}_{\mathbf{s}_{p}}=\widetilde{P}_{\mathbf{s}_{p}}$ into (12):

where $\mathbf{D}_{k}^{\mathrm{SU}}=\begin{bmatrix}\mathbf{A}&\mathbf{f}_{k}^{\mathrm{SU}}\\ (\mathbf{f}_{k}^{\mathrm{SU}})^{\mathrm{H}}&{c}_{k}^{\mathrm{SU}}\end{bmatrix}$, $\mathbf{E}_{k}^{\mathrm{SU}}=\begin{bmatrix}\bm{\Phi}\\ (\mathbf{H}_{\mathrm{S}}\mathbf{G}\mathbf{w}_{k})^{\mathrm{H}}\end{bmatrix}$, and $\widetilde{P}_{\mathbf{s}_{p}}=\sum_{k=1}^{K}\mathrm{Tr}\Big{(}\mathbf{s}_{p}^{\mathrm{T}}\mathbf{U}^{\mathrm{SU}}_{k}\mathbf{s}_{p}\Big{)},p=\{\hat{q},\tilde{q}\}$.

For ease of presentation, we define a set $\mathcal{A}=\Big{\{}\mathbf{f}_{k}^{\mathrm{SU}},\mathbf{U}_{k}^{\mathrm{SU}},{c}_{k}^{\mathrm{SU}},\mathbf{f}_{k,j}^{\mathrm{PU}},\mathbf{U}_{k,j}^{\mathrm{PU}},{c}_{k,j}^{\mathrm{PU}},\mathbf{A},\gamma_{\hat{q}},\Gamma_{\tilde{q}},\xi_{\hat{q}}^{\gamma},\xi_{\tilde{q}}^{\Gamma},\epsilon_{\hat{q}}^{\gamma 1},\epsilon_{\tilde{q}}^{\Gamma 1},\epsilon_{\hat{q}}^{\gamma 2},$ $\epsilon_{\tilde{q}}^{\Gamma 2}\Big{\}}$, which includes all introduced auxiliary optimization variables. Now, the optimization problem in (11) can be equivalently transformed to the following