Multi-Functional Reconfigurable Intelligent Surface: System Modeling and Performance Optimization

As shown in Fig. 1, each element of the MF-RIS can operate in two modes: energy harvesting mode (H mode) and signal relay mode (S mode). By flexibly adjusting the circuit connection, each element can switch between the H mode and S mode. The elements operating in H mode harvest RF energy from the incident signal, and convert it to direct current (DC) power for supporting the operation of the MF-RIS. The energy harvesting circuit contains the following components[22, 23]:

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  An impedance matching network consisting of a factor resonator is invoked to guarantee maximum power transmission from the element to the rectifier block.

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  An RF-DC conversion circuit rectifies the available RF power into DC voltage.

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  Capacitors are used to deliver the current to the energy storage device, or as a short-term reserve when RF energy is unavailable.

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  A power management block decides whether to store the converted electricity energy or use it immediately for signal reflection, refraction, and amplification.

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  Energy storage devices (e.g., rechargeable batteries and super capacitors) are used to store energy. Whenever the harvested energy exceeds consumption, any excess is stored for future use, thus achieving self-sustainability.

For other elements operating in S mode, the incident signals are divided into two parts by manipulating the electric and magnetic currents. One part is reflected to the reflection half-space and the other is refracted to the other refraction half-space. With the aid of a microcontroller unit, these elements leverage the harvested energy to sustain the operation of phase-shifting and amplifier circuits. Therefore, the proposed MF-RIS does not need any external power supply in principle. The schematic diagrams of the reflection and refraction amplifiers are also shown in Fig. 1, where the operational amplifier-based current-inverting converters are used to generate the reflected and refracted signals with desired amplification. Regarding practical implementation of MF-RIS, there have been many research contributions on the prototype design of signal reflection[24], refraction[25, 26], amplification[15, 27], and wireless power transfer[28, 29, 30, 31]. These existing prototypes can be used to implement the proposed MF-RIS.

To characterize the signal model of MF-RIS, we consider an MF-RIS with $M$ elements, indexed by $\mathcal{M}=\{1,2,\cdots,M\}$. Let $s_{m}$ denote the signal received by the $m$-th element. Each element cannot simultaneously work in both H and S modes. The signals harvested, reflected, and refracted by the $m$-th element are modeled as $y_{m}^{h}=(1-\alpha_{m})s_{m}$, $y_{m}^{r}=\alpha_{m}\sqrt{\beta_{m}^{r}}e^{j\theta_{m}^{r}}s_{m}$, and $y_{m}^{t}=\alpha_{m}\sqrt{\beta_{m}^{t}}e^{j\theta_{m}^{t}}s_{m}$, respectively, where $\alpha_{m}\in\{0,1\}$, $\theta_{m}^{r},\theta_{m}^{t}\in[0,2\pi)$, and $\beta_{m}^{r},\beta_{m}^{t}\in\left[0,\beta_{\max}\right]$ denote the energy harvesting coefficient, the reflective and refractive phase shifts, and the reflective and refractive amplitude coefficients, respectively. Here, $\alpha_{m}=1$ implies that the $m$-th element operates in S mode, while $\alpha_{m}=0$ implies that it works in H mode, and $\beta_{\rm max}\geq 1$ denotes the amplification factor. The energy consumed by the amplifier should not exceed the maximum available energy that can be applied by the MF-RIS, i.e., $\beta_{m}^{r}+\beta_{m}^{t}\leq\beta_{\rm max}$. The reflective and refractive coefficients are modeled as $\boldsymbol{\Theta}_{r}={\rm{diag}}(\alpha_{1}\sqrt{\beta_{1}^{r}}e^{j\theta_{1}^{r}},\alpha_{2}\sqrt{\beta_{2}^{r}}e^{j\theta_{2}^{r}},\cdots,\alpha_{M}\sqrt{\beta_{M}^{r}}e^{j\theta_{M}^{r}})$ and $\boldsymbol{\Theta}_{t}={\rm{diag}}(\alpha_{1}\sqrt{\beta_{1}^{t}}e^{j\theta_{1}^{t}},\alpha_{2}\sqrt{\beta_{2}^{t}}e^{j\theta_{2}^{t}},\cdots,\alpha_{M}\sqrt{\beta_{M}^{t}}e^{j\theta_{M}^{t}})$, respectively, where $\alpha_{m}\in\{0,1\}$, $\beta_{m}^{r},\beta_{m}^{t}\in[0,\beta_{\rm max}]$, $\beta_{m}^{r}+\beta_{m}^{t}\leq\beta_{\rm max}$, and $\theta_{m}^{r},\theta_{m}^{t}\in[0,2\pi)$.

The design variables and constraints for SF-, DF-, and MF-RIS are summarized in Table II. We observe that, SF- and DF-RIS can be regarded as special cases of the proposed MF-RIS. For example, when $\alpha_{m}\!=\!1$ and $\beta_{\rm max}\!=\!1$, MF-RIS reduces to the STAR-RIS in [9]; when $\alpha_{m}=1$, $\beta_{\rm max}\!=\!1$, $\beta_{m}^{t}\!=\!0$, and $\theta_{m}^{t}\!=\!0$, MF-RIS reduces to the reflecting-only RIS in [1].

We consider an MF-RIS assisted downlink NOMA network, where an $N$-antenna BS serves $J$ single-antenna users with the aid of an MF-RIS consisting of $M$ elements. We denote the reflection (refraction) spatial direction as $r$ ($t$). The spatial direction set and the user set are denoted by $\mathcal{K}\!=\!\{r,t\}$ and $\mathcal{J}\!=\!\{1,2,\cdots,J\}$, respectively. We denote $\mathcal{J}_{k}\!=\!\{1,2,\cdots,J_{k}\}$ as the set of users located in direction $k$ and $\mathcal{J}_{r}\cup\mathcal{J}_{t}\!=\!\mathcal{J}$. For notation simplicity, we index user $j$ in direction $k$ by $U_{kj}$. Furthermore, we consider a three-dimensional (3D) Cartesian coordinate system with the locations of the BS, MF-RIS, and user $U_{kj}$ being $\mathbf{w}_{b}\!=\![x_{b},y_{b},z_{b}]^{\rm T}$, $\mathbf{w}\!=\![x,y,z]^{\rm T}$, and $\mathbf{w}_{kj}\!=\![x_{kj},y_{kj},0]^{\rm T}$, respectively. In practice, due to the limited coverage of MF-RIS, its deployable region is also limited. Denote by $\mathcal{P}$ the predefined region for MF-RIS deployment. Then the following constraint should be satisfied:

where $\left[x_{\rm min},x_{\rm max}\right]$, $\left[y_{\rm min},y_{\rm max}\right]$, and $\left[z_{\rm min},z_{\rm max}\right]$ denote the candidate ranges along the $X$-, $Y$- and $Z$-axes, respectively.

To characterize the performance upper bound that can be achieved by MF-RIS, we assume that perfect channel state information of all channels is available. Since the BS and the MF-RIS are usually deployed at relatively high locations, the line-of-sight (LoS) links can be exploited for them. Therefore, similar to existing RIS works[7, 8], we adopt Rician fading for all channels. For instance, the channel matrix $\mathbf{H}\in\mathbb{C}^{M\times N}$ between the BS and the MF-RIS is given by

where $L_{bs}$ is the distance-dependent path loss, and $\hat{\mathbf{H}}$ is composed of the array response and small-scale fading. Specifically, $h_{0}$ is the path loss at the reference distance of 1 meter (m), $d_{bs}$ is the link distance between the BS and the MF-RIS, and $\kappa_{0}$ is the corresponding path loss exponent. As for the small-scale fading, $\beta_{0}$ is the Rician factor, and $\mathbf{H}^{\rm NLoS}$ is the non-LoS component that follows independent and identically distributed (i.i.d.) Rayleigh fading. Assuming that the MF-RIS is placed parallel to the $Y\!-\!Z$ plane and its $M$ elements form an $M_{y}\times M_{z}\!=\!M$ uniform rectangular array, the LoS component $\mathbf{H}^{\rm{LoS}}$ is expressed as[32]

where $\otimes$ denotes the Kronecker product, $\lambda$ is the carrier wavelength, and $d$ is the antenna separation. Here, $\varphi_{r}$, $\vartheta_{r}$, $\varphi_{t}$, and $\vartheta_{t}$ represent the vertical and horizontal angle-of-arrivals, and the vertical and horizontal angle-of-departures of this LoS link, respectively.

The channel vectors from the BS to user $U_{kj}$ and from the MF-RIS to user $U_{kj}$, denoted by $\mathbf{h}_{kj}^{\mathrm{H}}\in\mathbb{C}^{1\times N}$ and $\mathbf{g}_{kj}^{\mathrm{H}}\in\mathbb{C}^{1\times M}$, are generated by a process similar to obtaining $\mathbf{H}$, and are given by

Using the NOMA protocol, the BS transmits the superimposed signal by exploiting multiple beamforming vectors, i.e., $\mathbf{s}=\sum_{k\in\mathcal{K}}\mathbf{f}_{k}\sum_{j\in\mathcal{J}_{k}}\sqrt{p_{kj}}s_{kj}$. Here, $\mathbf{f}_{k}$ is the transmit beamforming vector for direction $k$, satisfying $\sum\nolimits_{k\in\mathcal{K}}\lVert\mathbf{f}_{k}\lVert^{2}\leq P_{\rm BS}^{\max}$, where $P_{\rm BS}^{\max}$ denotes the maximum transmit power of the BS. Moreover, $p_{kj}$ is the power allocation factor of user $U_{kj}$ with $\sum_{j\in\mathcal{J}_{k}}p_{kj}=1$, and $s_{kj}\!\sim\!\mathcal{CN}(0,1)$ denotes the modulated data symbol, which is independent over $k$. Therefore, the signal received at user $U_{kj}$ is given by[33]

where $\bar{k}=r$, if $k=t$; and $\bar{k}=t$, if $k=r$, $\mathbf{n}_{s}\sim\mathcal{C}\mathcal{N}(\mathbf{0},\sigma_{s}^{2}\mathbf{I}_{M})$ denotes the amplification noise introduced at the MF-RIS with per-element noise power $\sigma_{s}^{2}$, and $n_{kj}\sim\mathcal{C}\mathcal{N}(0,\sigma_{u}^{2})$ denotes additive white Gaussian noise (AWGN) at user $U_{kj}$ with power $\sigma_{u}^{2}$. In addition, $\bar{\mathbf{h}}_{kj}=\mathbf{h}_{kj}^{\rm H}+\mathbf{g}_{kj}^{\rm H}\boldsymbol{\Theta}_{k}\mathbf{H}$ represents the combined channel vector from the BS to user $U_{kj}$. For conventional passive RISs, the term $\mathbf{g}_{kj}^{\mathrm{H}}\boldsymbol{\Theta}_{k}\mathbf{n}_{s}$ is negligibly small compared to the AWGN at user $U_{kj}$ and thus usually omitted. However, such noise is amplified by the amplification unit in our considered MF-RIS and thus can no longer be ignored.

Following the NOMA protocol, all users employ successive interference cancellation (SIC) to detect the signal and remove interference[4]. We assume that the users in direction $k$ are ranked in an ascending order according to the equivalent combined channel gains, expressed as

where $k\in\mathcal{K}$, $j\in\mathcal{J}_{k}$, $l\in\mathcal{L}_{k}\!=\!\{j,j+1,\cdots,J_{k}\}$, and $P_{kj}\!=\!\sum_{i=j+1}^{J_{k}}p_{ki}$. The SIC condition in (6) can be equivalently transformed into the following inequality:

By subtracting the term $p_{kj}|\bar{\mathbf{h}}_{kj}\mathbf{f}_{k}|^{2}|\bar{\mathbf{h}}_{kl}\mathbf{f}_{k}|^{2}P_{kj}$ from both sides of (7) and dividing by $p_{kj}$, we obtain the following inequality for all $k,j,l$:

We observe from (8) that the SIC condition is independent of the power allocation coefficients $\{p_{kj}\}$.

Based on (8), for any users $U_{kj}$ and $U_{kl}$ satisfying $j\leq l$, the achievable rate for user $U_{kl}$ to decode the intended signal of user $U_{kj}$ is given by

To guarantee the success of SIC, the achievable signal-to-interference-plus-noise ratio (SINR) at user $U_{kl}$ to decode the signal of user $U_{kj}$ for all $j\leq l$ should be no less than the SINR at user $U_{kj}$ to decode its own signal. Thus, we have the following SIC decoding rate constraint:

Define the energy harvesting coefficient matrix of the $m$-th element as

Then, the RF power received at the $m$-th element is given by

where the expectation operator $\mathbb{E}(\cdot)$ is over $\mathbf{s}$ and $\mathbf{n}_{s}$.

In order to capture the dynamics of the RF energy conversion efficiency for different input power levels, a non-linear energy harvesting model is adopted[34]. Accordingly, the total power harvested at the $m$-th element is given by

where $\Upsilon_{m}$ is a logistic function with respect to (w.r.t.) the received RF power $P_{m}^{\rm RF}$, and $Z\geq 0$ is a constant determining the maximum harvested power. Constant $\Omega$ is introduced to ensure a zero-input/zero-output response for H mode, with constants $a>0$ and $q>0$ capturing the joint effects of circuit sensitivity limitations and current leakage.

To ensure energy self-sustainability, the power consumed by the proposed MF-RIS should not exceed the power harvested. The power consumption of the proposed MF-RIS is mainly caused by the operation of phase shifters, amplifiers, power conversion circuits, and the output power. Other sources of power consumption, such as powering the impedance matching and mode switching circuits, are negligible in comparison[20, 21, 19]. Given that the MF-RIS has $2\sum\nolimits_{m\in\mathcal{M}}\alpha_{m}$ phase shifters, $2\sum\nolimits_{m\in\mathcal{M}}\alpha_{m}$ amplifiers, and $M-\sum\nolimits_{m\in\mathcal{M}}\alpha_{m}$ power conversion circuits in operation, we have the following energy self-sustainability constraint:

where $P_{b}$, $P_{\rm DC}$, and $P_{\rm C}$ denote the power consumed by each phase shifter, the DC biasing power consumed by the amplifier, and the power consumed by the RF-to-DC power conversion circuit, respectively. Here, $\xi$ is the inverse of the amplifier efficiency, and $P_{\rm O}=\sum_{k\in\mathcal{K}}\big{(}\sum_{k^{\prime}\in\mathcal{K}}\lVert\boldsymbol{\Theta}_{k}\mathbf{H}\mathbf{f}_{k^{\prime}}\lVert^{2}$ $+\sigma_{s}^{2}\lVert\boldsymbol{\Theta}_{k}\lVert_{F}^{2}\big{)}$ represents the output power of the MF-RIS.

Our goal is to maximize the sum-rate of all users while maintaining self-sustainability of the MF-RIS by jointly optimizing the power allocation, transmit beamforming at the BS, and the coefficient matrix and 3D location of the MF-RIS. The optimization problem is formulated as

where $\mathcal{R}_{\rm MF}$ is the feasible set for MF-RIS coefficients, with $\mathcal{R}_{\rm MF}\!=\!\{\alpha_{m},\beta_{m}^{k},\theta_{m}^{k}|\alpha_{m}\in\{0,1\},\beta_{m}^{k}\in[0,\beta_{\max}],\sum\nolimits_{k\in\mathcal{K}}\beta_{m}^{k}$ $\leq\beta_{\rm max},\theta_{m}^{k}\in[0,2\pi),\forall m,k\}$. Constraint (15f) represents the power allocation restriction, (15f) ensures that the total transmit power at the BS cannot exceed the power budget $P_{\rm BS}^{\max}$, and (15f) guarantees that the achievable data rate of user $U_{kj}$ is above the quality-of-service (QoS) requirement $R_{kj}^{\rm min}$. Constraints (15f) and (1) specify the allowable ranges of MF-RIS coefficients and locations, respectively, and (8) determines the SIC decoding order of NOMA users. In addition, constraint (10) ensures successful SIC decoding, and (14) guarantees the energy self-sustainability of the MF-RIS.

The sum-rate maximization problems studied in existing works on self-sustainable RIS[20] and STAR-RIS[33] can be regarded as special cases of problem (15). However, their results are not applicable to solving (15) due to the following new challenges introduced by MF-RIS: 1) the objective function (15a) and constraints (8), (10), (14), and (15f) involve closely coupled variables; 2) the adopted non-linear energy harvesting model makes constraint (14) intractable, compared to the linear model in [20]; 3) due to the signal amplification, additional RIS noise needs to be considered in the objective function (15a) and constraints (8), (10), (14), and (15f), which complicates the resource allocation problem; 4) the binary energy harvesting coefficients result in combinatorial constraints (8), (10), (14), (15f), and (15f), which makes (15) an MINLP problem. Therefore, the formulated problem (15) for the MF-RIS is more challenging to solve as compared to those for existing RISs in [20] and [33].

Before solving (15), we transform it into a more tractable form. First, we observe that constraint (10) is a necessary condition for inequality (8), as (8) is equivalent to (6). This observation shows that under the proposed decoding order, the SIC condition is guaranteed, which is consistent with the conclusions obtained in existing NOMA works such as [33] and [35]. As a result, removing constraint (10) does not affect the optimality of (15) when (8) is satisfied. Hence, (10) is removed from (15) in the following.

Next, to deal with the non-convex constraint (8), we introduce slack variables as $A_{kj}=|\bar{\mathbf{h}}_{kj}\mathbf{f}_{k}|^{-2}$, $B_{kj}=|\bar{\mathbf{h}}_{kj}\mathbf{f}_{\bar{k}}|^{2}+\sigma_{s}^{2}\lVert\mathbf{g}_{kj}^{\mathrm{H}}\boldsymbol{\Theta}_{k}\lVert^{2}+\sigma_{u}^{2}$, and $\Gamma_{kj}=A_{kj}^{-1}B_{kj}^{-1}$. Then, (8) is equivalently transformed into

As for the energy self-sustainability constraint (14), it is difficult to directly observe and handle due to the non-linear energy harvesting model based on the logistic function. Therefore, we first substitute the terms in (13) into (14), and equivalently rewrite (14) in the following form:

where $\mathcal{W}\!=\!2(P_{b}+P_{\rm DC})\sum\nolimits_{m\in\mathcal{M}}\alpha_{m}\!+\!(M-\sum\nolimits_{m\in\mathcal{M}}\alpha_{m})P_{\rm C}$. The complex right-hand-side (RHS) of (17) and the non-convex expression in (12) make (17) difficult to deal with. By introducing slack variables $\zeta_{m}\!=\!P_{m}^{\rm{RF}}$ and $\mathcal{C}_{m}\!=\!1+\exp(-a(\zeta_{m}-q))$, we further recast (17) as

Constraints (16d) and (18a) are non-convex due to their RHSs. Here, we exploit the successive convex approximation (SCA) technique to tackle them. Specifically, the lower bounds of their RHSs at the feasible point $\{A_{kl}^{(\ell)},B_{kl}^{(\ell)},\mathcal{C}_{m}^{(\ell)}\}$ in the $\ell$-th iteration are, respectively, given by

As a result, by defining $\bar{W}=\frac{(\mathcal{C}^{\rm lb}-M\Omega)Z}{(1-\Omega)\xi}-\frac{\mathcal{W}}{\xi}$ and $\Delta_{0}=\{A_{kj},B_{kj},\Gamma_{kj},\mathcal{C}_{m},\zeta_{m}\}$, problem (15) is equivalently transformed into the following one:

To start with, we focus on the optimization of $\{p_{kj}\}$ with given $\{\mathbf{f}_{k},\boldsymbol{\Theta}_{k},\mathbf{w}\}$. Since the inter-cluster interference is independent of $\{p_{kj}\}$, the power allocation problem can be decomposed into two subproblems[35]. For direction $k$, the optimization problem is formulated as

We divide the power allocation coefficient $p_{kj}$ into two parts, $\bar{p}_{kj}$ and $\triangle p_{kj}$, where $p_{kj}=\bar{p}_{kj}+\triangle p_{kj}$, with $\bar{p}_{kj}$ denoting the minimum power allocation coefficient for user $U_{kj}$ to satisfy the QoS constraint (15f) and $\triangle p_{kj}$ denoting the power increment allocated to user $U_{kj}$. Then, based on SIC decoding, the optimal power allocation coefficients can be obtained by the following lemma and proposition.

Proposition 1 shows that the optimal power allocation coefficient, $p_{kj}^{\star}$, can be divided into two parts, $\bar{p}_{kj}^{\star}$ and $\triangle p_{kj}^{\star}$, where $p_{kj}^{\star}=\bar{p}_{kj}^{\star}+\triangle p_{kj}^{\star}$. Specifically, $\bar{p}_{kj}^{\star}$ maintains the QoS constraint of user $U_{kj}$, $\triangle p_{kJ_{k}}^{\star}$ maximizes the data rate of user $U_{kJ_{k}}$, while for other users, $\triangle p_{kj}^{\star}$ compensates for the SINR loss caused by the SIC interference. This is due to the fact that: 1) the users are ordered according to their equivalent channel gains; 2) at the optimum, improving the rate of one user comes at the cost of decreasing the rate of other users.

With given $\{p_{kj},\boldsymbol{\Theta}_{k},\mathbf{w}\}$, we aim to solve the transmit beamforming vector $\mathbf{f}_{k}$. Problem (20) is still difficult to solve directly due to the non-concave objective function (20a) and the non-convex constraint (15f). To this end, we introduce auxiliary variables $Q_{kj}$ and $C_{kj}$, satisfying $Q_{kj}=R_{j\to j}^{k}$ and $C_{kj}=|\bar{\mathbf{h}}_{kj}\mathbf{f}_{k}|^{2}P_{kj}+B_{kj}$. The objective function (15a) is then transformed into

In addition, we obtain the following new constraints:

The SCA technique is employed to handle the non-convex constraint $Q_{kj}\leq\log_{2}\big{(}1+p_{kj}A_{kj}^{-1}C_{kj}^{-1}\big{)}$. Specifically, a lower bound of its RHS in the $\ell$-th iteration is expressed as

Substituting (29)-(31) into (20), the transmit beamforming optimization problem is written as