Robust Power Allocation for Integrated Visible Light Positioning and Communication Networks

In the positioning subframe, the UE estimates the RSS based on the received signals from LEDs. Specifically, for $t\in\left[{0,{T_{\rm{p}}}}\right]$, the $i$-th LED transmits the positioning symbol ${{s_{{\mathrm{p}},i}}\left(t\right)}$, where $-A\leq{{s_{{\mathrm{p}},i}}\left(t\right)}\leq A$, $\mathbb{E}\left\{{{{s_{{\mathrm{p}},i}}\left(t\right)}}\right\}=0$, $\mathbb{E}\left\{{s_{{\mathrm{p}},i}^{2}\left(t\right)}\right\}=\varepsilon$, $A>0$ and $\varepsilon>0$. For brevity, we drop the time index $t$ throughout the paper, where ${{s_{{\mathrm{p}},i}}\left(t\right)}$ is denoted by ${{s_{{\mathrm{p}},i}}}$. Hence, the transmitted positioning signal ${{x_{{\mathrm{p}},i}}}$ of the $i$-th LED is given as

where ${{P_{{\mathrm{p}},i}}}$ indicates the allocated positioning power of the $i$-th LED, and ${I_{{\rm{DC}}}}>0$ denotes the direct current (DC) bias.

At the UE side, the received light is detected by the PD and usually travels via the line of sight (LOS) and diffuse links, and a DC blocking circuit is adopted to filter the DC component. The received positioning signal $y_{{\mathrm{p}},i}$ from the $i$-th LED can be expressed as

where $g_{i}$ is the channel gain between the $i$-th LED and the UE, and ${n_{{\mathrm{p}},i}}$ is the zero-mean Gaussian noise $\mathcal{N}\left({0,\sigma_{\mathrm{p}}^{2}}\right)$.

Generally, the channel gain $g_{i}$ can be decomposed as

where $g_{i,\mathrm{LOS}}$ and $g_{i,\mathrm{diffuse}}$ are the channel gains of the LOS and diffuse links between the $i$-th LED and the UE, respectively. For the LOS link, the channel gain $g_{i,\mathrm{LOS}}$ is given by[31]

where $m\triangleq\left.-\ln 2\middle/\ln\left(\cos\left(\phi_{\frac{1}{2}}\right)\right)\right.$ is the Lambertian index of the LED, $\phi_{\frac{1}{2}}$ denotes the semi-angle, $A_{\mathrm{R}}$ is the effective area of PD, $\eta_{\mathrm{c}}$ is the electric-optical conversion coefficient, $\eta_{\ell}$ is the optical-electric conversion coefficient, ${\psi_{{\rm{FoV}}}}$ indicates the FoV of PD, ${d_{i}}$ denotes the distance from the $i$-th LED to the UE, ${\phi_{i}}$ denotes the angle of irradiance of the $i$-th LED, and ${\psi_{i}}$ denotes the angle of the PD incidence from the $i$-th LED.

On the other hand, the CSI of the diffuse links is always unknown, and the gain of the LOS link is significantly higher than that of the diffuse link [32, 33]. Therefore, to simplify the analysis, we assume that the diffuse links can be ignored, and only the LOS link needs to be considered, i.e., $g_{i}=g_{i,\mathrm{LOS}}$.

Finally, according to the frequency components of the received positioning signal, the UE can detect which LED is in the UE’s FoV and estimate the RSS. Let $\tilde{\mathcal{M}}\subset\mathcal{M}$ denote the set of the received LED’s index, and $\mathbf{p}_{y}\left(\mathbf{u},\left\{\mathbf{v}_{i}\right\}_{i\in\tilde{\mathcal{M}}}\right)\triangleq\left[p_{y,i},...\right]^{T}\in\mathbb{R}^{\tilde{M}}$ denote the power vector of the received signals $\left\{y_{\mathrm{p},i}\right\}_{{i\in\tilde{\mathcal{M}}}}$ from the $\tilde{M}$ received LEDs to the UE, where $p_{y,i}=\mathbb{E}\left\{y_{\mathrm{p},i}^{2}\right\}$. Without loss of generality, we assume that $P_{\mathrm{p},i}=0,i\notin\tilde{\mathcal{M}}$.

When at least 3 non-collinear LEDs can be received, using the RSS-based positioning method, the UE’s location estimation can be transformed into a RSS-related estimation problem. For example, utilizing the least squares method[34], the positioning result of the UE can be solved by $\hat{\mathbf{u}}=\arg\min_{\hat{\mathbf{u}}}\left\|\mathbf{p}_{y}\left(\mathbf{u},\left\{\mathbf{v}_{i}\right\}_{i\in\tilde{\mathcal{M}}}\right)-\mathbf{p}_{\hat{y}}\left(\hat{\mathbf{u}},\left\{\mathbf{v}_{i}\right\}_{i\in\tilde{\mathcal{M}}}\right)\right\|^{2},$ where $\mathbf{p}_{\hat{y}}\left(\hat{\mathbf{u}},\left\{\mathbf{v}_{i}\right\}_{i\in\tilde{\mathcal{M}}}\right)$ denotes the corresponding received power vector at the UE’s Location $\hat{\mathbf{u}}\in\mathbb{R}^{3}$. Thus, the positioning error ${{\mathbf{e}}_{\mathrm{p}}}$ is defined as

Generally, the positioning error ${{\mathbf{e}}_{\mathrm{p}}}$ is assumed to follow two typical probability distribution models: Gaussian distribution and arbitrary distribution.

• •

  Gaussian distribution: If only the mean and variance are known, the Gaussian distribution can maximize the entropy of the uncertain parameter without additive constraints. This model has been adopted in [35, 36, 37] because the CRLB is asymptotically achieved by the maximum-likelihood (ML) estimator even for finite data size[38, 39].

• •

  Arbitrary distribution: The location estimators based on approximate ML[40, 41], and multidimensional scaling (MDS) [42] can also achieve the CRLB. However, the estimation errors do not necessarily follow a Gaussian distribution[43, 35]. In this case, although the distribution of the position error ${{\bf{e}}_{\rm{p}}}$ is unavailable, its mean and variance also can be assumed to be known [44, 35, 42].

Based on the RSS $\mathbf{p}_{y}\left(\mathbf{u},\left\{\mathbf{v}_{i}\right\}_{i\in\tilde{\mathcal{M}}}\right)$ and the signal $\left\{x_{\mathrm{p},i}\right\}_{i\in\tilde{\mathcal{M}}}$, we can select the $i^{*}$-th LED to transmit the downlink data, which is assumed to correspond to the maximum channel gain, i.e.,

and the corresponding LED location is ${{\mathbf{v}}_{{i^{*}}}}={\left[{{x_{{i^{*}}}},{y_{{i^{*}}}},{z_{{i^{*}}}}}\right]^{T}}$.

If only the LOS link is considered, according to the channel model (4), the estimated CSI $\hat{g}_{i^{*}}$ between the ${i^{*}}$-th LED and the UE is given by

where $\mu=\frac{{\left({m+1}\right){A_{\mathrm{R}}}{\eta_{\mathrm{c}}}{\eta_{\mathrm{\ell}}}}}{{2\pi}}$.

Due to the positioning error ${{\bf{e}}_{\rm{p}}}$, the channel estimation (7) is imperfect. The corresponding estimated CSI error $\Delta{g}_{i^{*}}$ between the ${i^{*}}$-th LED and the UE is defined as follows

Based on (7) and (8), the channel estimation error $\Delta g_{i^{*}}$ is given as

Therefore, $\Delta{g}_{i^{*}}$ is a function of ${{\bf{e}}_{\rm{p}}}$, i.e., $\Delta g_{i^{*}}=f\left(\mathbf{e}_{\mathrm{p}}\right)$ .

Based on the result of positioning and channel estimation, the UE can solve the expected power allocation of each LED to optimize the performance of the integrated VLPC system, and the power allocation result can be fed back to the central controller by the uplink. The designed robust power allocation scheme in this paper will be presented in the following section.

During the downlink data subframe, the $i^{*}$-th LED transmits the data symbol ${{s_{{\mathrm{c}}}}}$ to the UE, where $-A\leq{s_{\mathrm{c}}}\leq A$, $\mathbb{E}\left\{{{s_{\mathrm{c}}}}\right\}=0$, and $\mathbb{E}\left\{{s_{\mathrm{c}}^{2}}\right\}=\varepsilon$. Then, the transmitted data signal ${x_{\mathrm{c}}}$ of the ${i^{*}}$-th LED is given by

where ${P_{\mathrm{c}}}$ indicates the allocated power of the ${i^{*}}$-th LED. In addition, because the $i^{*}$-th LED simultaneously transmits the data and positioning signal, the practical transmitted signal can be represented by

Then, the received data signal ${y_{\mathrm{c}}}$ from the ${i^{*}}$-th LED is given as

where ${n_{\mathrm{c}}}\sim\mathcal{N}\left({0,\sigma_{\mathrm{c}}^{2}}\right)$ is the received noise, and ${n_{\mathrm{c}}}$ is independent of $\left\{{{n_{{\mathrm{p}},i}}}\right\}_{i=1}^{M}$.

To ensure that the transmitted signals of VLC are nonnegative, i.e., $x_{{\rm{p}},i}\geq 0$ and ${x_{i^{*}}}\geq 0$, the positioning signal power ${{P_{{\mathrm{p}},i}}}$ in (1) and the communication signal power ${P_{\mathrm{c}}}$ in (10) are, respectively, constrained by

For eye safety and practical illumination requirements, the maximum optical power of VLC signals should also be limited, i.e., ${x_{{\rm{p}},i}}\leq P_{\rm{o}}^{\max},{x_{i^{*}}}\leq P_{\rm{o}}^{\max}$, where $P_{\rm{o}}^{\max}$ denotes the maximum optical power of each LED. According to (1) and (10), ${{P_{{\mathrm{p}},i}}}$ and ${P_{\mathrm{c}}}$ are, respectively, limited by

Due to the limited power budget of the practical electrical circuit, the average electrical powers of the signal $x_{\rm{p}}$ and signal $x_{i^{*}}$ are also constrained, i.e.,

where $P_{\rm{e}}^{\mathrm{max}}$ denotes the maximum electrical transmitted power.

By combining constraints (13), (14) and (15), ${P_{{\mathrm{p}},i}}$ and ${P_{\mathrm{c}}}$ are constrained by

where $\bar{P}_{\mathrm{p}}\triangleq\min\left\{{\frac{{I_{{\mathrm{DC}}}^{2}}}{{{A^{2}}}},\frac{{P_{\mathrm{e}}^{\mathrm{max}}-I_{{\mathrm{DC}}}^{2}}}{\varepsilon},\frac{{{{\left({P_{\mathrm{o}}^{\max}-{I_{{\mathrm{DC}}}}}\right)}^{2}}}}{{{A^{2}}}}}\right\}$.

Meanwhile, considering the limited load capability in practical circuits, the total power of a VLPC integrated system should be constrained, i.e.,

where $P_{\mathrm{total}}$ denotes the maximum total power of the VLPC integrated system.

In this paper, we adopt the CRLB as the performance metrics for quantifying the localization accuracy of the UE [45, 46, 47]. The CRLB can be achieved by the unbiased estimator since VLC links inherently offer high SNR due to the short transmission distance with the dominant LOS path.

Specifically, we first fix the height of the UE, and focus on the two-dimensional location estimation. Let $\bm{\mathbf{u}}={\left[{{x_{\mathrm{u}}},{y_{\mathrm{u}}}}\right]^{T}}$ denote the arbitrary UE location vector, and the vertical height $z_{u}$ is a known constant perfectly, i.e., $z_{u}=\hat{z}_{u}$. Based on the received signal (2), the log-likelihood function of the received signal ${y_{\mathrm{p},i}}$ is given as

where $k$ is a constant and is independent of $\bm{\mathbf{u}}$.

Moreover, let ${{\mathbf{J}}_{\bm{\mathbf{u}}}}\left({{{\mathbf{p}}_{\mathrm{p}}}}\right)$ denote the Fisher Information matrix (FIM) of ${\bm{\mathbf{u}}}$, which is a function of the positioning power ${{\bf{p}}_{\rm{p}}}={\left[{{P_{{\rm{p}},1}},\ldots,{P_{{\rm{p}},M}}}\right]^{T}}$. Specifically, the FIM ${{\mathbf{J}}_{\bm{\mathbf{u}}}}\left({{{\mathbf{p}}_{\mathrm{p}}}}\right)$ is given as

where

The derivations of (21) are given in Appendix A. Thus, the variance of the zero-mean positioning error ${{\mathbf{e}}_{\mathrm{p}}}$ is lower bounded by the CRLB [45], i.e.,

Let ${R_{\rm{c}}}$ denote the achievable rate of the UE in the downlink data subframe. Based on the received signal (12), ${R_{\rm{c}}}$ is lower bounded by

where the inequality (24b) holds because of the entropy power inequality (EPI), and (24c) holds since ${s_{\rm{c}}}$ follows the ABG distribution that can achieve the maximum differential entropy [48]. Here, the ABG distribution of signal ${s_{\rm{c}}}$ is given by

where the parameters $\alpha$, $\beta$, $\gamma$ are the solutions of the following equations

where ${\rm{erf}}\left(x\right)\triangleq\frac{2}{{\sqrt{\pi}}}\int_{0}^{x}{{e^{-{t^{2}}}}\,\mathrm{d}t}$ is the error function.

The CSI estimation error $\Delta g_{i^{*}}$ affects the achievable rate ${R_{\rm{c}}}$. Furthermore, since $\Delta{g}_{i^{*}}=f\left({{{\mathbf{e}}_{\mathrm{p}}}}\right)$ and ${{\bf{e}}_{\rm{p}}}$ depend on the positioning signal power $\left\{{{P_{{\rm{p}},i}}}\right\}$, the achievable rate ${R_{\rm{c}}}$ not only depends on the communication signal power ${P_{\mathrm{c}}}$, but also depends on the positioning signal power $\left\{{{P_{{\rm{p}},i}}}\right\}$. Therefore, there exists an optimal tradeoff between communication power ${P_{\mathrm{c}}}$ and positioning power ${{P_{{\mathrm{p}},i}}}$ for the integrated VLPC system design.

Under the minimum rate requirement, the optical and electrical power constraints, the optimal VLPC power allocation to minimize the CRLB of VLP with perfect CSI can be formulated as follows:

where $\bar{r}$ denotes the minimum rate requirement. After replacing the constraint (29b) with a more stringent constraint $R_{\mathrm{L}}\geq\bar{r}$, problem (29) can be rewritten as a convex programming problem, which can be solved by standard optimization solvers such as CVX [49]. In addition, (29) also corresponds to the nonrobust design ignoring the coupling between the positioning error $\mathbf{e}_{\mathrm{p}}$ and the estimated CSI error $\Delta g$. The optimal communication power and positioning power are denoted by $P_{\text{c}}^{*}$ and $\mathbf{p}_{\text{p}}^{*}$, respectively.

Considering the relationship between $\Delta g$ and $\mathbf{e}_{\mathrm{p}}$, we propose the outage chance constraint to handle the minimum rate requirement (29b). When $\mathbf{e}_{\mathrm{p}}$ follows the Gaussian distribution, the corresponding chance-constrained VLPC programming problem can be stated as

where $P_{\mathrm{out}}$ denotes the maximum tolerable outage probability.

The robust integrated VLPC design problem (30) is nonconvex and computationally intractable. The main challenge lies in the chance constraint (30b), which does not admit closed-form expressions. In order to handle the chance-constrained problem (30), we first reformulate the chance constraint (30b). Based on the lower bound of the achievable rate (24c), the probability constraint (30b) can be conservatively transformed into the following constraint

Specifically, the constraint $R_{\mathrm{L}}\geq\bar{r}$ can be equivalently reformulated as

where $\delta\triangleq{\left({\frac{{{e^{1+2\left({\alpha+\gamma\varepsilon}\right)}}{\mu^{2}}{{\left({{z_{{i^{*}}}}-{z_{\mathrm{u}}}}\right)}^{2\left({m+1}\right)}}}}{{2\pi W\sigma_{\mathrm{c}}^{2}\left({{2^{{\frac{\bar{R}}{W}}}}-1}\right)}}}\right)^{\frac{1}{{{m+3}}}}}$.

Then, by substituting ${\mathbf{u}}={\mathbf{\hat{u}}}+{{\mathbf{e}}_{\mathrm{p}}}$ into (32), we have

Moreover, the positioning error ${{\bf{e}}_{\rm{p}}}$ can be rewritten as $\mathbf{e}_{\mathrm{p}}=\mathbf{J}_{\mathbf{u}}^{-\frac{1}{2}}\left(\mathbf{p}_{\mathrm{p}}\right)\mathbf{\hat{e}}_{\mathrm{p}}$, where ${{\mathbf{\hat{e}}}_{\mathrm{p}}}\sim\mathcal{N}\left({{\mathbf{0}},{\mathbf{I}}}\right)$. Then, the chance constraint (30b) can be reformulated as

where $\mathbf{B}\triangleq\mathbf{J}_{\mathbf{u}}^{-1}\left(\mathbf{p}_{\mathrm{p}}\right)$, $\mathbf{b}\triangleq\mathbf{J}_{\mathbf{u}}^{-\frac{1}{2}}\left(\mathbf{p}_{\mathrm{p}}\right)\left(\mathbf{\hat{u}}-\mathbf{v}_{i^{*}}\right)$, and ${\delta_{b}}=\delta P_{\mathrm{c}}^{\frac{1}{m+3}}-{{{\left\|{{\bf{\hat{u}}}-{{\bf{v}}_{{i^{*}}}}}\right\|}^{2}}}$.

Furthermore, to reformulate the chance constraint (34) into a deterministic form, we invoke the Bernstein-type inequality in Lemma $1$.

Lemma $1$[50] [51]: Let $L=\mathbf{x}^{T}\mathbf{B}\mathbf{x}+2\mathbf{x}^{T}\mathbf{b}$, where ${\mathbf{B}}\in{{\mathbf{R}}^{N\times N}}$ is a real symmetric matrix, ${\mathbf{b}}\in{\mathbb{R}^{N\times 1}}$ and ${\bf{x}}\sim{\mathcal{N}}\left({0,{\bf{I}}}\right)$. Then, for any $\eta\geq 0$, we have

where ${\lambda^{+}}\left({\mathbf{B}}\right)=\max\left\{{{\lambda_{\max}}\left({\mathbf{B}}\right),0}\right\}$ and ${\lambda_{\max}}\left({\mathbf{B}}\right)$ is the maximum eigenvalue of matrix ${\mathbf{B}}$.

According to the Bernstein-type inequality, the chance constraint (34) can be conservatively transformed into the following constraint

where $\eta=-\ln\left({{P_{{\rm{out}}}}}\right)$. Furthermore, constraint (36) can be equivalently reformulated as

where $\omega$ and $\rho$ are slack variables.

Note that constraint (37a) is convex, but constraints (37b) and (37c) are nonconvex because the matrix inverse is non-convexity-preserving. To tackle this issue, constraints (37b) and (37c) can be transformed to convex forms by exploiting the eigenvalue and singular value properties.

Specifically, the left-hand side of constraint (37b) can be approximated as

where $\sigma_{\mathrm{max}}\left(\cdot\right)$ denotes the maximum singular value, and the first inequality is due to $\sqrt{a^{2}+b^{2}}\leq a+b,a\geq 0,b\geq 0$, and the second inequality is because the norm is equal to the sum of all singular values. Because the FIM is invertible and semidefinite, we have

where $\lambda_{\min}\left(\cdot\right)$ denotes the minimum eigenvalue of the matrix. Thus, an upper bound on the left-hand side of constraint (37b) can be derived, and constraint (37b) can be rewritten with a convex form as follows

Meanwhile, because the positive semidefinite matrix means that its minimum eigenvalue is nonnegative, constraint (37c) can be transformed to a concave form as follows

Thus, problem (30) can be reformulated as

Problem (47) is a convex semidefinite program (SDP), which can be solved by standard optimization solvers such as CVX[49].

In this subsection, we investigate a more practical robust VLPC design scenario, where the central controller has no prior knowledge of the distribution of the position error ${{\bf{e}}_{\rm{p}}}$ except for its first and second-order moments, i.e., only the mean and variance of ${{\bf{e}}_{\rm{p}}}$ are known. Specifically, although the distribution of ${{\bf{e}}_{\rm{p}}}$ is arbitrary, the positioning error variance can achieve the CRLB, i.e., $\mathbb{E}\left\{\mathbf{e}_{\mathrm{p}}\mathbf{e}_{\mathrm{p}}^{T}\right\}=\mathbf{J}_{\mathbf{u}}^{-1}\left(\mathbf{p}_{\mathrm{p}}\right)$, and the mean of ${{{\bf{e}}_{\rm{p}}}}$ is zero, i.e., $\mathbb{E}\left\{{{{\bf{e}}_{\rm{p}}}}\right\}=0$.

With the arbitrary distributed ${{\bf{e}}_{\rm{p}}}$, we aim to minimize the CRLB of VLP by optimizing the power allocation subject to the VLC chance constraint and power constraints. Mathematically, the robust VLPC problem can be formulated as follows:

Problem (48) appears to be more challenging than problem (30) since less information about the distribution of ${{\bf{e}}_{\rm{p}}}$ is known. To reformulate the intractable chance constraints to computationally tractable constraints, (30b) can be equivalently expressed as

Furthermore, the chance constraint (49) can be transformed into a distributionally robust chance constraint, which is given by

where $\inf_{\mathbb{P}\in\mathcal{P}}\operatorname{Pr}\left\{\cdot\right\}$ denotes the probability lower bound under the probability distribution $\mathbb{P}$ and $\mathcal{P}$ is called the ambiguity set, which includes all the possible position error distributions.

Lemma $2$[52, 53]: Consider a continuous loss function $L:\mathbb{R}^{k}\to\mathbb{R}$ that is concave or quadratic in $\bm{\xi}$. The distributionally robust chance constraint is equivalent to the worst-case constraint, which is given by

where $\operatorname{\mathbb{P}-CVaR}_{\epsilon}\left\{L\left(\bm{\xi}\right)\right\}$ is denoted as the CVaR of $L\left(\bm{\xi}\right)$ at the threshold $\epsilon$ with respect to $\mathbb{P}$, defined as

Moreover, $\mathbb{R}$ is the set of real numbers and $\left(z\right)^{+}=\max\left\{0,z\right\}$, and $\beta\in\mathbb{R}$ is an auxiliary variable introduced by the CVaR.

Lemma $3$ [52, 53]: Let $L\left(\bm{\xi}\right)=\bm{\xi}^{T}\mathbf{Q}\bm{\xi}+\mathbf{q}^{T}\bm{\xi}+q^{0}$ be a quadratic function of $\bm{\xi}$, $\forall\bm{\xi}\in\mathbb{R}^{n}$. The worst-case CVaR can be computed as

where $\mathbf{M}$ and $\beta\in\mathbb{R}$ are the auxiliary variables, and $\mathbf{\Omega}$ is a matrix defined as

where $\bm{\mu}\in\mathbb{R}^{n}$ and $\mathbf{\Sigma}\in\mathbb{S}^{n}$ are the mean and covariance matrix of random vector $\bm{\xi}$, respectively.

The arbitrary distributed positioning error $\mathbf{e}_{\mathrm{p}}$ makes the chance constraint lower bound intractable. However, a CVaR-based method can overcome this challenge effectively, which is known as a good convex approximation of the worst-case chance constraint[52, 53]. As is described by Lemmas 2 and 3, for the continuous quadratic function $L\left(\hat{\mathbf{e}}_{\mathrm{p}}\right)\triangleq\hat{\mathbf{e}}_{\text{p}}^{T}\mathbf{B}\hat{\mathbf{e}}_{\text{p}}+2\mathbf{b}^{T}\hat{\mathbf{e}}_{\text{p}}-\delta_{b}$, the distributionally robust chance constraint (50) can be made equivalent to the worst-case CVaR constraint as follows:

where $\mathbf{M}$ and $\beta\in\mathbb{R}$ are two auxiliary variables, and $\mathbf{\Omega}=\begin{bmatrix}\mathbf{I}&\mathbf{0}\\ \mathbf{0}^{T}&1\end{bmatrix}$.