Joint design of ordered QR precoding and SIC detection for MIMO VLC systems

Congcong Wang; Chunyan Feng; Yang Yang; Caili Guo; Bowen Jia

doi:10.1364/OE.470468

1. Introduction

Visible light communication (VLC) has attracted significant attention recently due to its abundant license-free spectrum, non-electromagnetic interference and low power consumption [1]. In VLC, light-emitting diodes (LEDs) are used to achieve the dual function of lighting and communication. However, the limited modulation bandwidth of LEDs may degrade the overall data rate of VLC systems [2]. To circumvent this challenge, several promising techniques have been proposed, such as multiplexing and multiple-input multiple-output (MIMO) techniques [3–5]. In practice, multiple light-emitting diodes (LEDs) are equipped to satisfy the requirement of luminous intensity, which provides the hardware basis for the implementation of MIMO in VLC systems [3].

Due to the Lambert emission pattern of LEDs and the intensity modulation and direct detection (IM/DD) communications, MIMO VLC systems tend to have a high spatial correlation [6,7], which brings challenges to the separation and data detection of each spatial data stream. To solve this issue, various detection techniques have been investigated [6,8]. The optimal detection rule is the maximum likelihood (ML) criterion [8], which performs detection by traversing all the possible signal combinations. However, the computational complexity of ML increases exponentially with the number of antennas and the modulation order, which is difficult to be applied in practice [2,8]. In contrast, linear detection algorithms such as zero-forcing (ZF) detection or the minimum mean square error (MMSE) detection have low complexity [3,4]. However, these linear detection algorithms cannot obtain the full diversity gain and perform much worse than ML detection [9].

To improve the detection performance, the linear detectors are combined with the ordered successive interference cancellation (OSIC) technique, such as ZF-OSIC [5] and MMSE-OSIC [10], which can still achieve lower complexity than the ML detector. However, the inherent drawback of OSIC is that it may suffer from the error propagation issue when the power difference between simultaneous transmissions is not significant [11]. It has been shown that the system performance can be significantly improved by exploiting the channel state information (CSI) at the transmitter [12,13], which is known as the joint transceiver design. In particular, an MMSE-OSIC receiver with singular-value decomposition (SVD)-based index precoding was proposed in [13]. Although it achieves better performance than traditional OSIC schemes, a common issue of this scheme and the aforementioned OSIC schemes is that they have to perform reordering and matrix inversion for each detected symbol, which may lead to unaffordable complexity when a large number of symbols are transmitted. Therefore, it is worth investigating a low-complexity and high-performance SIC scheme for MIMO VLC systems.

Besides, there is plenty of prior art on MIMO VLC systems, including precoding [14], maximum ratio combining (MRC) [15] and angular diversity receiver (ADR) [16]. In these works, MIMO VLC systems with one receiver composed of multiple PDs are considered, and they may not work for multi-user mobile scenarios. For multi-user MIMO VLC systems, the work in [17] investigated the performance of dirty paper coding (DPC), channel inversion (CI), and block diagonalization (BD) precoding techniques. The work in [1] proposed an $L$-pulse position modulation ($L$-PPM) MIMO-based multi-user NOMA-VLC system and derived the closed-form expressions of error probability of this system with perfect and imperfect SIC. However, in these works, the optical power constraint is ignored, which has a trade-off between electrical power within the limited dynamic range of LEDs.

The main contribution of this work is a joint design of ordered QR decomposition precoding and SIC detection (OQR-SIC) scheme for MIMO VLC systems. To our best knowledge, this is the first work that uses the ordered QR decomposition and electrical power allocation at the transmitter to alleviate the error propagation issue of SIC detection at the receiver, which achieves an excellent compromise between complexity and performance. In particular, the proposed OQR-SIC first utilizes an OQR decomposition algorithm to decompose the channel matrix into an orthogonal matrix and an upper triangular matrix with decreasing diagonal elements, where a permutation matrix is employed to permute columns of the channel matrix in the QR decomposition. Due to the nonlinear electro-optical conversion characteristics of LEDs and the inherent error propagation issue of SIC detection, the power allocation of transmitted signals must be carefully designed. Thus, with the OQR decomposition matrices, we further formulate a sum-rate maximization problem under dimming control, electrical power and reliable SIC detection constraints to optimize the power allocation. Simulation results demonstrate that the proposed OQR-SIC achieves 2 dB and 9.8 dB signal-to-noise ratio (SNR) gains compared to conventional QR-SIC in $4 \times 4$ and $9 \times 9$ MIMO VLC systems, respectively, when the bit error rate is ${10^{ - 3}}$.

The remainder of this paper is organized as follows. The system model is described in Section 2. The OQR-SIC scheme is presented in Section 3. Simulation and numerical results are presented in Section 4. Section 5 draws some important conclusion.

Notations: Capital boldface letters such as ${\bf {A}}$ represent matrices, small boldface letters such as ${\bf {a}}$ represent vectors. ${\rm {diag}}\left \{ {\bf {a}} \right \}$ denotes the diagonal matrix created from ${\bf {a}}$. ${\left ( {\bf {A}} \right )^{\rm {T}}}$ denotes the transpose of matrix ${\bf {A}}$. ${\left \| \cdot \right \|_1}$ denotes the $1$ norm. ${{\bf {0}}}$ denotes the all-zeros matrix, and ${{\bf {I}}_N}$ denotes the $N \times N$ identity matrix.

2. System model

In this section, a MIMO VLC system with QR precoding and SIC detection (QR-SIC) is described. As shown in Fig. 1, a system model with ${N_{\rm {T}}}$ LEDs and ${N_{\rm {R}}}$ users is considered, where each user is equipped with one photodetector (PD). At the transmitter, the binary data is first modulated into a normalized modulated symbol vector ${{\bf {x}}} = {\left [ {x_1,x_2, \ldots,x_{{N_{\rm {R}}}}} \right ]^{\rm {T}}}$, where ${x_r},\forall r \in \left \{ {1,2, \ldots,{N_{\rm {R}}}} \right \}$ denotes the normalized $M$-PAM symbol intended for the $r$-th user (i.e., the amplitude range of ${x_r}$ is within $\left [ { - 1,1} \right ]$). Then, a precoding matrix ${\bf {P}} \in {{\cal R}^{{N_{\rm {T}}} \times {N_{\rm {R}}}}}$ obtained by QR decomposition is employed to alleviate the channel correlation, and a DC biasing vector ${\bf {d}} \in {{\cal R}^{{N_{\rm {T}}} \times 1}}$ is added to achieve dimming control. The transmitted signal vector can be expressed as

(1)$${{\bf{s}}} = {\bf{P}}{{\bf{x}}} + {\bf{d}},$$

where ${{\bf {s}}} = {\left [ {s_1,s_2, \ldots,s_{{N_{\rm {T}}}}} \right ]^{\rm {T}}}$. Note that the QR decomposition of the transpose of the channel matrix ${{\bf {H}}^{\rm {T}}}$ can be expressed as ${{\bf {H}}^{\rm {T}}} = {\bf {QR}}$, where ${\bf {Q}} \in {{\cal R}^{{N_{\rm {T}}} \times {N_{\rm {R}}}}}$ is a column orthogonal matrix, and ${{\bf {R}}} \in {{\cal R}^{{N_{\rm {R}}} \times {N_{\rm {R}}}}}$ is an upper-triangular matrix. Then, we have ${\bf {H}} = {{\bf {R}}^{\rm {T}}}{{\bf {Q}}^{\rm {T}}}$. Since the CSI can be obtained by channel estimation at the receiver [18] and then fed back to the transmitter, we assume both the transmitter and receiver have perfect CSI and define the precoding matrix in QR-SIC as

(2)$${\bf{P}} = {\bf{QB}},$$

where ${\bf {B}} = {\rm {diag}}\left [ {{\lambda _1},{\lambda _2}, \ldots,{\lambda _{{N_{\rm {R}}}}}} \right ]$ is a diagonal matrix that is used to control the power of transmitted signals. Due to the dynamic range constraints of LEDs, $s_t,\ t = 1,2, \ldots,{N_{\rm {T}}}$ is required to be within $\left [ {{I_{\rm {L}}},{I_{\rm {H}}}} \right ]$, where ${I_{\rm {L}}}$ and ${I_{\rm {H}}}$ denote the minimum and maximum forward current allowed by LEDs, respectively. Thus, ${\bf {P}}$ has to satisfy

(3)$${\left\| {{{\bf{P}}_{t,:}}{{\bf{x}}}} \right\|_1} \le \Delta I,$$

where $\Delta I = \min \left \{ {{I_{\rm {H}}} - {d_t},{d_t} - {I_{\rm {L}}}} \right \}$, and ${d_t}$ denotes the $t$-th element of ${\bf {d}}$.

We assume that each LED obeys Lambertian beam distribution, and then the channel gain between the $r$-th user and the $t$-th LED is given by

(4)$$\begin{aligned}{h_{r,t}} = \left\{ {\begin{array}{ll} {\frac{{(m + 1)A}}{{2\pi {{d_{r,t}}^2}}}T(\psi )g(\psi ){{\cos }^m}(\phi )\cos (\psi )}, & {0 < \psi < {\Psi _{\rm{C}}}}\\ {0}, & {\psi > {\Psi _{\rm{C}}}}, \end{array}} \right. \end{aligned}$$

with the Lambert order $m = - \frac {{\ln 2}}{{\ln (\cos {\Phi _{1/2}})}}$, where ${\Phi _{1/2}}$ is the transmitter semiangle (at half power). In addition, $\phi$ is the irradiance angle, $\psi$ is the incidence angle, ${\Psi _{\rm {C}}}$ is the receiver field of vision (FOV) semiangle, $A$ is the detector area, $d_{r,t}$ is the distance between the $i$-th PD and the $j$-th LED, $T(\psi )$ is the gain of optical filter, and $g(\psi )$ denotes the gain of optical concentrator, which is given in [6]. By gathering the channel gains ${h_{r,t}}$ for $r = 1, \ldots,{N_{\rm {R}}}$ and $t = 1, \ldots,{N_{\rm {T}}}$, we can obtain a channel matrix ${\bf {H}} \in {{{\cal R}}^{{N_{\rm {R}}} \times {N_{\rm {T}}}}}$ with ${h_{r,t}}$ in the $r$-th row and the $t$-th column.

Fig. 1. The system model of MIMO VLC with QR-SIC.

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The received signal vector can be expressed as

(5)$${{\bf{y}}} = {\bf{H}}{{\bf{s}}} + {{\bf{n}}},$$

where ${{\bf {n}}} \in {{\cal R}^{{N_{\rm {R}}} \times 1}}$ is the noise vector whose elements are real-valued Gaussian variables with zero mean and variance $\sigma _n^2$. After removing the bias, the effective signal can be expressed as

(6)$$\begin{aligned} {\bf{e}} &= {\bf{y}} - {\bf{Hd}} = {\bf{HPx}} + {\bf{n}} = {{\bf{R}}^{\rm{T}}}{\bf{Bx}} + {\bf{n}} \\ &= \left[ {\begin{array}{cccc} {{\gamma _{1,1}}} & 0 & \cdots & 0\\ {{\gamma _{2,1}}} & {{\gamma _{2,2}}} & \cdots & 0\\ \vdots & \vdots & \ddots & \vdots \\ {{\gamma _{{N_{\rm{R}}},1}}} & {{\gamma _{{N_{\rm{R}}},2}}} & \cdots & {{\gamma _{{N_{\rm{R}}},{N_{\rm{R}}}}}} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{\lambda _1}} & 0 & \cdots & 0\\ 0 & {{\lambda _2}} & \cdots & 0\\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & {{\lambda _{{N_{\rm{R}}}}}} \end{array}} \right]\left[ {\begin{array}{c} {{x_1}}\\ {{x_2}}\\ \vdots \\ {{x_{{N_{\rm{R}}}}}} \end{array}} \right] + \left[ {\begin{array}{c} {{n_1}}\\ {{n_2}}\\ \vdots \\ {{n_{{N_{\rm{R}}}}}} \end{array}} \right]. \end{aligned}$$

Benefiting from the upper triangular structure of ${\bf {R}}$, the SIC detector can detect the symbol sequentially based on backward propagation. To successfully perform SIC detection at the receiver, the power of transmitted signals must be properly allocated as [19]

(7)$${\lambda _r}{\gamma _{r,r}} \ge {\lambda _{r + 1}}{\gamma _{r + 1,r + 1}},r = 1,2, \ldots {N_{\rm{R}}} - 1.$$

Specifically, the first element of ${\bf {e}}$ is given by

(8)$${e_1} = {\gamma _{1,1}}{\lambda _1}{x_1} + {n_1},$$

where ${\gamma _{1,1}}$ is the element at the first row and first column of ${{\bf {R}}^{\rm {T}}}$. According to (8), the estimated value of $x_1$ can be obtained by hard decision as

(9)$$\widehat {{x_1}} = {\cal {Q}}\left( {\frac{{{e_1}}}{{{\gamma _{1,1}}{\lambda _1}}}} \right).$$

In (9), ${\cal {Q}}\left ( \cdot \right )$ denotes the quantization function as follow

(10)$$\begin{aligned}Q\left( x \right) = \left\{ \begin{array}{l} \frac{m}{{2M - 1}},x \le m + 1\\ \frac{{m + 2}}{{2M - 1}},x > m + 1, \end{array} \right. \end{aligned}$$

where $1 \le m \le 2M - 1$. With the estimated $\widehat {x_i},1 \le i < r$, the value of ${x_r}$ can be estimated as

(11)$$\widehat {x_r} = {\cal {Q}}\left( {\frac{1}{{{\gamma _{r,r}}{\lambda _r}}}\left( {{e_r} - \sum_{i = 1}^{r - 1} {{\gamma _{r,i}}{\lambda _i}\widehat {{x_i}}} } \right)} \right),$$

where ${e_r}$ denotes the $r$-th element of ${\bf {e}}$. Finally, we can obtain the detected symbol vector as $\widehat {\bf {x}} = {\left [ {\widehat {{x_1}},\widehat {{x_2}}, \ldots,\widehat {{x_{{N_{\rm {R}}}}}}} \right ]^{\rm {T}}}$. It can be observed that SIC obtains the detection signal by successively detecting, recovering and removing the interference signal with maximum power, in which the accuracy of the former detected symbols affect that of later detected ones, leading to error propagation. In particular, the interference generated by error propagation can be expressed as

(12)$${I^{{\rm{SIC}}}} = \lambda _r^2\sum_{i = 1}^{r - 1} {\gamma _{r,i}^2{{\left| {{x_i} - \widehat {{x_i}}} \right|}^2}},$$

where ${{x_i} - \widehat {{x_i}}}$ denotes the residual interference of the $i$-th transmitted signal, which can be approximated by a Gaussian distribution according to the central limit theorem [20]. From (12) and (6), we can observe that ${I^{{\rm {SIC}}}}$ is determined by the upper triangular matrix ${\bf {R}}$ and the power allocation matrix ${\bf {B}}$. Thus, in the next section, the ordered QR decomposition and power allocation are investigated.

3. Proposed OQR-SIC

This section proposes an OQR-SIC scheme, in which both the ordering of the columns of the channel matrix before QR decomposition and the power allocation of the transmitted signal are optimized to achieve a superior detection performance with low complexity. First, an OQR decomposition algorithm is designed to decompose the channel matrix into an orthogonal matrix and an upper triangular matrix with decreasing diagonal elements. Then, based on the decomposition matrices, the power allocation problem is formulated as a sum-rate maximization problem under constraints of dimming control, electrical power and reliable SIC detection.

3.1 OQR decomposition algorithm

To optimize the SIC detection order, an OQR decomposition algorithm is proposed in this subsection. Since different decomposition matrices can be obtained by permuting the corresponding columns of channel matrix before the QR decomposition, which may affect the detection order of SIC, we propose an OQR decomposition algorithm, where a permutation matrix ${\bf {\Pi }}$ is employed to permute columns of ${{\bf {H}}}$ in the QR decomposition.

First, to balance the effects of noise and inter-channel interference (ICI), we define an extended channel matrix in line with the MMSE criterion as

(13)$$\overline {\bf{H}} = {\left[ {{{\bf{H}}^{\rm{T}}}\bf{\Pi },{\sigma _n}{{\bf{I}}_{{N_{\rm{R}}}}}} \right]^{\rm{T}}},$$

and then the QR decomposition of the extend channel matrix can be expressed as

(14)$$\overline {\bf{H}} = \left[ {\begin{array}{l} {{{\bf{H}}^{\rm{T}}}\bf{\Pi }}\\ {{\sigma _n}{{\bf{I}}_{{N_{\rm{R}}}}}} \end{array}} \right] = \overline {\bf{Q}} \overline {\bf{R}} = \left[ {\begin{array}{l} {{{\bf{Q}}_a}}\\ {{{\bf{Q}}_b}} \end{array}} \right]\overline {\bf{R}},$$

where $\overline {\bf {R}}$ is an ${N_{\rm {R}}} \times {N_{\rm {R}}}$ upper triangular matrix, and $\overline {\bf {Q}} \in {\cal {R}^{\left ( {{N_{\rm {R}}} + {N_{\rm {T}}}} \right ) \times {N_{\rm {R}}}}}$ is a column orthogonal matrix, which is further decomposed into an ${{N_{\rm {T}}} \times {N_{\rm {R}}}}$ matrix ${{\bf {Q}}_a}$ and an ${{N_{\rm {R}}} \times {N_{\rm {R}}}}$ matrix ${{\bf {Q}}_b}$. In particular, the OQR algorithm starts by finding the column with the largest norm of $\overline {\bf {H}}$, and ${\gamma _{1,1}}$ is simply the norm of the column vector ${\overline {\bf {Q}} _{:,1}}$. Then, the algorithm orthogonalizes ${\overline {\bf {Q}} _{:,i}},i = 2, \ldots,{N_{\rm {R}}}$ with respect to the normalized vector ${\overline {\bf {Q}} _{:,1}}$ and computes the first row of ${\overline {\bf {R}}}$. After that, ${\gamma _{2,2}}$ and ${\overline {\bf {R}} _{2,:}}$ are calculated based on the remaining ${N_{\rm {R}}} - 1$ columns of ${\overline {\bf {Q}}}$. This process continues until ${\gamma _{{N_{\rm {R}}},{N_{\rm {R}}}}}$ and ${\overline {\bf {R}} _{{N_{\rm {R}}},:}}$ are obtained. The OQR decomposition algorithm is summarized in Algorithm 1. Note that the OQR decomposition algorithm only needs to calculate the column norms once at the beginning and can be easily updated afterward.

Algorithm 1. OQR decomposition algorithm

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Form (14) and Algorithm 1, we have

(15)$$\begin{aligned}{\overline {\bf{Q}} ^{*{\rm{T}}}}\overline {\bf{H}} = &\left[ {{\bf{Q}}_a^{*{\rm{T}}},{\bf{Q}}_b^{*{\rm{T}}}} \right]\left[ {\begin{array}{l} {{{\bf{H}}^{\rm{T}}}{\bf{\Pi} ^*}}\\ {{\sigma _n}{{\bf{I}}_{{N_{\rm{R}}}}}} \end{array}} \right] = {\overline {\bf{R}} ^*} \\ \Leftrightarrow &{\bf{Q}}_a^{*{\rm{T}}}{{\bf{H}}^{\rm{T}}}{\bf{\Pi} ^*} = {\overline {\bf{R}} ^*} - {\sigma _n}{\bf{Q}}_b^{*{\rm{T}}} \\ \Leftrightarrow &{\bf{\Pi} ^{*{\rm{T}}}}{\bf{HQ}}_a^* = {\overline {\bf{R}} ^{*{\rm{T}}}} - {\sigma _n}{\bf{Q}}_b^*. \end{aligned}$$

Then, the optimal ${\bf {Q}}$ can be obtained as ${\bf {Q}^*} = {{\bf {Q}}_a^*}$, and the detection vector can be expressed as

(16)$${\bf{e}} = {\bf{\Pi} ^{*{\rm{T}}}}{\bf{HPx}} + {\bf{\Pi} ^{*{\rm{T}}}}{\bf{n}} = {\overline {\bf{R}} ^{*{\rm{T}}}}{\bf{Bx}} - {\sigma _n}{\bf{Q}}_b^{*}{\bf{Bx}} + {\bf{n}} = {\overline {\bf{R}} ^{*{\rm{T}}}}{\bf{Bx}} + \widetilde {\bf{n}},$$

where $\widetilde {\bf {n}} = - {\sigma _n}{\bf {Q}}_b^*{\bf {Bx}} + {\bf {\Pi } ^*}{\bf {n}}$, and ${\sigma _n}{\bf {Q}}_b^{*}{\bf {Bx}}$ denotes residual interference that cannot be eliminated by SIC detection [21]. With the above OQR decomposition algorithm, an ${\overline {\bf {R}}}^*$ matrix with ${\gamma _{r,r}} \ge {\gamma _{r + 1,r + 1}}$ is obtained. However, to successfully perform SIC detection at the receiver, power constraint of SIC (7) should be satisfied. Thus, the power allocation of the transmitted signal must be further optimized.

3.2 Optimization of power allocation

This subsection further optimizes the power allocation matrix ${\bf {B}}$. In particular, the power allocation problem is formulated as a sum-rate maximization problem under the constraints of dimming control, total electrical power and reliable SIC detection constraints. The original problem is non-convex due to the non-convexity of the optimization function. Thus, we convert the original problem into a convex problem based on the Lagrangian function and fractional programming theory, which can be further solved by CVX toolbox.

We first define the mean square error of estimated symbols as $\varepsilon = {\rm {E}}\left [ {{{\left | {{x_i} - \widehat {{x_i}}} \right |}^2}} \right ]$, which can be obtained as the variance of ${I^{{\rm {SIC}}}}$. Then, the signal-to-interference plus noise ratio (SINR) of the $r$-th user can be expressed as

(17)$${\rm{SIN}}{{\rm{R}}_r} = \frac{{\sigma _s^2{{\left( {{\gamma _{r,r}}{\lambda _r}} \right)}^2}}}{{\varepsilon \sigma _s^2\sum\limits_{i = 1}^{r - 1} {{{\left( {{\gamma _{r,i}}{\lambda _i}} \right)}^2}} + \sigma _n^2}},$$

and the corresponding data rate of the $r$-th user can be expressed as

(18)$${R_r} = \frac{1}{2} {{{\log }_2}\left( {1 + \frac{2}{{\pi e}}{\rm{SIN}}{{\rm{R}}_r}} \right)}.$$

With the obtained ${\overline {\bf {R}}}^*$, a sum rate maximization problem under the dimming control and electrical power constraints is formulated as

(19)$$\arg \mathop {\max }_{\bf{B}} \sum_{r = 1}^{{N_{\rm{R}}}} {\frac{1}{2}{{\log }_2}\left( {1 + \frac{2}{{\pi e}}\frac{{\sigma _s^2{{\left( {{\gamma _{r,r}}{\lambda _r}} \right)}^2}}}{{\varepsilon \sigma _s^2\sum\limits_{i = 1}^{r - 1} {{{\left( {{\gamma _{r,i}}{\lambda _i}} \right)}^2}} + \sigma _n^2}}} \right)}$$

(19a)$$\rm{s.\;t.}\;\; \left\| {{{\bf{P}}_{t,:}}} \right\|_{1} \le \Delta I,$$

(19b)$${\lambda _r} \ge \frac{1}{{{\gamma _{r,r}}}}\left( {{\gamma _{r + 1,r + 1}}{\lambda _{r + 1}}} \right),r = 1,2, \ldots {N_{\rm{R}}} - 1 ,$$

(19c)$$\sum_{r = 1}^{{N_{\rm{R}}}} {{\lambda _r}} \le P_{{\rm{elc}}}^{{\rm{max}}} ,$$

where $P_{{\rm {elc}}}^{{\rm {max}}}$ denotes the maximum electrical power consumption of the system. Note that (19a) is the dimming control constraint, (19b) is the constraint of reliable SIC detection, and (19c) is the total electrical power constraint. Since the objective function is non-concave, the original problem (19) cannot be solved directly. Thus, we introduce extra variables ${\alpha _r},r = 1, \ldots,{N_{\rm {R}}}$ to transform the original problem into an equivalent convex problem as

(20)$$\begin{aligned} \arg \mathop {\max }_{{\bf{B}},\left\{ {{\alpha _r}} \right\}} \sum_{r = 1}^{{N_{\rm{R}}}} {\frac{1}{2}{{\log }_2}\left( {1 + \frac{2}{{\pi e}}{\alpha _r}} \right)} \end{aligned}$$

(20a)$$\begin{aligned}\rm{s.\;t.}\;\; \left\| {{{\bf{P}}_{t,:}}} \right\|_{1} \le \Delta I, \end{aligned}$$

(20b)$$\begin{aligned}\;\;\;\;\;\;\;{\lambda _r} \ge \frac{1}{{{\gamma _{r,r}}}}\left( {{\gamma _{r + 1,r + 1}}{\lambda _{r + 1}}} \right),r = 1,2, \ldots {N_{\rm{R}}} - 1 , \end{aligned}$$

(20c)$$\begin{aligned}\;\;\;\;\;\;\;\sum_{r = 1}^{{N_{\rm{R}}}} {{\lambda _r}} \le P_{{\rm{elc}}}^{{\rm{max}}} , \end{aligned}$$

(20d)$$\begin{aligned}\;\;\;\;\;\;\;{\alpha _r} \le \frac{{\sigma _s^2{{\left( {{\gamma _{r,r}}{\lambda _r}} \right)}^2}}}{{\varepsilon \sigma _s^2\sum_{i = 1}^{r - 1} {{{\left( {{\gamma _{r,i}}{\lambda _i}} \right)}^2}} + \sigma _n^2}} . \end{aligned}$$

It can be observed that with a fixed ${\bf {B}}$, the problem (20) is a convex problem with respect to $\left \{ {{\alpha _r}} \right \}$. The Lagrangian function with dual variables ${\mu _r},r = 1, \ldots,{N_{\rm {R}}}$ for (20d) can be expressed as

(21)$${L_{\bf{B}}}\left( {\left\{ {{\alpha _r}} \right\},\left\{ {{\mu _r}} \right\}} \right) = \sum_{r = 1}^{{N_{\rm{R}}}} {\frac{1}{2}{{\log }_2}\left( {1 + \frac{2}{{\pi e}}{\alpha _r}} \right)} - {\mu _r}\sum_{r = 1}^{{N_{\rm{R}}}} {\left( {{\alpha _r} - \frac{{\sigma _s^2{{\left( {{\gamma _{r,r}}{\lambda _r}} \right)}^2}}}{{\varepsilon \sigma _s^2\sum\limits_{i = 1}^{r - 1} {{{\left( {{\gamma _{r,i}}{\lambda _i}} \right)}^2}} + \sigma _n^2}}} \right)}.$$

According to the first-order condition of convexity $\frac {{\partial {L_{\bf {B}}}\left ( {\left \{ {{\alpha _r}} \right \},\left \{ {{\mu _r}} \right \}} \right )}}{{\partial {\alpha _r}}} = 0$, the optimal $\mu _r$ can be obtained as

(22)$$\mu _r^* = \frac{1}{{\ln 2\left( {\pi e + 2\alpha _r^*} \right)}},$$

where

(23)$$\alpha _r^* = \frac{{\sigma _s^2{{\left( {{\gamma _{r,r}}{\lambda _r}} \right)}^2}}}{{\varepsilon \sigma _s^2\sum\limits_{i = 1}^{r - 1} {{{\left( {{\gamma _{r,i}}{\lambda _i}} \right)}^2}} + \sigma _n^2}}.$$

Substituting (22) into (21), the optimization problem can be rewritten as

(24)$$\arg \mathop {\max }_{\bf{B}} \sum_{r = 1}^{{N_{\rm{R}}}} {\frac{1}{2}{{\log }_2}\left( {1 + \frac{2}{{\pi e}}\alpha _r^*} \right)} - \sum_{r = 1}^{{N_{\rm{R}}}} {\frac{1}{{\ln 2}}\frac{{\alpha _r^*\left( {\varepsilon \sigma _s^2\sum_{i = 1}^{r - 1} {{{\left( {{\gamma _{r,i}}{\lambda _i}} \right)}^2}} + \sigma _n^2} \right) - 1}}{{\pi e\left( {\varepsilon \sigma _s^2\sum\limits_{i = 1}^{r - 1} {{{\left( {{\gamma _{r,i}}{\lambda _i}} \right)}^2}} + \sigma _n^2} \right) + 2\sigma _s^2{{\left( {{\gamma _{r,r}}{\lambda _r}} \right)}^2}}}}$$

(24a)$$\rm{s.\;t.}\;\; \left\| {{{\bf{P}}_{t,:}}} \right\|_{1} \le \Delta I,$$

(24b)$${\lambda _r} \ge \frac{1}{{{\gamma _{r,r}}}}\left( {{\gamma _{r + 1,r + 1}}{\lambda _{r + 1}}} \right),r = 1,2, \ldots {N_{\rm{R}}} - 1 ,$$

(24c)$$\sum_{r = 1}^{{N_{\rm{R}}}} {{\lambda _r}} \le P_{{\rm{elc}}}^{{\rm{max}}} .$$

According to the fractional programming theory [22], the problem (24) is equivalent to

(25)$$\begin{aligned} \arg \mathop {\max }_{{\bf{B}},\left\{ {{\theta _r}} \right\}} \sum_{r = 1}^{{N_{\rm{R}}}} {\frac{1}{2}{{\log }_2}\left( {1 + \frac{2}{{\pi e}}\alpha _r^*} \right)} - \frac{1}{{\ln 2}}\sum_{r = 1}^{{N_{\rm{R}}}} {\left[ {2{\theta _r}\sqrt {\alpha _r^*\left( {\varepsilon \sigma _s^2\sum_{i = 1}^{r - 1} {{{\left( {{\gamma _{r,i}}{\lambda _i}} \right)}^2}} + \sigma _n^2} \right) - 1} } \right.} \end{aligned}$$

(25a)$$\begin{aligned}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left. { - \theta _r^2\left( {\pi e\left( {\varepsilon \sigma _s^2\sum_{i = 1}^{r - 1} {{{\left( {{\gamma _{r,i}}{\lambda _i}} \right)}^2}} + \sigma _n^2} \right) + 2\sigma _s^2{{\left( {{\gamma _{r,r}}{\lambda _r}} \right)}^2}} \right)} \right] \end{aligned}$$

(25b)$$\begin{aligned}\rm{s.\;t.}\;\; \left\| {{{\bf{P}}_{t,:}}} \right\|_{1} \le \Delta I, \end{aligned}$$

(25c)$$\begin{aligned}\;\;\;\;\;\;\;{\lambda _r} \ge \frac{1}{{{\gamma _{r,r}}}}\left( {{\gamma _{r + 1,r + 1}}{\lambda _{r + 1}}} \right),r = 1,2, \ldots {N_{\rm{R}}} - 1 , \end{aligned}$$

(25d)$$\begin{aligned}\;\;\;\;\;\;\;\sum_{r = 1}^{{N_{\rm{R}}}} {{\lambda _r}} \le P_{{\rm{elc}}}^{{\rm{max}}} , \end{aligned}$$

where the optimal $\theta _r$ is given by

(26)$$\theta _r^* = \frac{{\sqrt {{\alpha _r}\left( {\varepsilon \sigma _s^2\sum\limits_{i = 1}^{r - 1} {{{\left( {{\gamma _{r,i}}{\lambda _i}} \right)}^2}} + \sigma _n^2} \right) - 1} }}{{\pi e\left( {\varepsilon \sigma _s^2\sum\limits_{i = 1}^{r - 1} {{{\left( {{\gamma _{r,i}}{\lambda _i}} \right)}^2}} + \sigma _n^2} \right) + 2\sigma _s^2{{\left( {{\gamma _{r,r}}{\lambda _r}} \right)}^2}}}.$$

With the obtained optimal $\left \{ {\theta _r^*} \right \}$, the objective function in (25) is concave and the constraints are linear. Therefore, the optimal power allocation matrix ${{\bf {B}}^*}$ can be solved by the CVX toolbox. Finally, the optimal OQR precoding can be obtained as ${\bf {P}} = {\bf {Q}}_a^*{{\bf {B}}^*}$.

3.3 Complexity and BER analysis

In this subsection, we further analyze the computational complexity of OQR-SIC and conventional detection schemes and the bit error rate (BER) of the considered MIMO VLC system. Without loss of generality, we express the computation complexity in terms of the number of floating-point operations (FLOPs) [23]. Table 1 presents the numbers of FLOPs required by OQR-SIC and its baseline schemes. It can be observed from Table 1 that the number of FLOP required by MMSE-OSIC is $O\left ( \frac {9}{4}N_{\rm {T}}^4 + \frac {4}{3}N_{\rm {T}}^3{N_{\rm {R}}} \right )$, which is much higher than other schemes. Besides, comparing the coefficient of the highest term $N_{\rm {R}}^2{N_{\rm {T}}}$ in other baseline schemes, we can observe that the complexity of the proposed OQR-SIC is lower than that of the MMSE or SVD scheme, while it is slightly higher than that of QR-SIC. This is because OQR-SIC needs extra ordering operations compared to QR-SIC.

Table 1. Numbers of FLOPs required by OQR-SIC and its baseline schemes

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In line with [26], the pairwise error probability (PEP) can be calculated as

(27)$$PEP = {\rm{P}}\left( {{\bf{x}} \to {\bf{e}}\left| {\bf{HP}} \right.} \right) = Q\left( {\sqrt {\frac{{{T_s}}}{{4\sigma _n^2}}\left\| {{\bf{HP}}\left( {{\bf{x}} - {\bf{e}}} \right)} \right\|_{\rm{F}}^2} } \right),$$

where ${{\bf {x}} \to {\bf {e}}}$ denotes the receiver mistakes the transmitted signal vector ${\bf {x}}$ for another vector ${\bf {e}}$, ${{T_s}}$ denotes the symbol duration in seconds, and $Q\left ( a \right ) = \frac {1}{{\sqrt {2\pi } }}\int _a^{ + \infty } {\exp \left ( {\frac {{ - {t^2}}}{2}} \right )} dt$ is the Q function. Then, the BER of the system can be approximated by union bound methods, and it is given by

(28)$$BER \le \frac{1}{{{M^{{N_{\rm{T}}}}}{{\log }_2}{M^{{N_{\rm{T}}}}}}}\sum_{k = 1}^{{M^{{N_{\rm{T}}}}}} {\sum_{j = 1}^{{M^{{N_{\rm{T}}}}}} {{d_{{\bf{HP}}}}\left( {{b_k},{b_j}} \right)} } Q\left( {\sqrt {\frac{{{T_s}}}{{4\sigma _n^2}}\left\| {{\bf{HP}}\left( {{\bf{x}} - {\bf{e}}} \right)} \right\|_{\rm{F}}^2} } \right),$$

where ${{d_{{\bf {HP}}}}\left ( {{b_k},{b_j}} \right )}$ denotes the Hamming distance of the two bit assignments ${{b_k}}$ and ${{b_j}}$ of the signal vectors ${\bf {x}}$ and ${\bf {e}}$, i.e., ${d_{{\bf {HP}}}}\left ( { \cdot, \cdot } \right )$ denotes the number of bit errors when erroneously detecting ${\bf {e}}$ at the receiver instead of the actually transmitted signal vector ${\bf {s}}$.

4. Simulation and numerical results

This section presents comprehensive simulation and numerical results to validate the communication performance and complexity of the proposed schemes. We consider a MIMO VLC system with ${N_{\rm {R}}}$ users and ${N_{\rm {T}}}$ LEDs. To fully utilize multiplexing gains of the system, we assume that the number of PDs is equal to the number of LEDs, i.e., ${{N_{\rm {T}}} = {N_{\rm {R}}}}$. Please note that by combining the proposed OQR-SIC scheme with the existing scheduling schemes [27] or antenna selection schemes [28], the proposed scheme can also be employed in cases where ${{N_{\rm {T}}} \ne {N_{\rm {R}}}}$. Note that in our considered system, all users are distributed in the room randomly, and all LEDs are uniformly placed in the ceiling. For instance, when ${N_{\rm {T}}} = 4$, the inter LED distance is 1.67 m, and the positions of LEDs are (1.67 m, 1.67 m, 2.5 m), (1.67 m, 3.33 m, 2.5 m), (3.33 m, 1.67 m, 2.5 m) and (3.33 m, 3.33 m, 2.5 m). Unless specified otherwise, the system parameters are given in Table 2.

Figure 2 presents the simulated BERs of OQR-SIC and its baseline schemes, when ${N_{\rm {T}}} = {N_{\rm {R}}} = 4$. We can observe that compared with the baseline schemes, the proposed OQR-SIC scheme performs best in terms of BER among all schemes. This is because end-to-end optimization is achieved by jointly optimizing the SIC detection order and precoding in OQR-SIC. Specifically, from Fig. 2, when ${\rm {BER}} = {10^{ - 3}}$, OQR-SIC achieves 1.3 dB, 2 dB, 4.9 dB and 7.2 dB SNR gains compared to MMSE-OSIC, QR-SIC, SVD and MMSE, respectively.

To verify the advantages of the proposed OQR-SIC scheme under channels with high spatial correlation, we further compare BERs of OQR-SIC and its baseline schemes when ${N_{\rm {T}}} = {N_{\rm {R}}} = 9$ in Fig. 3. It can be observed that the BER performance of all schemes is degraded compared to that in Fig. 2 due to the higher channel correlation. Besides, in Fig. 3, OQR-SIC still has the best performance among all schemes, and it achieves a higher performance improvement than the QR-SIC scheme. In particular, when ${\rm {BER}} = {10^{ - 3}}$, OQR-SIC achieves 2.8 dB, 9.8 dB, 11.5 dB and 12.8 dB SNR gains compared to MMSE-OSIC, QR-SIC, SVD and MMSE, respectively.

Figure 4 illustrates the sum data rate of OQR-SIC and its baseline schemes in a scenario with relatively low spatial correlation where ${N_{\rm {T}}} = {N_{\rm {R}}} = 4$ is considered. It can be observed that OQR-SIC achieves the highest data rate of all schemes. This is because the power distribution in OQR-SIC is optimized with the goal of maximizing the sum rate under the constraints of dimming control, total electrical power and reliable SIC detection. In particular, OQR-SIC shows 6.18 bps/Hz, 9.72 bps/Hz, 13.16 bps/Hz and 20.65 bps/Hz improvements in terms of sum data rate compared to MMSE-OSIC, QR-SIC, SVD and MMSE, respectively, when the SNR is 30 dB.

To further investigate the sum data rate of OQR-SIC and its baseline schemes under channels with high spatial correlation, Fig. 5 illustrates the sum data rate of these schemes when ${N_{\rm {T}}} = {N_{\rm {R}}} = 9$. It can be observed that the sum data rates of all schemes improves compared to Fig. 4, this is because the transmission power increases with the number of LEDs. Besides, in Fig. 5, OQR-SIC still achieves the highest data rate among all schemes, which shows 1.77 bps/Hz, 7.4 bps/Hz, 16.99 bps/Hz and 23.77 bps/Hz improvements in terms of sum data rate compared to MMSE-OSIC, QR-SIC, SVD and MMSE, respectively, when the SNR is 30 dB.

To evaluate the computational complexity of OQR-SIC and its baseline schemes, we further compare the number of FLOPs for all schemes in Fig. 6 with the assumption of ${N_{\rm {T}}} = {N_{\rm {R}}}$. It can be observed that MMSE-OSIC requires much higher computational complexity than the other schemes. Besides, since OQR-SIC involves ordering and power allocation operations, it requires slightly higher computational complexity than QR-SIC. In particular, when ${N_{\rm {T}}} = {N_{\rm {R}}} = 10$, MMSE-OSIC has to perform $3.75 \times {10^4}$ FLOP calculations, while the other schemes require around ${10^3}$ FLOP calculations.

Fig. 2. Simulated BERs of OQR-SIC and its baseline schemes when ${N_{\rm {T}}} = {N_{\rm {R}}} = 4$.

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Fig. 3. Simulated BERs of OQR-SIC and its baseline schemes when ${N_{\rm {T}}} = {N_{\rm {R}}} = 9$.

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Fig. 4. Sum data rates of OQR-SIC and its baseline schemes when ${N_{\rm {T}}} = {N_{\rm {R}}} = 4$.

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Fig. 5. Sum data rates of OQR-SIC and its baseline schemes when ${N_{\rm {T}}} = {N_{\rm {R}}} = 9$.

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Fig. 6. Computational complexity comparison of OQR-SIC and its baseline schemes.

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Table 2. Simulation parameters

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5. Conclusion

This work has proposed an OQR-SIC scheme for MIMO VLC systems to achieve superior communication performance with low complexity. The performance degradation due to the error propagation in SIC detection is analyzed. Based on the analysis, both the ordering of the columns of the channel matrix before QR decomposition and the power allocation of the transmitted signal have been optimized to alleviate the error propagation. The superiority of OQR-SIC have been demonstrated by simulation and numerical results. The results have also verified that the proposed OQR-SIC can harvest more performance gains under high spatial correlation channels. The proposed OQR-SIC scheme is a promising candidate for MIMO VLC systems.

Funding

The Key Technology Research Project of Jiangxi Province (20213AAE01007); National Natural Science Foundation of China (61871047); National Natural Science Foundation of China (61901047); Beijing University of Posts and Telecommunications (CX2021203).

Disclosures

The authors declare that there are no conflicts of interest related to this article.

Data availability

Data underlying the results presented in this paper are not available at this time but may be obtained from the authors upon reasonable request.

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Name of Parameters	Values
Room size	5 m $\times$ 5 m $\times$ 2.5 m
Detector physical area of a PD, $A$	1 $c m^{2}$
Refractive index, $n_{r}$	1.5
Gain of an optical filter, $T (ψ)$	1
Maximum current, $I_{H}$	0.35 A
Receiver FOV semiangle, $Ψ_{1 / 2}$	$60^{\circ}$
Semiangle of LEDs radiation, $Φ_{1 / 2}$	$70^{\circ}$
Maximum transmitted electrical power, $P_{e l c}^{max}$	$N_{T}$
Data rate,	2 bits/symbol

Name of Parameters	Values
Room size	5 m $\times$ 5 m $\times$ 2.5 m
Detector physical area of a PD, $A$	1 $c m^{2}$
Refractive index, $n_{r}$	1.5
Gain of an optical filter, $T (ψ)$	1
Maximum current, $I_{H}$	0.35 A
Receiver FOV semiangle, $Ψ_{1 / 2}$	$60^{\circ}$
Semiangle of LEDs radiation, $Φ_{1 / 2}$	$70^{\circ}$
Maximum transmitted electrical power, $P_{e l c}^{max}$	$N_{T}$
Data rate,	2 bits/symbol

Joint design of ordered QR precoding and SIC detection for MIMO VLC systems

Abstract

1. Introduction

2. System model

3. Proposed OQR-SIC

3.1 OQR decomposition algorithm

3.2 Optimization of power allocation

3.3 Complexity and BER analysis

4. Simulation and numerical results

5. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (6)

Tables (3)

Equations (42)

Optics Express

Algorithm	Required numbers of FLOPs
OQR-SIC	$\underset{O Q R d e c o m p o s i t i o n}{\underset{⏟}{\frac{3}{2} (N_{T}^{2} - N_{T}) N_{R}}} + \underset{P o w e r a l l o c a t i o n}{\underset{⏟}{N_{T} N_{R}}} + \underset{N u l l i n g}{\underset{⏟}{N_{T} N_{R}}}$
QR-SIC	$(N_{T}^{2} - N_{R}) N_{R} + 2 N_{T} N_{R}$ [24]
MMSE-OSIC	$\frac{9}{4} N_{T}^{4} + \frac{4}{3} N_{T}^{3} N_{R} + O (N_{T}^{3}) + \frac{1}{2} (N_{T}^{2} N_{R} + N_{T} N_{R} + N_{T}^{2} + N_{T}) + N_{T} N_{R}$ [24]
MMSE	$2 N_{T}^{2} N_{R} + N_{T}^{2} + N_{T} N_{R}$ [25]
SVD	$4 N_{R}^{2} N_{T} + N_{T}^{2} + 2 N_{T} N_{R}$ [23]