Capacity Maximized Linear Precoder Design for Spatial-Multiplexing MIMO VLC Systems

This paper proposes a novel linear precoder design based on the singular value decomposition (SVD) for multi-input multi-output visible light communications (MIMO VLC). We provide an analytical expression on power allocation subject to the function of maximizing the lower bound of capacity according to the optical wireless communication channel model. The expression is rather different from the radio frequency (RF) communications due to the non-negativity of the transmitted signals in VLC. Furthermore, we design an adaptive bit loading scheme for sub-streams with consideration of the tremendous gain difference among subchannels. Performances of the proposed adaptive power and bits allocation strategy are evaluated in two traditional MIMO VLC scenarios, i.e., a practical $2\times 2$ indoor scenario and a classical $4\times 4$ system that is generated according to the MIMO model. In simulations, the techniques of unipolar M-level pulse amplitude modulation (M-PAM) and spatial-multiplexing (SMP) are employed, and the performance comparison with other SVD-based linear precoders is also given. Simulation results show that the proposed approach can effectively improve the spectral efficiency and guarantee the bit error rate (BER) performance under the same constraints of aggregate optical power budget and non-negativity included by the intensity modulation. It indicates that the strategy of uneven bits and power allocation with full consideration of the ill-conditioned characteristics of subchannels performs better than that based on equal bits transmission in VLC systems.


I. INTRODUCTION
Due to its low cost, high power efficiency, low electromagnetic interference [1] and use of license-free spectrum [2], visible light communication (VLC) has become a potential candidate for wideband communications. As white lightemitting diode (LED) have limited modulation bandwidth (about 150 MHz [3]) and illumination devices commonly consist of multiple LEDs, multi-input multi-output (MIMO) schemes have been considered to achieve high data rates and/or high reliability without consuming extra time and spectrum resources [4], [5]. It has been shown that multi-The associate editor coordinating the review of this manuscript and approving it for publication was Xingwang Li . stream transmissions with capacities up to Gbits/s [6], [7] can be achieved by MIMO VLC over line-of-sight (LOS) channels [8].
To achieve highly reliable multi-stream transmissions in MIMO VLC, the transceiver needs to be adapted to the channel state information (CSI). Precoding has been widely studied for traditional radio-frequency MIMO communications under different constraints and performance criteria. However, VLC faces an additional design constraint that the modulated signal must be non-negative as the information bits are intensity-modulated. Typically, a direct current (DC) bias is added to the modulated signal to ensure non-negativity signaling [10]. Other constraints such as illumination control [11] and power variation [12] may also be considered.
Channel inversion-based precoding schemes are considered for the MIMO VLC in [13]. These schemes feature low design complexity, but may lead to high power consumption when the channel matrix is ill-conditioned [10]. By contrast, the convex optimization-based designs require an iterative solution to optimization problems. Precoding based on singular value decomposition (SVD), which has been widely considered for RF communications, also attracts significant interests for point-to-point MIMO VLC [2], [14], [17]. For example, Gong et al. designed a bit and power allocation method for SVD-based precoding in color-division VLC systems [2], in which different transmitted colors are incorporated. Park et al. introduced a precoding method that maximizes the transmission rate [5], in which various modulation sizes were applied for different substreams under specific bit error rate (BER) requirements. An alternative SVD-based precoder that optimizes the capacity of the electrical channel without considering the non-negativity constraint [17]. More recently, a SVD-based precoder minimizing the total mean squared error (MSE) was designed in [14], which was not necessarily optimal for BER as the system performance is determined by the subchannel with the least gains. Besides, some other precoder designs for multiple-user downlink transmissions with decision feedback equalizer can also be found in [15], [16].
It is necessary to allocate power according to subchannel gain and reduce the effect on the subchannel in the worst condition. In this work, we design an SVD-based low-complexity scheme for point-to-point MIMO VLC and provide an analytical expression for power and bits allocation subject to the function of maximizing the lower bound of capacity. In the scheme, the optical wireless communication (OWC) channel model with intensity modulation/direct detection (IM/DD) is considered, in which the lower bound on the capacity is rather different from RF communications [18]. The scheme can improve the achievable rates under the constraints of nonnegativity signaling and aggregate optical power budget. The given approach can achieve adaptive modulation for multistream transmissions under a given transmission rate, by allocating fewer bits to the worse subchannels and more bits to the better subchannels. Simulation results demonstrate that the proposed scheme can effectively improve the performance in terms of BER compared with some recently proposed SVDbased precoder designs.

II. SYSTEM MODEL
We consider a spatial-multiplexing (SMP) point-to-point MIMO VLC system with N t LED emitters and N r photodiode detectors, which is illustrated in Fig. 1.
As one of the spatial diversity MIMO techniques, SMP can effectively overcome the fading effect caused by atmospheric turbulences in VLC systems [8], [9]. With all LED emitters transmitting independent data streams simultaneously, as shown in Fig.2, SMP provides higher spectral efficiency in comparison to Repetition Coding (RC) and Spatial Modulation (SM). In SMP, N t LED emitters transmit N t independent  data streams simultaneously. Spectral efficiency of N t × log 2 (M ) bit/s/ Hz can be obtained which is better than SM of log 2 (N t ) + log 2 (M ) bit/s/Hz and RC of log 2 (M ) bit/s/Hz.
The optical channel with IM/DD is assumed. A N t × 1 transmitted signal x = [x 1 , x 2 , · · · , x N t ] T is written as where F ∈ C N t ×N t is the precoding matrix, s = [s 1 , s 2 , · · · , s N t ] T consists of N t independent, modulated symbols with zero mean and a variance of σ 2 s , and d is a positive offset to guarantee the non-negativity of the intensitymodulated signal. We assume an M -level pulse amplitude modulation (M -PAM) with Gray mapping is employed, in which the symbols are uniformly distributed in the range of denotes half of the maximum distance between constellation points.
The following two constraints are considered for designing F and d in the MIMO VLC system: denotes the mathematical expectation. Given the total transmission power P t , we have N t i=1 p i ≤ P t . Since s has a zero mean and p i = d i , the total power constraint can be written as Let = [ , , · · · , ] T , the non-negativity constraint can be rewritten as [5] d − abs(F) ≥ 0, where abs(·) denotes taking the entry-wise absolute value. In order to minimize the total transmission power, d should satisfy which shows that the offset d is determined only by F and . As such, the average power constraint can be written as Typically, LEDs are installed in close proximity and can be jointly driven by a same baseband hardware and electronic driver. The path differences among multiple links are only some centimeters, and the temporal delays are just several nanoseconds. Consequently, the time dispersion between multiple transmission paths is negligible. In addition, the links are generally static or slow time variation under the indoor scenarios. Therefore, we assume perfect knowledge of the channel in the model.
Given the channel matrix H ∈ C N r ×N t with N r ≥ N t , the received signal can be represented as The noise vector n ∈ C N r ×1 denoting the ambient shot light noise and thermal noise can be modeled as an additive white Gaussian process with zero mean and variance σ 2 n , which is independent of the signals and assumed as the primary noise impairment. Furthermore, the intensity of the firstorder reflections on the surfaces (walls) is in the range of 10 (−10) I (assuming Lambertian reflectors with reflectivity of 1), and the path loss of higher-order is even more significant. Therefore, the diffuse transmission portion induced by reflected links is ignored, and only line-of-sight (LOS) paths are considered. The N r × N t channel matrix H is given by where h n r n t denotes the wireless link between the emitter n t and the receiver n r . Each LED has an average power of several tens of mW and is concentrated within a semiangle of 15 • -30 • . At the receiver, a silicon positive-intrinsic-negative (PIN) photodiode is employed to detect the signal, whose responsibility peak lies between 380 nm and 780 nm. A variety of practical LOS paths can be modeled reasonably with a generalized Lambertian radiant intensity [8]. It has been shown in [1] that the rate of the reflected light is low enough compared with the LOS light, and the performance of the system is mainly depends on the LOS light. Hence, the gain of link h n t ,n r in channel matrix H can be expressed as [1], [8] h n t ,n r = where φ denotes the angle of emergence to the emitter axis, and ψ is the angle of incidence to the receiver axis. Besides, d represents the distance between the emitter and detector. A is the detection area and the coefficient κ is κ = −ln (2) ln cos 1 2 .
For example, 1 2 = 60 • ( Lambertian transmitter) corresponds to κ = 1, while 1 2 = 15 • corresponds to κ = 20. Assuming that the offset d is known at the receiver, equalization can be performed after removing the offset from the received signal, i.e., y = y r − Hd = HFs + n.
At the receiver, the maximum-likelihood (ML) detection is considered. Thus, the decoder decides the constellation vector s that with the least Euclidean distance between the actual received signal vector and all potential received signals, that isŝ where p y is the probability density function of y conditioned on signal s and channel H. F represents the Frobenius norm.
A. SVD-BASED PRECODER DESIGN Similar to RF communications, various precoders can be applied to MIMO VLC. In this section, we present a SVDbased precoder for MIMO VLC that incorporates the nonnegativity constraint of the transmitted signal and optimizes the lower bound of the achievable capacity. We then propose an adaptive bit loading scheme for minimizing the error rate for fixed-rate uncoded PAM transmissions. In the following analysis, both the linear equalization and perfect knowledge of the channel matrix H at the transmitter and receiver are assumed.
Let the SVD of the channel matrix H be where U ∈ R N r ×N r and V ∈ R N t ×N t are the left and right singular vector matrix, respectively, and the N r × N t diagonal matrix = diag(λ 1 , λ 2 , · · · , λ N t ) consists of singular values. We assume that the precoder matrix F = [f 1 , f 2 , · · · , f N t ] follows the structure The received signal y is first preprocessed by a linear operator U, yielding where n U T n follows the same distribution as n. As is a diagonal matrix, the mutual interferences between the different transmitted symbols in s are eliminated. Therefore, single-tap equalization can be used to recover s.

B. PRECODER DESIGN
The precoder design aims to optimize the performance under the constraints of the aggregate power budget and the nonnegativity of the modulated signal. Applying the structure in (13) on the precoder may lead to a suboptimal solution but also reduces the design complexity. Assuming N r = N t for simplicity, the system in (14) is equivalent to N t parallel, scalar subchannels as Clearly, the scaling factors φ k in the precoder design influences the effective signal-to-noise ratio (SNR) in the corresponding subchannel and can be viewed as the power allocation factors.
We assume that the input to each subchannel s k is uniformly distributed in [− , ]. Without loss of generality, let = 1 for notational simplicity. Then for the k-th subchannel, a lower bound on the capacity is given by [18] where e is the natural number. The constraint on the total power is 1 T abs(V )1 = P t , which depends on the singular vector matrix. As the average transmitted optical power of P t in VLC is given by not the typical time average of |x (t) | 2 in radio frequency (RF) communications. The difference results in dissimilar power constraints tr( T ) = P t and power allocation (Equ.18 ) in comparison with RF systems.
We propose to maximize the total capacity under the total power constraint as Define where v jk is the (j, k)-th entry of V. In order to find the optimized power allocation, we resort to the Lagrangian method. Define where η is the Lagrange multiplier. Letting ∂L({φ k },η) ∂φ k = 0, the extreme value points satisfy There are two solutions to (21), which correspond to the extreme points of the power allocation. It can be easily seen that if we choose the solution where µ 2η ln2 , the allocated power φ k decreases as λ k increases, in contrast to the common waterfilling solutions. We choose the other solution in our power allocation. By solving the above equation, we propose the following power allocation scheme where µ is chosen such that the total power constraint is satisfied, i.e., Thus, the power allocation can be obtained by (23).

C. ADAPTIVE MODULATION
In order to evaluate the effectiveness of the proposed design, we consider a scenario with uncoded transmissions with PAM modulations and a given total transmission rate of B T bits/symbol. The order 2 b k of the PAM modulation for each data stream is determined according to the lower bounds C k of the achievable capacities in (16). This can be done by using the Algorithm 1 given in the table, where b k denotes the transmission rate of the stream k, · denotes taking the integer part of a number, and the vector j records the indexes of the sorted entries.

III. SIMULATION RESULTS
The performance of the proposed capacity maximized strategy is demonstrated in this section. The Monte Carlo method is used to evaluate the performance in terms of BER and capacity. Several hundreds of thousands of bits are transmitted under different channel conditions, and the BER is calculated by evaluating the practical error bits received at the detectors. The capacity depends on actual achieved spectral efficiency as well, which not only relies on the precoder design, spatial diversity scheme, but also is influenced by the noise variance, subchannel gains, and transmission rate. Furthermore, results for comparison with that of schemes in [5], [14], [16] are given, and differences between the strategies are also analyzed.
It has been found that several LED arrays would result in channel matrices with unacceptably high condition numbers even in massive 16 × 16 MIMO VLC systems (which is considered ''massive'' in VLC [19]). Hence, we focus on the traditional MIMO VLC system with 2 × 2 and 4 × 4 scenarios. Noting that dealing with high spatial channel correlation and inter-channel interference (ICI) will inevitably increase the system complexity. Here, we still consider the classical precoder and receiver design for traditional MIMO VLC scenarios.
As LED arrays are typically installed in a compact type and can be driven by a same baseband and electronic hardware, we assume the system in exact synchronization. In addition, the intensity of the first-order reflections on the surfaces (walls) is in the range of 10 −10 I (assuming Lambertian transmitters with reflectivity of 1). The path loss of higher-order reflections is even larger. Therefore, the diffuse transmission portion induced by reflected links is ignored. In simulations, the LEDs are oriented down-towards, and the detectors are oriented up-towards.
Performance of a practical symmetrical 2×2 indoor MIMO VLC communication and an asymmetrical 4×4 scenarios are evaluated. Without loss of generality, in the 4 × 4 system, the channel matrix is generated according to the MIMO model used in [5], [10].
In a room size of 3 m × 3 m × 2.5 m, we established a 2 × 2 (N r = 2 and N t = 2 ) MIMO VLC scenario. The LED arrays are centered in the middle of the room and oriented down-towards at the height of z = 2.5 m, and the detectors According to a practical LOS OWC system implemented within the European Union (EU) project OMEGA [21], [22], the transmitter semiangle at half of power is assumed to be 15 • , and FOV semiangle of the receiver is assumed to be 15 • . The simulation parameters are listed in Tab. 1.
The channel gain of each LOS link is calculated according to Equ.9, which is given by The BER and capacity of the 2 × 2 MIMO VLC system is shown in Fig. 4 and Fig.5, respectively. By employing M -PAM and SMP, 4 bits/symbol transmission rate is obtained in the system. From Fig. 4, it can be obviously found that the proposed scheme achieves the best bit transmission accuracy, whereas the method in [16] obtains the largest number of error bits. In addition, when the noise variance becomes smaller, the gap between different schemes gets narrower. This indicates that system performance is significantly influenced by the SNR. Hence, it is vital to allocate power appropriately at the transmitter. In the SVD-based scheme [14] and the geometric mean decomposition (GMD) [16],  equal bit loading among all subchannels is applied. This is in contrast to the proposed precoder, in which different amounts of bits are allocated to substreams with given total transmission bits. Combined with the bit allocation in our scheme, the subchannel with smaller gain will be allocated less power, or even no power when the gain is extremely small under the ill-conditioned scenarios. Thus, better performance can be achieved by fully considering the subchannel conditions.
Besides, at σ n = 0.01 and P t = 3 W, BER of the proposed design is 2.5 × 10 −5 , which is lower than 2.54 × 10 −2 in [14] and 3.5 × 10 −3 in [16]. Whereas with P t = 2 W, BER of the proposed method is 3.9 × 10 −3 , and the gap between the other two designs is narrower. It shows that performance superiority becomes more evident due to the higher transmission power.
From the capacity comparison in Fig. 5, it can be seen that the given strategy achieves better channel capacity under various noise situations. Besides, the same trend as BER can be found that the performance difference gets more obvious when the aggregate power increases. In addition, the three methods have nearly the same capacity when the noise variance is sufficiently low. This suggests the proposed scheme may perform better when combining with the capacity-approaching error control codes.
We also evaluate the performance of a system with larger asymmetrical antennas. In a 4 × 4 MIMO VLC system, we compare the proposed method with the SVD-based design in [14] and [5], which was also designed based on the SVD precoding and adaptive modulation size for subchannels. Without loss of generality, the channel matrix is generated according to the MIMO model in [10], which can also be calculated with the geometric position of transceiver. The generated channel matrix is expressed as As stated before, the imbalanced bit allocation is employed in the design. Fig. 6 illustrates the modulated symbols of each stream with rates allocated as [4,2,1,1]. It is found that different streams have various modulation sizes in the proposed scheme. Fig. 7 shows the transmitted signals after precoding and adding a DC offset, which is non-negative and could be transmitted in optical channels with IM/DD.
The performances of the uncoded BER and lower bounds on capacity with different noise variances and transmission powers are investigated. The results of a 4 × 4 MIMO VLC system with a total transmission rate of B T = 8 bits/symbol are shown in Fig. 8 and Fig. 9. It is found that the proposed scheme outperforms the SVD-based design in [14] under the same total power budget.
With P t = 10 W, as shown in Fig. 8, the BER of the proposed design is 4.9×10 −4 , which is about 2.7 times better than the design in [14] when σ 2 n = 0.02. In terms of the achievable capacity (Fig. 9), it also outperforms the SVDbased scheme. At the point of σ 2 n = 0.04, the capacity of the 63906 VOLUME 8, 2020 FIGURE 7. The transmitted signal x after applying precoding and adding the offset to the signal s in Figure 2. The average transmission power is P t = 10 W and σ 2 n = 0.01. proposed scheme is about 1.5 times of that in [14]. And with the decrease of the noise variance, the capacity gap between the two schemes becomes narrower. Generally, it is essential to apply adaptive modulation sizes in subchannels according to their channel conditions. Thus, the spectral efficiency can be improved by making full use of channel gains when compared with the schemes based on equal loading among all subchannels. Note that in [5], the rate maximization problem under specific BER requirement is formulated with a SVD precoding method and adaptive modulation size. Performance comparisons are also implemented for a 4×2 channel matrix considered in [5] with a total transmission rate of B T = 8 bits/symbol. Fig. 10 presents the comparison results among the proposed scheme and the precoding designs in [5], [16]. With the same total transmission bits and σ n = 1, BER performance versus various power of these methods is given. In the simulation, both the methods in our design and [5], the adaptive  modulation are applied according to the difference of subchannel conditions. In contrast, equal bits are allocated for all subchannels in [16]. BER is calculated according to the error bits obtained at the receiver. From Fig. 10, we can see that even though the strategies in the proposed design and [5] apply the same adaptive bit loading technique, the proposed precoding achieves better BER performance under different aggregate powers, which is due to the different target functions and bit allocation methods. In [5], the power allocation aims at minimum the mean square error (MMSE). In contrast, power is allocated for achieving the maximum capacity in the proposed method, and the modulation size of each subchannel is proportional to the square of the singular value rather than the singular value in [5]. In the VLC system, the channel condition number is typically large [19], [20], which makes a significant difference in gain among subchannels. In the proposed scheme, the performance is improved by avoiding the negative influence of the ill-conditioned subchannel and fully utilizing the subchannels with larger channel gain.

IV. CONCLUSION
We have proposed a novel linear precoder design based on SVD of the channel matrix for the spatial-multiplexing MIMO VLC system. Under the constraints of the total power budget and positive transmission signal, a power allocation scheme that maximizes the capacity is designed, which realizes adaptive PAM transmissions on different subchannels. Simulation results demonstrate that the proposed scheme can effectively improve the performance in terms of BER and the channel capacity. Furthermore, it is vital to pay attention to the property of the large condition number of VLC channels, which means a tremendous difference in gain among subchannels. In addition, applying adaptive modulation size according to subchannel gain can significantly improve the spectral efficiency. However, employing adaptive modulation will inevitably increase the system complexity, especially in a massive MIMO scenario, which should be further investigated in future research.