Performance of Optical Mobile Communications with User Mobility and Multiple Light Sources

Optical mobile communication (OMC) is a recently proposed optical wireless communication concept aiming to provide very high-speed data rate optical wireless links for multiple and, in general, distributed mobile users. Previous work analyzed the rate performance of a two-user OMC system without user mobility. This paper extends the rate analysis to multiple users with mobility. The scenario of employing multiple light sources with possible user grouping is also considered. User mobility and multiple light sources lead to new challenges on the system design which are addressed for broadcast downlink communication in this work. Simulations show that user mobility decreases the rate, and the way of how to utilize multiple light sources has great impact on the performance. In particular, simultaneous power division usage of multiple light sources through user grouping and power allocation brings almost no gain as compared with the case of single light source. On the other hand, time division usage of multiple light sources is capable of compensating for the hardware deficiency and thus increasing the rate greatly. It is found that OMC is not only superior to the conventional scheme with nonadjustable channel gains but also outperforms free space optical scheme at high signal-to-noise ratio region.


Introduction
Increasing the network throughput for diverse applications is always one of the main objectives of mobile communication systems. In the fifth-generation (5G) mobile communication system [1], the data rate can be boosted by three ways in the physical layer. Firstly, new spectrum with higher frequency, i.e., the millimeter wave band, is allocated to provide high-speed data rate links [2,3]. Secondly, spatial multiplexing gain or the multiuser diversity gain is exploited, with massive multiple input multiple output (MIMO) or small cells to provide manyfold increase in data rate compared to conventional techniques with few antennas or large cells [4,5]. In particular, with massive MIMO, the base station can generate multiple narrow beams for each user such that the emitted energy from transmit antennas can be conveyed much precisely, gaining very high spectral and energy efficiency. The third way is to utilize new capacityapproaching channel codes [6,7] and nonorthogonal multiple access techniques [8,9]. As the standardization process of 5G phase 2 has come to closure, researchers have begun to shift their research attention to the next-generation or the sixth-generation (6G) mobile communication system [10]. Although there is no official definition for 6G, some common views on it have been gradually reached [11][12][13]. An important view is that 6G will exploit full spectrum, spanning from radio frequency and millimeter wave to terahertz, visible light, and even higher frequency bands, to support full coverage communications ranging from terrestrial to ocean surface, air, and even space. It can be foreseen that high frequency electromagnetic wave including optical band will be utilized by 6G to achieve unprecedented data rate. To our understanding, the optical band will not be used merely as the wireless backhaul or in fiber links. More desirably, it can be utilized to support fiber-like wireless access for multiple mobile users in the radio access network, which may be termed as optical mobile communication (OMC) [14]. OMC may trigger many new and exciting applications such as mobile virtual reality. It also provides a handy carrier for future wireless quantum direct communication in the optical band with coherent processing [15].
OMC can be regarded as an alternative implementation of massive MIMO in two aspects [10]. Firstly, OMC may exploit one or more spatial light modulators (SLMs) for beam steering. Each SLM consists of a large number (possibly a few million) of optical elements such as liquid crystals which can be viewed as massive antennas. Secondly, the energy is concisely distributed and delivered to desired users with very thin laser beams, and there is little leakage in other spatial directions. Nonetheless, the optical band devices are not bulky and of low power consumption. Therefore, OMC is an alternative substitute of massive MIMO in the optical band. As a special variant of optical wireless communication (OWC), OMC also faces many physical limitations, e.g., the optical beam may be easily blocked by obstacles like buildings or attenuated in adverse weather. Therefore, it is more likely that OMC will find its role as a complementary technique for existing mainstream wireless communications to provide very high-speed peak data rate links in some future mobile communication scenarios.
It should be noted that there are several other OWC techniques that can support mobile access for multiple users utilizing light spectrum, including infrared communication (IR), visible light communication (VLC), and free-space optical (FSO) communication [16]. For example, [17,18] studied the throughput performance of mobile users in VLC systems. The mobility is guaranteed by utilizing wide half-power semiangle light emitting diodes (LEDs) as light sources. Multiple light sources is used to further enhance the light coverage probability. In other words, wide beams are projected in VLC to support mobility, which is also the case in IR. OMC differs from VLC and IR in that it concentrates the optical power on the receiver side, which is more energy efficient. In addition, the signal processing in LEDbased VLC is inherently noncoherent, whereas OMC can be both noncoherent or coherent, depending on how to perform modulation in the laser source. In this work, coherent processing is considered. OMC is also different from conventional FSO communication in that it is designed to accommodate multiuser mobile communication, where the users may not be covered by a single wide beam spot, and the number and locations of users may change from time to time. Therefore, flexible beam splitting and steering are the key enabling function and feature of OMC system, which bring larger achievable rate region than current OWC schemes.
Realizing OMC is quite challenging, and many technical and engineering issues must be resolved. For instance, due to cost and scalability concerns, it is impractical to equip a large number of laser devices to accommodate multiple users. Instead, flexible beam splitting and steering for a single laser beam must be dealt with. [19] proposed to use liquid crystal SLM for beam adaptation among other potential techniques such as digital micromirror device (DMD) to support infrastructure-to-vehicle (I2V) communication. Experimental results verified the feasibility of beam adaptation. Rate region for unicast downlink communication was derived for a two-user system with orthogonal and nonorthogonal multiple access techniques. However, the rate region only depicted the scenario of motionless users. [20,21] investigated more practical scenarios with user mobility for coherent and noncoherent transmissions, respectively. Many new factors, such as the beam length, the communication time (ON-state), and the power split ratio, must be configured in a joint manner to achieve the best potential performance. This paper extends the results in [20] to multiple light sources which bring new design freedom and challenges. As OMC is still in its infancy, in this work, we focus on the information theoretic aspects of the system. Although coherent processing is much more challenging in system implementation as compared with noncoherent schemes, we study the system performance under this assumption to unveil the maximum potential gain of OMC over the conventional schemes and to identify the key factors that affect the gain. In fact, considerable advancements have been achieved in recent years for coherent optical wireless access [22][23][24], even in communication with satellite [25]. Thus, OMC with coherent transmission and processing can be expected in the future.
Resource allocation is among the key issues in other OWC schemes with user mobility, such as in VLC [26][27][28]. In those systems, the optimal resource allocation is in general hard to derive, and suboptimal solutions were proposed. Similar situations are found in this work; however, the task in OMC is systematically different from those works, since the light sources, including SLM, have very different physical models as compared to other systems. Therefore, the optimization process is fundamentally different from existing works.
With this regard, this paper comprehensively investigates the achievable rate performance of broadcast OMC system with multiple mobile users. The contributions of this paper are summarized as follows: (i) This paper considers two scenarios of broadcast OMC. As a basis, in the first scenario, a single light source with multiple mobile users is considered. In this scenario, the impacts of user mobility and the switch time of SLM on the achievable rate performance is the key issue, which has not been addressed previously. In the second scenario, besides user mobility, multiple light sources are involved and share the total transmit power. The main challenges include how to efficiently coordinate the usage of multiple light sources and the associated optimal resource allocation strategies, which are also not studied in current literature (ii) To address the issue in the first scenario, we set the reflected beams to be rectangular to provide consistent communication links as users move. The beam 2 Wireless Communications and Mobile Computing length for each user is optimally selected according to user's speed. Then, the optimal period of the ON-state of SLM and the optimal power allocation ratios among users are jointly optimized. Based on the analysis, the rate region of a two-user system is depicted, and then, the maximum common rate is obtained (iii) To address the challenges in the second scenario, we distinguish and study power division usage of multiple light sources and time division usage of multiple light sources, respectively. In power division mode, users are grouped, and each group is served by a light source. A simple user grouping strategy as well as a user group power allocation algorithm is proposed. The user grouping strategy is proved to be optimal at low signal-to-noise ratio (SNR) regime. In time division mode, light sources take turns to serve all users. The optimal serving period of each light source is derived in closed form. It is found that time division usage of multiple light sources is more beneficial as compared to power division usage of multiple light sources as well as two benchmark schemes, i.e., the conventional scheme with nonadjustable channel gains and the FSO scheme The remaining of this paper is organized as follows. Section 2 provides the system model with a single light source whose beam is adapted by an SLM. The corresponding performance is analyzed in Section 3. Then, Section 4 and Section 5 investigate how to exploit multiple light sources in power division and time division manners, respectively. Various numerical results are provided in Section 6. Finally, conclusions are drawn in Section 7 with some discussions on future research directions.

System Model with a Single Light Source
This section reviews briefly the principle of OMC with a single light source and exemplifies it in I2V broadcast communication with user mobility.
2.1. OMC Principle. The basic approach of OMC is to split a single narrow optical beam generated by a laser device into multiple wider beams and steer those beams to desired and in general different spatial directions, realized by a beam splitting and steering device. In [19], SLM was employed to accomplish the task of flexible beam splitting and steering. Other options were also discussed therein. SLM can be viewed as an optical massive MIMO device which is capable of finely tuning the phase of the wave front of an input laser beam. By properly setting the control or address signal for the SLM, the wave front of the input laser beam will be manipulated into a hologram signal by the SLM. This hologram is then transformed to an image and projected onto a larger area by a lens system. The hologram is fully determined by the address signal fed to the SLM, and it also determines the image through the mapping of two-dimensional fractional Fourier transformation. Therefore, the image can be tuned flexibly and usually consists of several beam spots for user coverage in OMC.
Mathematically, the above operations are equivalent to splitting an optical signal xðtÞ of power P into K beams with possibly different crosssection shapes. Beam k serves user k and can be represented by x k ðtÞ = ffiffiffiffi ffi α k p xðtÞ. It has power P k which is a fraction of the total power, i.e., P k = α k P, where 0 ≤ α k ≤ 1 and ∑ K k=1 α k = 1 without considering the power loss in the beam splitting phase. The key feature of OMC lies in that K and α k are tunable, which is in contrast with the conventional beam adaptation technique based on the commonly used beam splitter, since beam splitter can only split a beam into two with a fixed power ratio after it has been fabricated. This implies that the communication channel in OMC is tunable or controllable to a large extent. Based on the above mathematical principle, the aim of this paper is to analyze the potential performance of OMC and compare it with existing schemes. Implementation issues, including the beam adaptation schemes and demo, positioning and coherent transmitting and receiving techniques, and their feasibilities, can be found in [19,29,30]. Improved or new implementations will be developed in future works.

OMC in I2V
Broadcasting with User Mobility. I2V communication is among the key enabling technologies for realizing Internet of Vehicles (IoV) and Internet of Things (IoT). In this paper, I2V broadcasting scenario is considered to reveal the potential of OMC, which is shown in Figure 1. Herein, the transmitter side is attached on the infrastructure such as a lamp post, which is usually within 10 meters in height. The wavelength of the light source does not affect the signal model; thus, infrared light can be utilized in this scenario. However, it should be noted that OMC can be applied in broader applications such as in industrial environment where human eye safety is not a concern. The transmitter generates an image which consists of several beam spots for multiple vehicles (users) in a certain area after the image is projected by a lens system. The beams bear the same information that is common for all users. Here, we denote the original electrical domain information as ffiffiffi P p x, where x is the random information symbol that has zero mean and unit variance, and P is the power of the signal. Each beam may be allocated with different power and crosssection shapes according to the user mobility behaviours. Suppose there are K users in total and user k has a speed of v k and is allocated power P k = α k P in its associated beam. The receiver can be attached on the roof of the vehicle to receive the broadcasted information when it is covered by a beam.
An important limitation of adopting SLM for beam adaptation is that the image cannot be changed continuously in time. In contrast, to change from a current image to another, a switch time Δt to fully reload the new address signal and regenerate the new image is required for the SLM. Commercially available SLMs have a minimum switch time in the order of several milliseconds. This hardware phenomena prevents the most efficient beam adaptation scheme that 3 Wireless Communications and Mobile Computing the receiver aperture is persistently covered and tracked by a very narrow beam. Instead, the beams generated by a realistic SLM have to be wide enough to support continuous link coverage with user mobility for a communication ON-state of duration T. When the vehicles move out of the current beam coverage area, switch is triggered to regenerate a new beam pattern to recover the links, resulting in an OFFstate of duration Δt. Figure 2 illustrates the beam adaptation process for user k. Initially at time t = 0, a beam is generated to cover the expected path of the receiver aperture. The shape of the beam is set as a rectangular with length L k and survives for a time duration T. The width of the beam is a bit larger than the size of the receiver aperture, which is assumed to be a square of length D for simplicity. At time t = T, the SLM enters a switch state, and the beam diminishes for a time duration of Δt. Then, at t = T + Δt, a new beam is established and the communication link recovers.
With the above description, the received downlink electrical domain signal of user k after coherent receiving can be written as where R rec is the responsivity of the receiver detector, P LO,k denotes the power of the local oscillator, and z k is the noise component. With coherent detection, the main contribution of the aggregate noise z k in (1) comes from the shot noise in the local oscillator. According to [31], z k can be modeled as complex Gaussian with zero mean and variance σ 2 n . The coefficient c k accounts for several nonideal factors affecting the link budget, including the loss in signal propagation, the never-collected power owing to the larger width of the beam over the receiver aperture length D, and the possible power loss due to the pointing error of the aperture toward the incident beam direction. Since the communication distance is relatively short in the studied scenario for laser beams, for simplicity and without loss of generality, c k is set as 1 in this paper, as both the fading effect and power attenuation due to atmospheric turbulence and distancedependent power decay are negligible in such a limited range [32,33]. On the other hand, if atmospheric turbulence and distance-dependent power decay are considered in long distance communication, they must be reflected in c k . This is beyond the scope of this paper and will be studied in the future. R rec ffiffiffiffiffiffiffiffiffi ffi P LO,k p is also assumed to be unity here. Thus, the effective received SNR is given by α k PD/L k σ 2 n . Finally, we note that x k in (1) is effective only in the time duration of ½0, min fT, ðL k − DÞ/v k g over a full time cycle of ðT + Δ tÞ, where ðL k − DÞ/v k is the maximum time duration for the receiver aperture to be fully covered by a beam spot as it moves forward. Therefore, according to Shannon's capacity formula [34], the effective achievable rate (with unit of bits/s/Hz) of user k, averaged in the whole time duration of ðT + ΔtÞ, can be expressed as which is related to a bunch of parameters. In particular, L k , α k , and T are not hardware determined and must be optimized according to other fixed parameters, such as Δt and v k . The optimization goal is to maximize the minimum value of fR k g K k=1 for a given total power P. The case of single light source is considered in Section 3, while the case of multiple light sources will be studied in Sections 4 and 5.

Rate Maximization
3.1. Beam Length Optimization. In this section, we investigate the best option for the beam length L k . This issue arises from the fact that a larger L k results in a longer communication time, but a reduced instantaneous power, while a smaller L k results in a shorter communication time, but a more concentrated received power. It is thus desired to optimize L k for rate maximization. To do this, we first note that L k ≥ D for a full coverage of the receiver aperture. Moreover, ðL k − DÞ should be no larger than v k T; otherwise, there will be a portion of ðL k − D − v k TÞ/L k transmitted power that will never be collected for usage. Note that in practical system, it is desired that a longer L k than assumed here is configured for robustness purpose. Thus, L k ≤ v k T + D and The problem is to find the optimal L k ∈ ½D, v k T + D to maximize R k in (3). Note the fact that which indicates that R k in (3) is an increasing function of L k . Therefore, the optimal L k must be L * k = v k T + D, and the corresponding rate is reduced to  Wireless Communications and Mobile Computing where γ = P/σ 2 n is defined as the transmit SNR. 3.2. Rate Region of Two-User System. In this subsection, we investigate the rate region of a two-user system by optimizing the power split ratios α 1 , α 2 , ⋯, α K and the communication time T. The result provides a simple yet relatively complete picture of OMC and reveals its potential gain with "controllable" channel over conventional fixed channel. Firstly, note that the maximum rate of user k is achieved only when α k = 1, i.e., which is still a function of T. Here, ≜ denotes definition. There is no analytical expression for the optimal T. However, we have the following proposition.
Proof. see Appendix A. Therefore, any efficient line search method can be used to obtain the optimal T very quickly. Assume that the maximal rate is given bỹ achieved at T = T * k , for k = 1, 2. Note that T * k and thusR * k ð v k Þ are only related to a specific k. In addition, in Proposition 1, α k = 1 is set only to help to find a vertex of the rate region boundary. In general case, α k s are less than 1 to ensure all users have positive rates, as will be elaborated in Section 3.3. The problem of finding a specific point ðr 1 , r 2 Þ on the boundary of the rate region can now be formulated as s:t: where r 1 is a given value in the range of [0,R * 1 ðv 1 Þ]. Using the first constraint, we obtain Then, we can transform the objective function to wherȇ which is also a function of T. Again, we have the following proposition.
Proof. α 2 in (10) is a function of T and is always no greater than 1 for T > 0. It can be shown that the second-order derivative of α 2 ðTÞ with respect to T is always positive, and thus, α 2 ðTÞ is a concave function. When T approaches 0 and infinity, α 2 ðTÞ approaches negative infinity. Therefore, there must exist two points T L and T U , The remaining proof is similar to that for Proposition 1 and is omitted here. Assuming the optimalȒ 2 ðT, v 2 Þ is given by whereT is related to both users' parameters, we can obtain all pairs of ðr 1 , r 2 Þ that constitute the boundary of the rate region.

Maximum Rate.
In broadcast applications, we are more concerned with the minimum rate of all users and want to maximize this minimum rate. In OMC, this task is easy to realize since the power ratios α k , k = 1, 2, ⋯, K, are adjustable: regardless of the moving speeds, higher rate users should "slice" a certain portion of their allocated power to the low rate users until all users have the same data rate.

Wireless Communications and Mobile Computing
We refer to the rate of equilibrium as the maximum rate R * ðvÞ here, which is just the maximum of the minimum rate among all users, where v = ½v 1 , v 2 ,⋯,v K T . To find R * ðvÞ and its associated optimal power allocation ratios and communication time T, we set which brings α k = ððv k T + DÞ/ðv 1 T + DÞÞα 1 . Combining ∑ K k=1 α k = 1, the optimal power allocation ratios can be found as which is a function of T as well as all users' speeds through v = ∑ K k=1 v k . Generally speaking, a user with high mobility should be allocated more power to achieve the rate balance. Substitute (14) into (13), we get which has the same structure as (6) and is thus unimodal according to Proposition 1. Again, the optimal T * that maximizes R * ðvÞ in (15) has no analytical expression and has to be obtained through numerical methods.

Power Division Usage of Multiple SLMs
As the number of users gets larger and their speeds vary in a wider range, setting a common communication period T * for all users may decrease the overall achievable rate. A natural remedy is to employ multiple laser sources and, correspondingly, multiple SLMs to serve the users. In this scenario, we consider a single base station consisting of multiple laser sources. Each laser source is associated with a single SLM, forming a laser-SLM pair. On one hand, a laser-SLM pair will serve a group of users as if in the single SLM case; however, all laser sources in the base station must share a total transmit power budget. On the other hand, different SLMs serving different user groups can be configured with unequal communication periods now, leading to new design freedom and potential performance gain. For notational simplicity, in the following, we use multiple SLMs to represent multiple laser-SLM pairs. We refer to this scenario as power division usage of multiple SLMs. The key problems include how to group the users and allocate powers among user groups.

Problem Statement.
Assume there are N available SLMs in the OMC system, where we only consider the nontrivial case 1 < N < K. Each SLM is allocated a fraction of total power given by β n P, where 0 < β n < 1, ∑ N n=1 β n = 1, and serves a group of users whose indices form a set K n with cardinality jK n j, where jK n j satisfies 1 ≤ jK n j < K, ∑ N n=1 j K n j = K. Each group of users behave just like the single SLM case in Section 3, i.e., through proper power allocation among those users, denoted as α n m β n P, where α n m , m = 1, 2 , ⋯, | K n | , is the allocated power ratio of the mth user in this group satisfying ∑ jK n j m=1 α n m = 1; they eventually approach a maximum rate for some optimal T * n that is common for all users in this group. Different groups in general have different optimal T * n . Therefore, for ease of user scheduling, it is assumed here that each SLM serves only a single group of users and each group of users are served by a single SLM.
The problem is to find the optimal grouping strategy fK n g N n=1 at the transmitter side, along with optimal power allocation among groups and users, i.e., fβ n g N n=1 and fα n m g |K n | m=1 , n = 1, 2, ⋯, N, as well as optimal transmission time fT * n g N n=1 for each group, to maximize the common rate of all users.

Optimal Power Allocation for Given User Grouping.
Firstly, we investigate how to optimally allocate power to each group and user for a given user grouping result. For users in group n, their rate should be the same by proper power allocation. Therefore, according to Section 3.3, the optimal power allocation ratios are given by which results in the rate of this group as R n T n , β n ð Þ= T n T n + Δ t log 2 1 + β n Dγ The remaining problem can be formulated as s:t: Proposition 3. There is a unique solution to problem (P2) for a given user grouping result.
Proof. Due to the unimodal property of R n ðT n , β n Þ with respect to T n for any given β n , the optimal T n is a function of β n and must be the point which maximizes R n ðT n , β n Þ. Otherwise, a portion of power is not fully exploited and can be sliced and distributed to all users to further increase the common rate. DenotingR n ðβ n Þ = max T n R n ðT n , β n Þ, the constraints are changed toR n ðβ n Þ =R 1 ðβ 1 Þ, ∀n, and ∑ N n=1 β n = 1. AsR n ðβ n Þ is an increasing function of β n , there exists only a single β n that satisfies the constraintR n ðβ n Þ 6 Wireless Communications and Mobile Computing =R 1 ðβ 1 Þ, i.e., β n is a bijective function of β 1 and can be denoted as β n = B n ðβ 1 Þ. Furthermore, when β 1 increases, R 1 ðβ 1 Þ increases and so does β n , which means B n ðβ 1 Þ is an increasing function of β 1 . As a consequence, Bðβ 1 Þ = β 1 + ∑ N n=2 B n ðβ 1 Þ is an increasing function of β 1 , and thus, Bð β 1 Þ = 1 has only a single (and thus optimal) solution β * 1 , implying there is also a single solution for fβ n , T n g N n=1 , which proves Proposition 3.
Based on the above proof, we can conclude that the maximum common rate can be obtained by searching the optimal β 1 such that Bðβ 1 Þ = 1 is satisfied. Noting Bðβ 1 Þ is a monotonic increasing function, here, we propose an efficient and optimal power allocation method to find the maximum common rate, which is shown in Algorithm 1.
In the procedure of Algorithm 1, step 11 forces the rate of all groups to be equal by slicing down the power of the groups that has larger rate than R min . While this guarantees the rate of all groups to be equal, it does not fully utilize the available power. Therefore, step 13 lifts up the power split ratio β n for all groups through normalization. We can observe that the minimum group rate is increasing for each additional iteration. Upon convergence, the two constraints in problem (P2) are automatically satisfied within numerical precision, indicating the resultant solution is optimal according to Proposition 3. Note that Algorithm 1 only involves a series of one-dimensional line search for unimodal functions instead of exhaustive search, which is efficient.

Suboptimal User Grouping.
In this subsection, we investigate the remaining issue of user grouping. The number of all feasible grouping results is referred to as Stirling's number of the second kind [35], denoted by SðK, NÞ, which grows very fast as K increases and N is not close to 1 nor K. For instance, Sð16, 4Þ ≈ 1:71 × 10 8 and Sð20, 8Þ ≈ 1:52 × 10 13 . This means the brute-force searching method is impractical here even for a medium K. Therefore, to assess the performance gain of using multiple SLMs, we have to seek some suboptimal strategies that have affordable computational complexities. In this section, we propose a heuristic user grouping strategy. The intuition is that users with lower speeds consume much less power such that we may allocate more power to poor users with higher speeds to improve their performance. Specifically, the proposed user grouping strategy is as follows: assuming the K users are sorted by their speeds: v 1 ≤ v 2 ≤ ⋯≤v K , the user grouping strategy is given by K n = fng, for n = 1, 2, ⋯, N − 1, and K N = fN, N + 1,⋯,K − 1, Kg, which needs no specific calculations. Proposition 4. The proposed user grouping strategy is asymptotically optimal in the low SNR regime.
Proof. see Appendix B.

Time Division Usage of Multiple SLMs
As will be seen through simulations in 6.3, quite unexpectedly, power division usage of multiple SLMs, even in the extreme case of K SLMs, does not bring noticeable performance gain as compared with the case of single SLM. Therefore, it is desired to further find better means to fully exploit the additional SLMs to enhance the performance in practice. In power division usage of multiple SLMs, each user will experience link break during the OFF-state of its serving SLM, which lasts at least Δt seconds. This is an important factor that limits the rate performance. Motivated by this, we consider time division usage of multiple SLMs in which the SLMs take turns to communicate with all users such that Input: fK n g N n=1 (with associated parameters K, N, jK n j, v n , n = 1, 2, ⋯, N), Δ t , γ, D, and precision ε. Output: the maximum common rate R * and its associated power allocation fβ * n g N n=1 , fα n m g jK n j m=1 , n = 1, 2, ⋯, N. Initialize: β n = 1/N, n = 1, 2, ⋯, N, tol = 10ε. 1: while tol > ε do 2: for n = 1 : N do 3: Set α n m using (16);
Algorithm 1: Proposed group and user power allocation scheme for given user grouping 7 Wireless Communications and Mobile Computing the users always receive light power without break. Figure 3 shows the states of two SLMs.
Basically, each SLM's ON-state will just cover other's OFF-state in an alternating manner. The period of OFFstate must be at least Δt. In each SLM's ON-state, it occupies the full power and serves all users with proper power allocation but without user grouping. Then, with a similar approach in 3.3 the maximum common rate can be rewritten as The constraints are T 1 ≥ Δt and T 2 ≥ Δt to cover the minimum switch time. Generalizing to N SLMs, the rate changes to where T = ∑ N n=1 T n and the constraints are ∑ l≠n T l = T − T n ≥ Δt, n = 1, ⋯, N, which means each SLM's OFF-state period will be fully covered by adding all the other SLM's ON-state periods. Denote a n = T n /T, ∀n, the rate becomes with constraints ∑ N n=1 a n = 1, a n > 0, and Tð1 − a n Þ ≥ Δt, ∀n. It can be shown that R * ðvÞ is maximized when T = ðN/ðN − 1ÞÞΔt and a n = 1/N, ∀n, and the rate is given by which is an increasing function of N for N ≥ 2. When N approaches to infinity, R * ðvÞ achieves its upper bound as It is worth noting that this upper bound can also be achieved with a single SLM system if the switch time Δt approaches to zero, as indicated by (5). Therefore, (24) implies that utilizing unlimited number of SLMs completely eliminates the adverse effect of switch time due to hardware limitation. However, in practice, it is more appropriate to use as less SLMs as possible to reduce the implementation difficulty in coordinating different SLMs and to improve the robustness of the system. From this perspective, N = 2 would be a good tradeoff between complexity and performance. In this case, the rate is given by It can be inferred from (25) that when v ⟶ 0 with motionless users, or Δt ⟶ 0 with advancement in hardware production, the rate can also approach its upper bound R UB even there are only two SLMs.
Since additional SLMs are utilized to reduce the adverse effect of switch time, the rate performance is expected to outperform that of power division usage of SLMs described in 4. In addition, it avoids user grouping and the corresponding power allocation among groups, which has much lower system complexity and can be viewed as a stronger candidate for practical OMC implementations.

Numerical Results
This section provides various numerical results of the rate maximization in OMC and compares them with other benchmark schemes. Subsections 6.1 to 6.2 provide results with single SLMs, while subsection 6.3 illustrates the results with multiple SLMs. Two benchmark schemes are considered here.
(1) The first benchmark scheme can also split and steer the beam but with fixed power allocation ratios (thus, this scheme is denoted as "fixed-PA" in the legends) for different users; thus, its effective channel gains are nonadjustable as compared with OMC. We assume equal power allocation here. This scheme may represent the conventional beam adaption scheme using beam splitter but with enhanced capability of steering the beams to different users freely. The purpose is to evaluate the gain brought by adaptive power allocation according to users' speeds in OMC (2) The second benchmark scheme is an FSO scheme (denoted as "FSO" in the legends) in time division mode, in which the single beam can track a single user precisely at any time interval with all power allocated to and focused on that user. Users are served in a time division manner with zero switch time. Thus, the rate for such an idealized system is simply given by When there are multiple SLMs, the rate of user group n is given by  Figure 3: Illustration of the state changes in time division usage of two SLMs. 8 Wireless Communications and Mobile Computing If we treat jK n j, n = 1, 2, ⋯, N, as continuous variables, it would not be hard through convex optimization to prove that the optimal |K n | is given by K/N and the optimal β n is given by 1/N. The corresponding rate is given by for N ≤ K, which can be viewed as an upper bound of the practical case in which |K n | , n = 1, 2, ⋯, N, are integers. Note that when N = K, the rate of FSO in (28) achieves its upper bound which is identical with the upper bound rate of OMC given by (24).
In the following, the size of the receiver aperture is set as D = 1 cm, which can be realized with large-aperture PD or through light concentration by a lens. The switch time of the SLM depends on the specific device and can be expected to improve in the future. In the following simulations, Δt = 0:1 ms, 1 ms, and 10 ms are involved to reveal its impact on the performance. 6.1. Rate Region of Two-User System. First, the rate region of a two-user system is given here. Figure 4 shows the rate region at SNR = 20 dB with different Δt. The speeds of the two users are set as v 1 = 20 Km/h and v 2 = 80 Km/h. Several facts can be observed in Figure 4. Firstly, in OMC and the fixed-PA schemes, the user with lower speed may achieve a larger maximum rate, which complies with the intuition. Secondly, with controllable channel management, the OMC scheme entitles much larger rate region than the fixed-PA scheme. In fact, in the fixed-PA scheme, the feasible rate pairs constitutes a trajectory rather than a region. Thirdly, Δt has great impact on the rate performance. With decreased Δt, the rate region is expanded. This indicates that, with advancement in manufacturing of SLM entitling much shorter switch time, higher rate OMC communication can be expected in the future. In fact, it can be seen that with smaller switch time, part of the rate region of OMC lies outside of that of the FSO scheme.  13)) as a function of T is plotted in Figure 5 with different other parameters. It can be seen that R k is indeed unimodal with respect to T. Again, Δt has important impact on the rate. The optimal T * shrinks with decreased Δt, indicating the tendency of more concentrated power in this case. In addition, SNR also Rate with optimized power ratio (bits/s/Hz) =10 dB, Δt = 0.1 ms =10 dB, Δt =1 ms =10 dB, Δt =10 ms =20 dB, Δt =0.1 ms =20 dB, Δt =1 ms =20 dB, Δt =10 ms 9 Wireless Communications and Mobile Computing plays a key role here. To see it more clearly, Figure 6 shows how the optimal T, in the normalized form as T * /Δt, changes with SNR. In general, a larger SNR results in a larger T * . Finally, it is observed from Figure 6 that the number of users K does not have significant impact on the optimal T * .

User Grouping and Power Allocation with Multiple
SLMs. In this subsection, we simulate the performance of user grouping and power allocation with multiple SLMs. The speeds of users are set as uniformly distributed in the range of 70~120 Km/h. Δt = 1 ms switch time is assumed here.
Firstly, we show the convergence behaviour of the proposed user and group power allocation scheme described in Algorithm 1. Figure 7 shows how the rate converges with the number of iterations for different number of groups and numerical precisions. K = 16 is assumed here, and the precision is defined as the normalized ratio ðR max − R min Þ/R max , where R max and R min denote the maximum and minimum per group rates (c.f. step 6 in Algorithm 1) in an iteration, respectively. The results are averaged over 500 random speed realizations. It can be seen from Figure 7 that the rate converges quickly with the number of iterations, indicating the effectiveness of the proposed power allocation algorithm.
Secondly, based on our proposed user grouping strategy, we show the potential gain of using multiple SLMs in power division usage mode, which is denoted by "OMC/PDU" in the legend, where "PDU" stands for power division usage. Figure 8 shows the maximum common rate with different system configurations. The results are also averaged over 500 random speed realizations. The number of users is fixed as K = 16, and N = 2 and 4 groups are tested. Several OMC schemes are compared here. For the proposed user grouping strategy, the results are labeled as "proposed UG" where "UG" stands for user grouping. Similarly, the scheme with random user grouping is labeled as "random UG." When N = 1, there is no user grouping, and the results constitute a lower bound. When N = K, every user forms a single-  user group, and the result can be viewed as performance upper bound of power division usage mode. Power allocations in those schemes are all based on Algorithm 1. Performance of the fixed-PA scheme is also plotted here. Quite surprisingly, it can be seen from Figure 8 that although OMC is indeed superior to the fixed-PA scheme with noticeable gain, multiple SLMs with user grouping does not bring desirable performance gain, i.e., the lower bound and the upper bound are very close to each other. Nonetheless, by zoomed view, the gain of the proposed user grouping strategy over the random user grouping strategy can be still clearly observed, which validates the suboptimality analysis in Section 4. This seemingly discouraging result may be a good thing for engineering, as it suggests we only need a single SLM! Beside the aforementioned schemes, the performance of time division usage of SLM discussed in Section 5 is also included in Figure 8 for comparison. All the curves obtained by this scheme is labeled by "OMC/TDU," where "TDU" stands for time division usage of multiple SLMs. Moreover, performance of the FSO scheme based on (28) is evaluated for comparison. Finally, the common upper bound of OMC/TDU and FSO is also plotted in the figure. It can be seen that with time division usage of multiple SLMs, the rate of OMC can be improved greatly compared with user grouping strategy with power division usage of SLMs. In addition, the rate increases with the number of SLMs and approaches to its upper bound. However, N = 2 can collect a fairly large portion of the benefit. Further increasing N brings only incremental gain. This confirms that N = 2 is a good complexity-performance tradeoff. On the other hand, for smaller N, OMC is also superior to the FSO scheme at high SNR region, which proves the effectiveness of the core idea of OMC, i.e., using SLMs to distribute power simultaneously.

Conclusions and Future Works
This paper investigated the rate performance of OMC system with user mobility through analysis and numerical examples. Due to the switch time of SLM, the communication must take place in periods separated by the minimum switch time. The optimal communication time and the corresponding rate were analyzed in this paper. It was found that mobility reduces the achievable rate in general. The rate loss can be compensated if the switch time of SLM can be reduced. Moreover, as OMC is capable of flexibly adjusting the power split ratios to generate a controllable channel, its rate performance is much better than the conventional scheme with fixed power split ratios. Then, two schemes integrating multiple SLMs, i.e., power division usage of multiple SLMs and time division usage of multiple SLMs, were studied in detail, respectively. It was found through simulations that power division usage of multiple SLMs brought little rate gain as compared to the single SLM case. On the other hand, time division usage of SLMs can compensate for the hardware deficiency of SLMs and boost the performance to a large extent. Compared with benchmark schemes, OMC is not only superior to the conventional scheme with fixed power allocation ratios but also outperforms FSO at high SNR regime.
Following this work and the above conclusions, there are many new issues that are worth studying in the future. For instance, even advanced SLM with reduced switch time is used in the future, the communication time T can still not be very small; otherwise, the localization precision requirement would be very stringent. A practical approach with lower costs is to use intensity modulation, which would have very different signal models and derivations. Another research point is to study the unicast communication scenario with user mobility and multiple light sources, which is more attractive for future applications and more challenging. In addition, integration with uplink transmission can be an important topic from a system point of view. Demonstration of prototype would be of key importance to evaluate how far this new technology can go. ðA:1Þ To proveR k ðT, v k Þ is unimodal with respect to T, it is sufficient to show ð∂R k ðT, v k ÞÞ/∂T has only a single root in ð0, ∞Þ, sinceR k ðT, v k Þ > 0 and lim ðA:2Þ Define b k = v k Δt and s k = Dγ/ðv k T + DÞ, then s k ∈ ð0, γÞ and (A.2) can be reformulated as Denoting hðs k Þ as the left-hand side of (A.3), the original problem is equivalent to show that hðs k Þ has only a single root on s k ∈ ð0, γÞ. As lim s k ⟶0 hðs k Þ < 0 and lim s k ⟶γ hðs k Þ > 0, we can conclude that hðs k Þ has at least one root. The first-order derivative of hðs k Þ is given by which is an increasing function of b k for any fixed s k . In particular, h′ðs k ; 0Þ = 2Dðγ − s k Þ > 0 for s k ∈ ð0, γÞ. Therefore, h ′ ðs k ; b k Þ > 0 in its domain of definition with respect to s k , for any given b k ≥ 0. This means hðs k Þ is a monotonically increasing function of s k . Therefore, hðs k Þ has and 11 Wireless Communications and Mobile Computing only has one root over s k ∈ ð0, γÞ, which finally leads to the conclusion thatR k ðT, v k Þ is unimodal.

B. Proof of Proposition 4
We first investigate a special case where N = 2 to reveal the behaviour of proposed user grouping strategy. According to Section 3.3, for any user group, the maximum rate is related to the number of users in this group and the summation of their speeds. We rewrite (17) by where k n = jK n j, v is the sum speed of all the K users, ððv/ 2Þ − Δ v Þ and ððv/2Þ + Δ v Þ, with 0 < Δ v ≤ ðv/2Þ, denote the sum speeds of the two groups after user grouping, respectively. The maximum common rate of the two groups is given by with constraints β 1 + β 2 = 1 and R 1 ðT 1 , β 1 , Δ v Þ = R 2 ðT 2 , β 2 , Δ v Þ. By treating Δ v as a continuous variable, we have the following lemma.
Proof of Lemma B.1. For notational simplicity, we replace the base-2 logarithmic function by the natural logarithmic function and take the following variable changes: t n ≜ ðT n /ΔtÞ, v ≜ ðvΔ t /2DγÞ,k n = ðk n /γÞ,Δ v = ðΔ v Δ t /DγÞ, where 0 <Δ v ≤ v. The rate pairs in (B.1) can be reformulated as The optimization variables are t i and β i , i = 1, 2. The known quantities areṽ,Δ v ,k 1 , andk 2 . When γ is very small, the rates can be approximated as It can be shown that the maximum values of the rates are given by which are achieved at t * 1 = ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðk 1 /ðṽ −Δ v ÞÞ q and t * 2 = ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðk 2 /ðṽ −Δ v ÞÞ q , respectively. The maximum common rate of the two user groups is given by with constraints of R * 1 ðβ 1 ,Δ v Þ = R * 2 ðβ 2 ,Δ v Þ and β 1 + β 2 = 1. The solution of problem (B.6) can be readily obtained as ðB:7Þ which is an increasing function ofΔ v for 0 <Δ v ≤ṽ. This in turn proves Lemma B.1. Now, we consider optimality only in the low SNR regime. When N = 2, Lemma B.1 immediately leads to Proposition 4 since ðv/2Þ − v 1 is the maximum feasible value of Δ v for given speed sets. When N > 2, we consider any other potential optimal user grouping strategy given by K′ n , n = 1, 2, ⋯, N, excluding the trivial cases in which fK′ n g N n=1 is just a permutation of fK n g K n=1 . In fK ′ n g N n=1 , there must exist at least one such user group that this group consists of more than one user, and at the same time, this group contains at least one of the users with indices 1, 2, ⋯, N − 1. Otherwise, the grouping strategy is the same with fK n g K n=1 . Locating a specific user m, m ∈ f1, 2,⋯,N − 1g, such that user m is in a user group, denoted by K ′m which had more than one user, but for any l, l < m, user l forms a single-user group. As m < N, there is at least one other group, denoted by K ′m +1 , that does not contain users 1, 2, ⋯, m. If we regroup the users in K ′m and K ′m +1 , according to Lemma B.1, the optimal strategy is to isolate user m in a single-user group, and all the other users form the other group, as v m is the lowest speed. This is in contradiction with the preassumption that the grouping is already optimal. Therefore, Proposition 4 is proved.

Data Availability
The data is available from the corresponding author upon request.

Disclosure
Part of this paper was published in the proceeding of 2019 IEEE Global Communications Conference.

Conflicts of Interest
The authors declare that there is no conflict of interest regarding the publication of this paper. 12 Wireless Communications and Mobile Computing