Performance Comparison of Repetition Coding MIMO Optical Wireless Communications with Distinct Light Beams

In current optical wireless communications (OWC) oriented research works, the various multiple input multiple output (MIMO) techniques are introduced and utilized to enhance the coverage performance. Objectively, this Lambertian light beam based MIMO research paradigm neglects the light beam diversity and the potential performance gains. In this work, the distinct non-Lambertian light beams of commercially available light emitting diodes (LEDs) were adopted to configure MIMO OWC links. Specifically, the homogenous and heterogeneous non-Lambertian MIMO configurations were constituted in typical indoor scenarios. Moreover, applying the repetition coding (RC) MIMO algorithm with low complexity, a spatial coverage performance comparison was made between the several above mentioned non-Lambertian configurations and the well-discussed Lambertian MIMO configuration. Numerical results illustrate that the homogeneous NSPW light beam configuration could provide a more than 30 dB average signal to noise ratio (SNR), while the achievable average SNR of the heterogeneous light beam configuration was up to 28.77 dB. On the other side, the counterpart of the Lambertian configuration achieved just about 27.00 dB. Objectively, this work paves the fundamental and essential way for the further design and optimization of MIMO OWC in this novel light beam dimension.


Introduction
As one vital enabling technology of next generation mobile networks, i.e., beyond 5G or 6G, optical wireless communication (OWC) is earning overwhelming discussion from the academic and research communities [1][2][3][4][5]. More importantly, OWC possesses several unique characteristics, including abundant optical spectrum resources, inherent security, no spectrum regulation, and immunity to severe radio frequency (RF) interference [3][4][5]. Although OWC could reuse ubiquitous illumination infrastructure, the original coverage performance of OWC was severely limited by the spatial beam characteristics of light emitting diodes (LED), which is the key component of OWC transmitters [1][2][3][4].
On the other side, for improving the link reliability and transmission capacity, various multiple input multiple output (MIMO) techniques have been aggressively applied in the OWC domain exploring degree of spatial freedom [6][7][8][9]. Nevertheless, it must be noted that the current MIMO OWC research paradigm usually assumes that the LED sources follow well-discussed Lambertian radiation patterns with quite low complexity.
Thanks to developments in solid state lighting, there are many commercially available LED sources rendering distinct non-Lambertian light beams [10][11][12][13][14]. As a matter of fact, for deriving customized lighting performance, the reflection cups and secondary lens are elaborately attached to the original LED modules by the manufacturers. Some pioneering works have elementarily identified the significant single input single output (SISO) channel characteristics induced by actual non-Lambertian light beams [15][16][17][18]. Specifically, in [15] the non-circular symmetric optical beam was introduced to the cells planning of visible light communication networks. In [16], for focusing more optical signal power to the user terminals, one optical beam-switchable access point was proposed using asymmetrical emission beams. In [17], a modified version of the Monte Carlo-based non-deterministic modeling scheme was systematically proposed for flexibly modeling practical non-Lambertian source radiation patterns in optical wireless channel characterization. In [18], the link coverages of outdoor VLC with actual LED street luminaries were modeled and evaluated. Nevertheless, current non-Lambertian OWC performance analysis is still limited to single output configurations and has not been sufficiently extended to MIMO configurations.
In the following reported work, motivated by the above analysis, the distinct non-Lambertian light beams were further explored in repetition coding (RC) MIMO OWC transmitter configurations. It should be noted that in RC MIMO transmission, the same data stream is sent via each of the transmitters [9]. Moreover, the non-Lambertian light beam based homogeneous and heterogeneous MIMO OWC configurations were proposed, and the relevant indoor coverage performance was modeled and estimated in typical indoor scenarios. In addition, the effect of access point (AP) spacing on coverage performance was estimated under a superior non-Lambertian configuration. It should be noted that, in this work, all concerned non-Lambertian beam characteristics were based on the prestigious and fundamental modeling work for representative and distinct actual LED beams of I. Moreno et al., in [11]. Therefore, the expected sufficient generalization could be satisfied through the whole following reported work.

General Lambertian Light Beams
In current MIMO OWC performance analysis, LED sources usually follow a simple Lambertian light beam pattern. The above pattern assumption means that the radiation intensity is just one cosine function of the viewing angle and could be given as: where m L is the order of Lambertian emission and φ is the irradiance angle relative to the optical axis of the optical source [19]. The Lambertian order m L is given by: where φ 1/2 is the half power angle of the Lambertian beam [10,19]. Specifically, in Figure 1a, the generalized 3D Lambertian light beam is illustrated for clarity.

Typical Non-Lambertian Light Beams
According to the reported measurement results, the non-Lambertian light beams were numerically fitted. In this work, without a loss of generality, two typical non-Lambertian light beams were considered. Specifically, these light beams were modeled from the LUXEON Rebel LED and the NSPW345CS LED, respectively.
In the LUXEON Rebel light beam case, the spatial radiation intensity could be profiled by the sum of two Gaussian functions [11]: where the coefficient values of Gaussian functions are identified as where α denotes the azimuth angle, the coefficient values of Gaussian functions are given as

Channel Model
In the well-known Lambertian beam configuration, the OWC direct current (DC) channel gain includes line of sight (LOS) and non-line-of-sight (NLOS) components. In simplified analyses, the NLOS components are usually ignored. Therefore, the respective channel gain could be given as [10,19] where AR denotes the physical area of the photo detector (PD) of the receiver, dmn denotes the distance between the nth optical access point (AP) and the mth PD of the receiver,

Typical Non-Lambertian Light Beams
According to the reported measurement results, the non-Lambertian light beams were numerically fitted. In this work, without a loss of generality, two typical non-Lambertian light beams were considered. Specifically, these light beams were modeled from the LUXEON Rebel LED and the NSPW345CS LED, respectively.
In the LUXEON Rebel light beam case, the spatial radiation intensity could be profiled by the sum of two Gaussian functions [11]: where the coefficient values of Gaussian functions are identified as g LUX 11 = 0.76, g LUX 21 = 0 • , g LUX 31 = 29 • , g LUX 12 = 1.10, g LUX 22 = 45 • , and g LUX 32 = 21 • . Correspondingly, the 3D display of LUXEON Rebel LB light beam is shown in Figure 1b.
As for the NSPW345CS case, the spatial radiation intensity should also be profiled by one sum of two Gaussian functions [11]: where α denotes the azimuth angle, the coefficient values of Gaussian functions are given as g NSPW

Channel Model
In the well-known Lambertian beam configuration, the OWC direct current (DC) channel gain includes line of sight (LOS) and non-line-of-sight (NLOS) components. In simplified analyses, the NLOS components are usually ignored. Therefore, the respective channel gain could be given as [10,19]: where A R denotes the physical area of the photo detector (PD) of the receiver, d mn denotes the distance between the nth optical access point (AP) and the mth PD of the receiver, denotes the optical concentrator gain at the receiver with an internal refractive index n RI , φ mn is the irradiance angle from the nth optical AP to the mth PD, θ mn is the incidence angle from the nth optical AP to the mth PD, and θ FOV is the field of view (FOV) at the receiver. Apparently, the OWC DC channel gain is closely related to the optical beam radiation pattern. Respectively, in the LUXEON Rebel beam case, the channel gain between the nth optical AP and the mth photo detector of the receiver should be given by: Similarly, in the NSPW345CS beam case, the DC channel gain should be replaced by: where α mn denotes the azimuth angle of the mth photo detector to the nth optical AP.

Repetition Coding MIMO OWC Transmission System
For one typical MIMO OWC system with four distributed APs and one receiver with four PDs, the output signal vector of the receiver is described by [8,9]: where the element H Lam ij of H could be calculated specifically via (5). Under such a homogeneous Lambertian light beam configuration, the typical indoor scenario is shown in Figure 2a.
If the all Lambertian light beams are replaced by the LUXEON Rebel light beams, the respective non-Lambertian homogeneous MIMO OWC configuration is constituted. Accordingly, the channel gain matrix H is renewed by: where the element H LUX ij of H could be calculated specifically by (6). Under such a homogeneous LUXEON Rebel light beams configuration, the typical indoor scenario is shown in Figure 2b.   H of H could be specifically calculated by (7). For clarity, typical indoor scenario of this homogeneous non-Lambertian configuration is shown Figure 2c,d from the side and top view, respectively. As shown in Figure 2d, the origi azimuth angle of four APs is 45°, 135°, 225°, and 315°, respectively. Following the work of Harald Haas [9], for convenience, the signal to noise ra (SNR) of MIMO OWC applying the same repetition coding (RC) MIMO algorithm w low complexity could be calculated as: Similarly, as for the NSPW345CS light beam, the respective non-Lambertian homogeneous MIMO OWC configuration could be constituted as well. In this case, the channel gain matrix H is given by: where the element H NSPW ij of H could be specifically calculated by (7). For clarity, the typical indoor scenario of this homogeneous non-Lambertian configuration is shown in Figure 2c,d from the side and top view, respectively. As shown in Figure 2d, the original azimuth angle of four APs is 45 • , 135 • , 225 • , and 315 • , respectively.
Following the work of Harald Haas [9], for convenience, the signal to noise ratio (SNR) of MIMO OWC applying the same repetition coding (RC) MIMO algorithm with low complexity could be calculated as: where r denotes the responsivity of the PDs, I TX denotes the emitted power of all transmitters, N 0 denotes the noise power, N t = 4 denotes the number of the APs, and N r = 4 denotes the number of the PDs of the receiver. In a well-lit environment, the shot noise is one dominant contributor to the signal disturbance at the receiver. Then, the noise power could be calculated as [19]: where q denotes the electron charge in coulombs, I bg denotes the current due to the background light, B denotes the system modulation bandwidth, K b denotes the Boltzmann constant, T denotes the absolute temperature and R f denotes the feedback resistance of the transimpedance amplifier (TIA) [19]. As for the potential heterogeneous light beams configurations, two specific configurations were investigated. Firstly, for the inter-AP heterogeneous light beam configuration, the NSPW345CS non-Lambertian light beams were utilized to replace part of the Lambertian homogeneous configuration. Specifically, in the investigated MIMO OWC system with distributed APs, the two APs with non-Lambertian light beams were placed along the diagonal of the ceiling, as shown in Figure 3a. Typically, the emitted power of each LED chip, i.e., LED element, was quite limited. To satisfy the general lighting function, the LED source, also known as the LED array, is usually composed of numerous LED elements. For the intra-AP heterogeneous configuration, the LED elements with distinct beam characteristics were adopted to constitute the LED array. For convenience of analysis, two types of LED elements were included in the investigated intra-AP heterogeneous configuration. For consistency with the abovementioned homogeneous NSPW345CS configuration, the original azimuth angle of the two NSPW345CS APs was 45 • and 225 • , respectively. Moreover, the MIMO OWC channel gain matrix of this inter-AP heterogeneous configuration could be given as: where the element H Lam ij and H NSPW ij of H could be specifically calculated by Equations (5) and (7), respectively. Secondly, for the intra-AP heterogeneous light beams configuration, the NSPW345CS non-Lambertian light beam was added to each AP of the above Lambertian homogeneous configuration, as shown in Figure 3b. For a fair comparison, the emitted optical power for each AP was equally divided between the included NSPW345CS and Lambertian light beams. Essentially, in this intra-AP heterogeneous configuration, the MIMO channel gain matrix H had to be renewed as:  (5) and to H NSPW ij in Equation (7).

Numerical Evaluation
For the following section, the SNR spatial distribution of the discussed homogeneous and heterogeneous MIMO OWC system was evaluated and compared. In addition, for a fair comparison, all concerned light beam intensities were normalized within the following reported work. Furthermore, Table 1 summarizes the main parameters for this work [10,11,19].

Numerical Evaluation
For the following section, the SNR spatial distribution of the discussed homogeneous and heterogeneous MIMO OWC system was evaluated and compared. In addition, for a fair comparison, all concerned light beam intensities were normalized within the following reported work. Furthermore, Table 1 summarizes the main parameters for this work [10,11,19]. It should be noted that all the following results were obtained from simulations and the average operation of results was avoided using the emitted average power and noise statistical variation. As shown in Figure 4a, for the conventional homogeneous Lambertian light beam configuration, the MIMO achievable SNR spatial dynamic range was about 20.23-31.15 dB while the average SNR of this conventional configuration was about 27.00 dB.
NSPW345CS light beam MIMO configuration. In particular, the respective achieva SNR spatial dynamic range was elevated to about 22.27-35.61 dB, while the average S of this non-Lambertian configuration was increased to 30.10 dB, as shown in Figure  Therefore, an up to about 3.10 dB average SNR gain was obtained by this homogene NSPW345CS MIMO configuration, compared to the previous Lambertian configurati For all compared three homogeneous configuration, the SNR spatial distributi presented intuitionally axial symmetry. In addition, the respective maximum MIMO S gain was up to 4.46 dB greater compared to the homogeneous Lambertian configurati As for the proposed inter-AP heterogeneous light beam configuration, the SNR spa distribution presented obvious asymmetry, as shown in Figure 5a. In this situation, S peaks of about 34.64 dB appeared at the receivers under the two NSPW345CS APs on receiver plane. The average SNR of this inter-AP heterogeneous configuration was ab 28.58 dB; the gain was about 1.58 dB compared to the original Lambertian configuration  On the other side, in the homogeneous distributed LUXEON Rebel light beam MIMO configuration, the achievable SNR spatial dynamic range was reduced to about 21.03-32.35 dB, while the average SNR of this non-Lambertian configuration was reduced to about 25.62 dB, as shown in Figure 4b. Compared to the above Lambertian configuration, an up to 1.38 dB average SNR loss was induced by this homogeneous LUXEON Rebel MIMO configuration.
By contrast, a significant SNR enhancement could be provided by the homogeneous NSPW345CS light beam MIMO configuration. In particular, the respective achievable SNR spatial dynamic range was elevated to about 22.27-35.61 dB, while the average SNR of this non-Lambertian configuration was increased to 30.10 dB, as shown in Figure 4c. Therefore, an up to about 3.10 dB average SNR gain was obtained by this homogeneous NSPW345CS MIMO configuration, compared to the previous Lambertian configuration. For all compared three homogeneous configuration, the SNR spatial distributions presented intuitionally axial symmetry. In addition, the respective maximum MIMO SNR gain was up to 4.46 dB greater compared to the homogeneous Lambertian configuration.
As for the proposed inter-AP heterogeneous light beam configuration, the SNR spatial distribution presented obvious asymmetry, as shown in Figure 5a. In this situation, SNR peaks of about 34.64 dB appeared at the receivers under the two NSPW345CS APs on the receiver plane. The average SNR of this inter-AP heterogeneous configuration was about 28.58 dB; the gain was about 1.58 dB compared to the original Lambertian configuration. Unlike the above inter-AP heterogeneous configuration, the SNR spatial distribution of the intra-AP heterogeneous configuration did not exhibit asymmetry anymore, as shown in Figure 5b. In this case, the respective achievable SNR spatial dynamic range was changed to about 22.89-33.65 dB, while the average SNR of this intra-AP heterogeneous configuration was increased to 28.77 dB. Correspondingly, the achieved average SNR gain was about 1.77 dB compared to the benchmark Lambertian configuration. Moreover, the respective maximum receiver SNR was 33.65 dB, while an up to 2.50 dB maximum MIMO coverage SNR gain could be derived from this light beam heterogeneity.
To compare the statistical characteristics of SNR distributions in the three homogeneous configurations and two heterogeneous configurations considered, the SNR cumulative distribution function (CDF) curves are described in Figure 6. For up to 80% of the receiver positions in the Lambertian configuration, the SNR was more than 24.87 dB, while the counterparts of the homogeneous LUXEON Rebel configuration and the homogeneous NSPW345CS configurations changed to 22.89 dB and 26.88 dB, respectively. Therefore, the proposed homogeneous NSPW345CS configuration is capable of dramatically improving the weak coverage domain in the indoor application scenario.
homogeneous NSPW345CS configurations changed to 22.89 dB and 26 respectively. Therefore, the proposed homogeneous NSPW345CS configuration is of dramatically improving the weak coverage domain in the indoor application s As for the proposed inter-AP and intra-AP heterogeneous light beam configu in up to 80% of receiver positions the SNR was more than 25.58 dB and 25 respectively. Therefore, there is an obvious performance tradeoff between the Lambertian and NSPW345CS light beam homogeneous configurations.
For identifying the AP spacing effect on the SNR CDF curve, the spacing was from 3 m to 1 m with a 0.5 m step in the NSPW345CS homogeneous configur should be noted that the size of the environment was always the same dur investigations of different LED APs spacing settings. Under different APs settings, the LED APs were always on the vertices of a square centered on the geo center of the ceiling. As shown in Figure 7, the fluctuation extent increased to m 40 dB at the 1 m AP spacing compared to the original 12.88 dB at the 3 m AP spa the same time, the maximum SNR gradually increased to 44.30 dB from 34.56 dB.    As for the proposed inter-AP and intra-AP heterogeneous light beam configurations, in up to 80% of receiver positions the SNR was more than 25.58 dB and 25.67 dB, respectively. Therefore, there is an obvious performance tradeoff between the original Lambertian and NSPW345CS light beam homogeneous configurations.

Conclusions
For identifying the AP spacing effect on the SNR CDF curve, the spacing was reduced from 3 m to 1 m with a 0.5 m step in the NSPW345CS homogeneous configuration. It should be noted that the size of the environment was always the same during the investigations of different LED APs spacing settings. Under different APs spacing settings, the LED APs were always on the vertices of a square centered on the geometrical center of the ceiling. As shown in Figure 7, the fluctuation extent increased to more than 40 dB at the 1 m AP spacing compared to the original 12.88 dB at the 3 m AP spacing. At the same time, the maximum SNR gradually increased to 44.30 dB from 34.56 dB.

Conclusions
The coverage performance of MIMO VLC systems based on Lambertian and non-Lambertian light beams was evaluated in typical indoor scenarios. This investigation shows that the different light beam configurations could significantly influence the MIMO SNR spatial distribution. As for the intra-AP heterogeneous light beam configuration, the mean SNR was enhanced to about 28.77 dB from about 27.00 dB in the conventional homogeneous Lambertian light beam configuration. Moreover, an up to 2.50 dB maximum MIMO coverage SNR gain could be derived from this light beam heterogeneity, while the counterpart of the homogeneous NSPW light beam configuration further increased to 4.46 dB.