An Exact Error Analysis of Multi-User RC/MRC Based MIMO-NOMA-VLC System With Imperfect SIC

Visible Light Communication (VLC) has appeared as a breakthrough technology for beyond 5G networks by delivering a broad license free spectrum. In this paper, multiple input multiple output (MIMO) technique is implemented with non-orthogonal multiple access (NOMA-VLC) system in order to develop a robust error-free high rate network for a multi-user scenario. In particular, a generalized multi-user MIMO-NOMA-VLC system with <inline-formula> <tex-math notation="LaTeX">$M\times N$ </tex-math></inline-formula> line-of-sight (LOS) links per user is proposed which adopts on-off keying (OOK) modulation for data transmission. A novel closed form expression of probability of error for the proposed system is derived considering a practical scenario of imperfect successive interference cancellation (SIC). This paper also presents the detailed analysis of a three user <inline-formula> <tex-math notation="LaTeX">$2\times 2$ </tex-math></inline-formula> MIMO-NOMA-VLC system and an exact closed form expression of probability of error of this system is derived. The three user <inline-formula> <tex-math notation="LaTeX">$2\times 2$ </tex-math></inline-formula> MIMO-NOMA-VLC system completely outperforms the single input single output (SISO) NOMA-VLC system with same set of parameters. The error performance of the proposed system enhances as the number of photo detectors (PDs) at each user increases. A three user <inline-formula> <tex-math notation="LaTeX">$2\times 2$ </tex-math></inline-formula> MIMO-NOMA-VLC system ensures the best performance at the power allocation co-efficient of 0.3. The error performance of the proposed system is investigated for the parameters of interest, i.e., distance between the transmitting LEDs and distance between the receiving PDs at each user terminal. The derived probability of error expressions are verified with the simulation results.


I. INTRODUCTION
It is expected that there will be 13.1 billion wireless connected devices by the year 2023 due to increase in global connectivity over 70 percent of population [1]. Thus accommodation of such a huge traffic is almost impossible for the radio frequency spectrum. Visible light communication (VLC) offers a license-free wide spectrum of 430 THz to 790 THz and thus emerges as a breakthrough technology for beyond 5G networks [2]. VLC offers a military grade security as being an optical wireless technique. Besides, VLC offers several merits such as straightforwardness in implementation, prolonged life, no eye hazard, low power operation and no electromagnetic interference and hence, finds various popular applications in the areas such as Li-Fi, radiation The associate editor coordinating the review of this manuscript and approving it for publication was Bong Jun David Choi . sensitive areas, optical-camera communication, indoor positioning system, and vehicular communication, etc. [3]- [5].
In recent times, many multiple access techniques, i.e., code division multiple access (CDMA), orthogonal frequency division multiple access (OFDMA) and interleave division multiple access (IDMA), have been applied with VLC systems to attain high rate and low error performance [5]- [7]. Recently, a new multiple access technique, i.e., non-orthogonal multiple access (NOMA), attracted the researchers by offering high spectral and energy efficiency, enhanced capacity and superior outage probability [8].
In NOMA, the users are multiplexed in power domain using multiuser superposition transmit (MUST) technique [9]. For decoding propose, a multi-user detection technique, i.e., successive interference cancellation (SIC), is used at the receiver. The performance of multi user downlink NOMA is investigated in [10], however, it is suggested in [11] that for better practical suitability, NOMA should be implemented with two users. In [12], it is shown that for large user count, NOMA attains higher system capacity than that of time division multiple access (TDMA) over VLC channel. An experimental demonstration of single carrier (SC) based NOMA-VLC system is presented in [13] which offers an enhanced error performance compared to OFDM-VLC system. The symbol error rate (SER) performance of NOMA-VLC system with square M-QAM modulation is studied in [14]. The error performance of NOMA-VLC system with different modulation techniques, i.e., M-PSK, M-QAM and M-PAM, is analyzed and compared in [15]. In [16], the BER performance of NOMA-VLC system with on-off keying (OOK) modulation is studied using maximum likelihood (ML) detection. The BER Analysis of NOMA-VLC System with Imperfect SIC and CSI is presented in [17]. In [18], it is shown that constellations partitioning coding (CPC) technique can mitigate the effect of imperfect SIC in QAM modulated NOMA-VLC system.
Multiple input multiple output (MIMO) technique is widely used in VLC systems in order to improve the system capacity and coverage [19]- [21]. In [19], performance of both non-imaging and imaging MIMO-VLC system is analyzed and compared. The error performance of angular diversity receiver based MIMO-VLC system is analyzed in [20]. In [21], various MIMO techniques are applied to VLC systems and their performance are compared. However, a very limited research work investigates the performance of MIMO based NOMA-VLC system [22], [23]. In [22], a new power allocation strategy for MIMO-NOMA-VLC system is presented which improves the sum-rate to a large extent. A novel precoder design for non-linear VLC channel is proposed in [23] to improve the performance of MIMO-NOMA-VLC system.
Most of the research works reported in literature focus on the outage and sum rate analysis of MIMO-NOMA-VLC system [22], [23]. However, to the best of our knowledge, no research work which investigates the error performance of MIMO-NOMA-VLC system is reported in the literature.
In this paper, the error performance of multi-user downlink MIMO-NOMA-VLC system is analyzed over line-of-sight (LOS) channel. Specifically, OOK modulation is used for signal transmission and a fixed power allocation strategy is considered for NOMA implementation. The main contributions of the paper are enlisted below • In particular, a generalized MIMO-NOMA-VLC system with K users is considered and without loss of generality, the performance analysis of k th user, 1 ≤ k ≤ K , is presented. Thereafter, a three user MIMO-NOMA-VLC system is adopted to investigate the error performance of the users in detail.
• In order to improve the error performance, MIMO technique is implemented with the NOMA-VLC system. Specifically, repetition coding (RC) is employed at the transmitter and maximum ratio combining (MRC) is adopted at the receiver to get the benefit of diversity combining.
• The exact closed form expressions of the probability of error of the generalized multi-user MIMO-NOMA-VLC system are derived in the presence of perfect and imperfect SIC. Further, this analysis is extended for a three user MIMO-NOMA-VLC system.
• The error performance of the proposed multi user MIMO-NOMA-VLC system is compared with that of single input single output (SISO) NOMA-VLC system of [16].
• The performance of the proposed system is investigated for the various parameters of interest. The derived closed form expressions are verified with the simulation results.
The remaining paper is organized as follows: Section II describes the system model. Section III and Section IV present the performance analysis of generalized MU-MIMO-NOMA-VLC system and three user MIMO-NOMA-VLC system, respectively. The results are discussed in Section V and Section VI concludes the paper.

II. SYSTEM MODEL
An intensity modulation and direct detection (IM/DD) based indoor multi-user downlink MIMO-NOMA-VLC system with N LED's as transmitter and M photo detector's (PD's) at each user as receiver is under consideration. The block diagram of the considered system is shown in Fig. 1. A room with typical dimension of (d, d, h) m is considered. The transmitter is located at the center of the roof wherein, the n th LED, i.e., 1 ≤ n ≤ N , is located with the co-ordinates of (x n , y n , h) m, and the receivers are located on the floor with the co-ordinates x k m , y k m , h 1 m, 1 ≤ k ≤ K , 1 ≤ m ≤ M , where h 1 is the height of receivers above the ground. In this analysis only LOS path is considered, as the power received through Non-LOS path is much lower than that of LOS path [19], [24], [25]. The direct current (DC) channel gain of LOS path between n th LED, 1 ≤ n ≤ N , and m th PD, 1 ≤ m ≤ M , of the k th user, 1 ≤ k ≤ K , is given by [2], [26] where A m is the active area of m th PD, d mn is the distance between n th LED and m th PD, φ mn is the angle of emission from n th LED to m th PD, ψ mn is angle of incidence at m th PD from n th LED, T (ψ mn ) is the optical filter gain of the m th PD from n th LED, g (ψ mn ) = n 2 /sin 2 ψ c ,0 ≤ ψ mn ≤ ψ c , is the gain of optical concentrator with refractive index n and ψ c is the field of view (FOV) of the PD and R (φ mn ) is the radiation intensity of the LED with Lambertian pattern and can be given as [26] R (φ mn ) =  where = −ln 2/ln cos φ 1/2 is the Lambertian order related to the semi-angle at half power of LED, φ 1/2 .
The M × N channel matrix for MIMO-NOMA-VLC system at the k th user is given by where h k mn , 1 ≤ m ≤ M , 1 ≤ n ≤ N , is defined in (1).

A. REPETITION CODING (RC)
At the transmitter, RC is used to gain the benefit of transmit diversity. In RC, the signal is transmitted from all the LEDs simultaneously [27]. Thus, in RC, At the receiver, the optical power emitted from multiple LEDs is combined constructively. Thus, the equivalent received optical power at m th PD is given as where P t is the total transmit power, P avg = P t /N is the average transmit power per LED and h k m = N n=1 h k mn is the equivalent channel gain at the m th PD.
Thus, the received signal at m th PD of the k th user is given by where R p is the responsivity of PD, x is the transmitted signal from LED, n k m is the additive white Gaussian noise (AWGN) at m th PD of the k th user with zero mean and σ 2 n variance. As n k m consists of shot and thermal noise, σ 2 n can be written as σ 2 are the variance of shot and thermal noise, respectively, q is charge of electron, W is bandwidth of noise which is equal to bandwidth of modulation, I b = 5100µA is background current under direct exposure to sunlight, P m is equivalent received optical power at the m th PD and I 2 = 0.562 is noise bandwidth factor, k is Boltzmann's constant, T a = 295K is absolute temperature, G = 10 is open loop voltage gain, η = 112pF/cm 2 is fixed capacitance of PD per unit area, A is PD's active area, = 1.5 is channel noise factor of FET, g m = 30mS is transconductance of FET, and I 3 = 0.0868 [1], [28].

B. MAXIMUM RATIO COMBINING (MRC)
At the receiver, MRC is used to gain the benefit of receiver diversity. In MRC, the received signals are co-phased followed by weight assignment and addition. The weights are assigned according to the knowledge of channel gain of the respective PD's at k th user so that the output SNR can be maximized. Thus, the output of MRC combiner at k th user is given by where w k m = h k m is the assigned weight at m th PD of k th user and n k is weighted noise at k th user.

C. NOMA-VLC SIGNALING SCHEME
Without loss of generality, the ordering of the users is performed according to the sum of M optical channel gains received by different PD's at a particular user as According to the NOMA principle, for successful implementation of SIC, the user with the lower channel gain is assigned with a higher optical transmission power. In this paper, fixed power strategy is considered in which the power assigned to k th user is P k = αP k+1 , where α is the power allocation co-efficient with 0 < α < 1 and K k=1 P k = P avg .
Using NOMA principle, the LED transmits the OOK modulated symbols, s l , l ∈ {1, . . . , K }, intended for K users after superimposing them in power domain as x = K l=1 2P l s l , where factor 2 is used to keep the average transmit power as P avg because there is no signal transmission for bit '0'.

III. PERFORMANCE ANALYSIS
The received signal at k th user after the MRC combining is given by For the simplicity of representation, equivalent channel gain at the k th user terminal is considered as The probability of error for the direct detection of the j th , k ≤ j ≤ K , user at k th user terminal can be written as where s i 1 , s i 2 , . . . , s i j is the i th combination of the symbols of users and i ranges from 0 to 2 j − 1. For equi-probable occurrence of symbols, the probability of error is given by where y k is defined in (6) and is a Gaussian distributed random variable with the probability density P l s l and σ 2 k = h k σ 2 n . It is to be noted that the threshold for symbol detection of j th user at k th user terminal is i th k j = 2R p P j h k /2, and thus, the expression for the probability of error using (8) can be given as The decoding of extreme far user does not require the involvement of SIC and hence, the probability of error for the extreme far user can be obtained by substituting, k = j = K in (9).
The decoding of all users except the extreme far user can be performed with the implementation of SIC. Thus, the output signal at the k th user, k = K , is given by where y k is Gaussian distributed with mean m k = P l s l and variance σ 2 k = h k σ 2 n , and s l is the decoded symbol of the l th user, (k + 1) ≤ l ≤ K . The probability of error of the k th user is given by where v denotes all the possible error combinations at the k th user which includes error in self-decoding and the errors generated during the SIC implementation for (k + 1) to K th user. The self-decoding error at k th user is arised, when s k = s k and is denoted as e Similarly, the correct self-decoding at k th user is denotes as e For further solution of (11), two cases may be possible; perfect SIC decoding and imperfect SIC decoding.

A. PERFECT SIC DECODING
For perfect SIC decoding, s l = s l , where (k + 1) ≤ l ≤ K . Thus the probability of error at the k th user for this case is given by substituting v = 1 in (11) as where P k l e is the probability of error of the l th user, (k + 1) ≤ l ≤ K , decoded at the k th user terminal and P k j e k=j is the probability of error of the self-decoding at the k th user terminal.
Eq. (12) can be further solved by substituting the appropriate values in (9).

B. IMPERFECT SIC DECODING
For imperfect SIC decoding, s l = s l , where (k + 1) ≤ l ≤ K . Thus, the probability of error at the k th user for this case is given by substituting 2 ≤ v ≤ 2 K −k in (11) as where P e is the probability of error of the v th error combination at the k th user, and can be given as where i denotes the number of possible symbol combinations. The error probability of the k th user can be obtained using (11), (12) and (14) as

IV. ANALYSIS OF THREE USER SCENARIO
This section presents the detailed probability of error analysis for a three user MIMO-NOMA-VLC system. In particular, three users, i.e., U 1 , U 2 and U 3 , are considered as shown in Fig. 2. The users are ordered according to their equivalent channel gains as U 1 > U 2 > U 3 . The fixed power allocation is adopted for NOMA implementation as discussed in Section II (C).

A. PERFORMANCE OF EXTREME FAR USER (U 3 )
The probability of error for U 3 can be calculated directly by substituting k = j = 3 in (9) as It is to be noted that there will be eight possible combinations of symbols at U 3 . Thus, an exact probability of error expression for U 3 can be obtained using (16) as where ϒ 3 = R p √ h 3 /σ n .

B. PERFORMANCE OF MIDDLE USER (U 2 )
The probability of error for the middle user, i.e., k = 2, can be obtained using (15) as where P e In perfect SIC case, U 3 is decoded without an error at U 2 and hence, the interference related to U 3 will be cancelled out perfectly. Thus, the probability of error in this case can be given using (12) as where P 2 2 e is the probability of error of self-decoding at U 2 and P 2 3 e is the probability of error of U 3 decoded at U 2 . The probabilities P 2 2 e and P 2 3 e can be obtained directly by substituting the appropriate value of k and j in (9), and are given in (20) and (21), respectively, where ϒ 2 = R p √ h 2 /σ n .
In imperfect SIC case, U 3 is decoded with an error and accordingly, an additional interference related to incorrect decoding of U 3 , i.e., s 3 = s 3 , is added to the received signal as in (10). Thus, the probability of error in this case can be calculated using (14) as P e (2) By substituting the symbol values in different symbol combinations, and using direct detection method, Eq. (22) can be further solved as in (23), as shown at the bottom of the page.

C. PERFORMANCE OF NEAREST USER (U 1 )
The probability of error for the nearest user, i.e., k = 1, can be obtained using (15) as where P e (1) where P 1 1 e is the probability of error of self-decoding at U 1 , P 1 2 e and P 1 3 e are the probability of error for incorrect decoding of U 2 and U 3 at U 1 , respectively. The probabilities P 1 1 e , P 1 2 e and P 1 3 e can be obtained directly by substituting the appropriate value of k and j in (9), and are given in (26), (27) and (28), respectively, where ϒ 1 = R p √ h 1 /σ n .
For imperfect SIC case, the probability of error can be written using (13) as where I 1 represents the probability of error of the combination when U 1 , U 2 and U 3 are decoded incorrectly, correctly and incorrectly, respectively, I 2 represents the probability of error when U 1 , U 2 and U 3 are decoded incorrectly, incorrectly and correctly, respectively, and I 3 represents the probability of error when U 1 , U 2 and U 3 are decoded incorrectly, incorrectly and incorrectly, respectively. The derivation of I 1 , I 2 and I 3 are given below. Using (14) and (29), I 1 can be written as P e By substituting the symbol values in different symbol combinations, and using direct detection method, Eq. (30) can be further solved as in (31), as shown at the bottom of the next page.
Using (14) and (29), I 2 can be written as P e P e

VOLUME 9, 2021
By substituting the symbol values in different symbol combinations, and using direct detection method, Eq. (32) can be further solved as in (33), as shown at the bottom of the page.

V. RESULT AND DISCUSSION
The derived probability of error expressions are evaluated numerically and compared with the simulation results. Table 1 shows the system configuration and simulation parameters considered for this analysis. A three user scenario is considered where the users, i.e., U 1 , U 2 and U 3 are located at the co-ordinates (3m, 3m, 0.8m), (3.5m, 1.5m, 0.8m) and (1m, 1m, 0.8m), respectively. Fixed power allocation strategy is considered for NOMA implementation with α = 0.2, unless otherwise stated. Since the channel gains of MU-MIMO-NOMA-VLC system are in the order of 10 −5 , the probability of error curves will experience an offset of 100dB with respect to the received SNR.
In Fig. 3, the probability of error curves for a three user 2 × 2 MIMO-NOMA-VLC system are plotted against the SNR. The probability of error for all the three users improves as SNR increases. The error performance of U 3 is superior to that of U 1 and U 2 , which is well justified as higher power is allocated to U 3 for successful SIC implementation. The figure also compares the error performance of 2 × 2 MIMO-NOMA-VLC system with that of single input single output (SISO) NOMA-VLC system. For SISO-NOMA-VLC system, the location co-ordinates of LED are considered as (2.5m, 2.5m, 3m) and the co-ordinates of U 1 , U 2 and U 3 are considered as (3m, 3m, 0.8m), (3.5m, 1.5m, 0.8m) and (1m, 1m, 0.8m), respectively. The 2×2 MIMO-NOMA-VLC system outperforms the later given that both systems have adopted the identical power allocation strategy. The simulation results are in close agreement with that of analytical results.
In Fig. 4, the probability of error curves for a three user M × N MIMO-NOMA-VLC system is plotted against the SNR for varying M , i.e., number of PD's at each user terminal. For the analysis, system is considered with a fixed number of LED's, i.e., N = 2, and M = 2, 3 and 4. For M = 2, the location coordinates of PD's are given in Table 1. For the case of M = 3 and 4, the location coordinates of U 1 , U 2 and U 3 are given in Table 2. The figure shows an improvement in error performance of U 1 , U 2 and U 3 as M increases. The reason behind this improvement is the increment in equivalent channel gain at the output of MRC combiner which results from the addition of more LOS paths between the LED and PD.
(35) VOLUME 9, 2021  of D Tx , i.e., D Tx = 0.2m, 0.4m, 2m, are considered for performance investigation. The figure shows that the probability of error of U 3 improves whereas, the probability of error of U 1 and U 2 degrades as D Tx increases. It is to be noted that increment/decrement in D Tx results in the change in location coordinates of transmitting LEDs and consequently, the channel gains between LEDs and PDs are affected. However, a significant change in error performance can only be observed when D Tx is sufficiently large. Table 3 investigates the effect of D Rx , i.e., distance between PDs at each user, on the error performance of three user 2 × 2 MIMO-NOMA-VLC system. For analysis purpose, four different values of D Rx , i.e., D Rx = 0.1m, 0.4m, 0.6m and 0.8m, are considered. The probability of error values given in Table depicts that the error performance of the users except U 1 improve when D Rx increases. Further, at a given  SNR, U 3 shows best performance followed by U 2 and U 1 . Furthermore, the performance of the system improves as SNR increases. Fig. 7 compares the performance of proposed three user 2 × 2 MIMO-NOMA-VLC system with SISO-NOMA-VLC system of [16]. For the comparison, the system model of [16] is adopted, and accordingly, the co-ordinates of LEDs and PDs of MIMO-NOMA-VLC system are redefined. Specifically, for 2 × 2 MIMO-NOMA-VLC system, the location coordinates of LEDs are considered as (2m × 1.95m × 3m) and (2m × 2.05m × 3m), and the location coordinates of PDs for U 1 , U 2 and U 3 are considered to be PD 1 (2.3m × 2.25m × 1.25m) and PD 2 (2.3m × 2.35m × 1.25m) , PD 1 (2.4m × 2.35m× 1.25m) and PD 2 (2.4m × 2.45m × 1.25m) , and PD 1 (2.5m× 2.45m × 1.25m) and PD 2 (2.5m × 2.55m × 1.25m) , respectively. The figure shows that the proposed 2 × 2 MIMO-NOMA-VLC system completely outperforms the SISO-NOMA-VLC system presented in [16]. Further, the figure shows that the error performance of U 3 is superior to that of U 1 and U 2 for both the systems.

VI. CONCLUSION
In this paper the error performance of MU-MIMO-NOMA-VLC system is analyzed in the presence of perfect and imperfect SIC. In particular, RC and MRC techniques are adopted at the transmitter and receiver, respectively, in order to improve the performance of NOMA-VLC system. The closed form expressions for the probability of error of the MU-MIMO-NOMA-VLC users are derived over LOS channel. It is concluded that a three user 2 × 2 MIMO-NOMA-VLC system outperforms the SISO-NOMA-VLC system with same set of parameters. Further, an improvement in the error performance of a three user M × N MIMO-NOMA-VLC system is observed as number of PDs, i.e., M, increases. Furthermore, the best error performance of a three user 2 × 2 MIMO-NOMA-VLC system is observed at α = 0.3, however, beyond this range of α, the error performance of the system deteriorates significantly. It is also concluded that a significant improvement/degradation in the error performance of the users is observed only at large D Tx . On the contrary, the variation in D Rx does not has much effect on the error performance of the users.