RoFSO system based on BCH and RS coded BPSK OFDM for 5G applications in smart cities

The radio over the free space optical (RoFSO) communication system has become a popular research topic in 5G communication in recent years. Atmospheric turbulence typically degrades the performance of the RoFSO system. Multiple input multiple output, aperture averaging, error-correcting codes, and robust modulation are standard mitigation techniques used to reduce the effects of atmospheric turbulence. In this paper, Reed Solomon (RS) and Bose-Chaudhuri-Hocquenghem (BCH) coded binary shift keying (BPSK) orthogonal frequency division multiplexing (OFDM) based RoFSO system is proposed for 5G applications. We introduced RS and BCH coding techniques for the first time in this proposed RoFSO system, and achieved an average bit error rate (ABER) of 10-6\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$10^{-6}$$\end{document}, at 40 dB, 17 dB, and 4 dB carrier to noise plus distortion ratio (CNDR) for the uncoded, RS coded, and BCH coded systems, respectively, under weak turbulence conditions. That is, when compared to the uncoded system, the proposed RS and BCH coded system provide transmit power gains of 13 dB and 34 dB, respectively. The ABER performance of the proposed coded RoFSO system is investigated and compared to an uncoded system under various turbulence, weather, and pointing error cases. In all turbulence regimes, weather conditions, and pointing error scenarios, the BCH coded system outperforms the RS coded and uncoded systems. The proposed RS and BCH coded system is energy efficient and may be useful in 5G implementation.


Introduction
One of the most important research areas in wireless communication is radio over freespace optical (RoFSO) communication. It offers huge data rate (100Mb/s-10Gb/s) of communication over shorter ranges (few Kms) Chadha 2011), phase (Kanno et al. 2011), code (Al-Khafaji et al. xxxx), polarization (Tang et al. 2012), and wavelength (Zhou et al. 2014) have been explored by researchers. In Ref. Gupta et al. (2019), the author investigates the performance of a Bose-Chaudhuri-Hocquenghem (BCH) coded FSO link with a Gamma-Gamma turbulence model. A bit error rate performance of a convolutional coded OFDM-based FSO system over a Gamma-Gamma turbulence channel is analyzed in Ajewole et al. (2019). Recently, in 5G new radio applications to improve cell coverage and reduce Peak to average power ratio (PAPR), BPSK-based modulation schemes have been preferred over higher-order modulation schemes such as QPSK, 16-QAM, and others. In 5G new radio applications, a new modulation scheme, ∕2 -BPSK, is proposed and widely used for uplink data channel and control channel transmissions (Khan et al. 2020;Borges et al. 2021;Choi et al. 2021).
In this paper, we have considered the OFDM-based RoFSO communication system with BPSK modulation having ECCs. We have compared the BER performance of BCH and Reed-Solomon (RS) coded RoFSO communication links with uncoded RoFSO links. We analyzed the performance of the proposed system over various weather, turbulence, scattering, and pointing error conditions.
The remainder of this work is structured as follows: The system model is presented in Sect. 2. The atmospheric channel model with pointing errors is described in Sect. 3. In Sect. 4, both ECCs BCH code and RS code details are discussed. The average bit error rate (ABER) of the proposed BPSK OFDM RoFSO system is derived for coded and uncoded cases in Sect. 5. In Sect. 6, numerical results are presented and compared for uncoded, BCH coded, and RS coded systems with different turbulence conditions, weather conditions, scattering conditions, and different misalignments. The work's conclusion has been outlined in Sect. 7.

System model
OFDM is a multicarrier modulation scheme that breaks large data streams into smaller ones and transmits them synchronously over narrowband subcarriers in the OFDM-FSO communication system. It uses other modulation schemes as its baseband, such as quadrature phase-shift keying (QPSK), BPSK, QAM, and so on, and is transmitted at a high frequency. In this paper, we have used the BPSK modulation scheme with OFDM RoFSO communication. Here the coding technique is added to improve the reliability of the system. System model block diagram shown in Fig. 1. In this figure FSO transmitter optics includes Laser diode, Mach-Zehnder modulator for E/O conversion and transmit aperture. FSO receiver optics includes receiving aperture and photodetector. We have considered the BCH and RS codes as ECCs for RoFSO communication. After conversion at the transmitter, the OFDM signal can be expressed as Wang et al. (2015).
where w n is the OFDM frequency, N is the total number of subcarriers, f c is the carrier frequency, X j is the complex transmitted data symbol, and T is the OFDM duration. The M-distributed FSO channel is used to send the S OFDM . Since the OFDM-based optical signal is transmitted through free space, it is attenuated and has noise added to it as it passes through the medium. The received power is given as Transmission (2010) at the receiving end.
Here I denote the channel attenuation due to path loss, atmospheric turbulence, and misalignment fading. Here n(t) is the additive white Gaussian noise (AWGN), added while traveling.
The instantaneous carrier to noise plus distortion ratio (CNDR) for a specific OFDM subcarrier is measured at the receiver's output as follows (Transmission 2010), m n represents the optical modulation index for each OFDM subcarrier and is the photodetector responsivity.
Considering the IMD noise as Gaussian distributed (Transmission 2010) then Eq. (3) can be expressed as Here the AV stands for the average value. From Eq. (4) the expected value of CNDR n can be expressed as Here E[I] is the expected value of I.

Channel model
We have considered the overall channel I which includes the atmospheric path loss I l , atmospheric turbulence I a , and pointing error I p . The overall channel model I is given by Wang et al. (2016)

Atmospheric loss
Beer-Lambert's law (Wang et al. 2016) is used to model Atmospheric loss and it is given by Here L stands for the link length and represents the attenuation coefficient. takes different values depending upon the weather conditions and optical wavelength used. We've used a 1550 nm wavelength optical signal in this work.

Atmospheric turbulence induced fading
In this work, atmospheric turbulence in the FSO channel has been considered to be M distributed. According to this model at the receiver, there are three components; line of sight (LOS) contribution U L , coupled to LOS contributions quasi-forward scattered component U C S , and scattered energy due to off-axis eddies U G S (Jurado-Navas et al. 2012). The power of components U L , U C S and U G S are Ω , 2 b 0 and 2(1 − )b 0 respectively. Here represents the coupling of power between the scattered component and LOS component. At = 0 the coupling is minimum and at = 1 it is maximum.
For the M-distribution turbulence model the probability density function (PDF) of the irradiance I a is given by Wang et al. (2016) where (6) I = I l I a I p Here is a scattering process positive parameter associated with the effective number of large-scale cells. is a natural number, allowing PDF to follow the actual observed data, resulting in a closed-form representation (General 2011). The , which is the average power from the coherent contributions. A represents the phase of the LOS component, and B represents the phase of coupled-to-LOS scatter components. The difference between A and B has been considered to be 90 • .

Pointing errors
Line of sight (LOS) communication is used by the FSO. As a result, the transmitter and receiver must be aligned. The performance of the FSO communication system is strongly affected by this alignment. Building sway, thermal expansion, and wind loads are some of the reasons for the misalignment between transmitter and receiver in this system (Outage xxxx), (Gappmair et al. 2010;Sandalidis et al. 2009). Beckmann and Rayleigh's distributions have been used to model the pointing errors. For pointing error, the irradiance PDF distribution is given as Krishnan et al. (2018) where A 0 = [erf ( )] 2 is the fraction of total received optical power.
is given by, Here a is the radius of the receiver aperture.. We have considered the receiver aperture diameter as 10 cm and trasnmitter aperture diameter as 8 cm. The transmitted optical signal has a beam width of w z at the distance L. Effective beam width at the receiver is given by 2 is the ratio between the effective beam width and the jitter standard deviation .

Combined channel model
The overall channel model for I is given as Wang et al. (2016) Here f I∕I a (I∕I a ) is conditional probability for considered turbulence state I a . It is given as for 0 ≤ I ≤ A 0 I a I l by substituting Eqs. (8) and (12) in Eq. (11) we have Overall channel model f I (I) obtained as below Wang et al. (2016) where B = Ω+ and G m,n p,q [.] is meijer G function

Error correcting codes
An appropriate channel code must be used to identify and correct errors caused by the channel. In this paper, we looked at the BCH code and the RS code.

BCH code
BCH codes come under binary cyclic ECCs, which are defined over the Galois field (GF). It can correct multiple bits in error Ding (2015). It is a type of linear block code that increases the signal's redundancy. In this case, k information bits are converted into an n-bit codeword with n-k redundant bits. The code rate (R), which is given by k/n, defines redundancy. In our work, we used a code rate (R) of less than one-third. We used the BCH code n = 31 , k = 11 . These BCH codes are ECCs with t bits. These codes have a bit length of n = 2 m − 1 in the GF ( 2 m ), where m is a positive integer ( m ⩾ 3 ). The field components are represented by 0, 1, , 2 , ...., 2m−2 . Here is the basic element in the field GF(2 m ) (Lin et al. 2001;MacWilliams et al. 1977). There are different conjugacy classes in which these elements are segregated. A minimal polynomial is (13) associated with every conjugacy class. A BCH code's generator polynomial is realised by finding the least common multiple of the minimal polynomials associated with the elements b , b+1 , ..., b+ −2 , where b is an integer ≥ 1 and is design distance, which equals 2t+1. The generator polynomial is computed as follows, Here M i (x) represents the minimal polynomial of the i th conjugacy class. Components of a particular conjugacy class have the same minimal polynomial. Various conjugacy classes consist of different minimal polynomials which are presented in Table 2  . By computing the product of minimal polynomial, we get the generator polynomial as Using the generator polynomial g(x), the generator matrix ( G 11×31 ) is estimated. The operation c 1×31 = s 1×11 G 11×31 is used to encrypt the data. This BCH-encoded data is modulated with an optical signal before being sent through the atmosphere.
The BCH encoder/decoder is shown in Fig. 2. The message bits are encoded using the BCH encoder and sent via the RoFSO channel as m. The BCH decoder receives the received message bits y and evaluates them as m'. As shown in Fig. 2  . there are three steps involved in decoding BCH code. The syndrome is computed first, and then the error position polynomial from the syndrome polynomial is developed using the Berlekamp-Massey algorithm. We get the error locations by finding the roots of the error location polynomial. After correcting the errors, an estimated information word is generated (Ding 2015).

Reed-solomon coding and decoding
RS code has great burst error-correcting capability . Sometimes in FSO communication, we encounter burst errors (Zhang et al. 2017). So, this RS ECC has been proposed to improve the integrity of information transmission.   10 , 20 , 9 , 18 M 1 (x) = 1 + x + x 2 + x 4 + x 5 7 , 14 , 28 , 25 , 19 M Page 9 of 16 18 The generator polynomial of RS-code having n-length over Galois Field GF(q), with error correcting capabilities t symbol is given by, g( , where is a primitive element of the GF(P m ), b ≥ 0 is any real positive integer and =2t+1. The code-word of n-length is depicted in polynomial as c(x) = c 0 + c 1 x + c 2 x 2 + ...c n−1 x n−1 and it has one to one correspondence to the message polynomial s(x) = s 0 + s 1 x + s 2 x 2 + ... + s k−1 x k−1 . In this work, we have taken n = 63 and k = 51 RS code over the field GF ( 2 6 ) for error detection/correction. Each symbol of the polynomial codeword is converted into six tuples of binary bits and transmitted via the FSO channel. The incoming bits are converted into 6-bit characters lying in the field GF ( 2 6 ). For ascertaining the location of induced errors Berlekamp-Massy and Chien algorithm is used, and for finding error magnitudes, Forney ′ s algorithm is used.

Analytical BER evaluation
Average bit error rate (ABER) for BPSK OFDM FSO system is given as Kaur et al. (2014) Conditional BER for BPSK OFDM system is given by Selvi and Murugesan (2012) Eq.18 can be approximated as Ajewole et al. (2019) Here N is number of subcarriers, CNDR j is carrier to noise plus distortion ratio, E b ∕N 0 is bit energy to noise ratio. After substituting Eqs. 14 and 19 in Eq. 17 , we get ABER as In Eq.20 replaced the erfc(.) with the suitable Meijer G function as given in Ajewole et al. (2019). After solving finally we get ABER as

BCH code
The upper bound on the probability of decoding error p d with a (n,k,t) BCH code over GF(2 m ) having t symbol error correcting capability is given by .
Where n is the block length, and P is the binary symmetric channel transition probability.

RS code
An error occurs when the decoded codeword is not same as the transmitted codeword. The probability of error for RS code is given by  Here P d is the upper bound BER associated with non-binary RS code, t is the symbol error correcting capability. P s is symbol error probability and, code rate r=k/n.

Result and discussion
The analytical BER results of the OFDM RoFSO system as a function of CNDR are described in this section. The optical signal wavelength is considered to be 1.55 m. We considered 'a' i.e. radius of the aperture of the receiver as 5 cm. We have taken different values of the refractive index structure parameter ( C 2 n ) for different turbulence conditions. C 2 n value is taken as 2 × 10 −14 m −2∕3 for weak turbulence , 4 × 10 −14 m −2∕3 for moderate turbulence and 8 × 10 −14 m −2∕3 for strong turbulence (Ninos et al. 2019). Different misalignment has been taken as in Kumar and Krishnan (2020). Figure 3 illustrates the BER performance comparison between uncoded, BCH coded, and RS coded RoFSO systems with various turbulence conditions. For plotting Fig. 3, we have considered very clear air condition. In all three turbulence condition, it is clear that the BCH coded RoFSO system outperform the uncoded and RS coded systems. For attaining BER of 10 −6 , BCH coded RoFSO system requires 5 dB CNDR in weak turbulence condition, but the uncoded RoFSO system requires more than 40 dB CNDR and RS Coded system require 17 dB CNDR. From Fig. 3, we can conclude that for weak turbulence RS Page 11 of 16 18 coded RoFSO system has performed better than the uncoded system, but for strong turbulence conditions, RS coded system has performed worse than the uncoded system. Figure 4 illustrates the BER performance comparison between uncoded, BCH coded, and RS coded RoFSO systems with various scattering conditions. For plotting Fig.4 we have considered very clear air condition and weak turbulence condition. It is visible that the BCH coded RoFSO system has performed better than the uncoded and RS coded RoFSO system in moderate ( = 0.5) and minimum scattering ( = 0.95) conditions. For maximum scattering ( = 0) condition, the BCH coded RoFSO system performed worse than the uncoded system for less than 15 dB CNDR, but if the system's CNDR is better than 15 dB, then BCH coded system performs better than the uncoded system. It is clear from the graph 4 that for moderate ( = 0.5) and maximum scattering ( = 0) RS coded RoFSO system has performed worse than the uncoded system. Only for the minimum scattering ( = 0.95) RS coded RoFSO system performs better than the uncoded system. Figure 5 shows the BER performance comparison between uncoded, BCH coded, and RS coded RoFSO systems with various weather conditions. For plotting Fig. 5,   Fig. 3 Average BER performance comparison of BCH and RS coded with uncoded OFDM ROFSO communication with different turbulence conditions weak turbulence is considered. The BCH-coded RoFSO system performs better than the uncoded and RS coded system for all three weather conditions. For achieving BER of 10 −6 in fog weather condition, the BCH-coded RoFSO system requires 15 dB CNDR, but the uncoded system requires more than 40 dB CNDR, and RS coded system requires 27 dB CNDR. From Fig. 5, it is concluded that for clear and haze weather conditions, the RS coded RoFSO system performs better than the uncoded system. For the fog weather condition, RS coded system performs better than the uncoded system for more than 12 dB CNDR. Figure 6 shows the BER performance comparison between uncoded, BCH coded, and RS coded RoFSO systems with two different Pointing error conditions weak misalignment and enhanced misalignment. Here weak turbulence and clear air weather conditions are considered. In Fig. 6 weak misalignment and enhanced misalignment represent weak pointing error and strong pointing error, respectively. From Fig. 6 it is clear that for both misalignments BCH coded RoFSO system has performed better than the uncoded and RS coded system. BER of 10 −7 is achieved for the enhanced misalignment of BCH coded RoFSO system with 9 dB CNDR and the same BER is achieved with 7.5 dB CNDR for weak misalignment. For achieving same BER performance 2.5 dB more power is required for enhanced misalignment condition.

Conclusion
In this work, we investigated the BER performance of the BPSK RoFSO system with two coding techniques BCH and RS. The average BER expression is obtained as a closed-form equation in terms of the Meijer-G function. In weak turbulence condition, Fig. 5 Average BER performance comparison of BCH and RS coded with uncoded OFDM ROFSO communication with different weather conditions BCH coded RoFSO system requires 5 dB CNDR for achieving BER of 10 −6 , and RS Coded system need 17 dB CNDR while uncoded RoFSO system requires more than 40 dB CNDR for attaining the same BER. For the enhanced misalignment condition, BCH coded RoFSO system requires 8 dB CNDR to achieve 10 −7 BER, RS coded system requires 20 dB CNDR, and the uncoded system requires 40 dB CNDR. BCH is found to be a better coding technique for BPSK OFDM based RoFSO system. We have considered different weather conditions and turbulence conditions for comparing both coding techniques.
Funding This work was supported by the Science and Engineering Research Board, Department of Science and Technology, Government of India, under Grant EEQ/2018/001107

Conflict of interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper