## Abstract

A low-density parity-check (LDPC) coded orbital angular momentum (OAM)-based uniform circular array (UCA) free space optical (FSO) system exploring linear equalization is investigated with channel estimation over the atmospheric turbulence fading channels. On the basis of the proposed system model, the least square (LS) channel estimator is adopted to obtain the channel state information (CSI) of this OAM-FSO system. Then, the average bit error ratio (ABER) expressions with *M*_{P}-ary phase shift keying (*M*_{P}PSK) modulation scheme are derived by ensemble average with the aid of the large number theorem. Besides, LDPC codes are applied in the simulation to improve the ABER performance, and subsequently the probability expressions of the estimated signals with zero forcing (ZF) and minimum mean squared error (MMSE) equalizers for LDPC decoder are achieved, respectively. Results show that the ABER performance of the OAM-FSO system degrades with increasing turbulence strengths. With ZF and MMSE equalization algorithms, the ABER performance is significantly enhanced with an increase in the number of receive antennas for considerable diversity gain. Furthermore, a substantial coding gain can be attained by LDPC codes in this OAM-FSO system, especially under strong turbulence condition. This work will benefit the research and development of OAM-FSO system.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

To progressively improve the spectral efficiency and data transmission capacity of the future free-space optical (FSO) communication, various properties of light, such as amplitude, wavelength, phase, and polarization, have been deeply exploited [1–3]. Driven by this urgent and increasing demand for support of high data rates to end-customers, more transmission methods are desired to be combined to expand their information conveyance capacity. The orbital angular momentum (OAM)-based FSO communication system has recently been proposed as a novel method for transmitting information over distances between transmitters and receivers using optical vortex beams; this method attracts considerable attention due to its higher information capacity, lower cost, and more secure data transmission compared with conventional communication methods [4,5]. Usually, OAM mode is associated with a donut shaped intensity distribution and helical shaped wavefront, which can be described as $\mathrm{exp}(\text{i}l\theta )$, where $\theta $ is the azimuthal angle around the propagation axis of the wave, and *l* is known as the topological charge of the OAM mode which can take any integer values. These OAM modes, which satisfy the orthogonality among beams, form an infinite-dimensional Hilbert space; thus, they can be used to effectively increase the information capacity of the classical optical communications by the encoding modulation or OAM-multiplexing technique [6–9]. Actually, the multiple-input multiple-output (MIMO) technique is another key technology for modern wireless communication that exhibits better transmission capacity, spectral efficiency, and link robustness than the single-input single-output (SISO) system. And MIMO is widely used in FSO systems in combination with other technologies, such as intensity modulation and direct detection (IM/DD), spatial diversity, and selection multiuser scheduling [10–14]. Recently, there is plainly a tendency to consider both OAM and MIMO in FSO communication system [15–17]. These two schemes both exploit the spatial degrees of freedom (DOFs) in different ways and could be utilized simultaneously under certain design trade-offs and limitations to help distribute the spatial DOFs of the system [18,19]. Nevertheless, the OAM mode will be seriously impacted by atmospheric turbulence and the existence of atmospheric turbulence has been one of the main impairments of the OAM-based MIMO FSO system due to the intensity fluctuation and beam wandering of OAM beams. These conditions further cause the power spreading of OAM modes. Therefore, the orthogonality among OAM modes is no longer maintained, and crosstalk is induced between FSO links [20,21]. Up to now, different methods [22–26] have been put forward to mitigate the impact of the atmospheric turbulence in OAM-based MIMO FSO systems such as equalization, spatial diversity and space-time coding. In [24], one scheme using MIMO equalization in OAM multiplexed FSO system was proposed since the OAM multiplexed communication is equivalent to the general MIMO link, and the numerical simulation results showed that this technique can alleviate the adverse effects of the atmospheric turbulence. In [25], a two OAM multiplexing-based general MIMO-FSO system employing spatial diversity combined with MIMO equalization was investigated, and the authors found that at least two OAM data channels can be recovered under both weak and strong turbulence using selection diversity assisted with MIMO equalization. In [26], space-time coding with channel estimation was adopted in OAM multiplexing-based general MIMO FSO system over atmospheric turbulence. The results indicated that both vertical Bell labs layered space-time (V-Blast) and space-time block codes (STBC) schemes can significantly improve the system performance by mitigating the distortions of atmospheric turbulence as well as additive white Gaussian noise (AWGN). Nevertheless, the OAM-FSO systems with MIMO architecture mentioned above are all based on the general antenna array. Given the orthogonality between OAM modes, a more suitable antenna array called uniform circular array (UCA) has recently been investigated in wireless OAM-based radio frequency (RF) communication systems due to its flexibility in radiating OAM beams with different charge numbers simultaneously [27–31]. These former studies found that in all situations, the optimum performance can be achieved when the UCA is adopted. In [31], an OAM-based UCA RF system was found to be capable of improving spectrum efficiency, and the beam steering was deemed effective in circumventing performance degradation caused by misalignment. Besides, channel coding is another effective method that improves the reliability of signal transmission in wireless communication through the addition of some redundant symbols in signals to automatically detect or correct errors introduced during transmission [32,33]. Among various channel coding methods, low-density parity-check (LDPC) codes are very promising error-correcting codes, which are normally used to enhance the system performance due to their achievements of near capacity limit and high coding gain [34,35]. In these years, LDPC coding has been introduced into the OAM-FSO communication system and is regarded as a valid method to compensate for performance degradation induced by atmospheric turbulence [36–38]. In [36], a LDPC coded OAM-based FSO transmission system was experimentally studied in the presence of atmospheric turbulence, and the results showed that the coding gains larger than 6.8 dB can be obtained at bit error rate (BER) of ${10}^{-4}$ for single OAM mode. In [37], a large-girth LDPC code was utilized to deal with remaining channel impairments in an OAM multiplexing-based FSO transmission system over strong atmospheric turbulence. In [38], a two-stage crosstalk mitigation method was proposed and experimentally demonstrated in an OAM-based FSO system over atmospheric turbulence through the combination of spatial offset and LDPC-coded non-uniform signaling. The authors demonstrated that in comparison with quadrature phase shift keying (QPSK) and 8-quadrature amplitude modulation (8-QAM) schemes, the LDPC-coded 5-QAM and 9-QAM are able to bring 1.1 dB and 5.4 dB performance improvements, respectively. Thus, a method combining the OAM-based UCA FSO system with LDPC codes can be foreseen to have substantial potential to better address the detrimental distortion of OAM-FSO system caused by atmospheric turbulence. However, no published work has focused on the performance of LDPC coded OAM-based UCA FSO systems over atmospheric turbulence with different equalization methods despite its importance in the design of real FSO communication systems.

Motivated by the above analyses, an LDPC coded OAM-based UCA FSO system with zero forcing (ZF) and minimum mean squared error (MMSE) equalizers is investigated over the atmospheric turbulence in this work. Based on the channel estimation carrying out least square (LS) algorithm, the channel state information (CSI) is obtained with the channel gain of this UCA MIMO system. Then, the expressions of the ensemble average bit error ratio (ABER) with two equalization schemes are mathematically derived with the help of the large number theorem. And the probability expressions of the estimated signal with ZF and MMSE equalizers for LDPC decoder are also achieved, respectively. After that, the ABER performances of the OAM-based UCA system over atmospheric turbulence are analyzed with different turbulence strengths, modulation orders and receive antenna numbers. Furthermore, LPDC codes are adopted to enhance the system performance with QPSK modulation scheme taken into account.

## 2. Theoretical model

#### 2.1 System model

Figure. 1 shows the configuration of the proposed LDPC coded OAM-based UCA FSO communication system with linear equalization over atmospheric turbulence. A signal stream at 1550 nm is generated and processed through a LDPC encoder firstly. The encoded data are then mapped to the phase shift keying (PSK) modulation formats. Subsequently, with a serial-to-parallel conversion (S/P), the data stream is divided into *M* parallel data streams which are interpolated with the constant-amplitude pilot information (PI), respectively. Further, the substreams are fed into the corresponding OAM generators with different OAM modes formed in a concentric circle, which ensures that their axes of propagation coincide. After transmission through the atmospheric turbulence simulated by several turbulence phase screens, the signals will be distorted and received by the *N* antennas placed in a circular array at the receiver, where the constant-amplitude pilot information is first extracted to obtain the CSI and the superposed signals are then sent to the equalizer which implements the ZF and MMSE algorithms to discard the interference from other signals. Finally, after a parallel-to-serial conversation (P/S), the desired signals will be retrieved by further signal processing including demodulation and LDPC decoding.

#### 2.2 CSI of the OAM-based UCA FSO system

As is known, the Laguerre-Gaussian (LG) beam, due to its simple generation and stable propagation, is recognized as one of the most well-known descriptions of vortex beam carrying OAM modes in optical wireless communication. The optical field distribution of a radial distance *r* with propagation distance *z* in the cylindrical coordinates can be expressed as [39,40]

*z*with ${\omega}_{0}$ being the beam waist radius of the zero-order Gaussian, ${z}_{R}=\pi {\omega}_{0}^{2}/\lambda $ is the Rayleigh range, and $\lambda $ is the optical wavelength. The term ${L}_{l}^{p}(\cdot )$ represents the generalized Laguerre polynomial.

*p*and

*l*, which denote the radial and angular mode numbers, respectively, determine the order of the mode in the form of 2

*p*+ |

*l*|. When

*p*=

*l*= 0, the LG field is defined as a zero-order Gaussian beam, i.e., TEM

_{00}mode. Without loss of generality,

*p*is assumed to be 0 in this work.

In the OAM-based FSO system, each data signal ${x}_{m}(t)$ is a modulated signal with the unified carrier characteristics at time *t*, where $1\le m\le M$. Each signal is fed into the OAM mode generator at each transmit antenna and the transmitted signal of the *m*^{th} transmit antenna at time *t* can be represented as

*M*transmit antennas with OAM generators are arranged in a circle. For simplicity, the

*m*

^{th}circular array antenna is assumed to produce the OAM mode with

*l*=

_{m}*m*, where

*m*= 1,2,3,…,

*M*. At the transmitter, the transmitted signal can be regarded as a superposition of

*M*OAM signals, which can be given as

As is known, the atmospheric turbulence is a medium whose refractive index randomly evolves over space and time. This causes OAM modes to be randomly distorted as propagating through the atmospheric turbulence. As a result, the signals carried by these OAM spatial modes also become deformed. With consideration of the influence of inner and outer scales, the modified Von Karman optical spectrum is adopted in this work to describe atmospheric turbulence. And its spectrum of fluctuations in the refractive index ${\Phi}_{n}(k)$ is given in [41,42] as

*k*denotes the spatial wavenumber.

*L*

_{0}is the outer scale of turbulence, that is, the largest eddy size formed by the injection of turbulent energy such as wind sheer.

*k*= 5.92/

_{l}*l*

_{0}, where the parameter

*l*

_{0}represents the inner scale of the turbulence, which is the size of the smallest turbulent eddy. The atmospheric turbulence in this simulation is treated as a finite number of discrete turbulence phase screens, the number of which depends on the transmission distance. With the help of Markov approximation and Fourier optics transformation, the transmission of the OAM modes through the atmospheric turbulence can be simulated by transmitting from the first phase screen to the last. After the LG beam propagates through the atmospheric turbulence, the sampled signal at

*n*receive antenna in the receiver end, where $1\le n\le N$, can be given by

^{th}*m*OAM mode. And $\text{exp}\left(\psi (r)\right)$ denotes the complex phase distortion after propagation through the phase screens [43].

^{th}With the assumption that the crosstalk and noise are mutually independent, according to Eq. (6), the expression of the output at the *n ^{th}* receive antenna has the form below:

*n*

^{th}receive antenna which can be expressed aswhere

*n*

_{0}is a selected reference antenna and assumed to be 1 for convenience.

Actually, the transmission coefficient of this OAM-based UCA FSO system includes two parts and it can be given as

where ${h}_{{C}_{n,m}}$ and ${h}_{A{T}_{n,m}}$ represent the channel gain and impairment of the*m*

^{th}transmitted signal caused by propagating over atmospheric turbulence, respectively. They are defined as below:where ${l}_{m}{\theta}_{n}$ denotes the phase of the

*m*

^{th}transmitted signal at the

*n*

^{th}receive antenna.

Further, the *N* × *M* channel matrix ${H}_{C}$ can be expressed as follows:

On the basis of channel estimation applying the LS algorithm [26], the turbulence distortion ${h}_{A{T}_{n,m}}$ can be obtained as

**H**can be expressed as

#### 2.3 ZF/MMSE equalizer and LDPC coding

Considering the investigated OAM-based UCA FSO system, this work assumes that each transmit antenna has equal power allocation. ZF and MMSE equalization schemes are exploited here to reduce the impairment induced by the turbulence and recover the transmitted signals. The ZF and MMSE equalization matrices can be obtained in [30] as follows:

withBy left multiplying the received signal vector**Y**by

**W**

*and*

_{ZF}**W**

*, the decision-point SNR of the*

_{MMSE}*n*

^{th}signal stream can be given aswithwhere $(\cdot )$

*is the (*

_{(n,n)}*n,n*)

^{th}element of the matrix, and

**I**is the

*N*×

*N*identity matrix.

Over the turbulence channels, the exact instantaneous ABER of *M*_{P}-ary phase shift keying (*M*_{P}PSK) over an AWGN channel is defined on the basis of [44] as

*M*

_{P}PSK OAM-based MIMO FSO system with UCA over atmospheric turbulence channel can be evaluated by ensemble average as

In addition, LDPC codes are adopted in this OAM-FSO system with QPSK modulation to further compensate for the degradation caused by atmospheric turbulence. This approach is adopted because LDPC codes have the error-correcting capability of approaching the Shannon limit. In this work, the random method is used to construct the check matrix and belief propagation algorithm is applied in the decoding process. In fact, it is quite essential to obtain the estimated symbol ${\tilde{x}}_{m}$ and its probability of the *m*^{th} transmit antenna symbol in LDPC decoder. In this OAM-based UCA FSO system with ZF and MMSE equalizers, the estimated symbol ${\tilde{x}}_{m}$ is achieved by multiplexing ${y}_{m}$ with the ZF and MMSE weight matrix, which can be given with the help of [45,46] as

*denotes the*

_{m}*m*

^{th}column of the matrix. ${\tilde{x}}_{m}(t)$ can be approximated as the output of an equivalent AWGN channelwhere ${\mu}_{m}$ for ZF and MMSE can be expressed aswithAnd ${z}_{m}$ for ZF and MMSE is a zero-mean complex Gaussian variable with variance aswithBased on this approximation, the probability of ${\tilde{x}}_{m}(t)$ of this UCA OAM-FSO system with ZF and MMSE methods can be written as follows:

## 3. Results and analysis

In this section, the ABER performances of the OAM-based UCA FSO system with ZF and MMSE equalizers are investigated theoretically over the atmospheric turbulence. The optical wavelength and beam radius ${\omega}_{0}$ of the LG beam are 1550 nm and 3 cm, respectively. And this OAM system is equipped with four transmit and receive antennas formed in a circle. The transmission distance *z* equals 1 km. Therefore, the atmospheric turbulence is approximated by 20 turbulence phase screens placed 50 m apart following the modified Von Karman optical spectrum. For weak, moderate and strong atmospheric turbulence strength, the corresponding refractive-index structure constant ${C}_{n}^{2}$ is equal to${10}^{-15}{m}^{-2/3}$, ${10}^{-14}{m}^{-2/3}$, and ${10}^{-13}{m}^{-2/3}$, respectively. And the LDPC codes adopted in this simulation have a code length of 512 with code rates of 0.25, 0.5, and 0.75.

In Fig. 2, the ABER performance versus SNR with ZF and MMSE equalization methods of the studied 4 × 4 FSO system is given under different turbulence conditions. As can be seen, the ABER performance of the proposed system degrades with the increase of the atmospheric turbulence strengths. For example, when the SNR equals 32 dB, the ABER values under weak, moderate and strong turbulence conditions for MMSE and ZF equalizations are approximately equal to $1.6\times {10}^{-4}$,$1.7\times {10}^{-3}$, $1.2\times {10}^{-2}$ and $4.7\times {10}^{-4}$, $5.2\times {10}^{-3}$, $3.7\times {10}^{-2}$, respectively. This is because the transmitted signals carried by OAM modes are mixed in the multiple channels and deteriorated by the atmospheric turbulence, thus leading to the amplitude attenuation and phase twist, which will further destroy the signals carried by the OAM modes. Besides, the ABER curve with ZF equalizer decreases slower than that with MMSE equalizer under different turbulence strengths, which indicates that the MMSE algorithm outperforms ZF algorithm. For instance, in order to achieve the ABER of $3.8\times {10}^{-3}$ (the forward error correction (FEC) threshold), the required SNRs are approximately equal to 45 dB, 34 dB, and 23 dB in the system with ZF equalization method under weak, moderate and strong turbulence conditions, respectively. Nevertheless, the required SNRs reduce to 38 dB, 28 dB, and 18 dB for MMSE equalization under weak, moderate and strong turbulence conditions, respectively. This is because the MMSE equalizer minimizes the mean square error between the transmitted symbols and the estimation of the receiver, while the ZF equalizer converts the joint decoding problem into several single stream decoding problems and thus neglects the correlated noise in the separated data streams. In addition, Figs. 2(b) and 2(c) present the constellations of the recovered signals with ZF and MMSE equalizers at the receiver with SNR equal to 30 dB, respectively. As can be observed, the constellation of the MMSE equalizer is better defined than that of ZF equalizer, further confirming the conclusion in Fig. 2(a).

Figure. 3 illustrates the ABER performances against the SNR under different atmospheric turbulence conditions with BPSK, QPSK and 8PSK modulations, respectively. As is found from Fig. 3(a)-3(c), the ABER values of this system with MMSE equalization are lower than those with ZF equalization for different modulation orders. As the modulation order increases, the ABER values increase for both equalization schemes. For example, in order to achieve the ABER of ${10}^{-4}$ in weak atmospheric turbulence regime, the corresponding required SNRs for BPSK-based ZF and MMSE systems are approximately equal to 28 dB and 36 dB, respectively. Whereas the required SNRs for QPSK and 8PSK modulations are about 34 dB, 38 dB, and 40 dB, 42 dB for ZF and MMSE schemes, respectively. This is because that the increase of modulation order is achieved by doubling the modulation carrier phase intervals, which will enhance the spectral efficiency and save the transmission bandwidth [10,26]. That is, for the equalization with two algorithms, with the increase of modulation order, the system performances are degraded at the expense of shortening the distance between two neighboring points of the signal constellation diagram, thus enhancing the sensitivity to noise and interference and leading to performance degradation.

Figure. 4 shows the ABERs of OAM-FSO system with QPSK modulation scheme under weak turbulence for different receive antennas with ZF and MMSE equalizers. It can be clearly seen that when the number of transmit antennas is fixed as 4, the ABER values have a significant decrease with the increasing receive antennas for both equalizers. For example, when the SNR is equal to 24 dB, in this system with four, five and six receive antennas, the corresponding ABER values with ZF equalizer are $5\times {10}^{-3}$, $5\times {10}^{-5}$, and $3\times {10}^{-7}$, respectively. While the corresponding ABERs with MMSE equalizer are ${10}^{-3}$, $8\times {10}^{-5}$ and $5\times {10}^{-7}$, respectively. This phenomenon is due to the fact that the increase of receive antennas will enhance the receive diversity, therefore leading to a better system performance. In addition, the ABER difference between ZF equalizer and the MMSE equalizer decreases with the increase of the receive antennas. For instance, when the SNR equals 18 dB, the differences of ABER values are $7.6\times {10}^{-3}$, $2.3\times {10}^{-4}$, and $8.1\times {10}^{-6}$ with four, five and six receive antennas, respectively. This is because that the MMSE equalizer is better than the ZF equalizer at alleviating noise, thus, the MMSE equalizer degenerates into the ZF equalizer when noise is mitigated [10].

In Fig. 5, the ABER performances of this OAM-based UCA FSO system with and without LDPC codes under atmospheric turbulence are presented with ZF and MMSE equalization algorithms. Here, the code rate is set to 0.5. As can be observed, a significant improvement of the ABER performance is achieved by LDPC codes for both algorithms under all turbulence conditions. For example, for the uncoded system, to achieve an ABER of ${10}^{-3}$, the required SNRs for ZF and MMSE equalization schemes are approximately 28 dB, 38 dB, 50 dB and 24 dB, 34 dB, 46 dB in weak, moderate and strong turbulence regimes, respectively. But for the LDPC coded system, the corresponding SNRs are only about 12 dB, 15 dB, 20 dB and 3 dB, 10 dB, 18 dB, respectively. That is because LDPC codes can achieve fairly high coding gain, which will improve the signal recovery capability [46,47]. Besides, when the LDPC codes are adopted, the ABER values of MMSE equalizer decline faster than ZF equalizer, that is, MMSE equalizer performs better than ZF equalizer in this LDPC coded OAM-based FSO system. Furthermore, the coding gain improvement becomes more apparent with the increasing turbulence strength for both ZF and MMSE equalization schemes. For example, the coding gains achieved at ABER of ${10}^{-3}$ are about 16 dB, 23 dB, 30 dB with ZF equalization and 21 dB, 24 dB, 28 dB with MMSE equalization for weak, moderate, strong turbulence conditions, respectively.

Figure. 6 shows the ABER performances of this LDPC-coded OAM-FSO with different code rates over the atmospheric turbulence. As can be found, the LDPC code rate has a significant impact on the ABER performance of both equalization methods under different turbulence conditions. With the increase of the code rate, the ABER values increase. For example, in Fig. 6(a) of weak turbulence condition, the required SNRs for ABER ${10}^{-6}$ with code rates of 0.25, 0.5, 0.75 for ZF equalizer and MMSE equalizer are approximately 12 dB, 15 dB, 20 dB, and 5 dB, 6 dB, 9 dB, respectively. The explanation for this is that when the length of LDPC code is fixed, the code with lower code rate contains more redundant information; thus, the capability of correcting the error induced by atmospheric turbulence and channel noise will become strong [47]. Besides, the coding gain improvement is more apparent when the code rate decreases from 0.75 to 0.5 than that when the code rate decreases from 0.5 to 0.25 under all turbulence conditions for both equalizers. In addition, the ABER performances of all code rates degrade with the increasing atmospheric turbulence strengths for both equalization schemes, and the ABER differences between ZF equalizer and MMSE equalizer decrease with the increase of the turbulence strengths for all code rates. For example, to achieve an ABER of ${10}^{-6}$ under strong turbulence regime, the SNR differences between MMSE and ZF equalizer are 11 dB, 9 dB, and 7 dB with code rates 0.75, 0.5 and 0.25 respectively. While under moderate and strong turbulence regimes, the corresponding SNR differences are 7 dB, 5 dB, and 3 dB and 1 dB, 1 dB, and 0.5 dB, respectively. The reason is that the influence of noise is greater in the weak turbulence regime than in the strong turbulence regime for this LDPC coded OAM-FSO system. Therefore, the advantage of the MMSE equalizer over the ZF equalizer for mitigating both channel crosstalk and noise is more apparent in weaker turbulence.

## 4. Conclusion

In this work, an LDPC coded OAM-based UCA FSO system exploring linear equalization with channel estimation over the atmospheric turbulence was studied in detail. In this system, the atmospheric turbulence was simulated by turbulence phase screens and LS channel estimator was used to obtain the CSI of this communication system. Then, with the help of the large number theorem, the ABER expressions with *M*_{P}PSK modulation were derived by ensemble average. Subsequently, LDPC codes were adopted to enhance the ABER performance of this OAM-FSO system. And the probability expressions of the estimated signals with ZF and MMSE equalizers for LDPC decoder were mathematically derived, respectively. The ABER performances of this system with both ZF and MMSE equalization algorithms were analyzed with different turbulence strengths, modulation orders, and the receive antenna numbers, respectively. The results showed that the ABER performance degrades with the increasing turbulence strengths and modulation orders. However, the ABER performance improves pronouncedly with the increasing receive antenna numbers. Furthermore, LDPC codes can significantly enhance the ABER performance of the presented OAM-FSO system.

## Funding

The Key Research and Development Program of Shaanxi Province (Grant No. 2017ZDCXL-GY-06-02); Fundamental Research Funds for the Central Universities (Grant No. JB160105); 111 Project of China (Grant No. B08038).

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