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 MP-ary phase shift keying (MPPSK) 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 exp(ilθ), where θ 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 104 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.

 figure: Fig. 1

Fig. 1 The proposed LDPC coded OAM-based UCA FSO system with ZF/MMSE equalizer combined with channel estimation

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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]

Ul(r,θ,z)=Al(r,z)exp(ilθ),
where (r,θ,z) represents the cylindrical coordinates, and Al(r,z) is given as
Al(r,z)=2p!π(p+|l|)!1ω(z)[2rω(z)]|l|exp[r2ω2(z)]Lpl[2r2ω2(z)]×exp[ikr2z2(z2+zR2)]exp[i(2p+|l|+1)tan1zzR],
where ω(z)=ω01+(z/zR)2 is the beam radius at z with ω0 being the beam waist radius of the zero-order Gaussian, zR=πω02/λ is the Rayleigh range, and λ is the optical wavelength. The term Llp() 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 2p + |l|. When p = l = 0, the LG field is defined as a zero-order Gaussian beam, i.e., TEM00 mode. Without loss of generality, p is assumed to be 0 in this work.

In the OAM-based FSO system, each data signal xm(t) is a modulated signal with the unified carrier characteristics at time t, where 1mM. Each signal is fed into the OAM mode generator at each transmit antenna and the transmitted signal of the mth transmit antenna at time t can be represented as

Xm(r,θ,z)=xm(t)Um(r,θ,z)=xm(t)Am(r,z)exp(ilmθ).
M transmit antennas with OAM generators are arranged in a circle. For simplicity, the mth circular array antenna is assumed to produce the OAM mode with lm = 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

XTx(r,θ,t)=m=1Mxm(t)Um(r,θ,z)=m=1Mxm(t)Am(r,z)exp(ilmθ).

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 Φn(k) is given in [41,42] as

Φn(k)=0.033Cn2(k2+1/L02)116exp(k2/kl2),
where Cn2 is the structure constant of the refractive-index, which represents the turbulence strength. k denotes the spatial wavenumber. L0 is the outer scale of turbulence, that is, the largest eddy size formed by the injection of turbulent energy such as wind sheer. kl = 5.92/l0, where the parameter l0 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 nth receive antenna in the receiver end, where 1nN, can be given by
Rn(r,θ,t)=m=1Mxm(t)Um'(r,θ,z)=m=1Mxm(t)Am'(r,z)exp(ilmθn)exp(ψ(r)),
where Um'(r,θ,z) is the twisted optical field after propagation over the atmospheric turbulence for this OAM-based UCA FSO system. Am'(r,z) is the distorted amplitude of the mth OAM mode. And exp(ψ(r)) denotes the complex phase distortion after propagation through the phase screens [43].

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

yn(r,θ,t)=m=1Mxm(t)Um'(r,θ,z)+wn=m=1Mxm(t)Am'(r,z)exp(ilmθn)exp(ψ(r))+wn,
where wn denotes the zero-mean circular symmetric complex AWGN with the same variance σn2=N0/2 over all links. Let hn,m denotes the channel gain, which can be written as
hn,m=m=1MAm'(r,z)exp(ψ(r))exp(ilmθn),
where m=1Mexp(ilmθn) is the channel gain from the circular receive antenna array. θn is the azimuthal position of the nth receive antenna which can be expressed as
θn=2π(nn0)N,
where n0 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

hn,m=hCn,mhATn,m,
where hCn,m and hATn,m represent the channel gain and impairment of the mth transmitted signal caused by propagating over atmospheric turbulence, respectively. They are defined as below:
hCn,m=exp(ilmθn),
where lmθn denotes the phase of the mth transmitted signal at the nth receive antenna.

hATn,m=Am'(r,z)Am(r,z)exp(ilm(θ'θ)=Am'(r,z)Am(r,z)exp(ψ(r)).

Further, the N × M channel matrix HC can be expressed as follows:

HC=(exp(il1θ1)exp(il2θ1)exp(ilmθ1)exp(il1θ2)exp(il2θ2)exp(ilmθ2)exp(il1θn)exp(il2θn)exp(ilmθn)).

On the basis of channel estimation applying the LS algorithm [26], the turbulence distortion hATn,m can be obtained as

hATn,m=(Am(r,z)exp(ilmθ))1(m=1MAm'(r,z)exp(ψ(r))+wn).
By substituting Eqs. (11) and (14) into Eq. (10), the channel matrix can also be given as
hn,m=exp(ilmθn)(Am(r,z)exp(ilmθ))1(m=1MAm'(r,z)exp(ψ(r))+wn).
Therefore, the CSI H can be expressed as

H=(exp(il1θ1)(A1(r,z)exp(il1θ))1(m=1MAm'(r,z)exp(ψ(r))+w1)exp(ilmθ1)(Am(r,z)exp(ilmθ))1(m=1MAm'(r,z)exp(ψ(r))+w1)exp(il1θn)(Am(r,z)exp(ilmθ))1(m=1MAm'(r,z)exp(ψ(r))+wn)exp(ilmθn)(Am(r,z)exp(ilmθ))1(m=1MAm'(r,z)exp(ψ(r))+wn)).

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:

WZF=(HHH)1HH
with
WMMSE=(HHH+σn2I)1HH.
By left multiplying the received signal vector Y by WZF and WMMSE, the decision-point SNR of the nth signal stream can be given as
γZF,n=γ[(HHH)1](n,n)1
with
γMMSE,n=γ((HHH+σn2I)1)(n,n)1,
where ()(n,n) is the (n,n)th element of the matrix, and I is the N × N identity matrix.

Over the turbulence channels, the exact instantaneous ABER of MP-ary phase shift keying (MPPSK) over an AWGN channel is defined on the basis of [44] as

Pe(γ)=1max(log2MP,2)erfc(γsinπMP),
where erfc() is the complementary error function. Then, by substituting Eqs. (17) and (18) into Eq. (21) and with the help of the large number theorem, the approximate ABER of an MPPSK OAM-based MIMO FSO system with UCA over atmospheric turbulence channel can be evaluated by ensemble average as
Pe,ZF(γ)=1max(log2MP,2)erfc(γ[(HHH)1](n,n)1sinπMP)
with

Pe,MMSE(γ)=1max(log2MP,2)erfc(γ((HHH+σn2I)1)(n,n)1sinπMP).

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 x˜m and its probability of the mth transmit antenna symbol in LDPC decoder. In this OAM-based UCA FSO system with ZF and MMSE equalizers, the estimated symbol x˜m is achieved by multiplexing ym with the ZF and MMSE weight matrix, which can be given with the help of [45,46] as

x˜ZF,m=((HHH)1HH)myZF,m=xZF,m+((HHH)1HH)mwm
with
x˜MMSE,m=((HHH+σn2I)1Hm)HyMMSE,m=((HHH+σn2I)1Hm)H(Hx+wm),
where ()m denotes the mth column of the matrix. x˜m(t) can be approximated as the output of an equivalent AWGN channel
x˜m=μmxm+zm,
where μm for ZF and MMSE can be expressed as
μZF,m=xZF,m(t)
with
μMMSE,m=((HHH+σn2I)1Hm)HHm.
And zm for ZF and MMSE is a zero-mean complex Gaussian variable with variance as
σZF,m2=((HHH)1HH)m2
with
σMMSE,m2=μMMSE,m(1μMMSE,m).
Based on this approximation, the probability of x˜m(t) of this UCA OAM-FSO system with ZF and MMSE methods can be written as follows:
PZF,m(x˜m/xm)1πσZF,m2exp(|x˜mxm|2σZF,m2)
with

PMMSE,m(x˜m/xm)1πσMMSE,m2exp(|x˜mμMMSE,mxm|2σMMSE,m2).

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 ω0 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 Cn2 is equal to1015m2/3, 1014m2/3, and 1013m2/3, 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 1.6×104,1.7×103, 1.2×102 and 4.7×104, 5.2×103, 3.7×102, 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 3.8×103 (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: Fig. 2

Fig. 2 (a) ABER performances of OAM-based UCA FSO system with ZF and MMSE equalizers under different atmospheric turbulence strengths. The constellation of the recovered QPSK signals propagated in the proposed system under weak turbulence condition (b) with ZF equalizer, (c) with MMSE equalizer, respectively.

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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 104 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: Fig. 3

Fig. 3 ABER performances of OAM-based UCA FSO system with ZF and MMSE equalizers under (a) weak, (b) moderate, and (c) strong turbulence conditions using BPSK, QPSK and 8PSK modulations

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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 5×103, 5×105, and 3×107, respectively. While the corresponding ABERs with MMSE equalizer are 103, 8×105 and 5×107, 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 7.6×103, 2.3×104, and 8.1×106 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].

 figure: Fig. 4

Fig. 4 ABER performances comparison for different receive antenna numbers under weak turbulence considering ZF and MMSE equalizers.

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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 103, 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 103 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: Fig. 5

Fig. 5 ABER performances of uncoded and LDPC coded QPSK OAM-based UCA FSO systems over the atmospheric turbulence with ZF and MMSE equalizers (The straight and dashed lines represent the ABER values of uncoded and LDPC coded FSO system, respectively. The lines in black, red and blue color represent the ABER values of this system under weak, moderate, strong turbulence, respectively. The lines with hollow and solid squares represent the system with ZF and MMSE equalizer, respectively).

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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 106 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 106 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.

 figure: Fig. 6

Fig. 6 ABER performances of LDPC coded QPSK OAM-based UCA FSO systems considering ZF and MMSE equalizers with different LDPC code rates under (a) weak, (b) moderate, and (c) strong turbulence conditions, respectively. (The lines in black, blue and red color represent the ABER values with code rates 0.25, 0.5 and 0.75, respectively. The lines with hollow and solid squares represent the FSO system with ZF and MMSE equalizers, respectively)

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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 MPPSK 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).

References and links

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3. X. Zhang, Y. He, Y. Cai, M. Su, X. Zhou, Y. Chen, S. Chen, Y. Xiang, L. Chen, C. Su, Y. Li, and D. Fan, “Coherent separation detection for orbital angular momentum multiplexing in free-space optical communications,” IEEE Photonics J. 9(3), 1–11 (2017).

4. T. Doster and A. T. Watnik, “Machine learning approach to OAM beam demultiplexing via convolutional neural networks,” Appl. Opt. 56(12), 3386–3396 (2017). [CrossRef]   [PubMed]  

5. J. Peng, L. Zhang, K. Zhang, and J. Ma, “Channel capacity of OAM based FSO communication systems with partially coherent Bessel–Gaussian beams in anisotropic turbulence,” Opt. Commun. 418, 32–36 (2018). [CrossRef]  

6. S. Li, S. Chen, C. Gao, A. E. Willner, and J. Wang, “Atmospheric turbulence compensation in orbital angular momentum communications: Advances and perspectives,” Opt. Commun. 408, 68–81 (2018). [CrossRef]  

7. X. Hu, Q. Zhao, P. Yu, X. Li, Z. Wang, Y. Li, and L. Gong, “Dynamic shaping of orbital-angular-momentum beams for information encoding,” Opt. Express 26(2), 1796–1808 (2018). [CrossRef]   [PubMed]  

8. L. Li, G. Xie, Y. Ren, N. Ahmed, H. Huang, Z. Zhao, P. Liao, M. P. J. Lavery, Y. Yan, C. Bao, Z. Wang, A. J. Willner, N. Ashrafi, S. Ashrafi, M. Tur, and A. E. Willner, “Orbital-angular-momentum-multiplexed free-space optical communication link using transmitter lenses,” Appl. Opt. 55(8), 2098–2103 (2016). [CrossRef]   [PubMed]  

9. Y. Ren, Z. Wang, P. Liao, L. Li, G. Xie, H. Huang, Z. Zhao, Y. Yan, N. Ahmed, A. Willner, M. P. J. Lavery, N. Ashrafi, S. Ashrafi, R. Bock, M. Tur, I. B. Djordjevic, M. A. Neifeld, and A. E. Willner, “Experimental characterization of a 400 Gbit/s orbital angular momentum multiplexed free-space optical link over 120 m,” Opt. Lett. 41(3), 622–625 (2016). [CrossRef]   [PubMed]  

10. S. Yong, J. Kim, W. Yang, and C. Kang, MIMO-OFDM Wireless Communications with MATLAB (Wiley, 2010).

11. Z. Zhang, S. Zheng, Y. Chen, X. Jin, H. Chi, and X. Zhang, “The capacity gain of orbital angular momentum based multiple-input-multiple-output system,” Sci. Rep. 6(1), 25418 (2016). [CrossRef]   [PubMed]  

12. A. García-Zambrana, C. Castillo-Vázquez, and B. Castillo-Vázquez, “Outage performance of MIMO FSO links over strong turbulence and misalignment fading channels,” Opt. Express 19(14), 13480–13496 (2011). [CrossRef]   [PubMed]  

13. C. Abou-Rjeily, “On the optimality of the selection transmit diversity for MIMO-FSO links with feedback,” IEEE Commun. Lett. 15(6), 641–643 (2011). [CrossRef]  

14. M. Qin, L. Chen, and W. Wang, “Generalized selection multiuser scheduling for the MIMO FSO communication system and its performance analysis,” IEEE Photonics J. 8(5), 1–9 (2016). [CrossRef]  

15. A. Wang, J. Wang, L. Zhu, and J. Liu, “Experimental demonstration of dense fractional orbital angular momentum (OAM) multiplexing with a channel spacing of 0.2 assisted by MIMO equalization,” in Proceedings of Asia Communications and Photonics Conference (OSA 2016), pp. AF1D.1. [CrossRef]  

16. H. Huang, Y. Cao, G. Xie, Y. Ren, Y. Yan, C. Bao, N. Ahmed, M. A. Neifeld, S. J. Dolinar, and A. E. Willner, “Crosstalk mitigation in a free-space orbital angular momentum multiplexed communication link using 4×4 MIMO equalization,” Opt. Lett. 39(15), 4360–4363 (2014). [CrossRef]   [PubMed]  

17. Z. Xu, “6×6 MIMO equalization assisted fractional orbital angular momentum (OAM) dense mode-division multiplexing (DMDM) for free-space optical communications,” in Proceedings of Asia Communications and Photonics Conference (OSA 2014), pp. AW3F.2. [CrossRef]  

18. Y. Ren, Z. Wang, G. Xie, L. Li, Y. Cao, C. Liu, P. Liao, Y. Yan, N. Ahmed, Z. Zhao, A. Willner, N. Ashrafi, S. Ashrafi, R. D. Linquist, R. Bock, M. Tur, A. F. Molisch, and A. E. Willner, “Free-space optical communications using orbital-angular-momentum multiplexing combined with MIMO-based spatial multiplexing,” Opt. Lett. 40(18), 4210–4213 (2015). [CrossRef]   [PubMed]  

19. M. Charnotskii, “Turbulence effects on fluctuations of the aperture-averaged orbital angular momentum,” J. Opt. Soc. Am. A 35(5), 702–711 (2018). [CrossRef]   [PubMed]  

20. C. Chen, H. Yang, S. Tong, and Y. Lou, “Changes in orbital-angular-momentum modes of a propagated vortex Gaussian beam through weak-to-strong atmospheric turbulence,” Opt. Express 24(7), 6959–6975 (2016). [CrossRef]   [PubMed]  

21. G. Liang, Y. Wang, Q. Guo, and H. Zhang, “Anisotropic diffraction induced by orbital angular momentum during propagations of optical beams,” Opt. Express 26(7), 8084–8094 (2018). [CrossRef]   [PubMed]  

22. Y. Ren, Z. Wang, G. Xie, L. Li, A. J. Willner, Y. Cao, Z. Zhao, Y. Yan, N. Ashrai, S. Ashrafi, R. Bock, M. Tur, and A. E. Willner, “Demonstration of OAM-based MIMO FSO link using spatial diversity and MIMO equalization for turbulence mitigation,” in Proceedings of Optical Fiber Communication Conference (OSA 2016), pp. Th1H.2. [CrossRef]  

23. Z. Xu, C. Gui, S. Li, J. Zhou, and J. Wang, “Fractional orbital angular momentum (OAM) free-space optical communications with atmospheric turbulence assisted by MIMO equalization,” in Proceedings of Advanced Photonics for Communications (OSA, 2014), paper JT3A.1.

24. L. Zou, L. Wang, C. Xing, J. Cui, and S. Zhao, “Turbulence mitigation with MIMO equalization for orbital angular momentum multiplexing communication,” in Proceedings of 8th International Conference on Wireless Communications & Signal Processing (WCSP) (IEEE 2016), paper Th1H.2. [CrossRef]  

25. Y. Ren, Z. Wang, G. Xie, L. Li, A. J. Willner, Y. Cao, Z. Zhao, Y. Yan, N. Ahmed, N. Ashrafi, S. Ashrafi, R. Bock, M. Tur, and A. E. Willner, “Atmospheric turbulence mitigation in an OAM-based MIMO free-space optical link using spatial diversity combined with MIMO equalization,” Opt. Lett. 41(11), 2406–2409 (2016). [CrossRef]   [PubMed]  

26. Y. Zhang, P. Wang, L. Guo, W. Wang, and H. Tian, “Performance analysis of an OAM multiplexing-based MIMO FSO system over atmospheric turbulence using space-time coding with channel estimation,” Opt. Express 25(17), 19995–20011 (2017). [CrossRef]   [PubMed]  

27. K. A. Opare, Y. Kuang, J. J. Kponyo, K. S. Nwizege, and P. I. Tebe, “The effect of receiver-side circular antenna arrays on bit error probability in a wireless line-of-sight OAM communication system,” in Proceedings of Advanced Computing and Communication Technologies (ACCT) (IEEE, 2015), pp. 614–620.

28. Y. Yuan, Z. Zhang, C. Ji, and H. Wu, “Capacity analysis of UCA-based OAM multiplexing communication system,” in Proceedings of Wireless Communications and Signal Processing (WCSP) (IEEE, 2015), pp. 1–5.

29. M. Lin, Y. Gao, P. Liu, and J. Liu, “Theoretical analyses and design of circular array to generate orbital angular momentum,” IEEE Trans. Antenn. Propag. 65(7), 3510–3519 (2017). [CrossRef]  

30. K. A. Opare and Y. Kuang, “Performance of an ideal wireless orbital angular momentum communication system using multiple-input multiple-output techniques,” in Proceedings of Telecommunications and Multimedia (TEMU) (IEEE, 2014), pp. 144–149.

31. R. Chen, H. Xu, M. Moretti, and J. Li, “Beam Steering for the Misalignment in UCA-Based OAM Communication Systems,” IEEE Wirel. Commun. Lett. 99, 1–4 (2018).

32. C. Fan, Principles of Communications (Beijing, 2010).

33. G. Ungerboeck, “Channel coding with multilevel/phase signals,” IEEE Trans. Inf. Theory 28(1), 55–67 (2003). [CrossRef]  

34. S. Zhao, L. Wang, L. Zou, L. Gong, W. Cheng, B. Zheng, and H. Chen, “Both channel coding and wavefront correction on the turbulence mitigation of optical communications using orbital angular momentum multiplexing,” Opt. Commun. 376, 92–98 (2016). [CrossRef]  

35. J. Zhou, Z. Xu, and J. Wang, “Performance evaluation of fractional orbital angular momentum (OAM) based LDPC-coded free-space optical communications with atmospheric turbulence,” in Proceedings of Asia Communications and Photonics Conference (ACPC) (OSA 2014), pp.AF3D.2. [CrossRef]  

36. Z. Qu and B. Ivan, F. Djordjevic, “Experimental evaluation of LDPC-coded OAM based FSO communication in the presence of atmospheric turbulence,” in Proceedings of International Conference of Telecommunication in Modern Satellite, Cable and Broadcasting Services (TELSIKS) (IEEE 2015), pp.117–122.

37. Z. Qu and I. B. Djordjevic, “500 Gb/s free-space optical transmission over strong atmospheric turbulence channels,” Opt. Lett. 41(14), 3285–3288 (2016). [CrossRef]   [PubMed]  

38. Z. Qu and I. B. Djordjevic, “Two-stage cross-talk mitigation in an orbital-angular-momentum-based free-space optical communication system,” Opt. Lett. 42(16), 3125–3128 (2017). [CrossRef]   [PubMed]  

39. M. Li, Z. Yu, and M. Cvijetic, “Influence of atmospheric turbulence on OAM-based FSO system with use of realistic link model,” Opt. Commun. 364, 50–54 (2016). [CrossRef]  

40. J. A. Anguita, M. A. Neifeld, and B. V. Vasic, “Turbulence-induced channel crosstalk in an orbital angular momentum-multiplexed free-space optical link,” Appl. Opt. 47(13), 2414–2429 (2008). [CrossRef]   [PubMed]  

41. S. Fu and C. Gao, “Influences of atmospheric turbulence effects on the orbital angular momentum spectra of vortex beams,” Photon. Res. 4(5), B1–B4 (2016). [CrossRef]  

42. J. D. Schmidt, Numerical Simulation of Optical Wave Propagation (2010).

43. S. M. Zhao, J. Leach, L. Y. Gong, J. Ding, and B. Y. Zheng, “Aberration corrections for free-space optical communications in atmosphere turbulence using orbital angular momentum states,” Opt. Express 20(1), 452–461 (2012). [CrossRef]   [PubMed]  

44. P. Wang, L. Zhang, L. Guo, F. Huang, T. Shang, R. Wang, and Y. Yang, “Average BER of subcarrier intensity modulated free space optical systems over the exponentiated Weibull fading channels,” Opt. Express 22(17), 20828–20841 (2014). [CrossRef]   [PubMed]  

45. P. Suthisopapan, K. Kasai, V. Imtawil, and A. Meesomboon, “Approaching capacity of large MIMO systems by non-binary LDPC codes and MMSE detection,” in Proceedings of International Symposium on Information Theory Proceedings (IEEE, 2012), pp. 1712–1716. [CrossRef]  

46. M. T. Hossain, I. Misra, J. J. Sadique, and S. E. Ullah, “Impact of various signal detection schemes in performance assessment of 5G compatible LDPC encoded GPQSM wireless communication system,” Electr. & Comput. Eng. 1(3), 72–80 (2017).

47. M. Duan, P. Wang, X. Liu, Y. Li, and W. Chen, and A. L, “ABER performance analysis of LDPC-coded OFDM FSO system under Málaga distribution considering atmospheric attenuation and pointing errors,” Appl. Opt. 57(19), 5505–5513 (2018).

References

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  1. J. Li, M. Zhang, D. Wang, S. Wu, and Y. Zhan, “Joint atmospheric turbulence detection and adaptive demodulation technique using the CNN for the OAM-FSO communication,” Opt. Express 26(8), 10494–10508 (2018).
    [Crossref] [PubMed]
  2. Q. Tian, L. Zhu, Y. Wang, Q. Zhang, B. Liu, and X. Xin, “The propagation properties of a longitudinal orbital angular momentum multiplexing system in atmospheric turbulence,” IEEE Photonics J. 10(1), 1–11 (2018).
    [Crossref]
  3. X. Zhang, Y. He, Y. Cai, M. Su, X. Zhou, Y. Chen, S. Chen, Y. Xiang, L. Chen, C. Su, Y. Li, and D. Fan, “Coherent separation detection for orbital angular momentum multiplexing in free-space optical communications,” IEEE Photonics J. 9(3), 1–11 (2017).
  4. T. Doster and A. T. Watnik, “Machine learning approach to OAM beam demultiplexing via convolutional neural networks,” Appl. Opt. 56(12), 3386–3396 (2017).
    [Crossref] [PubMed]
  5. J. Peng, L. Zhang, K. Zhang, and J. Ma, “Channel capacity of OAM based FSO communication systems with partially coherent Bessel–Gaussian beams in anisotropic turbulence,” Opt. Commun. 418, 32–36 (2018).
    [Crossref]
  6. S. Li, S. Chen, C. Gao, A. E. Willner, and J. Wang, “Atmospheric turbulence compensation in orbital angular momentum communications: Advances and perspectives,” Opt. Commun. 408, 68–81 (2018).
    [Crossref]
  7. X. Hu, Q. Zhao, P. Yu, X. Li, Z. Wang, Y. Li, and L. Gong, “Dynamic shaping of orbital-angular-momentum beams for information encoding,” Opt. Express 26(2), 1796–1808 (2018).
    [Crossref] [PubMed]
  8. L. Li, G. Xie, Y. Ren, N. Ahmed, H. Huang, Z. Zhao, P. Liao, M. P. J. Lavery, Y. Yan, C. Bao, Z. Wang, A. J. Willner, N. Ashrafi, S. Ashrafi, M. Tur, and A. E. Willner, “Orbital-angular-momentum-multiplexed free-space optical communication link using transmitter lenses,” Appl. Opt. 55(8), 2098–2103 (2016).
    [Crossref] [PubMed]
  9. Y. Ren, Z. Wang, P. Liao, L. Li, G. Xie, H. Huang, Z. Zhao, Y. Yan, N. Ahmed, A. Willner, M. P. J. Lavery, N. Ashrafi, S. Ashrafi, R. Bock, M. Tur, I. B. Djordjevic, M. A. Neifeld, and A. E. Willner, “Experimental characterization of a 400 Gbit/s orbital angular momentum multiplexed free-space optical link over 120 m,” Opt. Lett. 41(3), 622–625 (2016).
    [Crossref] [PubMed]
  10. S. Yong, J. Kim, W. Yang, and C. Kang, MIMO-OFDM Wireless Communications with MATLAB (Wiley, 2010).
  11. Z. Zhang, S. Zheng, Y. Chen, X. Jin, H. Chi, and X. Zhang, “The capacity gain of orbital angular momentum based multiple-input-multiple-output system,” Sci. Rep. 6(1), 25418 (2016).
    [Crossref] [PubMed]
  12. A. García-Zambrana, C. Castillo-Vázquez, and B. Castillo-Vázquez, “Outage performance of MIMO FSO links over strong turbulence and misalignment fading channels,” Opt. Express 19(14), 13480–13496 (2011).
    [Crossref] [PubMed]
  13. C. Abou-Rjeily, “On the optimality of the selection transmit diversity for MIMO-FSO links with feedback,” IEEE Commun. Lett. 15(6), 641–643 (2011).
    [Crossref]
  14. M. Qin, L. Chen, and W. Wang, “Generalized selection multiuser scheduling for the MIMO FSO communication system and its performance analysis,” IEEE Photonics J. 8(5), 1–9 (2016).
    [Crossref]
  15. A. Wang, J. Wang, L. Zhu, and J. Liu, “Experimental demonstration of dense fractional orbital angular momentum (OAM) multiplexing with a channel spacing of 0.2 assisted by MIMO equalization,” in Proceedings of Asia Communications and Photonics Conference (OSA 2016), pp. AF1D.1.
    [Crossref]
  16. H. Huang, Y. Cao, G. Xie, Y. Ren, Y. Yan, C. Bao, N. Ahmed, M. A. Neifeld, S. J. Dolinar, and A. E. Willner, “Crosstalk mitigation in a free-space orbital angular momentum multiplexed communication link using 4×4 MIMO equalization,” Opt. Lett. 39(15), 4360–4363 (2014).
    [Crossref] [PubMed]
  17. Z. Xu, “6×6 MIMO equalization assisted fractional orbital angular momentum (OAM) dense mode-division multiplexing (DMDM) for free-space optical communications,” in Proceedings of Asia Communications and Photonics Conference (OSA 2014), pp. AW3F.2.
    [Crossref]
  18. Y. Ren, Z. Wang, G. Xie, L. Li, Y. Cao, C. Liu, P. Liao, Y. Yan, N. Ahmed, Z. Zhao, A. Willner, N. Ashrafi, S. Ashrafi, R. D. Linquist, R. Bock, M. Tur, A. F. Molisch, and A. E. Willner, “Free-space optical communications using orbital-angular-momentum multiplexing combined with MIMO-based spatial multiplexing,” Opt. Lett. 40(18), 4210–4213 (2015).
    [Crossref] [PubMed]
  19. M. Charnotskii, “Turbulence effects on fluctuations of the aperture-averaged orbital angular momentum,” J. Opt. Soc. Am. A 35(5), 702–711 (2018).
    [Crossref] [PubMed]
  20. C. Chen, H. Yang, S. Tong, and Y. Lou, “Changes in orbital-angular-momentum modes of a propagated vortex Gaussian beam through weak-to-strong atmospheric turbulence,” Opt. Express 24(7), 6959–6975 (2016).
    [Crossref] [PubMed]
  21. G. Liang, Y. Wang, Q. Guo, and H. Zhang, “Anisotropic diffraction induced by orbital angular momentum during propagations of optical beams,” Opt. Express 26(7), 8084–8094 (2018).
    [Crossref] [PubMed]
  22. Y. Ren, Z. Wang, G. Xie, L. Li, A. J. Willner, Y. Cao, Z. Zhao, Y. Yan, N. Ashrai, S. Ashrafi, R. Bock, M. Tur, and A. E. Willner, “Demonstration of OAM-based MIMO FSO link using spatial diversity and MIMO equalization for turbulence mitigation,” in Proceedings of Optical Fiber Communication Conference (OSA 2016), pp. Th1H.2.
    [Crossref]
  23. Z. Xu, C. Gui, S. Li, J. Zhou, and J. Wang, “Fractional orbital angular momentum (OAM) free-space optical communications with atmospheric turbulence assisted by MIMO equalization,” in Proceedings of Advanced Photonics for Communications (OSA, 2014), paper JT3A.1.
  24. L. Zou, L. Wang, C. Xing, J. Cui, and S. Zhao, “Turbulence mitigation with MIMO equalization for orbital angular momentum multiplexing communication,” in Proceedings of 8th International Conference on Wireless Communications & Signal Processing (WCSP) (IEEE 2016), paper Th1H.2.
    [Crossref]
  25. Y. Ren, Z. Wang, G. Xie, L. Li, A. J. Willner, Y. Cao, Z. Zhao, Y. Yan, N. Ahmed, N. Ashrafi, S. Ashrafi, R. Bock, M. Tur, and A. E. Willner, “Atmospheric turbulence mitigation in an OAM-based MIMO free-space optical link using spatial diversity combined with MIMO equalization,” Opt. Lett. 41(11), 2406–2409 (2016).
    [Crossref] [PubMed]
  26. Y. Zhang, P. Wang, L. Guo, W. Wang, and H. Tian, “Performance analysis of an OAM multiplexing-based MIMO FSO system over atmospheric turbulence using space-time coding with channel estimation,” Opt. Express 25(17), 19995–20011 (2017).
    [Crossref] [PubMed]
  27. K. A. Opare, Y. Kuang, J. J. Kponyo, K. S. Nwizege, and P. I. Tebe, “The effect of receiver-side circular antenna arrays on bit error probability in a wireless line-of-sight OAM communication system,” in Proceedings of Advanced Computing and Communication Technologies (ACCT) (IEEE, 2015), pp. 614–620.
  28. Y. Yuan, Z. Zhang, C. Ji, and H. Wu, “Capacity analysis of UCA-based OAM multiplexing communication system,” in Proceedings of Wireless Communications and Signal Processing (WCSP) (IEEE, 2015), pp. 1–5.
  29. M. Lin, Y. Gao, P. Liu, and J. Liu, “Theoretical analyses and design of circular array to generate orbital angular momentum,” IEEE Trans. Antenn. Propag. 65(7), 3510–3519 (2017).
    [Crossref]
  30. K. A. Opare and Y. Kuang, “Performance of an ideal wireless orbital angular momentum communication system using multiple-input multiple-output techniques,” in Proceedings of Telecommunications and Multimedia (TEMU) (IEEE, 2014), pp. 144–149.
  31. R. Chen, H. Xu, M. Moretti, and J. Li, “Beam Steering for the Misalignment in UCA-Based OAM Communication Systems,” IEEE Wirel. Commun. Lett. 99, 1–4 (2018).
  32. C. Fan, Principles of Communications (Beijing, 2010).
  33. G. Ungerboeck, “Channel coding with multilevel/phase signals,” IEEE Trans. Inf. Theory 28(1), 55–67 (2003).
    [Crossref]
  34. S. Zhao, L. Wang, L. Zou, L. Gong, W. Cheng, B. Zheng, and H. Chen, “Both channel coding and wavefront correction on the turbulence mitigation of optical communications using orbital angular momentum multiplexing,” Opt. Commun. 376, 92–98 (2016).
    [Crossref]
  35. J. Zhou, Z. Xu, and J. Wang, “Performance evaluation of fractional orbital angular momentum (OAM) based LDPC-coded free-space optical communications with atmospheric turbulence,” in Proceedings of Asia Communications and Photonics Conference (ACPC) (OSA 2014), pp.AF3D.2.
    [Crossref]
  36. Z. Qu and B. Ivan, F. Djordjevic, “Experimental evaluation of LDPC-coded OAM based FSO communication in the presence of atmospheric turbulence,” in Proceedings of International Conference of Telecommunication in Modern Satellite, Cable and Broadcasting Services (TELSIKS) (IEEE 2015), pp.117–122.
  37. Z. Qu and I. B. Djordjevic, “500 Gb/s free-space optical transmission over strong atmospheric turbulence channels,” Opt. Lett. 41(14), 3285–3288 (2016).
    [Crossref] [PubMed]
  38. Z. Qu and I. B. Djordjevic, “Two-stage cross-talk mitigation in an orbital-angular-momentum-based free-space optical communication system,” Opt. Lett. 42(16), 3125–3128 (2017).
    [Crossref] [PubMed]
  39. M. Li, Z. Yu, and M. Cvijetic, “Influence of atmospheric turbulence on OAM-based FSO system with use of realistic link model,” Opt. Commun. 364, 50–54 (2016).
    [Crossref]
  40. J. A. Anguita, M. A. Neifeld, and B. V. Vasic, “Turbulence-induced channel crosstalk in an orbital angular momentum-multiplexed free-space optical link,” Appl. Opt. 47(13), 2414–2429 (2008).
    [Crossref] [PubMed]
  41. S. Fu and C. Gao, “Influences of atmospheric turbulence effects on the orbital angular momentum spectra of vortex beams,” Photon. Res. 4(5), B1–B4 (2016).
    [Crossref]
  42. J. D. Schmidt, Numerical Simulation of Optical Wave Propagation (2010).
  43. S. M. Zhao, J. Leach, L. Y. Gong, J. Ding, and B. Y. Zheng, “Aberration corrections for free-space optical communications in atmosphere turbulence using orbital angular momentum states,” Opt. Express 20(1), 452–461 (2012).
    [Crossref] [PubMed]
  44. P. Wang, L. Zhang, L. Guo, F. Huang, T. Shang, R. Wang, and Y. Yang, “Average BER of subcarrier intensity modulated free space optical systems over the exponentiated Weibull fading channels,” Opt. Express 22(17), 20828–20841 (2014).
    [Crossref] [PubMed]
  45. P. Suthisopapan, K. Kasai, V. Imtawil, and A. Meesomboon, “Approaching capacity of large MIMO systems by non-binary LDPC codes and MMSE detection,” in Proceedings of International Symposium on Information Theory Proceedings (IEEE, 2012), pp. 1712–1716.
    [Crossref]
  46. M. T. Hossain, I. Misra, J. J. Sadique, and S. E. Ullah, “Impact of various signal detection schemes in performance assessment of 5G compatible LDPC encoded GPQSM wireless communication system,” Electr. & Comput. Eng. 1(3), 72–80 (2017).
  47. M. Duan, P. Wang, X. Liu, Y. Li, and W. Chen, and A. L, “ABER performance analysis of LDPC-coded OFDM FSO system under Málaga distribution considering atmospheric attenuation and pointing errors,” Appl. Opt. 57(19), 5505–5513 (2018).

2018 (9)

J. Peng, L. Zhang, K. Zhang, and J. Ma, “Channel capacity of OAM based FSO communication systems with partially coherent Bessel–Gaussian beams in anisotropic turbulence,” Opt. Commun. 418, 32–36 (2018).
[Crossref]

S. Li, S. Chen, C. Gao, A. E. Willner, and J. Wang, “Atmospheric turbulence compensation in orbital angular momentum communications: Advances and perspectives,” Opt. Commun. 408, 68–81 (2018).
[Crossref]

X. Hu, Q. Zhao, P. Yu, X. Li, Z. Wang, Y. Li, and L. Gong, “Dynamic shaping of orbital-angular-momentum beams for information encoding,” Opt. Express 26(2), 1796–1808 (2018).
[Crossref] [PubMed]

J. Li, M. Zhang, D. Wang, S. Wu, and Y. Zhan, “Joint atmospheric turbulence detection and adaptive demodulation technique using the CNN for the OAM-FSO communication,” Opt. Express 26(8), 10494–10508 (2018).
[Crossref] [PubMed]

Q. Tian, L. Zhu, Y. Wang, Q. Zhang, B. Liu, and X. Xin, “The propagation properties of a longitudinal orbital angular momentum multiplexing system in atmospheric turbulence,” IEEE Photonics J. 10(1), 1–11 (2018).
[Crossref]

M. Charnotskii, “Turbulence effects on fluctuations of the aperture-averaged orbital angular momentum,” J. Opt. Soc. Am. A 35(5), 702–711 (2018).
[Crossref] [PubMed]

G. Liang, Y. Wang, Q. Guo, and H. Zhang, “Anisotropic diffraction induced by orbital angular momentum during propagations of optical beams,” Opt. Express 26(7), 8084–8094 (2018).
[Crossref] [PubMed]

R. Chen, H. Xu, M. Moretti, and J. Li, “Beam Steering for the Misalignment in UCA-Based OAM Communication Systems,” IEEE Wirel. Commun. Lett. 99, 1–4 (2018).

M. Duan, P. Wang, X. Liu, Y. Li, and W. Chen, and A. L, “ABER performance analysis of LDPC-coded OFDM FSO system under Málaga distribution considering atmospheric attenuation and pointing errors,” Appl. Opt. 57(19), 5505–5513 (2018).

2017 (6)

M. T. Hossain, I. Misra, J. J. Sadique, and S. E. Ullah, “Impact of various signal detection schemes in performance assessment of 5G compatible LDPC encoded GPQSM wireless communication system,” Electr. & Comput. Eng. 1(3), 72–80 (2017).

Z. Qu and I. B. Djordjevic, “Two-stage cross-talk mitigation in an orbital-angular-momentum-based free-space optical communication system,” Opt. Lett. 42(16), 3125–3128 (2017).
[Crossref] [PubMed]

Y. Zhang, P. Wang, L. Guo, W. Wang, and H. Tian, “Performance analysis of an OAM multiplexing-based MIMO FSO system over atmospheric turbulence using space-time coding with channel estimation,” Opt. Express 25(17), 19995–20011 (2017).
[Crossref] [PubMed]

M. Lin, Y. Gao, P. Liu, and J. Liu, “Theoretical analyses and design of circular array to generate orbital angular momentum,” IEEE Trans. Antenn. Propag. 65(7), 3510–3519 (2017).
[Crossref]

X. Zhang, Y. He, Y. Cai, M. Su, X. Zhou, Y. Chen, S. Chen, Y. Xiang, L. Chen, C. Su, Y. Li, and D. Fan, “Coherent separation detection for orbital angular momentum multiplexing in free-space optical communications,” IEEE Photonics J. 9(3), 1–11 (2017).

T. Doster and A. T. Watnik, “Machine learning approach to OAM beam demultiplexing via convolutional neural networks,” Appl. Opt. 56(12), 3386–3396 (2017).
[Crossref] [PubMed]

2016 (10)

C. Chen, H. Yang, S. Tong, and Y. Lou, “Changes in orbital-angular-momentum modes of a propagated vortex Gaussian beam through weak-to-strong atmospheric turbulence,” Opt. Express 24(7), 6959–6975 (2016).
[Crossref] [PubMed]

M. Qin, L. Chen, and W. Wang, “Generalized selection multiuser scheduling for the MIMO FSO communication system and its performance analysis,” IEEE Photonics J. 8(5), 1–9 (2016).
[Crossref]

L. Li, G. Xie, Y. Ren, N. Ahmed, H. Huang, Z. Zhao, P. Liao, M. P. J. Lavery, Y. Yan, C. Bao, Z. Wang, A. J. Willner, N. Ashrafi, S. Ashrafi, M. Tur, and A. E. Willner, “Orbital-angular-momentum-multiplexed free-space optical communication link using transmitter lenses,” Appl. Opt. 55(8), 2098–2103 (2016).
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Y. Ren, Z. Wang, P. Liao, L. Li, G. Xie, H. Huang, Z. Zhao, Y. Yan, N. Ahmed, A. Willner, M. P. J. Lavery, N. Ashrafi, S. Ashrafi, R. Bock, M. Tur, I. B. Djordjevic, M. A. Neifeld, and A. E. Willner, “Experimental characterization of a 400 Gbit/s orbital angular momentum multiplexed free-space optical link over 120 m,” Opt. Lett. 41(3), 622–625 (2016).
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Z. Zhang, S. Zheng, Y. Chen, X. Jin, H. Chi, and X. Zhang, “The capacity gain of orbital angular momentum based multiple-input-multiple-output system,” Sci. Rep. 6(1), 25418 (2016).
[Crossref] [PubMed]

Y. Ren, Z. Wang, G. Xie, L. Li, A. J. Willner, Y. Cao, Z. Zhao, Y. Yan, N. Ahmed, N. Ashrafi, S. Ashrafi, R. Bock, M. Tur, and A. E. Willner, “Atmospheric turbulence mitigation in an OAM-based MIMO free-space optical link using spatial diversity combined with MIMO equalization,” Opt. Lett. 41(11), 2406–2409 (2016).
[Crossref] [PubMed]

S. Zhao, L. Wang, L. Zou, L. Gong, W. Cheng, B. Zheng, and H. Chen, “Both channel coding and wavefront correction on the turbulence mitigation of optical communications using orbital angular momentum multiplexing,” Opt. Commun. 376, 92–98 (2016).
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[Crossref]

2015 (1)

2014 (2)

2012 (1)

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2008 (1)

2003 (1)

G. Ungerboeck, “Channel coding with multilevel/phase signals,” IEEE Trans. Inf. Theory 28(1), 55–67 (2003).
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[Crossref] [PubMed]

L. Li, G. Xie, Y. Ren, N. Ahmed, H. Huang, Z. Zhao, P. Liao, M. P. J. Lavery, Y. Yan, C. Bao, Z. Wang, A. J. Willner, N. Ashrafi, S. Ashrafi, M. Tur, and A. E. Willner, “Orbital-angular-momentum-multiplexed free-space optical communication link using transmitter lenses,” Appl. Opt. 55(8), 2098–2103 (2016).
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Y. Ren, Z. Wang, P. Liao, L. Li, G. Xie, H. Huang, Z. Zhao, Y. Yan, N. Ahmed, A. Willner, M. P. J. Lavery, N. Ashrafi, S. Ashrafi, R. Bock, M. Tur, I. B. Djordjevic, M. A. Neifeld, and A. E. Willner, “Experimental characterization of a 400 Gbit/s orbital angular momentum multiplexed free-space optical link over 120 m,” Opt. Lett. 41(3), 622–625 (2016).
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H. Huang, Y. Cao, G. Xie, Y. Ren, Y. Yan, C. Bao, N. Ahmed, M. A. Neifeld, S. J. Dolinar, and A. E. Willner, “Crosstalk mitigation in a free-space orbital angular momentum multiplexed communication link using 4×4 MIMO equalization,” Opt. Lett. 39(15), 4360–4363 (2014).
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Ashrafi, N.

Ashrafi, S.

Bao, C.

Bock, R.

Cai, Y.

X. Zhang, Y. He, Y. Cai, M. Su, X. Zhou, Y. Chen, S. Chen, Y. Xiang, L. Chen, C. Su, Y. Li, and D. Fan, “Coherent separation detection for orbital angular momentum multiplexing in free-space optical communications,” IEEE Photonics J. 9(3), 1–11 (2017).

Cao, Y.

Castillo-Vázquez, B.

Castillo-Vázquez, C.

Charnotskii, M.

Chen, C.

Chen, H.

S. Zhao, L. Wang, L. Zou, L. Gong, W. Cheng, B. Zheng, and H. Chen, “Both channel coding and wavefront correction on the turbulence mitigation of optical communications using orbital angular momentum multiplexing,” Opt. Commun. 376, 92–98 (2016).
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Chen, L.

X. Zhang, Y. He, Y. Cai, M. Su, X. Zhou, Y. Chen, S. Chen, Y. Xiang, L. Chen, C. Su, Y. Li, and D. Fan, “Coherent separation detection for orbital angular momentum multiplexing in free-space optical communications,” IEEE Photonics J. 9(3), 1–11 (2017).

M. Qin, L. Chen, and W. Wang, “Generalized selection multiuser scheduling for the MIMO FSO communication system and its performance analysis,” IEEE Photonics J. 8(5), 1–9 (2016).
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Chen, R.

R. Chen, H. Xu, M. Moretti, and J. Li, “Beam Steering for the Misalignment in UCA-Based OAM Communication Systems,” IEEE Wirel. Commun. Lett. 99, 1–4 (2018).

Chen, S.

S. Li, S. Chen, C. Gao, A. E. Willner, and J. Wang, “Atmospheric turbulence compensation in orbital angular momentum communications: Advances and perspectives,” Opt. Commun. 408, 68–81 (2018).
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X. Zhang, Y. He, Y. Cai, M. Su, X. Zhou, Y. Chen, S. Chen, Y. Xiang, L. Chen, C. Su, Y. Li, and D. Fan, “Coherent separation detection for orbital angular momentum multiplexing in free-space optical communications,” IEEE Photonics J. 9(3), 1–11 (2017).

Chen, W.

Chen, Y.

X. Zhang, Y. He, Y. Cai, M. Su, X. Zhou, Y. Chen, S. Chen, Y. Xiang, L. Chen, C. Su, Y. Li, and D. Fan, “Coherent separation detection for orbital angular momentum multiplexing in free-space optical communications,” IEEE Photonics J. 9(3), 1–11 (2017).

Z. Zhang, S. Zheng, Y. Chen, X. Jin, H. Chi, and X. Zhang, “The capacity gain of orbital angular momentum based multiple-input-multiple-output system,” Sci. Rep. 6(1), 25418 (2016).
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Cheng, W.

S. Zhao, L. Wang, L. Zou, L. Gong, W. Cheng, B. Zheng, and H. Chen, “Both channel coding and wavefront correction on the turbulence mitigation of optical communications using orbital angular momentum multiplexing,” Opt. Commun. 376, 92–98 (2016).
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Chi, H.

Z. Zhang, S. Zheng, Y. Chen, X. Jin, H. Chi, and X. Zhang, “The capacity gain of orbital angular momentum based multiple-input-multiple-output system,” Sci. Rep. 6(1), 25418 (2016).
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M. Li, Z. Yu, and M. Cvijetic, “Influence of atmospheric turbulence on OAM-based FSO system with use of realistic link model,” Opt. Commun. 364, 50–54 (2016).
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Djordjevic, I. B.

Dolinar, S. J.

Doster, T.

Duan, M.

Fan, D.

X. Zhang, Y. He, Y. Cai, M. Su, X. Zhou, Y. Chen, S. Chen, Y. Xiang, L. Chen, C. Su, Y. Li, and D. Fan, “Coherent separation detection for orbital angular momentum multiplexing in free-space optical communications,” IEEE Photonics J. 9(3), 1–11 (2017).

Fu, S.

Gao, C.

S. Li, S. Chen, C. Gao, A. E. Willner, and J. Wang, “Atmospheric turbulence compensation in orbital angular momentum communications: Advances and perspectives,” Opt. Commun. 408, 68–81 (2018).
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S. Fu and C. Gao, “Influences of atmospheric turbulence effects on the orbital angular momentum spectra of vortex beams,” Photon. Res. 4(5), B1–B4 (2016).
[Crossref]

Gao, Y.

M. Lin, Y. Gao, P. Liu, and J. Liu, “Theoretical analyses and design of circular array to generate orbital angular momentum,” IEEE Trans. Antenn. Propag. 65(7), 3510–3519 (2017).
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Gong, L.

X. Hu, Q. Zhao, P. Yu, X. Li, Z. Wang, Y. Li, and L. Gong, “Dynamic shaping of orbital-angular-momentum beams for information encoding,” Opt. Express 26(2), 1796–1808 (2018).
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S. Zhao, L. Wang, L. Zou, L. Gong, W. Cheng, B. Zheng, and H. Chen, “Both channel coding and wavefront correction on the turbulence mitigation of optical communications using orbital angular momentum multiplexing,” Opt. Commun. 376, 92–98 (2016).
[Crossref]

Gong, L. Y.

Guo, L.

Guo, Q.

He, Y.

X. Zhang, Y. He, Y. Cai, M. Su, X. Zhou, Y. Chen, S. Chen, Y. Xiang, L. Chen, C. Su, Y. Li, and D. Fan, “Coherent separation detection for orbital angular momentum multiplexing in free-space optical communications,” IEEE Photonics J. 9(3), 1–11 (2017).

Hossain, M. T.

M. T. Hossain, I. Misra, J. J. Sadique, and S. E. Ullah, “Impact of various signal detection schemes in performance assessment of 5G compatible LDPC encoded GPQSM wireless communication system,” Electr. & Comput. Eng. 1(3), 72–80 (2017).

Hu, X.

Huang, F.

Huang, H.

Imtawil, V.

P. Suthisopapan, K. Kasai, V. Imtawil, and A. Meesomboon, “Approaching capacity of large MIMO systems by non-binary LDPC codes and MMSE detection,” in Proceedings of International Symposium on Information Theory Proceedings (IEEE, 2012), pp. 1712–1716.
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Y. Yuan, Z. Zhang, C. Ji, and H. Wu, “Capacity analysis of UCA-based OAM multiplexing communication system,” in Proceedings of Wireless Communications and Signal Processing (WCSP) (IEEE, 2015), pp. 1–5.

Jin, X.

Z. Zhang, S. Zheng, Y. Chen, X. Jin, H. Chi, and X. Zhang, “The capacity gain of orbital angular momentum based multiple-input-multiple-output system,” Sci. Rep. 6(1), 25418 (2016).
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Kasai, K.

P. Suthisopapan, K. Kasai, V. Imtawil, and A. Meesomboon, “Approaching capacity of large MIMO systems by non-binary LDPC codes and MMSE detection,” in Proceedings of International Symposium on Information Theory Proceedings (IEEE, 2012), pp. 1712–1716.
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K. A. Opare, Y. Kuang, J. J. Kponyo, K. S. Nwizege, and P. I. Tebe, “The effect of receiver-side circular antenna arrays on bit error probability in a wireless line-of-sight OAM communication system,” in Proceedings of Advanced Computing and Communication Technologies (ACCT) (IEEE, 2015), pp. 614–620.

Kuang, Y.

K. A. Opare, Y. Kuang, J. J. Kponyo, K. S. Nwizege, and P. I. Tebe, “The effect of receiver-side circular antenna arrays on bit error probability in a wireless line-of-sight OAM communication system,” in Proceedings of Advanced Computing and Communication Technologies (ACCT) (IEEE, 2015), pp. 614–620.

K. A. Opare and Y. Kuang, “Performance of an ideal wireless orbital angular momentum communication system using multiple-input multiple-output techniques,” in Proceedings of Telecommunications and Multimedia (TEMU) (IEEE, 2014), pp. 144–149.

Lavery, M. P. J.

Leach, J.

Li, J.

R. Chen, H. Xu, M. Moretti, and J. Li, “Beam Steering for the Misalignment in UCA-Based OAM Communication Systems,” IEEE Wirel. Commun. Lett. 99, 1–4 (2018).

J. Li, M. Zhang, D. Wang, S. Wu, and Y. Zhan, “Joint atmospheric turbulence detection and adaptive demodulation technique using the CNN for the OAM-FSO communication,” Opt. Express 26(8), 10494–10508 (2018).
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Li, M.

M. Li, Z. Yu, and M. Cvijetic, “Influence of atmospheric turbulence on OAM-based FSO system with use of realistic link model,” Opt. Commun. 364, 50–54 (2016).
[Crossref]

Li, S.

S. Li, S. Chen, C. Gao, A. E. Willner, and J. Wang, “Atmospheric turbulence compensation in orbital angular momentum communications: Advances and perspectives,” Opt. Commun. 408, 68–81 (2018).
[Crossref]

Li, X.

Li, Y.

Liang, G.

Liao, P.

Lin, M.

M. Lin, Y. Gao, P. Liu, and J. Liu, “Theoretical analyses and design of circular array to generate orbital angular momentum,” IEEE Trans. Antenn. Propag. 65(7), 3510–3519 (2017).
[Crossref]

Linquist, R. D.

Liu, B.

Q. Tian, L. Zhu, Y. Wang, Q. Zhang, B. Liu, and X. Xin, “The propagation properties of a longitudinal orbital angular momentum multiplexing system in atmospheric turbulence,” IEEE Photonics J. 10(1), 1–11 (2018).
[Crossref]

Liu, C.

Liu, J.

M. Lin, Y. Gao, P. Liu, and J. Liu, “Theoretical analyses and design of circular array to generate orbital angular momentum,” IEEE Trans. Antenn. Propag. 65(7), 3510–3519 (2017).
[Crossref]

Liu, P.

M. Lin, Y. Gao, P. Liu, and J. Liu, “Theoretical analyses and design of circular array to generate orbital angular momentum,” IEEE Trans. Antenn. Propag. 65(7), 3510–3519 (2017).
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Liu, X.

Lou, Y.

Ma, J.

J. Peng, L. Zhang, K. Zhang, and J. Ma, “Channel capacity of OAM based FSO communication systems with partially coherent Bessel–Gaussian beams in anisotropic turbulence,” Opt. Commun. 418, 32–36 (2018).
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P. Suthisopapan, K. Kasai, V. Imtawil, and A. Meesomboon, “Approaching capacity of large MIMO systems by non-binary LDPC codes and MMSE detection,” in Proceedings of International Symposium on Information Theory Proceedings (IEEE, 2012), pp. 1712–1716.
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Misra, I.

M. T. Hossain, I. Misra, J. J. Sadique, and S. E. Ullah, “Impact of various signal detection schemes in performance assessment of 5G compatible LDPC encoded GPQSM wireless communication system,” Electr. & Comput. Eng. 1(3), 72–80 (2017).

Molisch, A. F.

Moretti, M.

R. Chen, H. Xu, M. Moretti, and J. Li, “Beam Steering for the Misalignment in UCA-Based OAM Communication Systems,” IEEE Wirel. Commun. Lett. 99, 1–4 (2018).

Neifeld, M. A.

Nwizege, K. S.

K. A. Opare, Y. Kuang, J. J. Kponyo, K. S. Nwizege, and P. I. Tebe, “The effect of receiver-side circular antenna arrays on bit error probability in a wireless line-of-sight OAM communication system,” in Proceedings of Advanced Computing and Communication Technologies (ACCT) (IEEE, 2015), pp. 614–620.

Opare, K. A.

K. A. Opare, Y. Kuang, J. J. Kponyo, K. S. Nwizege, and P. I. Tebe, “The effect of receiver-side circular antenna arrays on bit error probability in a wireless line-of-sight OAM communication system,” in Proceedings of Advanced Computing and Communication Technologies (ACCT) (IEEE, 2015), pp. 614–620.

K. A. Opare and Y. Kuang, “Performance of an ideal wireless orbital angular momentum communication system using multiple-input multiple-output techniques,” in Proceedings of Telecommunications and Multimedia (TEMU) (IEEE, 2014), pp. 144–149.

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J. Peng, L. Zhang, K. Zhang, and J. Ma, “Channel capacity of OAM based FSO communication systems with partially coherent Bessel–Gaussian beams in anisotropic turbulence,” Opt. Commun. 418, 32–36 (2018).
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M. Qin, L. Chen, and W. Wang, “Generalized selection multiuser scheduling for the MIMO FSO communication system and its performance analysis,” IEEE Photonics J. 8(5), 1–9 (2016).
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Ren, Y.

Y. Ren, Z. Wang, G. Xie, L. Li, A. J. Willner, Y. Cao, Z. Zhao, Y. Yan, N. Ahmed, N. Ashrafi, S. Ashrafi, R. Bock, M. Tur, and A. E. Willner, “Atmospheric turbulence mitigation in an OAM-based MIMO free-space optical link using spatial diversity combined with MIMO equalization,” Opt. Lett. 41(11), 2406–2409 (2016).
[Crossref] [PubMed]

Y. Ren, Z. Wang, P. Liao, L. Li, G. Xie, H. Huang, Z. Zhao, Y. Yan, N. Ahmed, A. Willner, M. P. J. Lavery, N. Ashrafi, S. Ashrafi, R. Bock, M. Tur, I. B. Djordjevic, M. A. Neifeld, and A. E. Willner, “Experimental characterization of a 400 Gbit/s orbital angular momentum multiplexed free-space optical link over 120 m,” Opt. Lett. 41(3), 622–625 (2016).
[Crossref] [PubMed]

L. Li, G. Xie, Y. Ren, N. Ahmed, H. Huang, Z. Zhao, P. Liao, M. P. J. Lavery, Y. Yan, C. Bao, Z. Wang, A. J. Willner, N. Ashrafi, S. Ashrafi, M. Tur, and A. E. Willner, “Orbital-angular-momentum-multiplexed free-space optical communication link using transmitter lenses,” Appl. Opt. 55(8), 2098–2103 (2016).
[Crossref] [PubMed]

Y. Ren, Z. Wang, G. Xie, L. Li, Y. Cao, C. Liu, P. Liao, Y. Yan, N. Ahmed, Z. Zhao, A. Willner, N. Ashrafi, S. Ashrafi, R. D. Linquist, R. Bock, M. Tur, A. F. Molisch, and A. E. Willner, “Free-space optical communications using orbital-angular-momentum multiplexing combined with MIMO-based spatial multiplexing,” Opt. Lett. 40(18), 4210–4213 (2015).
[Crossref] [PubMed]

H. Huang, Y. Cao, G. Xie, Y. Ren, Y. Yan, C. Bao, N. Ahmed, M. A. Neifeld, S. J. Dolinar, and A. E. Willner, “Crosstalk mitigation in a free-space orbital angular momentum multiplexed communication link using 4×4 MIMO equalization,” Opt. Lett. 39(15), 4360–4363 (2014).
[Crossref] [PubMed]

Sadique, J. J.

M. T. Hossain, I. Misra, J. J. Sadique, and S. E. Ullah, “Impact of various signal detection schemes in performance assessment of 5G compatible LDPC encoded GPQSM wireless communication system,” Electr. & Comput. Eng. 1(3), 72–80 (2017).

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X. Zhang, Y. He, Y. Cai, M. Su, X. Zhou, Y. Chen, S. Chen, Y. Xiang, L. Chen, C. Su, Y. Li, and D. Fan, “Coherent separation detection for orbital angular momentum multiplexing in free-space optical communications,” IEEE Photonics J. 9(3), 1–11 (2017).

Su, M.

X. Zhang, Y. He, Y. Cai, M. Su, X. Zhou, Y. Chen, S. Chen, Y. Xiang, L. Chen, C. Su, Y. Li, and D. Fan, “Coherent separation detection for orbital angular momentum multiplexing in free-space optical communications,” IEEE Photonics J. 9(3), 1–11 (2017).

Suthisopapan, P.

P. Suthisopapan, K. Kasai, V. Imtawil, and A. Meesomboon, “Approaching capacity of large MIMO systems by non-binary LDPC codes and MMSE detection,” in Proceedings of International Symposium on Information Theory Proceedings (IEEE, 2012), pp. 1712–1716.
[Crossref]

Tebe, P. I.

K. A. Opare, Y. Kuang, J. J. Kponyo, K. S. Nwizege, and P. I. Tebe, “The effect of receiver-side circular antenna arrays on bit error probability in a wireless line-of-sight OAM communication system,” in Proceedings of Advanced Computing and Communication Technologies (ACCT) (IEEE, 2015), pp. 614–620.

Tian, H.

Tian, Q.

Q. Tian, L. Zhu, Y. Wang, Q. Zhang, B. Liu, and X. Xin, “The propagation properties of a longitudinal orbital angular momentum multiplexing system in atmospheric turbulence,” IEEE Photonics J. 10(1), 1–11 (2018).
[Crossref]

Tong, S.

Tur, M.

Ullah, S. E.

M. T. Hossain, I. Misra, J. J. Sadique, and S. E. Ullah, “Impact of various signal detection schemes in performance assessment of 5G compatible LDPC encoded GPQSM wireless communication system,” Electr. & Comput. Eng. 1(3), 72–80 (2017).

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G. Ungerboeck, “Channel coding with multilevel/phase signals,” IEEE Trans. Inf. Theory 28(1), 55–67 (2003).
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Wang, D.

Wang, J.

S. Li, S. Chen, C. Gao, A. E. Willner, and J. Wang, “Atmospheric turbulence compensation in orbital angular momentum communications: Advances and perspectives,” Opt. Commun. 408, 68–81 (2018).
[Crossref]

Wang, L.

S. Zhao, L. Wang, L. Zou, L. Gong, W. Cheng, B. Zheng, and H. Chen, “Both channel coding and wavefront correction on the turbulence mitigation of optical communications using orbital angular momentum multiplexing,” Opt. Commun. 376, 92–98 (2016).
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Wang, P.

Wang, R.

Wang, W.

Y. Zhang, P. Wang, L. Guo, W. Wang, and H. Tian, “Performance analysis of an OAM multiplexing-based MIMO FSO system over atmospheric turbulence using space-time coding with channel estimation,” Opt. Express 25(17), 19995–20011 (2017).
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M. Qin, L. Chen, and W. Wang, “Generalized selection multiuser scheduling for the MIMO FSO communication system and its performance analysis,” IEEE Photonics J. 8(5), 1–9 (2016).
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Wang, Y.

G. Liang, Y. Wang, Q. Guo, and H. Zhang, “Anisotropic diffraction induced by orbital angular momentum during propagations of optical beams,” Opt. Express 26(7), 8084–8094 (2018).
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Q. Tian, L. Zhu, Y. Wang, Q. Zhang, B. Liu, and X. Xin, “The propagation properties of a longitudinal orbital angular momentum multiplexing system in atmospheric turbulence,” IEEE Photonics J. 10(1), 1–11 (2018).
[Crossref]

Wang, Z.

X. Hu, Q. Zhao, P. Yu, X. Li, Z. Wang, Y. Li, and L. Gong, “Dynamic shaping of orbital-angular-momentum beams for information encoding,” Opt. Express 26(2), 1796–1808 (2018).
[Crossref] [PubMed]

Y. Ren, Z. Wang, P. Liao, L. Li, G. Xie, H. Huang, Z. Zhao, Y. Yan, N. Ahmed, A. Willner, M. P. J. Lavery, N. Ashrafi, S. Ashrafi, R. Bock, M. Tur, I. B. Djordjevic, M. A. Neifeld, and A. E. Willner, “Experimental characterization of a 400 Gbit/s orbital angular momentum multiplexed free-space optical link over 120 m,” Opt. Lett. 41(3), 622–625 (2016).
[Crossref] [PubMed]

L. Li, G. Xie, Y. Ren, N. Ahmed, H. Huang, Z. Zhao, P. Liao, M. P. J. Lavery, Y. Yan, C. Bao, Z. Wang, A. J. Willner, N. Ashrafi, S. Ashrafi, M. Tur, and A. E. Willner, “Orbital-angular-momentum-multiplexed free-space optical communication link using transmitter lenses,” Appl. Opt. 55(8), 2098–2103 (2016).
[Crossref] [PubMed]

Y. Ren, Z. Wang, G. Xie, L. Li, A. J. Willner, Y. Cao, Z. Zhao, Y. Yan, N. Ahmed, N. Ashrafi, S. Ashrafi, R. Bock, M. Tur, and A. E. Willner, “Atmospheric turbulence mitigation in an OAM-based MIMO free-space optical link using spatial diversity combined with MIMO equalization,” Opt. Lett. 41(11), 2406–2409 (2016).
[Crossref] [PubMed]

Y. Ren, Z. Wang, G. Xie, L. Li, Y. Cao, C. Liu, P. Liao, Y. Yan, N. Ahmed, Z. Zhao, A. Willner, N. Ashrafi, S. Ashrafi, R. D. Linquist, R. Bock, M. Tur, A. F. Molisch, and A. E. Willner, “Free-space optical communications using orbital-angular-momentum multiplexing combined with MIMO-based spatial multiplexing,” Opt. Lett. 40(18), 4210–4213 (2015).
[Crossref] [PubMed]

Watnik, A. T.

Willner, A.

Willner, A. E.

S. Li, S. Chen, C. Gao, A. E. Willner, and J. Wang, “Atmospheric turbulence compensation in orbital angular momentum communications: Advances and perspectives,” Opt. Commun. 408, 68–81 (2018).
[Crossref]

L. Li, G. Xie, Y. Ren, N. Ahmed, H. Huang, Z. Zhao, P. Liao, M. P. J. Lavery, Y. Yan, C. Bao, Z. Wang, A. J. Willner, N. Ashrafi, S. Ashrafi, M. Tur, and A. E. Willner, “Orbital-angular-momentum-multiplexed free-space optical communication link using transmitter lenses,” Appl. Opt. 55(8), 2098–2103 (2016).
[Crossref] [PubMed]

Y. Ren, Z. Wang, G. Xie, L. Li, A. J. Willner, Y. Cao, Z. Zhao, Y. Yan, N. Ahmed, N. Ashrafi, S. Ashrafi, R. Bock, M. Tur, and A. E. Willner, “Atmospheric turbulence mitigation in an OAM-based MIMO free-space optical link using spatial diversity combined with MIMO equalization,” Opt. Lett. 41(11), 2406–2409 (2016).
[Crossref] [PubMed]

Y. Ren, Z. Wang, P. Liao, L. Li, G. Xie, H. Huang, Z. Zhao, Y. Yan, N. Ahmed, A. Willner, M. P. J. Lavery, N. Ashrafi, S. Ashrafi, R. Bock, M. Tur, I. B. Djordjevic, M. A. Neifeld, and A. E. Willner, “Experimental characterization of a 400 Gbit/s orbital angular momentum multiplexed free-space optical link over 120 m,” Opt. Lett. 41(3), 622–625 (2016).
[Crossref] [PubMed]

Y. Ren, Z. Wang, G. Xie, L. Li, Y. Cao, C. Liu, P. Liao, Y. Yan, N. Ahmed, Z. Zhao, A. Willner, N. Ashrafi, S. Ashrafi, R. D. Linquist, R. Bock, M. Tur, A. F. Molisch, and A. E. Willner, “Free-space optical communications using orbital-angular-momentum multiplexing combined with MIMO-based spatial multiplexing,” Opt. Lett. 40(18), 4210–4213 (2015).
[Crossref] [PubMed]

H. Huang, Y. Cao, G. Xie, Y. Ren, Y. Yan, C. Bao, N. Ahmed, M. A. Neifeld, S. J. Dolinar, and A. E. Willner, “Crosstalk mitigation in a free-space orbital angular momentum multiplexed communication link using 4×4 MIMO equalization,” Opt. Lett. 39(15), 4360–4363 (2014).
[Crossref] [PubMed]

Willner, A. J.

Wu, H.

Y. Yuan, Z. Zhang, C. Ji, and H. Wu, “Capacity analysis of UCA-based OAM multiplexing communication system,” in Proceedings of Wireless Communications and Signal Processing (WCSP) (IEEE, 2015), pp. 1–5.

Wu, S.

Xiang, Y.

X. Zhang, Y. He, Y. Cai, M. Su, X. Zhou, Y. Chen, S. Chen, Y. Xiang, L. Chen, C. Su, Y. Li, and D. Fan, “Coherent separation detection for orbital angular momentum multiplexing in free-space optical communications,” IEEE Photonics J. 9(3), 1–11 (2017).

Xie, G.

Y. Ren, Z. Wang, P. Liao, L. Li, G. Xie, H. Huang, Z. Zhao, Y. Yan, N. Ahmed, A. Willner, M. P. J. Lavery, N. Ashrafi, S. Ashrafi, R. Bock, M. Tur, I. B. Djordjevic, M. A. Neifeld, and A. E. Willner, “Experimental characterization of a 400 Gbit/s orbital angular momentum multiplexed free-space optical link over 120 m,” Opt. Lett. 41(3), 622–625 (2016).
[Crossref] [PubMed]

L. Li, G. Xie, Y. Ren, N. Ahmed, H. Huang, Z. Zhao, P. Liao, M. P. J. Lavery, Y. Yan, C. Bao, Z. Wang, A. J. Willner, N. Ashrafi, S. Ashrafi, M. Tur, and A. E. Willner, “Orbital-angular-momentum-multiplexed free-space optical communication link using transmitter lenses,” Appl. Opt. 55(8), 2098–2103 (2016).
[Crossref] [PubMed]

Y. Ren, Z. Wang, G. Xie, L. Li, A. J. Willner, Y. Cao, Z. Zhao, Y. Yan, N. Ahmed, N. Ashrafi, S. Ashrafi, R. Bock, M. Tur, and A. E. Willner, “Atmospheric turbulence mitigation in an OAM-based MIMO free-space optical link using spatial diversity combined with MIMO equalization,” Opt. Lett. 41(11), 2406–2409 (2016).
[Crossref] [PubMed]

Y. Ren, Z. Wang, G. Xie, L. Li, Y. Cao, C. Liu, P. Liao, Y. Yan, N. Ahmed, Z. Zhao, A. Willner, N. Ashrafi, S. Ashrafi, R. D. Linquist, R. Bock, M. Tur, A. F. Molisch, and A. E. Willner, “Free-space optical communications using orbital-angular-momentum multiplexing combined with MIMO-based spatial multiplexing,” Opt. Lett. 40(18), 4210–4213 (2015).
[Crossref] [PubMed]

H. Huang, Y. Cao, G. Xie, Y. Ren, Y. Yan, C. Bao, N. Ahmed, M. A. Neifeld, S. J. Dolinar, and A. E. Willner, “Crosstalk mitigation in a free-space orbital angular momentum multiplexed communication link using 4×4 MIMO equalization,” Opt. Lett. 39(15), 4360–4363 (2014).
[Crossref] [PubMed]

Xin, X.

Q. Tian, L. Zhu, Y. Wang, Q. Zhang, B. Liu, and X. Xin, “The propagation properties of a longitudinal orbital angular momentum multiplexing system in atmospheric turbulence,” IEEE Photonics J. 10(1), 1–11 (2018).
[Crossref]

Xu, H.

R. Chen, H. Xu, M. Moretti, and J. Li, “Beam Steering for the Misalignment in UCA-Based OAM Communication Systems,” IEEE Wirel. Commun. Lett. 99, 1–4 (2018).

Yan, Y.

L. Li, G. Xie, Y. Ren, N. Ahmed, H. Huang, Z. Zhao, P. Liao, M. P. J. Lavery, Y. Yan, C. Bao, Z. Wang, A. J. Willner, N. Ashrafi, S. Ashrafi, M. Tur, and A. E. Willner, “Orbital-angular-momentum-multiplexed free-space optical communication link using transmitter lenses,” Appl. Opt. 55(8), 2098–2103 (2016).
[Crossref] [PubMed]

Y. Ren, Z. Wang, P. Liao, L. Li, G. Xie, H. Huang, Z. Zhao, Y. Yan, N. Ahmed, A. Willner, M. P. J. Lavery, N. Ashrafi, S. Ashrafi, R. Bock, M. Tur, I. B. Djordjevic, M. A. Neifeld, and A. E. Willner, “Experimental characterization of a 400 Gbit/s orbital angular momentum multiplexed free-space optical link over 120 m,” Opt. Lett. 41(3), 622–625 (2016).
[Crossref] [PubMed]

Y. Ren, Z. Wang, G. Xie, L. Li, A. J. Willner, Y. Cao, Z. Zhao, Y. Yan, N. Ahmed, N. Ashrafi, S. Ashrafi, R. Bock, M. Tur, and A. E. Willner, “Atmospheric turbulence mitigation in an OAM-based MIMO free-space optical link using spatial diversity combined with MIMO equalization,” Opt. Lett. 41(11), 2406–2409 (2016).
[Crossref] [PubMed]

Y. Ren, Z. Wang, G. Xie, L. Li, Y. Cao, C. Liu, P. Liao, Y. Yan, N. Ahmed, Z. Zhao, A. Willner, N. Ashrafi, S. Ashrafi, R. D. Linquist, R. Bock, M. Tur, A. F. Molisch, and A. E. Willner, “Free-space optical communications using orbital-angular-momentum multiplexing combined with MIMO-based spatial multiplexing,” Opt. Lett. 40(18), 4210–4213 (2015).
[Crossref] [PubMed]

H. Huang, Y. Cao, G. Xie, Y. Ren, Y. Yan, C. Bao, N. Ahmed, M. A. Neifeld, S. J. Dolinar, and A. E. Willner, “Crosstalk mitigation in a free-space orbital angular momentum multiplexed communication link using 4×4 MIMO equalization,” Opt. Lett. 39(15), 4360–4363 (2014).
[Crossref] [PubMed]

Yang, H.

Yang, Y.

Yu, P.

Yu, Z.

M. Li, Z. Yu, and M. Cvijetic, “Influence of atmospheric turbulence on OAM-based FSO system with use of realistic link model,” Opt. Commun. 364, 50–54 (2016).
[Crossref]

Yuan, Y.

Y. Yuan, Z. Zhang, C. Ji, and H. Wu, “Capacity analysis of UCA-based OAM multiplexing communication system,” in Proceedings of Wireless Communications and Signal Processing (WCSP) (IEEE, 2015), pp. 1–5.

Zhan, Y.

Zhang, H.

Zhang, K.

J. Peng, L. Zhang, K. Zhang, and J. Ma, “Channel capacity of OAM based FSO communication systems with partially coherent Bessel–Gaussian beams in anisotropic turbulence,” Opt. Commun. 418, 32–36 (2018).
[Crossref]

Zhang, L.

J. Peng, L. Zhang, K. Zhang, and J. Ma, “Channel capacity of OAM based FSO communication systems with partially coherent Bessel–Gaussian beams in anisotropic turbulence,” Opt. Commun. 418, 32–36 (2018).
[Crossref]

P. Wang, L. Zhang, L. Guo, F. Huang, T. Shang, R. Wang, and Y. Yang, “Average BER of subcarrier intensity modulated free space optical systems over the exponentiated Weibull fading channels,” Opt. Express 22(17), 20828–20841 (2014).
[Crossref] [PubMed]

Zhang, M.

Zhang, Q.

Q. Tian, L. Zhu, Y. Wang, Q. Zhang, B. Liu, and X. Xin, “The propagation properties of a longitudinal orbital angular momentum multiplexing system in atmospheric turbulence,” IEEE Photonics J. 10(1), 1–11 (2018).
[Crossref]

Zhang, X.

X. Zhang, Y. He, Y. Cai, M. Su, X. Zhou, Y. Chen, S. Chen, Y. Xiang, L. Chen, C. Su, Y. Li, and D. Fan, “Coherent separation detection for orbital angular momentum multiplexing in free-space optical communications,” IEEE Photonics J. 9(3), 1–11 (2017).

Z. Zhang, S. Zheng, Y. Chen, X. Jin, H. Chi, and X. Zhang, “The capacity gain of orbital angular momentum based multiple-input-multiple-output system,” Sci. Rep. 6(1), 25418 (2016).
[Crossref] [PubMed]

Zhang, Y.

Zhang, Z.

Z. Zhang, S. Zheng, Y. Chen, X. Jin, H. Chi, and X. Zhang, “The capacity gain of orbital angular momentum based multiple-input-multiple-output system,” Sci. Rep. 6(1), 25418 (2016).
[Crossref] [PubMed]

Y. Yuan, Z. Zhang, C. Ji, and H. Wu, “Capacity analysis of UCA-based OAM multiplexing communication system,” in Proceedings of Wireless Communications and Signal Processing (WCSP) (IEEE, 2015), pp. 1–5.

Zhao, Q.

Zhao, S.

S. Zhao, L. Wang, L. Zou, L. Gong, W. Cheng, B. Zheng, and H. Chen, “Both channel coding and wavefront correction on the turbulence mitigation of optical communications using orbital angular momentum multiplexing,” Opt. Commun. 376, 92–98 (2016).
[Crossref]

Zhao, S. M.

Zhao, Z.

Zheng, B.

S. Zhao, L. Wang, L. Zou, L. Gong, W. Cheng, B. Zheng, and H. Chen, “Both channel coding and wavefront correction on the turbulence mitigation of optical communications using orbital angular momentum multiplexing,” Opt. Commun. 376, 92–98 (2016).
[Crossref]

Zheng, B. Y.

Zheng, S.

Z. Zhang, S. Zheng, Y. Chen, X. Jin, H. Chi, and X. Zhang, “The capacity gain of orbital angular momentum based multiple-input-multiple-output system,” Sci. Rep. 6(1), 25418 (2016).
[Crossref] [PubMed]

Zhou, X.

X. Zhang, Y. He, Y. Cai, M. Su, X. Zhou, Y. Chen, S. Chen, Y. Xiang, L. Chen, C. Su, Y. Li, and D. Fan, “Coherent separation detection for orbital angular momentum multiplexing in free-space optical communications,” IEEE Photonics J. 9(3), 1–11 (2017).

Zhu, L.

Q. Tian, L. Zhu, Y. Wang, Q. Zhang, B. Liu, and X. Xin, “The propagation properties of a longitudinal orbital angular momentum multiplexing system in atmospheric turbulence,” IEEE Photonics J. 10(1), 1–11 (2018).
[Crossref]

Zou, L.

S. Zhao, L. Wang, L. Zou, L. Gong, W. Cheng, B. Zheng, and H. Chen, “Both channel coding and wavefront correction on the turbulence mitigation of optical communications using orbital angular momentum multiplexing,” Opt. Commun. 376, 92–98 (2016).
[Crossref]

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[Crossref]

X. Zhang, Y. He, Y. Cai, M. Su, X. Zhou, Y. Chen, S. Chen, Y. Xiang, L. Chen, C. Su, Y. Li, and D. Fan, “Coherent separation detection for orbital angular momentum multiplexing in free-space optical communications,” IEEE Photonics J. 9(3), 1–11 (2017).

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R. Chen, H. Xu, M. Moretti, and J. Li, “Beam Steering for the Misalignment in UCA-Based OAM Communication Systems,” IEEE Wirel. Commun. Lett. 99, 1–4 (2018).

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S. Zhao, L. Wang, L. Zou, L. Gong, W. Cheng, B. Zheng, and H. Chen, “Both channel coding and wavefront correction on the turbulence mitigation of optical communications using orbital angular momentum multiplexing,” Opt. Commun. 376, 92–98 (2016).
[Crossref]

J. Peng, L. Zhang, K. Zhang, and J. Ma, “Channel capacity of OAM based FSO communication systems with partially coherent Bessel–Gaussian beams in anisotropic turbulence,” Opt. Commun. 418, 32–36 (2018).
[Crossref]

S. Li, S. Chen, C. Gao, A. E. Willner, and J. Wang, “Atmospheric turbulence compensation in orbital angular momentum communications: Advances and perspectives,” Opt. Commun. 408, 68–81 (2018).
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M. Li, Z. Yu, and M. Cvijetic, “Influence of atmospheric turbulence on OAM-based FSO system with use of realistic link model,” Opt. Commun. 364, 50–54 (2016).
[Crossref]

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S. M. Zhao, J. Leach, L. Y. Gong, J. Ding, and B. Y. Zheng, “Aberration corrections for free-space optical communications in atmosphere turbulence using orbital angular momentum states,” Opt. Express 20(1), 452–461 (2012).
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P. Wang, L. Zhang, L. Guo, F. Huang, T. Shang, R. Wang, and Y. Yang, “Average BER of subcarrier intensity modulated free space optical systems over the exponentiated Weibull fading channels,” Opt. Express 22(17), 20828–20841 (2014).
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X. Hu, Q. Zhao, P. Yu, X. Li, Z. Wang, Y. Li, and L. Gong, “Dynamic shaping of orbital-angular-momentum beams for information encoding,” Opt. Express 26(2), 1796–1808 (2018).
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J. Li, M. Zhang, D. Wang, S. Wu, and Y. Zhan, “Joint atmospheric turbulence detection and adaptive demodulation technique using the CNN for the OAM-FSO communication,” Opt. Express 26(8), 10494–10508 (2018).
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Figures (6)

Fig. 1
Fig. 1 The proposed LDPC coded OAM-based UCA FSO system with ZF/MMSE equalizer combined with channel estimation
Fig. 2
Fig. 2 (a) ABER performances of OAM-based UCA FSO system with ZF and MMSE equalizers under different atmospheric turbulence strengths. The constellation of the recovered QPSK signals propagated in the proposed system under weak turbulence condition (b) with ZF equalizer, (c) with MMSE equalizer, respectively.
Fig. 3
Fig. 3 ABER performances of OAM-based UCA FSO system with ZF and MMSE equalizers under (a) weak, (b) moderate, and (c) strong turbulence conditions using BPSK, QPSK and 8PSK modulations
Fig. 4
Fig. 4 ABER performances comparison for different receive antenna numbers under weak turbulence considering ZF and MMSE equalizers.
Fig. 5
Fig. 5 ABER performances of uncoded and LDPC coded QPSK OAM-based UCA FSO systems over the atmospheric turbulence with ZF and MMSE equalizers (The straight and dashed lines represent the ABER values of uncoded and LDPC coded FSO system, respectively. The lines in black, red and blue color represent the ABER values of this system under weak, moderate, strong turbulence, respectively. The lines with hollow and solid squares represent the system with ZF and MMSE equalizer, respectively).
Fig. 6
Fig. 6 ABER performances of LDPC coded QPSK OAM-based UCA FSO systems considering ZF and MMSE equalizers with different LDPC code rates under (a) weak, (b) moderate, and (c) strong turbulence conditions, respectively. (The lines in black, blue and red color represent the ABER values with code rates 0.25, 0.5 and 0.75, respectively. The lines with hollow and solid squares represent the FSO system with ZF and MMSE equalizers, respectively)

Equations (32)

Equations on this page are rendered with MathJax. Learn more.

U l ( r , θ , z ) = A l ( r , z ) exp ( i l θ ) ,
A l ( r , z ) = 2 p ! π ( p + | l | ) ! 1 ω ( z ) [ 2 r ω ( z ) ] | l | exp [ r 2 ω 2 ( z ) ] L p l [ 2 r 2 ω 2 ( z ) ] × exp [ i k r 2 z 2 ( z 2 + z R 2 ) ] exp [ i ( 2 p + | l | + 1 ) tan 1 z z R ] ,
X m ( r , θ , z ) = x m ( t ) U m ( r , θ , z ) = x m ( t ) A m ( r , z ) exp ( i l m θ ) .
X T x ( r , θ , t ) = m = 1 M x m ( t ) U m ( r , θ , z ) = m = 1 M x m ( t ) A m ( r , z ) exp ( i l m θ ) .
Φ n ( k ) = 0.033 C n 2 ( k 2 + 1 / L 0 2 ) 11 6 exp ( k 2 / k l 2 ) ,
R n ( r , θ , t ) = m = 1 M x m ( t ) U m ' ( r , θ , z ) = m = 1 M x m ( t ) A m ' ( r , z ) exp ( i l m θ n ) exp ( ψ ( r ) ) ,
y n ( r , θ , t ) = m = 1 M x m ( t ) U m ' ( r , θ , z ) + w n = m = 1 M x m ( t ) A m ' ( r , z ) exp ( i l m θ n ) exp ( ψ ( r ) ) + w n ,
h n , m = m = 1 M A m ' ( r , z ) exp ( ψ ( r ) ) exp ( i l m θ n ) ,
θ n = 2 π ( n n 0 ) N ,
h n , m = h C n , m h A T n , m ,
h C n , m = exp ( i l m θ n ) ,
h A T n , m = A m ' ( r , z ) A m ( r , z ) exp ( i l m ( θ ' θ ) = A m ' ( r , z ) A m ( r , z ) exp ( ψ ( r ) ) .
H C = ( exp ( i l 1 θ 1 ) exp ( i l 2 θ 1 ) exp ( i l m θ 1 ) exp ( i l 1 θ 2 ) exp ( i l 2 θ 2 ) exp ( i l m θ 2 ) exp ( i l 1 θ n ) exp ( i l 2 θ n ) exp ( i l m θ n ) ) .
h A T n , m = ( A m ( r , z ) exp ( i l m θ ) ) 1 ( m = 1 M A m ' ( r , z ) exp ( ψ ( r ) ) + w n ) .
h n , m = exp ( i l m θ n ) ( A m ( r , z ) exp ( i l m θ ) ) 1 ( m = 1 M A m ' ( r , z ) exp ( ψ ( r ) ) + w n ) .
H = ( exp ( i l 1 θ 1 ) ( A 1 ( r , z ) exp ( i l 1 θ ) ) 1 ( m = 1 M A m ' ( r , z ) exp ( ψ ( r ) ) + w 1 ) exp ( i l m θ 1 ) ( A m ( r , z ) exp ( i l m θ ) ) 1 ( m = 1 M A m ' ( r , z ) exp ( ψ ( r ) ) + w 1 ) exp ( i l 1 θ n ) ( A m ( r , z ) exp ( i l m θ ) ) 1 ( m = 1 M A m ' ( r , z ) exp ( ψ ( r ) ) + w n ) exp ( i l m θ n ) ( A m ( r , z ) exp ( i l m θ ) ) 1 ( m = 1 M A m ' ( r , z ) exp ( ψ ( r ) ) + w n ) ) .
W Z F = ( H H H ) 1 H H
W M M S E = ( H H H + σ n 2 I ) 1 H H .
γ Z F , n = γ [ ( H H H ) 1 ] ( n , n ) 1
γ M M S E , n = γ ( ( H H H + σ n 2 I ) 1 ) ( n , n ) 1 ,
P e ( γ ) = 1 m a x ( l o g 2 M P , 2 ) e r f c ( γ sin π M P ) ,
P e , Z F ( γ ) = 1 m a x ( l o g 2 M P , 2 ) e r f c ( γ [ ( H H H ) 1 ] ( n , n ) 1 sin π M P )
P e , M M S E ( γ ) = 1 m a x ( l o g 2 M P , 2 ) e r f c ( γ ( ( H H H + σ n 2 I ) 1 ) ( n , n ) 1 sin π M P ) .
x ˜ Z F , m = ( ( H H H ) 1 H H ) m y Z F , m = x Z F , m + ( ( H H H ) 1 H H ) m w m
x ˜ M M S E , m = ( ( H H H + σ n 2 I ) 1 H m ) H y M M S E , m = ( ( H H H + σ n 2 I ) 1 H m ) H ( H x + w m ) ,
x ˜ m = μ m x m + z m ,
μ Z F , m = x Z F , m ( t )
μ M M S E , m = ( ( H H H + σ n 2 I ) 1 H m ) H H m .
σ Z F , m 2 = ( ( H H H ) 1 H H ) m 2
σ M M S E , m 2 = μ M M S E , m ( 1 μ M M S E , m ) .
P Z F , m ( x ˜ m / x m ) 1 π σ Z F , m 2 exp ( | x ˜ m x m | 2 σ Z F , m 2 )
P M M S E , m ( x ˜ m / x m ) 1 π σ M M S E , m 2 exp ( | x ˜ m μ M M S E , m x m | 2 σ M M S E , m 2 ) .

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