1 Introduction

High speed, no electromagnetic interference (EMI), large bandwidth, unlicensed spectrum, low power requirement, high security, light weight, and low cost are some of the advantages of FSO link over radio frequency (RF) communication system. Many factors such as scattering, light reflection, atmospheric turbulence, gases, cloud cover, wind, rain, dust, fog, smoke, snow, and other non-ideal characteristics of the system can degrade the performance of FSO link [1,2,3,4]. FSO link technology was considered as a potential distributer of worldwide interoperability for microwave access (WiMAX) traffic in metro or access networks [5,6,7,8,9]. Cvijetic et al., in 2006, studied transmission of OFDM-WiMAX traffic over FSO link via subcarrier modulation with direct/heterodyne detection in [6, 7], respectively, where outage and symbol error performance was determined for both detection techniques. In 2006 [8], a multiple-input–multiple-output (MIMO) architecture for distributing WiMAX traffic using optical wireless technology is presented, where different protocols are considered. In 2008 [9], the 10 Gb/s first experimental demonstration of OFDM-FOS link is given in the presence of atmospheric effects, including variable power attenuation and signal fading. N. Zdravkovi et al., in 2016, presented outage analysis of Mixed FSO/WiMAX Link in [10]. A. Bekkali et al. [11], in 2010, presented the closed-form expressions for the OFDM-based turbulent radio-over-FSO (RoFSO) system and studied the transmission performance of the link. M. Sharma et al. [12, 13], in 2013 and 2014, have evaluated the capacity of MIMO-OFDM FSO communication system in the presence of intersymbol interference (ISI) for different values of weak atmospheric turbulence strength and studied the performance of spatially multiplexed MIMO-OFDM FSO communication system. The transmission performance of OFDM FSO system using dual-diversity reception under correlated lognormal distribution have studied by F. Bai et al. [14] in 2014. The theoretical reference to study coherent detection OFDM-FSO systems was given by Y. Wang et al. [15], in 2015. P. Kumar et al. [16], in 2015, compared the performance of OFDM and TDM signals over unlicensed turbulent FSO link. In 2016, the performance of multi-user OFDM-FSO link was improved with combined receive space diversity and transmit frequency diversity [17]. In OFDM-FSO link with individual orthogonal subcarriers, the multiple independent subsignals for main signal are modulated and multiplexed in RF domain. This RF-OFDM signal is converted into optical OFDM signal using laser diode (LD) and transmitted to the turbulent FSO channel. At the receiver end, this optical signal is received by photodetector (PD) diode that uses direct detection technique. The complex equalization is not required due to less distortion experienced in OFDM-FSO link [18]. Due to the saving in electrical bandwidth, ease of channel and phase estimation, and scalability to higher-order modulation [19], OFDM is used for FSO link in this paper.

In RF systems, the OFDM signal is bipolar and complex which cannot be applied to the LD because it cannot generate intensity variation for negative amplitude signals. Thus a positive and real RF-OFDM signal is required for IM/DD systems. To generate positive and real signal, there are different techniques used in the literature [20,21,22,23,24,25,26,27,28,29]: DCO-OFDM [20]; ACO-OFDM [21, 22]; combined asymmetrically clipped-DC-biased optical OFDM (ADO-OFDM) [23]; hybrid ACO-OFDM [24]; asymmetrically and symmetrically clipping optical OFDM (ASCO-OFDM) [25]; auto-bias controlled technique [26]; optimal DC-biasing for DCO-OFDM [27]. In DCO-OFDM technique, DC bias is applied to make the entire negative part of the signal positive. If sufficient DC bias is not added, the remaining negative part of the signal gets clipped and results in clipping noise. With DC bias addition, the average OFDM signal power is increased and that leads to enhancement in nonlinearity effects at the transmitter and optical channel. Thus, DCO-OFDM is not a power-efficient technique. ACO-OFDM is a power-efficient technique where additional DC bias to make the OFDM signal positive is not required. In ACO-OFDM, the complex symbols are mapped only on the odd subcarriers, whereas even subcarriers are set to zero. The generated bipolar signal is then clipped at zero to make it positive. This positive signal is only applied to the laser for direct detection. The clipping does not introduce any loss of information due to anti-symmetric property of ACO-OFDM. S. D. Dissanayak et al. [23], J. Armstrong et al. [28], D. Patel et al. [29], compared the performance of DCO-OFDM and ACO-OFDM and have shown that ACO-OFDM provides better system performance for short distance up to 75 km. ACO-OFDM using 16-QAM has the same normalized bandwidth as DCO-OFDM using 4-QAM but requires 4.7 dB less power than DCO-OFDM with 7 dB bias. In DCO-OFDM, the optimum value of DC bias depends on the QAM constellation. Large constellations would require more DC bias to eliminate the clipping noise. L. Chen et al. [30] characterized the nonlinear biasing and clipping effects on an IM/DD optical OFDM system. They have shown that nonlinear clipping distortion process can be modeled as a virtual relay channel. This nonlinear process can be modeled as a linear deterministic attenuation plus an uncorrelated random additive clipping noise in the time domain.

In FSO channel, scattering due to fog creates multiple propagation paths resulting in multiple reception of the transmitted signal. Due to multiple reception, temporal broadening (dispersion) of the transmitted signal arises, which creates ISI resulting into performance degradation [31,32,33]. This dispersion is linear distortion in the channel, but it is converted into nonlinear distortion after square-law-based detection at the receiver. This nonlinear distortion can reduce the effectiveness of linear equalization. The square-root-module (SRM) device, added after photodiode detector to compensate for square-law behavior, improves linear equalizer performance resulting in overall performance improvement of OFDM-FSO link [34,35,36].

An FSO system, based on line-of-sight (LoS), is concerned with the transmission of optical beam carrying information through free space. If there is obstacle between the users, then an event of link outage occurs. In this non-LoS (NLoS) situation, users cannot communicate directly with each other. To make possible the information exchange, a relay can be used, and with it, both the users can communicate with each other, and this scheme is known as traditional scheme (TS) [37]. The capacity or throughput or achievable rate of TS link is less since more time slots are required to make a end-to-end communication. To improve the information rate in relay network, various coding schemes are used such as digital network coding (DNC), analog network coding (ANC), physical layer network coding (PLNC) scheme, in RF-wireless communication system [37].

1.1 Related work

FSO systems with network coding have been studied in related works [38,39,40,41,42]. The related work [37] of S. C. Liew et al. gives an overview of PLNC, TC, DNC, and ANC schemes with an example where PLNC scheme can improve throughput by 100\(\%\) in passive optical network (PON). In wireless communication networks, to prevent harmful interference, scheduling of multiple transmission at the same time is avoided. Instead of avoiding interference, PLNC scheme exploits the use of interference to increase the network capacity. The related work [43] of S. Katti et al. proposed and implemented ANC, a less complex version of PLNC, where the received mixed signal at the relay is simply amplified and forwarded without any processing. In [37], the achievable rates, in symmetric exchange rates, of PLNC, DNC, ANC, and TS under equal power usages for all nodes are compared and it has been shown that PLNC performs better with high achievable rate as compared to other schemes. The related work [44, 45] of H. Gacanin et al. and T. Sjodin et al., give the study of OFDM link based on ANC with the perfect knowledge of channel state information. However, this ANC scheme cannot exploit the direct link between the two users, i.e., LoS link, even if it exists physically resulting in poorer performance despite of having high spectral efficiency and diversity. A three time-slotted analog network coding (3T-ANC) scheme has been proposed [46,47,48,49] that can utilize the direct link by allowing the transmission of two users in two different time slots and the relay transmission in third time slot. Consequently, the 3T-ANC scheme can improve the performance of a two-way relay channel in presence of direct LoS link.

1.2 Our contribution

In all related works [37, 43,44,45], a network coding scheme (such as DNC, ANC, and PLNC) is applied in wireless communication networks and have shown that PLNC performed better than ANC scheme, but, in optical communication point-of-view, PLNC is not cost efficient. In this work, the performance of OFDM-based FSO link, with various network coding schemes, is analyzed to make successful end-to-end communication with increased capacity of the link. In addition to this, we analyze the performance of ANC scheme in FSO network where both NLoS and LoS link are present at the same time. Normally ANC scheme is based on amplify-forward (AF) protocol with two time slot requirement for delivering the end-to-end information resulting into increased capacity of relay-assisted FSO link. In presence of both NLoS and LoS link in FSO network, we propose to use three time-slotted ANC scheme referred to as 3T-ANC scheme that can gives further improvement in receiver sensitivity and increase throughput of FSO link.

In ANC, the relay operation is based on AF protocol and in PLNC, the relay operation is based on decode-forward (DF) protocol resulting into reduced cost and complexity with ANC scheme as compared to PLNC scheme. ANC, with no optical-to-electrical (O/E) and electrical-to-optical (E/O) conversion for AF operation at relay, consumes less electrical power than PLNC where E/O and O/E is mandatory during DF operation at relay. Thus, ANC is energy- and cost- saving approach in communication network because it minimizes the number of optical–electrical–optical (O/E/O) conversion interface and electrical interface resulting into less energy consumption at the relay.

In this work, we have compared the performance of ACO-OFDM and DCO-OFDM FSO link in LoS situation. Also we have compared the ACO-OFDM and DCO-OFDM in NLoS+LoS with 3T-ANC scheme. The new results show that with 3T-ANC scheme, the ACO-OFDM technique is also power-efficient scheme where additional DC bias is not required to make the RF-OFDM signal positive.

The performance of OFDM-FSO link with different network coding schemes over turbulent FSO channel is compared in terms of achievable rate vs SNR and BER vs received power. In this work, Gamma-Gamma distribution is used to describe turbulent-induced-faded atmospheric channel. The paper is organized as follows: Various network coding schemes with 3T-ANC scheme are described in Sect. 2. The architecture of OFDM transmitter and receiver for optical signal transmission and reception over turbulent FSO link and FSO channel model is discussed in Sect. 3. The achievable rate and BER analysis are discussed in Sect. 4. Results are discussed in Sect. 5, and finally the conclusion is given in Sect. 6.

2 Network coding

2.1 Digital network coding

Consider a situation as shown in Fig. 1, where two users \(U_{1}\) and \(U_{2}\) wants to communicate each other via an FSO link. If the channel is half duplex and LoS event occurs, then two stages are required to exchange information between them.

Fig. 1
figure 1

Two-way network (without relay)

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In a situation when an obstacle is between them, communication link will breakdown. To solve this problem, relay-assisted NLoS-FSO link can be used where the relay N is added to facilitate communication as shown in Fig. 2. This kind of network is known as two-way relay network (TWRN), and this scheme is known as traditional scheme (TS) [37]. In this condition, four stages are required instead of two as in LoS situation, and hence, with this, there is 50\(\%\) degradation in throughput.

Fig. 2
figure 2

A two-way relay network (TWRN) or traditional scheme (TS)

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In TS, to avoid interference, the signals transmitted form \(U_{1}\) and \(U_{2}\) are separated in time or wavelength. If the relay is capable of mixing the two data packets together, the four time slots needed in TS to exchange information is reduced to three time slots as shown in Fig. 3. This TS with mixing capable relay is known as DNC scheme. Thus, DNC has throughput improvement of 33\(\%\) over TS or DNC has a coding gain of 1.3. In this, users \(U_{1}\) and \(U_{2}\) transmit their packets \(P_{1}\) and \(P_{2}\) during the first and second time slots, respectively. The relay receives data packets and performs XOR operation bitwise, i.e., \(P_{1,2}=f\left( P_{1},P_{2} \right) = P_{1} \oplus P_{2}\) (thus encodes two packets into one) and then broadcast it to both the users during third time slot. In DNC, the decode-forward operation is done at the relay, and thus O/E/O conversion is required at the relay.

Fig. 3
figure 3

Digital network coding (DNC)-based two-way relay network

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For simplicity, assume that users \(U_{1}\) and \(U_{2}\) transmits bits 0 and 1 during the first and second time slots, respectively. Relay perform XOR operation on bit 0 and 1 to create a mixed bit \((0 \oplus 1 = 1)\). This bit 1 is broadcasted. Now, at user end \(U_{1}\), XOR operation is performed on received broadcasted bit and transmitted bit itself i.e., \((1 \oplus 0 = 1)\). Thus, \(U_{1}\) will receive successfully the transmitted bit by \(U_{2}\). With this same procedure \(U_{2}\) can also recover the transmitted bit by user \(U_{1}\). Thus, in DNC scheme, users transmit sequentially, the relay mixes the content of the packets and broadcasts the mixed version.

Note that, both TWRN without network coding and with DNC, avoid simultaneous transmissions i.e., different time slots are for each user. After receptions at the relay, network coding is performed.

2.2 Physical layer network coding (PLNC)

In PLNC, only two time slots are required as compared to three time slots required in DNC scheme. In the first time slot, users \(U_{1}\) and \(U_{2}\) transmit their data to relay N. Based on the superimposed EM waves that carry packets of \(U_{1}\) and \(U_{2}\), i.e., \(P_{1}\) and \(P_{2}\), respectively, relay N computes \(P_{1,2}=P_{1} \oplus P_{2}\). In the second time slot, relay N broadcasts the \(P_{1,2}\) to both the users. Thus, there is 100\(\%\) improvement in throughput of canonical relay network. The signal broadcasted in second time slot in PLNC is the same as the signal transmitted in third time slot in DNC. The key difference of the two systems lies in how they derive \(P_{1,2}\). In DNC, \(P_{1}\) and \(P_{2}\) are separately transmitted by users \(U_{1}\) and \(U_{2}\); and relay N decodes \(P_{1}\) and \(P_{2}\) in order to form \(P_{1,2}\). In PLNC, \(P_{1,2}\) is derived from \(P_{1}+P_{2}\), which is the superimposed signal received in the first time slot. The superimposed signal is due to the network coding performed by nature and the relay transforms it to the XOR network coding function, i.e., \(P_{1,2}=f\left( P_{1}, P_{2}\right) \). At both users, \(U_{1}\) and \(U_{2}\), there must be a network decoding functions \(g_{1}\) and \(g_{2}\), such that \(P_{2}=g_{1}\left( f\left( P_{1}, P_{2}\right) ,P_{1}\right) \) and \(P_{1}=g_{2}\left( f\left( P_{1}, P_{2}\right) ,P_{2}\right) \), to extract their counterpart’s signal from \(P_{1,2}\) and self-information. PLNC can be divided into two types [50] based on whether the range of f is a finite or infinite set. Finite set PLNC include the XOR operation and known as simple PLNC; on the other hand, infinite set PLNC is known as ANC in which the relay N retains the additive mixing that occurs in the nature and amplifies the simultaneously received signals and noise and then broadcast it to both the users. In the case of ANC, the received signal is \(R_{s}=f(S_{1},S_{2})=h_{1}S_{0}+h_{2}S_{1}\), where \(h_{1}\) and \(h_{2}\) are the channel fading coefficients between user \(U_{1}\) to relay N and \(U_{2}\) to the relay N, \(S_{1}\) and \(S_{2}\) are the transmitted signals from user \(U_{1}\) and \(U_{2}\) to relay, respectively. In PLNC, DF operation is also performed at the relay, thus in case of FSO link, O/E/O conversion is also required at the relay that can increase the cost and complexity of the relay significantly.

2.3 Analog network coding (ANC)

Instead of avoiding interference at the relay, ANC utilizes it. The relay forwards received mixed signals, and the destinations use their information to cancel the self-interference and thus recover the desired signal. ANC is less complex version of PLNC, since the relay uses AF instead of DF protocol [37, 43,44,45]. The difference between ANC and PLNC, in optical communication point-of-view, is as follows: (1) ANC is based on AF protocol where the interfering signal of both the users by the channel is simply amplified with the help of optical amplifier, i.e., EDFA and forwarded to both the users. On the other hand, PLNC is based on DF protocol where the received signals from both the users is decoded and converted into bits and after applying XOR operation it is forwarded to both the users; (2) optical amplifier can directly amplify the optical signal without O/E conversion, and thus, there is no need for O/E/O conversion at the relay resulting into lower complexity and cost of the relay-assisted link. On the other hand, O/E/O conversion is required at the relay in PLNC, and thus, relay is equipped with both electrical and optical components resulting into higher complexity and cost of the relay-assisted link. The relay that require O/E/O conversion can be termed as opto-electronic relay (OER), and the relay without this requirement termed as optical relay (OR) ; (3) because of only EDFA at relay, ANC scheme consume less electrical power, whereas PLNC scheme consumes higher electrical power because of decoding operation and O/E/O conversion at the relay; (4) ANC performs AF operation on signals not on bits. On the other hand, PLNC performs decode-forward operation on bits not signals; (5) finite-field mapping is used in PLNC, whereas infinite-field mapping is used in ANC [50]; (6) PLNC is used when the uplink is good and ANC is used when downlink is good [37, 50].

ANC scheme is shown in Fig. 4, where two time slots are required to exchange information between two users. The first time slot is used by both users simultaneously to transmit their signals to the relay and this stage is termed as multiple access (MA) stage. The combined signal at the relay is broadcasted in second time slot, and this stage is termed as and termed as broadcast channel (BC) stage.

Fig. 4
figure 4

Analog network coding (ANC)-based two-way relay network

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2.4 Three time-slotted ANC (3T-ANC) scheme

The 3T-ANC scheme is shown in Fig. 5, where three time slots are needed. In the first time slot, user \(U_{1}\) broadcasts its signal to both relay N and node \(U_{2}\). In the second time slot, user \(U_{2}\) broadcasts its signal to both relay N and node \(U_{1}\). In the third time slot, relay N broadcasts its combined amplified signal back to the users. The protocol for 3T-ANC has been investigated in [47].

Fig. 5
figure 5

Three time-slotted analog network coding (3T-ANC)

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2.5 Mathematical model for ANC and 3T-ANC

Consider a two user nodes \(U_{1}\)- and \(U_{2}\)-based FSO system consisting an amplify-forward (AF) relay node N as shown in Fig. 6. The turbulent-induced faded atmospheric channels between the nodes are described by Gamma–Gamma distribution, and they are time invariant during transmission phases. Let \(h_{d}\) and \(h_{j}\) are the channel fading coefficients of direct LoS link between \(U_{1}\) and \(U_{2}\) at distance \(d_{d}\), and the relay link between \(U_{j}\) to N at distance \(d_{j}\) for j = 1, 2.

Fig. 6
figure 6

A relay-assisted FSO system model

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In ANC scheme, users \(U_{j}\) transmit their OFDM signal \(S_{j}(t)\) in first time slot simultaneously for j = 1, 2. The received signals at relay node N is given as

$$\begin{aligned} R_{N,1}(t)^{\mathrm{ANC}} = \sum _{j=1}^{2}h_{j}S_{j}(t)+n_{N,1}(t). \end{aligned}$$
(1)

It is noticeable here that a user cannot receive the signal from other during the first time slot.

In the second time slot, the transmitted relay signal, with transmit power \(P_{N}\), to both of the users is given as

$$\begin{aligned} T_{N,2}(t)^{\mathrm {ANC}}=G_{2}R_{N,1}(t)^{\mathrm {ANC}}. \end{aligned}$$
(2)

where \(G_{2}\) is the amplifying gain factor.

The signal received at \(U_{j}\) in the second time slot is given as

$$\begin{aligned} R_{U_{j},2}(t)^{\mathrm{ANC}}= & {} h_{j}G_{2}\left( h_{1}S_{1}(t)+h_{2}S_{2}(t)\right. \nonumber \\&\quad \left. +n_{N,1}(t)\right) +n_{U_{j},2}(t). \end{aligned}$$
(3)

After self-interference \((G_{2}h_{j}h_{j}S_{j}(t))\) cancelation, the received signal is given as

$$\begin{aligned} R_{U_{j},2}(t)^{\mathrm{ANC}}=G_{2} h_{j}h_{i}S_{i}(t)+G_{2}h_{j}n_{R,1}(t)+n_{U_{j},2}(t). \end{aligned}$$
(4)

where \((i,j)=(1,2),(2,1)\), \(n_{k,j}\sim \textit{N}(0,\sigma ^{2}) \) is the additive noise at node k in the jth time slot, \(\sim \) stands for ‘distributed as’, \(\textit{N}(0,\sigma ^{2})\) denotes the zero mean Gaussian distribution with \(\sigma ^{2}\) variance. Finally, user \(U_{j}\) detects the signal of user \(U_{i}\) from \(r_{U_{j},2}(t)^{\mathrm{ANC}}\).

In 3T-ANC, three time slots are required in completing the information exchange: user \(U_{j}\) broadcast their OFDM signal \(S_{j}(t)\) in the jth time slot for j = 1, 2, and the relay N broadcast their signal \(S_{3}(t)\) in the third time slot. The received signals, in the first time slot, at the relay node N and at user node \(U_{2}\) are given as

$$\begin{aligned} R_{N,1}(t)^{\text {3T-ANC}}=h_{1}S_{1}(t)+n_{N,1}(t). \end{aligned}$$
(5)
$$\begin{aligned} R_{U_{2},1}(t)^{\text {3T-ANC}}=h_{d}S_{1}(t)+n_{U_{2},1}(t). \end{aligned}$$
(6)

Similarly, the received signals at the relay node N and at user node \(U_{1}\) in the second time slot are given as

$$\begin{aligned} R_{N,2}(t)^{\text {3T-ANC}}=h_{2}S_{2}(t)+n_{N,2}(t). \end{aligned}$$
(7)
$$\begin{aligned} R_{U_{1},2}(t)^{\text {3T-ANC}}=h_{d}S_{2}(t)+n_{U_{1},2}(t). \end{aligned}$$
(8)

The relay signal, in the third time slot, is linear combination of two received signals and can be given as

$$\begin{aligned} S_{3}(t)^{\text {3T-ANC}}=G_{3}\left\{ R_{N,1}(t)^{\text {3T-ANC}}+R_{N,2}(t)^{\text {3T-ANC}}\right\} . \end{aligned}$$
(9)

where \(G_{3}\) is the amplifying gain factor.

The signal received at node \(U_{j}\) in the third time slot is given as

$$\begin{aligned} R_{U_{j},3}(t)^{\text {3T-ANC}}= & {} G_{3}h_{j}\left\{ h_{i}S_{i}(t)+h_{j}S_{j}(t)+(n_{N,1}+n_{N,2})\right\} \nonumber \\&\qquad +n_{U_{j},3}. \end{aligned}$$
(10)

After self-interference elimination, the received signal at node \(U_{j}\) is given as

$$\begin{aligned} R_{U_{j},3}(t)^{\text {3T-ANC}}= & {} G_{3}h_{j}h_{i}S_{i}(t)+G_{3}h_{j} \left( n_{N,1}+n_{N,2} \right) \nonumber \\&\qquad +n_{U_{j},3} . \end{aligned}$$
(11)

for \((i,j)=(1,2),(2,1)\). Now, this received signal is combined with Eq. (6) for \(j=2\) and Eq. (8) for \(j=1\) as the direct received signal is available. Performing the maximal ratio combining (MRC), the resultant end-to-end SNR from \(U_{i}\) to \(U_{j}\) can be obtained.

2.6 Multiple relay-assisted FSO link

To further improve the receiver sensitivity and achievable rate of 3T-ANC-based FSO link, one can use more than one relay nodes between two source nodes. A two-way relay FSO channel with n relay nodes is shown in Fig. 7. The combination of FSO link with multi-relay networks is termed as co-operative FSO (C-FSO) communication system which can be a promising technique to achieve the power reduction objective, since it could be used to enhance the system throughput and increase the coverage area without increasing transmission power.

Fig. 7
figure 7

C-FSO link with multiple relay

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3 System and channel model

3.1 Architecture of OFDM transmitter and receiver

The transmitter (Tx) and receiver (Rx) architecture are shown in Figs. 8 and 9. At the transmitter, after inverse fast Fourier transform (IFFT) and cyclic prefix (CP) addition, the time domain RF-OFDM signal is obtained. The CP addition helps to remove intercarrier interference and intersymbol interference. The optical OFDM signal is obtained, after parallel-to-serial (P/S) and digital-to-analog converter (DAC), using LD and Mach–Zehnder modulator (MZM) at optical transmitter [19]. This optical signal is transmitted through the FSO channel using FSO transmit antenna. At the receiver, the optical signal is converted into RF signal using photodetector [51]. The received signal is sampled with analog-to-digital converter (ADC). After S/P, fast Fourier transform (FFT), 4-QAM demodulation and finally P/S converter, the transmitted signal is recovered back. The orthogonality allows the use of FFT to reconstruct the transmitted signal at the receiver side and requires simple equalization [18].

Fig. 8
figure 8

Block diagram of OFDM transmitter architecture [17]

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Fig. 9
figure 9

Block diagram of OFDM receiver architecture [17]

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The time domain RF-OFDM signal is defined as

$$\begin{aligned} \text {S}_{j}(t)= \sqrt{P} \sum _{k=0}^{N_{\mathrm{sc}}-1} \text {s}_{j}(k) e^{j2\pi f_{k}t}; \quad j=1,2. \end{aligned}$$
(12)

where \(\text {S}_{j}(t)\) is the OFDM signal of user \(U_{j}\), \(P(=\frac{E_{s}}{T_{c}N_{\mathrm{sc}}})\) is the power coefficient, \(N_{\mathrm {sc}}\) is the total number of subcarriers. \(E_{s}\) is the data-modulated symbol energy, \(T_{c}\) is the sampling period, \(f_{k}\) should be equal to \(\frac{k-1}{T_{s}}\) to maintain the orthogonality between subcarriers, \(T_{s}\) is OFDM symbol duration and \(s_{k}\) is data symbol of kth subcarrier.

The OFDM signal \(\text {S}_{j}(t)\) is used to modulate the optical intensity of LD and transmitted through FSO channel. The laser optical power output from LD is given as [11, 52]

$$\begin{aligned} P_{\mathrm{T}}(t)= P_{\mathrm{av}} n_\mathrm{T} G_\mathrm{T} \times \left[ 1+m~\text {S}_{j}(t)\right] \mathrm{cos}(2\pi \nu t). \end{aligned}$$
(13)

where \(P_{\mathrm{av}}\) is the average transmitted optical power, \(G_\mathrm{T}\) is the transmitter antenna gain, \(n_\mathrm{T}\) is the optical transmitter efficiency, \(\nu \) is the optical carrier frequency and m is the modulation index.

The Friis transmission equation is used to define the received optical power by PD diode and can be given as [51]

$$\begin{aligned} P_\mathrm{{R}}(t)= P_\mathrm{{T}}(t)n_\mathrm{{R}}G_\mathrm{{R}}\left( \frac{\lambda }{4\pi L} \right) ^{2}\times 10^{(- \delta L /10)}. \end{aligned}$$
(14)

where \(n_\mathrm{{R}}\) is the optical receiver efficiency, \(G_\mathrm{{R}}\left( =\left( \frac{\pi D_\mathrm{{r}}}{\lambda }\right) ^{2}\right) \) is the gain of receiver antenna and \(D_\mathrm{{r}}\) is the receiver collecting lens aperture diameter.

The diode current can be obtained by putting Eq. (13) into Eq. (14) and is given as [11, 52]

Fig. 10
figure 10

FSO channel model

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$$\begin{aligned} i(t) = \mathfrak {R}G n_\mathrm{{T}}n_{\mathrm{R}}G_\mathrm{{T}}G_\mathrm{{R}}P_{\text {T-av}} \left( \frac{\lambda }{4\pi L} \right) ^{2} \times 10^{(- \delta L /10)}\nonumber \\ \times \left[ 1 + m~S(t)\right] \mathrm{cos}(2 \pi \nu t) + n_\mathrm{{opt}}. \end{aligned}$$
(15)

where \(\mathfrak {R}\) is the photodetector responsivity, G is the optical amplifier gain, \(n_\mathrm{{opt}}\) is the dominant noise present in the FSO communication laser link which is combination of relative intensity noise, thermal noise, and shot noise [53].

The bandwidth for OFDM signal is given as [19] \(B_{\mathrm{OFDM}}= \frac{2}{T_{s}} + \frac{N_{\mathrm{sc}}-1}{t_{s}}\), where \(t_{s}\) is observation time of IFFT window. For a data rate of 1 Gb/s, \(\frac{1}{9}\) overhead of CP, 110 data, and 18 pilot subcarriers, the required bandwidth of OFDM signal per user is 0.586 GHz. For TDM case, the required bandwidth, with 1 Gb/s data rate, is approximately 1 GHz. Thus, compared to TDM system, there is 41.4\(\%\) reduction in required bandwidth with OFDM system.

3.2 FSO channel characteristics

Figure 10 shows the statistical FSO channel model, where the laser signal transmission is affected by atmospheric losses (due to show, rain, fog etc.,), pointing losses, atmospheric turbulence, background noise or ambient light, beam divergence, and optical/window losses. The forward error correcting codes (turbo codes, LDPC), high transmit power, adaptive optics, MIMO, multi-beam hybrid FSO/RF system, and optical/electrical filtering help to mitigate adverse effect due to these factors. The eye safety due to laser radiation in FSO link is another challenge that can cause damage to eye and skin. The system can be within eye safety limit with operating it 1550 nm wavelength range and/or using class-1 laser.

The atmospheric turbulent channel, where the channel fading coefficient is multiplied by optical signal, can be characterized by multiplicative noise and defined as

$$\begin{aligned} R = hS+n. \end{aligned}$$
(16)

where R and S are the received and transmitted signals, respectively, h is the channel fading coefficient and n is additive white Gaussian noise (AWGN) with \(\sigma ^{2}\) variance.

The channel fading coefficient h consists the effect of varying weather condition such as rain, fog, snow, gases, etc., with creates atmospheric attenuation \((h_{a})\), and varying refractive index of air i.e., atmospheric turbulence which creates scintillation \((h_{s})\). Thus, \(h = h_{a}h_{s}\) and the Beer Lambert law is used to calculate the value of \(h_{a}(=e^{-\delta L})\) [54], where \(\delta \) is atmospheric attenuation coefficient defined by Kim model [55].

In modeling of optical channel, the Gamma–Gamma distribution is used for accurate estimation of probability density function (pdf) of the random signal [54], where the channel state \(h_{s}\) described as

$$\begin{aligned} f_{h_{s}}(h_{s}) = \frac{2(\alpha \beta )^{\frac{\alpha +\beta }{2}}}{\Gamma (\alpha )\Gamma (\beta )}h_{s}^{\frac{\alpha +\beta }{2}-1}K_{\alpha -\beta } \left( 2\sqrt{\alpha \beta h_{s}}\right) . \end{aligned}$$
(17)

where \(h_{s}>0\), \(K_{n}(.)\) is the second kind nth-ordered modified Bessel function, \(\Gamma (.)\) is the Gamma function, \(\alpha \) and \(\beta \) are the scintillation parameters defined for spherical wave as [54]

$$\begin{aligned} \alpha = \left[ \mathrm{exp}\left\{ \frac{0.49\beta _{0}^{2}}{\left( 1+0.56\beta _{0}^{12/5}\right) ^{7/6}}\right\} -1 \right] ^{-1}, \end{aligned}$$
(18)
$$\begin{aligned} \beta = \left[ \mathrm{exp}\left\{ \frac{0.51\beta _{0}^{2}}{\left( 1+0.69\beta _{0}^{12/5} \right) ^{5/6}}\right\} -1 \right] ^{-1}. \end{aligned}$$
(19)

where \(\beta _{0}^{2}\left( =0.5 C_{n}^{2}k^{7/6}L^{11/6}\right) \) is the Rytov variance for spherical wave, \(C_{n}^{2}\left( m^{-2/3} \right) \) is the refractive index structure parameter and \(k=2\pi /\lambda \) is the optical wave number.

The pdf of channel state h, with the deterministic nature of atmospheric attenuation \(h_{a}\), is given as [54]

$$\begin{aligned} f_{h}(h)=\left| \frac{\mathrm{d}}{\mathrm{d}h}\left( \frac{h}{h_{a}} \right) \right| f_{h_{s}}\left( \frac{h}{h_{a}}\right) . \end{aligned}$$
(20)

According to [56, Eq. (03.04.26.0008.01)], the modified Bessel function (in Eq. (17)) in terms of MeijerG-function is \(K_{n}(X)=\frac{1}{2}G_{0,2}^{2,0}\left( \frac{X^{2}}{4}\left| _{\frac{n}{2},-\frac{n}{2}}^{~~~ -} \right. \right) \). Thus, from Eqs. (17) and (20), the pdf of channel state h is defined as

$$\begin{aligned} f_{h}(h) = \frac{(\alpha \beta )^{\frac{\alpha +\beta }{2}}h^{\frac{\alpha +\beta }{2}-1}}{\Gamma (\alpha )\Gamma (\beta )h_{a}^{\frac{\alpha +\beta }{2}}}G_{0,2}^{2,0}\left( \alpha \beta \frac{h}{h_{a}} \left| _{\frac{\alpha -\beta }{2},\frac{\beta -\alpha }{2}}^{~~~~~-~~}\right. \right) . \end{aligned}$$
(21)

4 Achievable rate and bit error rate (BER)

4.1 Achievable rate

Let \(C_{j\longrightarrow N}\) be the uplink information capacity from node j to relay N and \(C_{N\longrightarrow j}\) be the downlink information capacity from relay N to node j. Thus, information capacity is given as

$$\begin{aligned} C_{j\longrightarrow N}&=\frac{1}{2}\text {log}_{2}\left( 1+\frac{P_{\mathrm{R}_{j \longrightarrow N}}}{\sigma ^{2}}\right) ,\nonumber \\ C_{N\longrightarrow j}&=\frac{1}{2}\text {log}_{2}\left( 1+\frac{P_{\mathrm{R}_{N \longrightarrow j}}}{\sigma ^{2}}\right) . \end{aligned}$$
(22)

where \(P_{\mathrm{R}_{j \longrightarrow N}}\) is the received power by relay N from node j and \(P_{\mathrm{R}_{N \longrightarrow j}}\) is the received power by node j from relay N, for \(j=1,2\).

Let \(t_\mathrm{{MA}}\ge 0\) be the fraction of time used in MA time slot during which the relay receives from user \(U_{j}\) and \(t_\mathrm{{BC}}=1-t_\mathrm{{MA}}\) the fraction of time used in BC time slot during which the relay transmit to user \(U_{j}\). The information rate from user \(U_{1}\) to user \(U_{2}\) and from user \(U_{2}\) to user \(U_{1}\) can be given as

(23)

To achieve symmetry for achievable rates in both direction, i.e., , homogeneous scenario is considered in which the same transmit power with same channel gains is used by both users and relay i.e., \(P_{j\longrightarrow N}=P_{N\longrightarrow j}=P_{N}=P\).

In LoS scheme, two time slots are needed with same amount of time in each transmission. Thus, the achievable rate can be given as [37]

(24)

With LoS scheme, this is an upper bound on achievable rates.

In TS scheme with link-by-link channel coding, four time slots are needed with same amount of time in each transmission. Thus, the quarter of the airtime is used in each transmission, and the achievable rate can be given as

(25)

In DNC, each transmission requires three time slots of same amount. Thus, the achievable rate can be given as

(26)

In 3T-ANC, each transmission requires three time slots of same amount. Thus, after performing maximal ratio combining (MRC) technique, SNR from \(U_{i}\) to \(U_{j}\) can be given as [49].

$$\begin{aligned} \mathrm{SNR}_{i\longrightarrow j}^{\text {3T-ANC}}&=P_{i\longrightarrow j}\dfrac{\left| h_{d}\right| ^{2} }{\sigma ^{2}}+\dfrac{G_{3}^{2}P_{i\longrightarrow N}\left| h_{i}\right| ^{2} \left| h_{j}\right| ^{2}}{G_{3}^{2}\left| h_{j}\right| ^{2}\sigma ^{2}+\sigma ^{2}}\nonumber \\&=P_{i\longrightarrow j}\alpha _{d}+\dfrac{G_{3}^{2}P_{i\longrightarrow N}\left| h_{i}\right| ^{2} \left| h_{j}\right| ^{2}}{G_{3}^{2}\left| h_{j}\right| ^{2}\sigma ^{2}+\sigma ^{2}}. \end{aligned}$$
(27)

where \(\alpha _{i}\left( =\frac{\left| h_{i}\right| ^{2}}{\sigma ^{2}}\right) \) is the normalized SNR of link i,

\(G_{3}\left( =\sqrt{\frac{P_{N}}{P_{i\longrightarrow N}\left| h_{i}\right| ^{2}+P_{j\longrightarrow N}\left| h_{j}\right| ^{2}+\sigma ^{2}}} \right) \) is the amplification gain factor to make the relay power \(P_{N}\).

After putting the value of \(G_{3}\) in Eq. (26), the end-to-end SNR is given as

$$\begin{aligned}&\mathrm{SNR}_{i\longrightarrow j}^{\text {3T-ANC}} =P_{i\longrightarrow j}\alpha _{d} ~+ ~\left\{ \dfrac{G_{3}^{2}P_{i\longrightarrow N}\left| h_{i}\right| ^{2} \left| h_{j}\right| ^{2}}{G_{3}^{2}\left| h_{j}\right| ^{2}\sigma ^{2}+\sigma ^{2}}\right\} \nonumber \\&\quad = P_{i\longrightarrow j}\alpha _{d}\nonumber \\&\qquad +\,\left\{ \dfrac{\left( \dfrac{P_{N}}{P_{i\longrightarrow N}\left| h_{i}\right| ^{2}+P_{j\longrightarrow N}\left| h_{j}\right| ^{2}+\sigma ^{2}}\right) P_{i\longrightarrow N}\left| h_{i}\right| ^{2} \left| h_{j}\right| ^{2}}{\left( \dfrac{P_{N}}{P_{i\longrightarrow N}\left| h_{i}\right| ^{2}+P_{j\longrightarrow N}\left| h_{j}\right| ^{2}+\sigma ^{2}}\right) \left| h_{j}\right| ^{2}\sigma ^{2}+\sigma ^{2}}\right\} \nonumber \\&\quad =P_{i\longrightarrow j}\alpha _{d} +\left\{ \dfrac{P_{i\longrightarrow N}P_{N}\alpha _{i}\alpha _{j}}{P_{i\longrightarrow N}\alpha _{i}+\left( P_{j\longrightarrow N}+P_{i\longrightarrow N}\right) \alpha _{j} +1}\right\} \nonumber \\&\quad =P+\dfrac{P^{2}}{3P+1}. \end{aligned}$$
(28)

The achievable rate can be given as

(29)

In ANC scheme where a direct LoS is not available, end-to-end SNR can be obtained by letting \(\alpha _{d}=0\) in Eq. (27). Thus, the SNR from \(U_{i}\) to \(U_{j}\) for ANC scheme can be given as [37, 49]

$$\begin{aligned} \mathrm{SNR}_{i\longrightarrow j}^{\mathrm{ANC}}= & {} \dfrac{P_{i\longrightarrow N}P_{N}\alpha _{i}\alpha _{j}}{P_{i\longrightarrow N}\alpha _{i}+\left( P_{j\longrightarrow N}+P_{i\longrightarrow N}\right) \alpha _{j} +1}\nonumber \\= & {} \dfrac{P^{2}}{3P+1}. \end{aligned}$$
(30)

As two time slots are required in ANC scheme, the achievable rate can be given as [37]

(31)

In PLNC scheme, two time slots are required with the same amount of time in each transmission. Thus, the achievable rate can be given as [37]

(32)

4.2 Bit-error-rate (BER) formulation

The average received electrical SNR per subcarrier \((\gamma )\) is given as [16]

$$\begin{aligned} \gamma&= \frac{\left( \mathfrak {R}m G G_{e} n_{\mathrm{T}}n_{\mathrm{R}}G_{\mathrm{T}}G_{\mathrm{R}}P_{\text {T-av}} \left( \frac{\lambda }{4\pi L} \right) ^{2}\times 10^{(- \delta L /10)} \right) ^{2}}{\langle i_\mathrm{{shot}}^2\rangle +\langle i_\mathrm{{ASE}}^2\rangle +\langle i_\mathrm{{DB}}^2\rangle +\langle i_\mathrm{{DS}}^2\rangle +\langle i_\mathrm{{Th}}^2\rangle +\langle i_\mathrm{{RIN}}^2\rangle }. \end{aligned}$$
(33)

where \(G_{e}\) is the receiver amplifier gain, \(\langle i_\mathrm{{shot}}^2\rangle \left( = 2eI_\mathrm{{ph}}B_{e}\right) \) is the receiver-generated mean-square shot noise, \(I_\mathrm{{ph}}\) is the primary photocurrent, e is electron charge and \(B_{e}\) is the receiver electrical bandwidth, \(\langle i_\mathrm{{ASE}}^2\rangle \) is the OA-generated mean-square shot noise which is the sum of shot-amplified spontaneous emission noise \(\langle i_{\text {shot-ASE}}^2\rangle \) and the signal-amplified spontaneous emission noise \(\langle i_{\text {S-ASE}}^2\rangle \), \(\langle i_{\mathrm{DB}}^2\rangle \left( = 2eI_{d}G_{e}B_{e}\right) \) is the mean-square value of bulk dark current noise with dark current \(I_{d}\), \(\langle i_{\mathrm{DS}}^2\rangle \left( = 2eI_{l}G_{e}B_{e}\right) \) is the mean-square value of surface dark current noise with surface leakage current \(I_{l}\), \(\langle i_\mathrm{{Th}}^2\rangle \left( = 4 G_{e}^{2} K_{b} T B_{e} F_{n}/R_{L}\right) \) is the mean-square value of thermal noise with receiver load \(R_{L}\) is receiver load, Boltzmann constant \(K_{b}\), noise figure \(F_{n}\), and absolute temperature T, \(\langle i_\mathrm{{RIN}}^2\rangle \) \((=\mathrm{RIN} I_\mathrm{{ph}}^{2}B_{e})\) is mean-square relative intensity noise with relative intensity noise \(\mathrm{RIN}(=-150~ \mathrm{dB/Hz})\).

The performance of any communication system can be analyzed from the important metric known as bit error rate (BER) which can be defined as [54, 57]

$$\begin{aligned}&\text {BER} =\int _{0}^{\infty }P_{e}(h)f_{h}(h)\mathrm{d}h\nonumber \\&\quad = \int _{0}^{\infty }P_{e}(h)\frac{(\alpha \beta )^{\frac{\alpha +\beta }{2}}h^{\frac{\alpha +\beta }{2}-1}}{\Gamma (\alpha )\Gamma (\beta )h_{a}^{\frac{\alpha +\beta }{2}}}G_{0,2}^{2,0}\left( \alpha \beta \frac{h}{h_{a}} \left| _{\frac{\alpha -\beta }{2},\frac{\beta -\alpha }{2}}^{~~~~~-~~}\right. \right) \mathrm{d}h .\nonumber \\ \end{aligned}$$
(34)

where \(P_{e}(h)\) is the conditional probability of error and for OFDM case it is given as [17]

$$\begin{aligned}&P_{e}(h)= \frac{2}{\text {log}_{2}(M)}\nonumber \\&\quad \times \left[ \left( 1-\frac{1}{\sqrt{M}} \right) \mathrm{erfc}\left( \sqrt{\frac{3\text {log}_{2}(M)}{\left( M-1\right) }\frac{\gamma h^{2}}{2}}\right) \right] .\nonumber \\ \end{aligned}$$
(35)

where \(\gamma h^{2}\) is the received instantaneous SNR, \(\gamma \) is the average end-to-end SNR.

According to [56, Eq. (06.27.26.0006.01)], the complementary error function in the form of MeijerG-function is \(\text{ erfc }\left( \sqrt{{X}}\right) =\frac{1}{\sqrt{\pi }}G_{1,2}^{2,0}\left( X\left| _{0,1/2}^{~~~1} \right. \right) \). Thus, The BER expression for M-QAM modulated OFDM-FSO link is defined as [17]

$$\begin{aligned} \mathrm {BER}= & {} {} \int _{0}^{\infty }P_{e}(h)\frac{(\alpha \beta )^{\frac{\alpha +\beta }{2}}h^{\frac{\alpha +\beta }{2}-1}}{\Gamma (\alpha )\Gamma (\beta )h_{a}^{\frac{\alpha +\beta }{2}}}G_{0,2}^{2,0}\left( \alpha \beta \frac{h}{h_{a}} \left| _{\frac{\alpha -\beta }{2},\frac{\beta -\alpha }{2}}^{~~~~~-~~}\right. \right) \mathrm{d}h \nonumber \\&=\frac{2\left( 1-\frac{1}{\sqrt{M}} \right) }{\sqrt{\pi }\text {log}_{2}(M)}\int _{0}^{\infty }G_{1,2}^{2,0}\left( \frac{3\text {log}_{2}(M)}{\left( M-1\right) }\frac{\gamma h^2}{2} \left| _{0,1/2}^{~~~-~~}\right. \right) \nonumber \\&\quad \times \,\frac{(\alpha \beta )^{\frac{\alpha +\beta }{2}}h^{\frac{\alpha +\beta }{2}-1}}{\Gamma (\alpha )\Gamma (\beta )h_{a}^{\frac{\alpha +\beta }{2}}}G_{0,2}^{2,0}\left( \alpha \beta \frac{h}{h_{a}} \left| _{\frac{\alpha -\beta }{2},\frac{\beta -\alpha }{2}}^{~~~~~-~~}\right. \right) \mathrm{d}h.\nonumber \\ \end{aligned}$$
(36)

Let, \(\left( \frac{\alpha \beta }{h_{a}} \right) =\omega \) and \(\frac{\alpha +\beta }{2} = \varpi \). The product of two G-functions is also Meijer G function [58, Eq. (21)]. So, Eq. (36) is given as [17]

$$\begin{aligned} \mathrm{BER}= & {} \frac{\left( 1-\frac{1}{\sqrt{M}} \right) }{\text {log}_{2}(M)} \times \frac{2^{\alpha + \beta }}{\pi ^{\frac{3}{2}} \Gamma (\alpha )\Gamma (\beta ) } \nonumber \\&\quad \times \,G_{5,2}^{2,4}\left( \frac{\frac{3\text {log}_{2}(M)}{(M-1)}8 \gamma h_{a}^{2}}{\left( \alpha \beta \right) ^2}\left| _{~~~~~~~~~0,{\frac{1}{2}}}^{\frac{1-\alpha }{2},1-\frac{\alpha }{2},\frac{1-\beta }{2},1-\frac{\beta }{2}, 1}\right. \right) .\nonumber \\ \end{aligned}$$
(37)

5 Results and discussion

To analyze the performance of LoS and NLoS-FSO communication link, the value of different parameters is shown in Table 1. The achievable rates of various schemes versus SNR with ACO-OFDM technique is shown in Fig. 11. As the SNR increases, the achievable rate gap between PLNC and LoS scheme decreases quickly. From Eqs. (32) and (24), it can be seen that . At low SNR, there is an noticeable gap between and which indicate that PLNC may not be the optimal scheme for low SNR. It is also clear that the performance of DNC scheme is not as good as that of PLNC scheme. From Eqs. (26) and (24), it can be seen that . Thus, the gap between achievable rates is 33\(\%\) across all SNR. In case of TS and from Eqs. (25) and (24), it can be seen that . Thus, the gap between achievable rates is 50\(\%\) across all SNR. From the above discussion, it is clear that there is 100 and 33\(\%\) throughput improvement with PLNC and DNC schemes, respectively compared to TS. ANC scheme does not give improved performance, as link-by-link channel-coded PLNC scheme gives, because of the noise is also amplified and transmitted to both the users node. 3T-ANC scheme gives improved performance compared to NLoS-TS, DNC, PLNC, ANC, and LoS scheme in both high- and low-SNR condition because, in addition with relay-assisted signal, there is direct signal also available at each node resulting into diversity gain. With MRC at each node, we can get the advantage of diversity and performance improvement as a result. With addition of diversity gain in performance 3T-ANC has no O/E/O conversion interface and electrical interface at the relay resulting in reduced cost and complexity. The achievable rate for PLNC, ANC, 3T-ANC schemes with ACO-OFDM technique are shown in Table 2. It is clear from this table that the performance gap for 3T-ANC relative to the PLNC scheme ranges from 38.4\(\%\) at 15 dB to 36.5\(\%\) at 20 dB SNR, respectively, and that for ANC ranges from 96.5\(\%\) at 15 dB to 69\(\%\) at 20 dB SNR, respectively.

Table 1 Parameters and their value
Full size table
Fig. 11
figure 11

Achievable rate versus SNR with various NC schemes

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The BER as a function of received power with ACO-OFDM technique for various network coding schemes is shown in Fig. 12. The performance of relay-assisted NLoS link, with PLNC scheme, is same as that of point-to-point LoS link. At BER of \(10^{-3}\), the received power with relay-assisted PLNC scheme is \({-}\)20 dBm. For relay-assisted NLoS link with ANC scheme, the received power is \({-}\)18.9 dBm and 1.1 dB degradation in receiver sensitivity as compared to PLNC scheme as a result. In the case of availability of both links, i.e., NLoS and LoS link, the receiver sensitivity is \({-}\)26 dBm with 3T-ANC scheme. Thus there is 7.1 and 6 dB improvement in receiver sensitivity with 3T-ANC compared to ANC and PLNC scheme, respectively.

The comparison between the error performance of ACO-OFDM and 7 dB-biased DCO-OFDM FSO link is shown in Fig. 13. The results show that with 3T-ANC scheme, the ACO-OFDM technique is also power-efficient scheme where additional DC bias is not required to make the RF-OFDM signal positive. At BER of \(10^{-3}\) with ACO-OFDM technique, there is about 2.7 and 3 dB improvement in receiver sensitivity in LoS and NLoS+LoS with 3T-ANC scheme compared to DCO-OFDM technique. Thus, the optical power budget for ACO-OFDM is higher compared to DCO-OFDM technique.

Table 2 Achievable rate with various NC scheme
Full size table
Fig. 12
figure 12

BER versus received power with various NC schemes

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Fig. 13
figure 13

BER versus received power with ACO/DCO-OFDM FSO link

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Fig. 14
figure 14

BER versus received power of multiple relayed 3T-ANC FSO link

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In Figs. 14 and 15 the performance of C-FSO link with more than one relay nodes is analyzed. The error performance of C-FSO link is improved with two relay nodes. One can observe that with increasing number of relay there is improvement in performance. However, the improvement is marginal for more than two number of relays, because each relay amplify the noise also. The increased number of relays can also increase the cost and complexity of the relay-assisted C-FSO link.

Fig. 15
figure 15

Achievable rate versus SNR with multiple relay

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From the above discussion it is clear that relay-assisted NLoS link can make a continuous information exchange, even if LoS is breakdown between two users, but at the cost of reduced achievable rate and error performance. PLNC gives better performance in NLoS condition but with expensive OER relay which require O/E/O conversion resulting in increased complexity at relay. ANC allow the use of cost-effective OR relay where O/E/O is not required resulting in reduced complexity and energy saving at the relay but performance is degraded since the noise is also amplified at the relay. On the other hand, when NLoS as well as direct LoS links are available, ANC with three time slot known as 3T-ANC gives improved performance compared to PLNC with energy saving and reduced cost and complexity. We have also shown that the optical power budget for ACO-OFDM is higher compared to DCO-OFDM technique. To further improve the performance C-FSO link with multiple relay nodes can be used, but with more than two relay nodes, there is marginal improvement as relay also amplify the noise in the link.

6 Conclusion

OFDM technique gives improved performance in FSO link because of its dispersion tolerance capacity, efficient utilization of electrical bandwidth, ease of channel and phase estimation, and scalability to higher-order modulation. The ACO-OFDM technique is more optical power efficient compared to DCO-OFDM as ACO-OFDM signal eliminates the need of an additional DC bias to make it unipolar. FSO link requires LoS condition between two users and in NLoS condition communication between them will breakdown. In this situation, to continue information exchange between two users with increased achievable rate and improved error performance, we have proposed the use of OR relay-assisted link with 3T-ANC scheme. Analytical results shows that PLNC gives better performance in NLoS condition but with expensive OER relay. ANC allow the use of cost-effective energy-saving relay but at the cost of performance degradation as noise is also amplified at the relay. On the other hand, when NLoS as well as direct LoS links are available, 3T-ANC gives improved performance compared to PLNC with energy saving, reduced, cost and complexity at the relay node. The C-FSO link with multiple relay nodes can be used to further improve the error performance of the link, but with more than two relay nodes, there is marginal improvement as relay also amplify the noise in the link.