Performance Evaluation and Enhancement of M-Ary DPPM Modulation for WDM-PON/FSO Systems Impaired by Atmospheric Turbulence, Interchannel Crosstalk, and ASE Noise

This paper analyzes and enhances the performance of moment generating function techniques, notably the Chernoff bound (CB) and modified Chernoff bound (MCB), is used to improve the bit-error-rate (BER) performance of an optically pre-amplified for the wavelength division multiplexing (WDM) based on the passive optical network (PON) free-space optical (FSO) communications in the presence of both atmospheric turbulence (AT), amplified spontaneous emission (ASE) noise, and interchannel crosstalk. In the absence of AT and ASE at a data rate of 2.5 Gbps on the 1550 nm wavelength, digital pulse-position modulation (DPPM) systems with coding level (M) of 2 provide about 2.9 dB improvement in average power over at a BER of (depending on the turbulence level) compared with an equivalent on-off keying (OOK) non-return-to-zero (NRZ) in the WDM-PON/FSO system while maintaining minimum bandwidth expansion to leverage the extended reach and enhanced user capacity and considered as a good solution to the bandwidth requirement for future access networks, with potential for higher data rate, improved data security. The receiver sensitivities of M-ary DPPM about 51.4 dBm ( ~ 21.9 photons/bit) (CB), and 51.5 dBm (21.4 photons / bit, MCB) can be achieved, which implies an improvement when compared with an OOK-NRZ system ( ~ 38 photons/bit) in the non-turbulent atmospheric condition. M-ary DPPM retains its sensitivity improvement over OOK even in the existence of crosstalk while predicting a lower power penalty of about 0.2–3.0 dB for weak turbulence at low coding level (M) 2 in WDM systems.


Introduction
One of the popular modulation schemes widely applied in free-space optical (FSO) communications is digital pulse-position modulation (DPPM) [1,2]. This scheme is well known to be attractive in different FSO environments including inter-satellite, atmospheric, and indoor wireless channels [1 3]. Apart from the power efficiency advantage, there is the additional advantage that there is no need to set and track a decision threshold in many DPPM systems [4,5]. DPPM has been proposed and intensively investigated for optical fiber systems [5, 6 8], however, DPPM is particularly attractive in an FSO channel relative to an optical fiber channel because the FSO channel is dispersion free [1,5,8]. The advantages of DPPM however do come at the expense of bandwidth expansion, but with a moderately low coding level, DPPM can combine with most multiplexing/multiple access techniques without considerable bandwidth expansion [8,9]. Also, some hybrids of DPPM with other modulation schemes such as phase-shift keying and frequency-shift keying have been proposed for point-to-point fiber communication systems [5,8,9] and could be alternative options to improve the DPPM bandwidth efficiency. With the availability of bandwidth-intensive services such as video on demand (VoD), Internet protocol television (IPTV), IP telephony, and interactive gaming/videoconferencing, there has been a rapid rise in bandwidth demand from users [5, 10 12]. In response to this increase in bandwidth demand, wavelength division multiplexing (WDM) and dense WDM (DWDM) systems have been investigated and/or deployed for optical fiber, atmospheric and indoor wireless optical networks [8, 9, 13 15]. WDM could also be applied in multiple user access networks. For example, WDM passive optical network (PON) is generally considered as a good solution to the bandwidth requirement for future access networks, with the potential for higher data rate, improved data security, and longer reach [8,9,13,16]. WDM system has application in both optical fiber and FSO systems [8, 9, 17 19]. With WDM PON, fixed wavelengths are assigned to each optical network unit (ONU), thus more fully exploiting the hightransmission bandwidth available in the optical domain and avoiding the synchronization and threshold acquisition required in the burst mode upstream of time-division multiplexing/time division multiple access (TDM/TDMA) systems [5,8,9,20,21]. The application of optical pre-amplification to overcome the impact of receiver thermal noise is one way of improving the receiver sensitivity of FSO systems. Aside from the optical gain, the optical pre-amplifier also generates amplified spontaneous emission (ASE) noise which, in turn, generates additional beat noises (spontaneous-spontaneous and signalspontaneous) in the electrical domain at the receiver. The overall electrical domain noise is non-Gaussian, although it has often been approximated as Gaussian in probability density functions (pdfs) used for describing binary signals dominated by ASE noise [22,23]. The moment generating function (MGF) represents a convenient statistical way of describing the signal plus ASE noise in a system employing an optically pre-amplified receiver while Chernoff bound (CB) and modified Chernoff bound (MCB) are techniques that use this description to obtain upper bounds upon the bit-error-rate (BER) [5, 8,9,24,25]. The reminder paper is organized as follows: Section 2 describes the system design and modeling for optically pre-amplified WDM DPPM in the FSO communication system. Section 3 introduces the biterror-rate (BER) analysis. Section 4 discusses the BER evaluation for hybrid WDM-PON /FSO systems Section 5 presents the calculation results and discussion. Section 6 summarizes our observations and discusses potential future extensions of the research.

System description and model
In the DPPM signal transmission scheme, a frame of duration equal to is divided into n equal time slots of length t s n ⁄ , where M is the coding level and equal to the number of data bits transmitted per DPPM frame, ⁄ is the equivalent on-off keying non-return-to-zero (OOK NRZ) bit period and is the data rate [9]. The maximum likelihood detection receiver is preferred for the best performance in DPPM FSO systems [1,8,9]. The decision circuitry is required to integrate over each slot in a frame and the decision is made by comparing the results and selecting the slot with the largest signal as the pulse position [4,5]. Therefore faster electronic processing speed is required compared to the threshold method used in OOK systems. A general WDM DPPM system that might require evaluation of crosstalk impact could include a fiber or free space (or hybrid) system and maybe in a point-to-point, multipoint-to-point, or PON configuration [9]. Different sources and levels of crosstalk could arise in a DWDM DPPM system depending on the link configuration. In most point-to-point systems with all signal wavelengths originating from the same place, the major source of crosstalk is imperfect optical band-pass filter (OBPF)/demultiplexer (demux) rejection and since most realistic systems will employ OBPF/demux with good rejection ratio [5, 8 10]. But in multipoint-to-point links such as upstream transmission in hybrid fiber free-space optical (HFFSO) systems as shown in Fig. 1 (a) or in PON where signals could experience asymmetric splitting loss, fiber and/or FSO attenuation, beam spreading, and coupling loss, signals at different wavelengths will arrive at the OBPF/demux at different power levels. A generic system structure that could be easily adapted to all the different scenarios above is shown in Fig. 1 (b) [9]. DPPM signals from different wavelengths are multiplexed and transmitted over an FSO link to a receiving lens (not shown They could also in principle arise from different physical locations as long as they can be collected and coupled effectively into the optical amplifier (OA) which is done by collimating them into a short fiber length at the amplifier input before being demultiplexed into different wavelengths for detection by a positive-intrinsic-negative (PIN) photodiode [9]. The optical preamplifier is just treated as a linear gain block generating noise as in Fig. 1 (b).

Figure 1
Structure for optically pre-amplified WDM/M-ary DPPM scheme of PON-HFFSO system: (a) Specific system architecture for crosstalk evaluation and (b) Generic receiver system. We reproduced it from our previous publication in [9].

M-ary DPPM scheme modeling
The analysis of crosstalk in a DPPM system requires some consideration to ensure that the different scenarios that could arise during frame reception are taken into account. For example, there may be an alignment of frames and slots or only slots between the signal and crosstalk (XT) as shown in Fig. 2 (a) and (b) respectively [5, 8,9]. In Fig. 2, the case where signal and crosstalk frames are aligned is referred to as frames aligned (FA) Fig. 2 (a), the case where the signal and crosstalk have only their slots aligned is referred to as only slots aligned (OSA) Fig. 2 (b) while the case where there is a misalignment between the signal and crosstalk slots is referred to as slots misaligned (SMis) Fig. 2 (c). The moment generating function (MGF) describing the random variable of the current where sig  {0, 1} depending on pulse transmitted or not, t is the duration of the crosstalk pulse overlap with the slot under consideration) for a general slot which contains ASE, possibly a signal pulse and possibly a single XT pulse (or some fraction of one) is derived using the same treatment as [5,9,24,26,27]. It is written as: where slots align with signal slots otherwise or , and for no crosstalk in the slot, . In addition, tr and are the DPPM rectangular pulse and the crosstalk pulse power respectively, both defined at the PD input, ⁄ , is the PD quantum efficiency, is lanck"s constant, ν and are the optical frequencies of the signal and crosstalk wavelengths respectively [5, 8 10], is the electron charge, o is the single polarization ASE power spectral density (PSD) at the amplifier output (and also at the PD input if demultiplexer nominal loss is neglected), and are the optical amplifier gain and noise figure respectively, is the product of spatial and temporal modes [2, 8,9], o is the demux channel optical noise bandwidth and is the number of ASE noise polarisation states. o is the ASE PSD at the photodetector at crosstalk wavelength and tr ⁄ is the signal-to-crosstalk ratio, fixed at the output of the demux.
The MGF has been modified to account for crosstalk-ASE beat noise assuming the crosstalk and the desired signal experiences the same ASE noise at the amplifier output [9,10,28]. The overall MGF including the zero-mean Gaussian thermal noise is given as [5,9] sig t s sig t s exp ( where th-is the DPPM thermal noise variance. Following [1, 2, 8 10], the means and variances of the random variables representing the integration over the slot that contains only the signal pulse, only crosstalk pulse, both signal and crosstalk pulses, and no pulses (i.e. empty slot) are derived from the overall MGF, through its first and second derivatives respectively, with s set equal to 0. They are respectively generally written as [5,9,10,29]   Given that each symbol has an equal probability of being transmitted in a slot, the probability that a symbol is successfully received in the presence of crosstalk where is the symbol error probability in the presence of crosstalk, and { } denote the number of crosstalk (of duration , or occurring in the signal pulse slot and signal frame respectively. Thus for single crosstalk case, { } while { } Following the same treatment as [1], one can write that: where represents the content of the non-signal slot ( ) and is the crosstalk overlap with the j th (empty) slot. Assuming that the random variables and are Gaussian, the For the Chernoff bound (CB) we have that the general form for random variable X and a fixed threshold is Thus P , and manipulation of this for the difference of two random variables implies that [5, 9,29] { ( ) } (7) For the modified Chernoff bound (MCB) [2, 5], ⁄ √ [5, 9,10]. Modifying this inequality for the difference of two random variables for ( ) and which both have the same thermal noise contribution then yields, [1,2,5,9,29] { ( ) } √ For the FA and OSA cases the symbol error probability in the presence of a specific crosstalk combination is written as [1,9,10] where and are the number of crosstalk of duration occurring in the signal frame and signal pulse slot respectively, if crosstalk hits signal pulse slot, otherwise t 0. Similarly, the symbol error probability in the presence of crosstalk for the slot misaligned (SMis) case is written as, where and are the number of crosstalk of duration , , occurring in the signal frame and signal pulse slot respectively, ̈ , or if crosstalk of duration or respectively hits the signal pulse slot, otherwise t 0.

Atmospheric turbulence model
Atmospheric scintillation occurs due to the temperature difference between the earth's surface and atmosphere induced refractive index changes of the air along with the optical link [8], causing rapid fluctuation of the received signal, deviation in the degree of coherence of the optical signal, and also poor BER [8,9]. These effects of turbulence, weak, moderate, and strong are characterized using the gammagamma (GG) distribution probability density function (pdf) is given as [5,8,9,29 where is the attenuation due to atmospheric turbulence for the signal (h sig ) or interfere h int , and are the effective numbers of large-scale and small-scale eddies of the scattering process, respectively, n is a modified Bessel function (second kind, order n), and is the gamma function [ 8 10]. The signal and interferer travel over physically distinct paths in the upstream and thus have uncorrelated turbulence; hence, their GG pdfs are each treated independently. The parameter and for plane-wave propagation for arbitrary aperture size are given as [5, 8 10, 29 34] Table 1 Typical parameters for characterizing weak-to saturated turbulence regimes [5, 8 10, 31, 35].

BER analysis
The number of other crosstalk combinations taking place in the signal frame increases with slot misalignment, with the specific analysis becoming upset. Considering Fig. 2 (c) with n + n = n+1. Fig. 3 below shows crosstalk plotted against the channel numbers using the WDM system. Here we determined crosstalk from the Eq. where, = number of hop, B = bit ratio of signal peak power, = detector resistance and = input power =effective adjacent and effective nonadjacent, N = number of channel, X= switch values, taken for Fig. 3, B =1, = 0.85, X = 1. It can be said that if we increase the hops number then crosstalk also increases. Another way, it can be analyzed that for a fixed number of crosstalks if we use more hops, the number of channel decrease and in another way, we can use more channels for fewer hops as seen in The overall BER in the presence of crosstalk for slots misaligned is calculated by computing up all the error contributions calculated from Eqs. (17) and (18)  As shown in Fig. 4, for values of m  100, the target BER of is achieved while the BER becomes worse for m  100, specifically for lower coding levels and higher crosstalk power. For definiteness, m 100 is used in the calculations, as seen in Fig.4, higher values of m do not show any significant change on the BER, but rather increases the computational time. The OSA case is recovered for m  1, although a higher received power than those used in Fig. 4 would be required to attain the BER of

Downstream analysis
For the downstream transmission, sig and int can be expressed as [5, 10,15] where is the responsivity of the PIN photodiode, i is the optical amplifier fixed-gain at the i-th relay (assuming that i . SNR is the electrical signal to noise ratio in the case of DPPM signaling, which is defined as [10,31] ( ) In (21) and u are receiver noise variances of signal currents in 0 and slots, which can be expressed as [15,31] * + * + where sig , int , z,r , ,r , z,c , ,c , are variances of signal shot noise, interferer shot noise, accumulated amplified background noise over multiple relays, accumulated amplified ASE noise over multiple relays (N), background noise at the destination, ASE noise resulted from the amplifier at the RN received at the destination, and receiver thermal noise, respectively [10,31].

BER evaluations for WDM-PON/FSO system
The BER is the key performance attribute commonly used for FSO communication systems analysis [5,36]. By making a GA assumption for the noise, a BER, conditioned on the instantaneous loss (or gain) state of the turbulent channel h t , is given as [5, 24] The optical signal power at the output of the optical amplifier (OA) is obtained as out in in in Now, the non-adaptive decision threshold, assumed set to a long term average received power at the PD, can be obtained by statistically averaging (31) over the atmospheric turbulence pdf and it is obtained as [2, 5, 8,9,15] i It is stressed that the method here is confined to a single wavelength system. Multiple wavelengths constitute a natural further development of this work. Under such circumstances, assuming that an OA is not to favor particular wavelength channels systematically, it will be necessary to ensure gain flatness at least in the small-signal regime. The value of i in Eq. (30) can be defined in such a way as to maintain the use of Q and adaptive thresholding and hence Eq. (32) in the non-adaptive thresholding process. In the adaptive case it varies with h t in the case of Eq. (32) it does not vary with h t . The average BER obtained by statistically averaging the conditioned BER over the turbulence pdf is given as [2,5,8,9,15] av For a non-amplified receiver system, G = 1, = 0 and then the receiver thermal noise is the dominant impairment (i.e. = = th ) [36]. The outage probability (OP) is thus obtained by integrating the joint pdf over the area that the instantaneous BER is greater than the BER target, and is written as; where is the region in in d , l space where inst d , l target. Equation (34) could be simplified further as, [15] where d l is the threshold instantaneous irradiance required to achieve a inst that is equal to the BER target and is dependent on the instantaneous irradiance of the interferer. The instantaneous BER of the system is a function of the instantaneous irradiances of the signal and interferer d , l , [5, 8,15]  i d d i l l is the desired signal plus interferer current at the decision instant in the receiver [15]. The electrical bandwidth ( e ) is set equal to ⁄ where is the data rate, R =ηq/ ν, η is the photodetector quantum efficiency, h is Planck's constant, ν is the optical frequency, q is the electron charge, is the single polarization ASE PSD, G and NF are the optical amplifier gain and noise figure respectively, is the demux channel optical noise bandwidth and m t is the number of ASE noise polarization states. The means and variances of the random variables, both are representing the integration over the slot that contains only the signal pulse, only crosstalk pulse, both signal and crosstalk pulses and no pulses (i.e. empty slot) are derived and, respectively, the general equations for the upstream transmission are derived in Eqs. ( Finally, in the last case the dependence is the following [8]: (58) The following values of the input parameters are reported in Table 2. The major challenge consists in taking integrals in the right hand sides of the Eqs. (56) and (58) properly. To find correct values for the integral limits in these equations, one should examine the dependence that is the following as written in [15].

Calculations results and discussion
The physical parameters used in the model are listed in Table 2. o is fixed by o ⁄ at the receiver with  1, i.e. assuming that the crosstalk and the accompanying ASE have been attenuated by the demultiplexer upon pairing to the desired signal photodetector. The same data rate is assumed for both crosstalk and signal [5,8]. The DPPM thermal noise variance is back-calculated using a bandwidth expansion factor such that th exp th where exp  ⁄ is the DPPM bandwidth growing factor and th = 7 A is obtained from a model of a PIN-field effect The physical parameters used in the model are listed in Table 2. o is fixed by o ⁄ at the receiver with  1, i.e. assuming that the crosstalk and the accompanying ASE have been attenuated by the demultiplexer upon pairing to the desired signal photodetector. The same data rate is assumed for both crosstalk and signal [5,8]. The DPPM thermal noise variance is back-calculated using a bandwidth expansion factor such that th exp th where exp  ⁄ is the DPPM bandwidth growing factor and th = 7 A is obtained from a model of a PIN-field effect transistor receiver with = 2.5 Gbps at BER of assuming a sensitivity of 23 dBm [5, 8 10, 36].
The demux (or OBPF) channel bandwidth is 80 GHz with 100 GHz adjacent channel spacing, this is about the same as those seen in [5,8,37,38] and will easily accommodate the slot rate of 45.7 GHz for maximum DPPM coding level of M = 7 analyzed [2]. Typical values for adjacent channel rejection ratio ranges from 20 dB to 30 dB [5, 8 10, 37 39].   9,29]. The MCB coincides with the GA at low gain, but shifts close to the CB at high gain as the ASE noise reduces the significance of the thermal noise. The GA on the other hand is seen to exceed the CB and MCB (which are upper bounds) at high gain with no crosstalk and in the presence of crosstalk. Fig. 6 shows the BER curves for CB, MCB, and GA at low gain optical amplifier (G = 8 dB) and G = 30 dB). Fig. 6 (a) shows the discrepancy between the K and the GG distribution. Fig. 6 (b) shows the BER curves for the high-gain (G = 30 dB) case using the same parameters as before to characterize the atmospheric turbulence regimes. Here the CB and MCB BER curves are almost matching, while the GA differs from both CB and MCB, even more as the turbulence strength increases [9,10]. Consideration of both Fig. 6 (a) and (b) indicates that the MCB with GG distribution is probably the most sensible approach for modeling optically pre-amplified FSO receiver in all atmospheric turbulence regimes. This is because the MCB gives a tighter bound than the CB especially when the contribution of the thermal noise is relatively high and because the GG distribution is reasonable over a whole range of turbulence conditions.    To further understand the single crosstalk system, consider Fig. 7 which shows the result of the power penalty as a function of fixed misalignment case. Each point in Fig. 7 presents the power penalty for the different fixed slot alignments (subcases) that are averaged to obtain the overall power penalty for the OSA case [5]. The penalty at = 8 corresponds to the penalty for FA. The best performance for the fixed slot alignments is attained at = 4, this is because the probability of no crosstalk impairing the signal frame is highest for such misalignment [5]. In Fig. 8, the BER curves are presented for two typical FSO optical link lengths i.e. fso = 1500 m and fso = 2000 m, using G = 30 dB, = 1 mm and M = 5 for NT with WT, MT, and ST regimes [9,10], respectively. Here it can be seen that the effect of turbulence becomes more severe for the longer optical link (recall that n is fixed, so the Rytov variance is where the length change impacts), for example, at target BER, the receiver sensitivity degrades by about 5 dB (WT), 12 dB (MT), and 8 dB (ST) as optical link length increase from 1500 m to 2000 m. Specifically, the effect of the optical link length gradually becomes less significant as the turbulence strength approaches very strong regimes [5,9,10,40] for NT condition, receiver sensitivities of about 50.25 dBm (~27.2 photons/bit) (GA), 51.4 dBm (~21.9 photons/bit) (CB) and 51.5 dBm (21.4 photons/ bit, MCB) can be achieved, which implies an improvement when compared to the fundamental limit (38 photons/bit) of non-turbulent optically pre-amplified OOK-NRZ as stated in [40] as shown in Fig. 8. The result of DPPM power penalty analyses for crosstalk M = 2 is compared with the power penalty for OOK in Fig. 9 for target BER of . DPPM predicts a reasonable penalty which is less than the OOK penalty for multiple crosstalks, even at low coding levels. The DPPM improvement in the power penalty becomes better as the number of crosstalk sources increases and as the coding level increases from M = 1 to 2. The FA is compared with OSA and simulation for M = 2 [5, 40]. Although the FA seems to overestimate the power penalty, the approximation gets better for M = 2. Also, it is computationally quicker than the other approaches and provides an upper bound for the system. In the absence of turbulence and at a data rate of 2.5 Gbps on the 1550 nm wavelength, DPPM systems with a coding level of 2 provide about 2 dB improvement in average power over an equivalent OOK NRZ system while maintaining minimum bandwidth expansion as seen in Fig. 9 [5, 8,9]. As a numerical method, it is necessary to judge and review the validity of the BERs versus average power at the optical amplifier input. Fig. 10 shows a comparison of the GA, CB, and MCB performance at high gain G = 27 dB [5, 29] and G = 30 dB [present] with a single crosstalk source and M = 2. The MCB coincides with the GA at low gain, but shifts close to the CB at high gain as the ASE noise reduces the significance of the thermal noise. The GA on the other hand is seen to exceed the CB and MCB (which are upper bounds) at high gain with no crosstalk and in the presence of crosstalk. The margin with which the GA exceeds the MCB and CB widens as the coding level and the noise equivalent bandwidth e of the DPPM receiver increases. This erratic behavior of the GA is well reported for both OOK and DPPM systems [2, 8,9,24]. In Fig. 11, the receiver sensitivity for each DPPM coding level (M = 1 6) is presented for NT with WT, and ST, respectively, using GA, CB, and MCB, G = 30.6 dB, fso =1500 m, 1 mm, 20 mm, and 50 mm, at target BER of that presented in [2]. At M = 5, G = 30.6 dB and =2.5 Gb/s, for NT condition, receiver sensitivities of about 50.53 dBm (~27.4 photons/bit) (GA), 51.49 dBm (~22 photons/bit) (CB), and 51.59 dBm (~21.5 photons/bit) (MCB) can be achieved, which implies an improvement when compared to the fundamental limit (38 photons/bit) of non-turbulent optically preamplified OOK-NRZ as stated in [9,40] as achieved in paper [2]. While the receiver sensitivity for each DPPM coding level (M = 1 6) is presented for NT with WT, and ST, respectively, using GA, CB, and MCB, G = 30 dB, fso =1500 m, 1 mm, 20 mm, and 50 mm, at target BER of [present]. Numerical results show for NT condition, receiver sensitivities of about 50.25 dBm (~27.2 photons/bit) (GA), 51.49 dBm (~22 photons/bit) (CB), and 51.56 dBm (~21.4 photons/bit) (MCB) can be achieved [present], which implies an improvement when compared to the fundamental limit (38 photons/bit) of non-turbulent optically preamplified OOK-NRZ as stated in [40]. To further reinforce this point, in Fig. 12, we show a plot of the required mean irradiance to achieve instantaneous BER performance target of and with fixed outage probability values of either or and the required mean irradiance to achieve average BER targets of and for a WDM-FSO system. From the result shown in Fig. 11, it is seen that with demux, = 35 dB and n = m ⁄ , the required mean irradiance to achieve a target instantaneous BER of with an outage probability target of is greater than the required mean irradiance to achieve a target instantaneous BER of with an outage probability target of for all FSO link length considered [5,8]. This shows that more transmitter power is required to improve OP for a given instantaneous BER target than to improve the instantaneous BER for a given OP target. From the result shown in Fig. 12 it is seen that with demux, = 35 dB a and n = m ⁄ , the required mean irradiance to achieve a target instantaneous BER of with OP target of is greater than the required mean irradiance to achieve a target instantaneous BER of with OP target of for all FSO link length considered [5,8]. This shows that more transmitter power is required to improve OP for a given instantaneous BER target than to improve the instantaneous BER for a given outage probability target. For example, at 2000 m in Fig. 12, an extra 2.66 d m cm is required to improve the instantaneous BER from to at a fixed OP of , while it takes an additional 14.23 d m cm to improve the OP from to at fixed instantaneous BER of [5,8]. Figure 13, shows the required optical power (dBm) for the upstream as a function of the refractive index structure constant ( n m ⁄ ) and interferer demux channel rejection demux, d at fso m. The result shows that for target BERs of , , and , to be met at all turbulence regimes, the system requires demultiplexer with an adjacent channel rejection greater or equal to 46 dB, 33 dB, and 17 dB, respectively. However, to meet a BER of would require high power (above 20 dB) [2,8]. With forward error correction (FEC) implemented in most recent practical systems, operation at BER of is becoming feasible and demultiplexers with 18 dB rejection are readily available [5,8]. Fig. 14 shows the power penalty (dB) for the upstream as a function of the refractive index structure constant ( n m ⁄ ) at n e m ⁄ and interferer demux channel rejection demux, d at fso m. The result shows that the power penalty for the DPPM system tends to be lower than that of the OOK system. As shown in Fig. 14, an interferer that is closer to the remote node (RN) causes more crosstalk to other users farther away, even at low ( n m ⁄ ) value. Thus, in the absence of power control, user positioning should be considered as an important design parameter. Next, Fig. 15 illustrates the BER of the proposed system using (4-pulse position modulation (PPM) versus s , for the downstream transmission with various turbulence strengths [5,8]. Within the total distance of 4 km, the BER performance constantly deteriorates when the turbulence becomes stronger (i.e., higher n ). With only one relay (N), the required s to attain BER of are 10.5 dBm, 7.5 dBm, and 4.5 dBm corresponding to n of m ⁄ , m ⁄ , and m ⁄ . With two relays, the BER performance is significantly improved compared to the case when relay (N = 1), the performance improvements when N = 2 are 8 dB, 7 dB, and 6 dB for n of m ⁄ , m ⁄ , and m ⁄ , respectively as shown in Fig. 15 [8].  (b)

Figure 9
Power penalty against signal-to-crosstalk ratio for OOK and DPPM (multiple crosstalks OSA and simulation) at BER =10 9 and OOK comparison with DPPM at M = 2.

Conclusions and future work
BER modeling for optically pre-amplified M-ary DPPM FSO process taking over atmospheric turbulence is investigated utilizing MGF-based approaches such as CB and MCB. Analyses of crosstalk for optically preamplified WDM DPPM systems are given for the GA, CB, and MCB. The FA case is found to somewhat show the worst power penalty. However, the efficiency penalty is justified by a significant reduction in calculation complexity. The approach using the OSA hypothesis calls a sensible penalty compared to the state with the FA hypothesis and hence presents a better depiction of a possible system. Furthermore, the MCB is introduced as the surest approach of evaluation as it does a higher upper bound than the CB and is further sensitive to the optical amplification, indeed still the GA is computationally quicker. The DPPM coding level M = 2 is a reasonable option for WDM DPPM free space and wireless systems because of its sensitivity improvement for a small bandwidth expansion over OOK, and when crosstalk is performing this present significantly supported to by a reduced power penalty relative to OOK. The presence of turbulence-accentuated crosstalk for the upstream transmission which somewhat reduces the relative distances between the RN and both the interferer and the desired user for a limited target BER and demultiplexer adjacent rejection ratio was established in the proceeds. BER modeling for optically pre-amplified of M-ary DPPM in the WDM FSO system operating over atmospheric turbulence is investigated using MGF-based techniques such as CB and MCB.

Compliance with ethical standards
Conflict of interest: The authors declare that there is no conflict of interest regarding the manuscript. The authors are responsible for the content and writing of this article. The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

CRediT author statement
Ebrahim E. Elsayed made substantial contributions to the design, analysis, characterization, conceptualization, methodology, software, data curation, writing-original draft, formal analysis, writing, visualization, investigation, and validation. Bedir B. Yousif participated in the conception and critical revision of the article for important intellectual content, supervision, project administration, data curation, and writing-review \& editing.
No funding for this work.

Author contributions
Ebrahim E. Elsayed made substantial contributions to the design, analysis, characterization, conceptualization, methodology, software, data curation, writing-original draft, formal analysis, writing, visualization, investigation, and validation. Bedir B. Yousif participated in the conception and critical revision of the article for important intellectual content, supervision, project administration, data curation, and writing-review \& editing.