Performance of M-PAM FSO communication systems in atmospheric turbulence based on APD detector

Haifeng Yao; Xiaolong Ni; Chunyi Chen; Bo Li; Xinmeng Zhang; Yi Liu; Shoufeng Tong; Zhi Liu; Huilin Jiang

doi:10.1364/OE.26.023819

1. Introduction

With the development of artificial intelligence and the Internet of Things, mass data transmission rates (such as 4k movies, HD aerial images, etc.) are getting faster. Compared to on-off-keying (OOK), each symbol of multi-level pulse amplitude modulation (M-PAM) represents log₂M bit. With higher spectral efficiency, M-PAM is a potential way to extend the reach of short-range links at high bit-rates. In addition, considering the cost constraints, M-PAM as one of intensity modulation and direct detection (IM/DD) is greatly appealing in optical communication [1–3].

Currently, free-space optical (FSO) communication is one of the hotspots for achieving high-speed transmission due to huge bandwidth, no electromagnetic interference (EMI), strong security and confidentiality [4, 5]. Most of the previous studies on M-PAM were based on fiber channel, see [1–3, 6–8]. Because, in the channel of FSO communication, there is the atmospheric turbulence which causes arrival angle fluctuation, beam drift, and scintillation in the process of optical propagation, owing to variations in the index of refraction caused by temperature fluctuation [9, 10]. Nevertheless, to the best of our knowledge, Hai-Han et al. have achieved over 100 m free-space link 64 Gb/s 4-PAM communication. The experiment condition can be considered “clear- air” turbulence, and the link distance is hundred-meter level. However, there was no detailed analysis about the theoretical model of 4-PAM FSO communication systems [11].

Besides, in [12], the performance of FSO systems based on the PIN photodiode have been studied. In these systems, there are amplified spontaneous emission (ASE), which dominates the receiver thermal and shot noises, and affect the communication quality of the FSO, hence the sensitivity of the FSO systems to the detector is very high. Because the sensitivity of avalanche photodiode (APD) detection is high, it has been used in the deep-space communication systems or the FSO communication systems, which has been described in [13,14]. Moreover, the literature [15] has compared the performance of PIN and APD, further indicating that the detection sensitivity of APD is high. Based on the above reasons, we employ the model of APD in M-PAM FSO communication systems. It is noteworthy that the BER expression of M-PAM in the free-space channel has been given in [16], however, the APD model was not considered, see [16, Eq. (10)]. Anshul Jaiswal et al. have deduced the average bit error rate (ABER) analytic expression of L-ary PAM through analyzing the channel model of Gamma-Gamma (G-G) distribution. This expression has limitations that there is no mathematical analysis of APD model, when estimating the BER of actual FSO communication systems, see [17, Eqs. (26)-(28)]. Therefore, it is significant theoretically and practically to analyze the BER performance of M-PAM FSO communication systems which based on APD detection model. In particular, this model can evaluate the communication performance of M-PAM in atmospheric turbulence and provide a reference for or the design of system parameter.

In this paper, we firstly propose a mathematical model for the performance of M-PAM FSO communication systems based on the APD. In the Section 2, the G-G atmospheric channel model is introduced. The BER performance model of M-PAM optical signal through atmospheric turbulence propagation which based on the received photons and optical intensity on the effective surface of the APD are given, respectively, in Section 3. In Section 4, a numerical analysis of 4-PAM is carried out to reveal the BER performance relationship of the atmospheric turbulence intensity, link lengths, optical wavelength, symbol transmission rate, temperature of APD, pointing errors (PEs) and signal to noise ratio (SNR). Finally, the conclusion is elaborated in Section 5.

2. Channel models

The propagation of optical waves in the free-space channel is affected by atmospheric turbulence. This characterization is described as random fluctuations of refractive index, resulting in random fluctuations of light intensity. The literature [18, see, pp. 341–402] has accurately modeled the light intensity distribution and compared the two models of lognormal and G-G distribution. The numerical simulation results show that the lognormal model does not fit the simulation data very well in the tails of the distribution. However, when calculating fade probabilities, the probability density function (PDF) tail near the origin that is the most interesting factor. The G-G model solves this problem perfectly, the parameters of this distribution are determined entirely by the atmospheric structure constant, the inner scale and outer scale of turbulence. Through the large-scale and small-scale irradiance fluctuations of the propagating optical wave, this model is more accurate than the lognormal distribution. It is defined as follows

f (〈 I 〉) = \frac{2 {(α β)}^{(α + β) / 2}}{Γ (α) Γ (β)} {〈 I 〉}^{(α + β) / 2 - 1} K_{α - β} (2 \sqrt{α β 〈 I 〉}), I > 0,

where K_v(·) and Γ(·) denotes Neumann function and gamma function, respectively. α and β depend upon the Rytov variance,

σ_{I}^{2} = 1.23 C_{n}^{2} κ^{7 / 6} L^{11 / 6}

, where

C_{n}^{2}

is the structure constant of atmospheric, which ranges from 10⁻¹⁷ m^−2/3 for weak turbulence to 10⁻¹⁵ m^−2/3 for medium level turbulence. L represents the free-space link distance, κ = 2π/λ is space wave number, λ denotes wavelength. 〈·〉 indicates normalization, and the normalized optical intensity can be defined as

〈 I 〉 = I / E (I) .

3. M-PAM model based APD

When M-PAM is implemented, during 0 ≤ t ≤ T_s, the optical signal can be expressed

s_{i} (t) = a_{i} g_{T} (t) \cos (2 π ν t), {i = 1, 2 \dots, M},

where T_s denotes the slot width of symbol, g_T(t) represents the rectangular pulse, and a_i is the i-th pulse amplitude. v = c/λ is optical frequency, where c represents light speed. The best received optical signal on the target surface of APD can be expressed as

\begin{array}{l} r = \int_{0}^{T_{s}} (s_{i} (t) + n_{w} (t)) \cdot f (t) d t \\ = a_{i} \sqrt{\frac{2}{E_{g}}} \int_{0}^{T_{s}} g_{T}^{2} (t) \cos^{2} (2 π ν t) d t + \int_{0}^{T_{s}} n_{w} (t) \cos (ν t) d t \\ = a_{i} \sqrt{\frac{E_{g}}{2}} + n, \end{array}

where

f (t) = \sqrt{\frac{2}{E_{g}}} g_{T} (t) \cos 2 π ν t

is the normalized basis function, s_i(t) can be written as s_i(t) = s_if(t). n is approximately additive white Gaussian noise (AWGN),

E_{g} = \int_{0}^{T_{s}} p_{g_{T}} (t) d t

denotes the energy of rectangular pulse, here

p_{g_{T}} (t)

is optical power, which can be defined as

P_{g_{T}} (t) = {\begin{matrix} \frac{E_{g}}{T_{s}} & 0 \leq T_{s} \\ 0 & o t h e r s . \end{matrix}

In the statistical models for direct detection, the average number of photons absorbed by the active surface of an APD illuminated with the optical signal s_i(t) in T_s seconds can be expressed as, here, we assume that the APD is operated at a reasonable bias voltage, see [14, pp. 226].

\begin{array}{l} \bar{K_{s}} = \bar{\frac{η}{h v} \int_{0}^{T_{s}} s_{i}^{2} (t) d t} \\ = \frac{η}{h v} \times \bar{\frac{E_{g} a_{i}^{2}}{2}} \\ = \frac{η}{h v} \times \frac{6 (\log_{2} M)}{M^{2} - 1} E_{b} \\ = \frac{η}{h v} \times \frac{6 (\log_{2} M)}{M^{2} - 1} T_{s} I, \end{array}

where

\bar{(\cdot)}

represents the mean function, η denotes quantum efficiency, h is Planck’s constant. E_b is the average bit energy of the optical signal, and the expression of 〈E_b〉 is given by

〈 E_{b} 〉 \approx A_{v} T_{s} I / E (E_{b}) = T_{s} 〈 I 〉,

here, A_v denotes the active surface area of detector. In [14, pp. 225–230] the Poisson distribution is suitable for conditions where the APD detector receives a relatively small number of particles, and the noise of the detector is relatively less. However, in actual communication working state, the APD has the AWGN. The probability density of the Poisson channel is approximately Gaussian when operating under high background conditions, i.e., the Poisson channel model of APD can approximate Gaussian distribution when the numbers of absorbed photons are much greater than one (K_s ≫ 1). For this relation, under the condition of the emitted optical signal, the expectation and variance of the received optical signal r can be expressed as Eqs. (8) and (9)

\begin{array}{l} E [r | s_{i}] = m_{0} - m_{1} \\ a_{i} G K_{s}, \end{array}

\begin{array}{l} D [r | s_{i}] = σ_{0}^{2} + σ_{1}^{2} \\ = [G^{2} F (a_{i} K_{s} + 2 K_{b}) + \frac{2 I_{s} T_{s}}{q} + \frac{2 κ_{T} T T_{s}}{q^{2} R_{L}}] \times 2 B T_{s}, \end{array}

where m is the mean of the current which generated by APD, here m₀ represents that there is no light on the surface of the APD, m₁ represents that there is light on the surface of the APD.

σ_{0}^{2}

represents the variance of the current which generated by APD when there is no light on the surface of the APD,

σ_{1}^{2}

is the variance of the current which generated by APD when there is light on the surface of the APD. In addition, G is the average APD gain, B represents the receiver bandwidth, K_b = ηI_bT_s/h_v denotes the average number of absorbed background photons, I_b, I_s and q are the background optical intensity, the surface leakage current of the APD and the electron charge. The other parameters are the “excess noise” factor of the APD F, the equivalent noise temperature of the device T, the Boltzmann’s constant κ_T, the load impedance R_L.

The likelihood function is obtained by combining Eqs. (8) and (9)

\begin{array}{l} p (r | s_{i}) = \frac{1}{\sqrt{4 π [G^{2} F (a_{i} K_{s} + 2 K_{b}) + \frac{2 I_{s} T_{s}}{q} + \frac{2 κ_{T} T T_{s}}{q^{2} R_{L}}] B T_{s}}} \times \\ \exp (- \frac{{(r - s_{i})}^{2}}{4 π [G^{2} F (a_{i} K_{s} + 2 K_{b}) + \frac{2 I_{s} T_{s}}{q} + \frac{2 κ_{T} T T_{s}}{q^{2} R_{L}}] B T_{s}}) . \end{array}

Assuming M = 2, under the maximum likelihood (ML) criterion, there is the optimal decision threshold is zero, i.e., when the prior probability P(s₁) = P(s₂) = 1/2, the optimal threshold of the decision domain is zero, which is shown as Fig. 1. Thus, the error probability of s₁(t) is given by

\begin{array}{l} P (e r r o r | s_{1}) = \int_{0}^{\infty} p (r | s_{1}) d r \\ = \frac{1}{2} e r f c (\frac{a_{i} G K_{s} q}{\sqrt{4 [G^{2} F q^{2} (a_{i} K_{s} + 2 K_{b}) + 2 I_{s} T_{s} q + \frac{2 κ_{T} T T_{s}}{R_{L}}] B T_{s}}}), \end{array}

where erfc(·) is the Gauss error function, the Eq. (11) can be also rewritten as, here Q(·) is the Q function,

P (e r r o r | s_{1}) = Q (\frac{a_{i} G K_{s} q}{\sqrt{2 [G^{2} F q^{2} (a_{i} K_{s} + 2 K_{b}) + 2 I_{s} T_{s} q + \frac{2 κ_{T} T T_{s}}{R_{L}}] B T_{s}}}) .

Using Eq. (12), the symbol error rate (SER) of 2-PAM can be calculated as

\begin{array}{l} P_{S E R} = P (s_{1}) P (e r r o r | s_{1}) + P (s_{2}) P (e r r o r | s_{2}) \\ = P (e r r o r | s_{1}) = P (e r r o r | s_{2}) \\ = Q (\frac{a_{i} G K_{s} q}{\sqrt{2 [G^{2} F q^{2} (a_{i} K_{s} + 2 K_{b}) + 2 I_{s} T_{s} q + \frac{2 κ_{T} T T_{s}}{R_{L}}] B T_{s}}}) . \end{array}

Similarly, under the maximum likelihood criterion, there is the prior probability P(s₁) = P(s₂) = ⋯= P(s_M) = 1/M. Therefore, the SER of M-PAM is given by

\begin{array}{l} P_{S E R} = P (s_{1}) P (e r r o r | s_{1}) + P (s_{2}) P (e r r o r | s_{2}) + \dots + P (s_{M}) P (e r r o r | s_{M}) \\ = P (e r r o r | s_{1}) = P (e r r o r | s_{2}) = \dots = P (e r r o r | s_{M}) \\ = \frac{2 (M - 1)}{M} Q (\frac{a_{i} G K_{s} q}{\sqrt{2 [G^{2} F q^{2} (a_{i} K_{s} + 2 K_{b}) + 2 I_{s} T_{s} q + \frac{2 κ_{T} T T_{s}}{R_{L}}] B T_{s}}}) . \end{array}

According to [8], the BER of M-PAM can be described as

P_{b} = \frac{2 (M - 1)}{M \log_{2} M} Q (\frac{a_{i} G K_{s} q}{\sqrt{2 [G^{2} F q^{2} (a_{i} K_{s} + 2 K_{b}) + 2 I_{s} T_{s} q + \frac{2 κ_{T} T T_{s}}{R_{L}}] B T_{s}}}) .

Fig. 1 Likelihood function of 2-PAM and schematic of optimal decision domain

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It is important to note that the receiver should employ a variety of compensation techniques to reduce the scintillation effect, such as adaptive optics, aperture averaging, ect. [19,20]. Here, it is assumed that the G-G model is suitable for the intensity obtained after the optical wave passes through the aperture smoothing, the effect of aperture averaging is not considered in this paper. Therefore, the ABER of the atmospheric channel can be expressed as

\begin{array}{l} \bar{P_{B E R}} = \int_{0}^{\infty} f (〈 I 〉) P_{b} (〈 I 〉) d 〈 I 〉 \\ = \frac{2 (M - 1)}{M \log_{2} M} \int_{0}^{\infty} \frac{2 {(α β)}^{(α + β) / 2}}{Γ (α) Γ (β)} {〈 I 〉}^{(α + β) / 2 - 1} K_{α - β} (2 \sqrt{α β 〈 I 〉}) \times \\ Q (\frac{6 η G q (\log_{2} M) T_{s} 〈 I 〉}{\sqrt{\begin{array}{l} 2 h v (M^{2} - 1) [G^{2} F q^{2} (6 η (\log_{2} M) T_{s} 〈 I 〉 \\ + 2 h v (M^{2} - 1) K_{b} + 2 h v (M^{2} - 1) (I_{s} T_{s} q + \frac{κ_{T} T T_{s}}{R_{L}})] B T_{s} \end{array}}}) d 〈 I 〉 . \end{array}

The Eq. (16) is a general expression, there is no analytical solution, so we define K_n as unified noise number of APD, which can be expressed as

K_{n} = 2 F K_{b} + \frac{2 (I_{s} T_{s} R_{L} q + κ_{T} T T_{s})}{G^{2} q^{2}} .

Assume that the noise bandwidth matches the duration of the slot, i.e., B= 1/2T_s, the Eq. (16) can be simplified as

\begin{array}{l} \bar{P_{B E R}} = \int_{0}^{\infty} f (〈 \bar{K_{s}} 〉) P_{b} (〈 \bar{K_{s}} 〉) d 〈 \bar{K_{s}} 〉 \\ = \frac{2 (M - 1)}{M \log_{2} M} \int_{0}^{\infty} \frac{2 {(α β)}^{(α + β) / 2}}{Γ (α) Γ (β)} {〈 \bar{K_{s}} 〉}^{(α + β) / 2 - 1} K_{α - β} (2 \sqrt{α β 〈 \bar{K_{s}} 〉}) \times \\ Q (\frac{6 η G q (\log_{2} M) 〈 \bar{K_{s}} 〉}{\sqrt{6 (M^{2} - 1) \log_{2} (M) F 〈 \bar{K_{s}} 〉 + {(M^{2} - 1)}^{2} K_{n}}}) d 〈 \bar{K_{s}} 〉, \end{array}

where

〈 \bar{K_{s}} 〉 = \bar{K_{s}} / E (\bar{K_{s}})

. The Eq. (16) is derived from the light intensity received from the target surface of APD, and the Eq. (18) describes the ABER model of photon number

\bar{K_{s}}

absorbed by the active surface of APD, the two mathematical models can be equivalent. Furthermore, these models show that besides the influence of the atmospheric turbulence channel, it is also mainly affected by the temperature, gain, modulation order and transmission rate of APD. It is critical to figure out what atmospheric turbulence conditions (i.e.,

C_{n}^{2}

,

σ_{I}^{2}

, etc.) are the M-PAM applicable to, and how the device parameters of APD affect the ABER performance of FSO communication systems. It must be pointed out that due to the complexity of the ABER expression, the optimization of the parameters cannot be obtained by calculating Eq. (16) or (18), but the numerical method can be used to obtain these optimal values, which will be reported in the next section.

4. Numerical analysis

From the beginning, the parameters were set by taking into account the actual working state of APD. For convenience and calculation, and without loss of generality, these parameters are set, i.e, R_L = 50 Ω, η = 0.75, F = G^1/2, I_s = 1 nA, A_v = πd²/4, where diameter d = 1.6 mm. Moreover, we consider that if the environmental conditions of communication are daytime or there are bright luminous objects in the field of view, in other words, there is the backlight, so assume I_b = 1 nW/m². In order to ensure that the M-PAM optical signal of the transmitter is ideal, the optical power $p_{g_{T}} (t)$ is assumed to be 1 W and the AWGN is set to 50 dB. In the atmospheric turbulence channel, it is the focus of our work to keep a high bit rate and find out the most appropriate M modulation method of PAM. Therefore, the ABER performance of different M-modulation schemes of PAM under the different APD gain is demonstrated in Fig. 2. With the M increases, the ABER performance tends to decrease. It can be concluded that 2-PAM has the best performance, as shown by the cyan diamond line.

Fig. 2 ABER as a function of G for M level PAM, where λ = 1550 nm, L = 1500 m, $C_{n}^{2} = 8 \times 10^{- 15} m^{- 2 / 3}$ , T = 300 K, Symbol rate = 1 GB/s.

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Compared to the Fig. 2, the Fig. 3 shows the performance of M-PAM optical signal propagation in the different atmospheric turbulence channel, here, the Fig. 3(a) is 2-PAM and (b) represents 4-PAM. We can draw the same conclusion that the ABER performance of 2-PAM is superior to 4-PAM in the same atmospheric turbulence channel, as depicted in Fig. 3(a). But it is not discussed in detail here, because 2-PAM is the limit case of M-PAM and it’s similar to OOK, and we focus on the ABER performance of the 4-PAM optical signal propagating in the atmospheric turbulence channel in this work. In Fig. 3(b), under the same gain conditions, the ABER performance becomes worse as the turbulence intensity increases. Especially when $C_{n}^{2} < 3 \times 10^{- 15} m^{- 2 / 3}$ , the ABER is lower than the 7% forward error correction (FEC) limit 3.8 × 10⁻³, which means the 4-PAM communication is suitable for weak turbulence [21]. Selecting an appropriate gain APD has a significant effect on reducing the ABER, which shows that the high gain of APD is better in practical operation. In Fig. 4, the 4-PAM communication performance of different link lengths is analyzed. Here, the intensity of the turbulence is set to 8 × 10⁻¹⁵ m^−2/3, that is, the scintillation index $σ_{I}^{2} = 1.1936$ when the link distance = 3000 m. The ABER becomes higher as the communication distance increases. At this time, the optimal communication distance is 1500 m or less, which is under the FEC limit of 3.8 × 10⁻³, it implies that the communication can be implemented. In fact, it also confirms that the PAM is more suitable for short-range communication which is experimentally presented in [7,22]. It is worth thinking about that the relays are used in the FSO communication systems in [23] due to the ABER becomes higher as link length increases, the principle for application of this device is the same as depicted in Fig. 4.

Fig. 3 ABER of the M-PAM as a function of G for the atmospheric structure constant $C_{n}^{2}$ , (a) and (b) represent 2-PAM and 4-PAM, respectively. Where λ = 1550 nm, L = 3000 m, T = 300 K, Symbol rate = 1 GB/s.

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Fig. 4 BER of the 4-PAM as a function of G for the link lengths L, where λ = 1550 nm, $C_{n}^{2} = 8 \times 10^{- 15} m^{- 2 / 3}$ , T = 300 K, Symbol rate = 1 GB/s.

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Particularly, it is worth noting that the parameters of the optical signal itself also affect the transmission of communication performance, such as the wavelength, symbol rate, et. al. The atmospheric attenuations of different wavelengths are not identical due to the “atmospheric window effect” [24]. Figure 5 depicts the ABER performances of the four wavelengths of the optical wave, where the symbol rate of the communication is defined as Symbol Rate = 1/T_s = 1 GB/s (or = 2 Gb/s). From Fig. 5, we can draw a conclusion that the 4-PAM ABER performance of 1550 nm wavelength is the best under the same APD gain condition. Moreover, the symbol transmission rate is an important parameter that directly affects the capacity of the transmitted data. With the continuous increase of 4-PAM optical signal transmission rate, the noise introduced is also increasing accordingly, which presents a challenge for the transmission of four equally spaced levels. However, under weak turbulence conditions, the results demonstrate that when the is gain of APD is above 100, the symbol transmission rate of 4-PAM greater than 3 GB/s (6 Gb/s) is feasible, because the ABER is lower than the FEC limit of 3.8 × 10⁻³. This relationship exposed in Fig. 6. In practical work, it must also be noted that it is an effective measure to reduce the received noise by increasing the gain of the APD. As shown in Figs. 2 and 6, it shows that as the gain of ADP increases, the ABER decreases, it is beneficial to improve the transmission rate. In addition, the temperature of APD is a key parameter, which will extremely affect the thermal noise of APD.

Fig. 5 ABER of the 4-PAM as a function of G for the wavelength λ, where L = 1500 m, $C_{n}^{2} = 8 \times 10^{- 15} m^{- 2 / 3}$ , T = 300 K, Symbol rate = 1 GB/s.

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Fig. 6 BER of the 4-PAM as a function of G for symbol rate, where λ = 1550 nm, L = 1500 m, $C_{n}^{2} = 8 \times 10^{- 15} m^{- 2 / 3}$ , T = 300 K.

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From the results of the simulation in Fig. 7, we can know that the APD exhibits some well-detected characteristics at room temperature of 300 K, but with the increase of APD temperature at work state, the ABER becomes worse. Especially when the temperature is above 500 K, it is more obvious.

Fig. 7 BER of the 4-PAM as a function of G for the temperature of APD K, where λ = 1550nm, L = 1500m, $C_{n}^{2} = 8 \times 10^{- 15}$ , Symbol rate = 1GB/s.

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Furthermore, an important factor cannot be ignored is that there is still the effect of PEs in the actual FSO communication systems, as described in [23,25–27]. Therefore, according to [28, see, Eq. (3)], we can get the probability density function of PEs, which can be written as

f_{P E s} (〈 I 〉) = \frac{ε^{2}}{A_{0}^{ε^{2}}} {〈 I 〉}^{ε^{2} - 1} .

Where A₀ = [erf (χ)]²,

ε^{2} = R_{e}^{2} / 4 σ_{s}^{2}

,

χ = \sqrt{π / 2 d} / R

is the PEs parameters, here d represents effective aperture diameter of the APD detector, R denotes the beam waist

R_{e} = {[\sqrt{π} e r f (χ) R^{2} / (2 χ e^{- χ^{2}})]}^{1 / 2}

is the equivalent beam width, and

σ_{s}^{2}

is the equivalent beam width. Similarly, under the combined effect of G-G distributed turbulence and PEs, we can derive the ABER expression based on APD according to Eqs. (1), (15) and (19). It can be described as

\begin{array}{l} \bar{P_{B E R - P E s}} = \int_{0}^{\infty} [f (〈 I 〉) P_{b} (〈 I 〉) f_{P E s} (〈 I 〉) d 〈 I 〉 \\ = \frac{2 (M - 1)}{M \log_{2} M} \frac{ε^{2}}{A_{0}^{ε^{2}}} \int_{0}^{\infty} \frac{2 {(α β)}^{(α + β) / 2}}{Γ (α) Γ (β)} {〈 I 〉}^{(α + β) / 2 + ε^{2} - 2} K_{α - β} (2 \sqrt{α β 〈 I 〉}) \times \\ Q (\frac{6 η G q (\log_{2} M) T_{s} 〈 I 〉}{\sqrt{\begin{array}{l} 2 h v (M^{2} - 1) [G^{2} F q^{2} (6 η (\log_{2} M) T_{s} 〈 I 〉 \\ + 2 h v (M^{2} - 1) K_{b} + 2 h v (M^{2} - 1) (I_{s} T_{s} q + \frac{κ_{T} T T_{s}}{R_{L}})] B T_{s} \end{array}}}) d 〈 I 〉 . \end{array}

From the Eq. (19), we can know that the severity of PE inversely depends upon the value of ε² and A₀. If ε² = 1 and A₀ = 1, there is $\bar{P_{B E R - P E s}} \equiv \bar{P_{B E R}}$ , i.e., there are no PEs in the M-PAM FSO communication systems. In order to better study the influence of pointing error constants on the system, we analyze the two different conditions of ε = 0.93 and ε = 1.2, where we assume that A₀ = 1, as depicted in Fig. 8. Compared to Fig. 3, the bit error rate in Fig. 8 is significantly worse under the same atmospheric turbulence conditions. Moreover, as the constant of the pointing error increases, the ABER is decreasing. It shows that we should reduce the PEs of the M-PAM FSO communication systems in actual operation.

Fig. 8 ABER-PEs of the M-PAM as a function of G for the atmospheric structure constant $C_{n}^{2}$ , (a) and (b) represent ε = 0.93 and ε = 1.2, respectively. Where A₀ = 1, λ = 1550 nm, L = 3000 m, T = 300 K, Symbol rate = 1 GB/s.

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However, in actual FSO communication systems, APD temperature control systems are usually designed to mitigate thermal noise, so, here, we can assume (κ_TTT_s)/R_L → 0. If the optical system is utilized to filter out backlight noise, i.e., K_b → 0 Thus, making the change of variable

ξ = \frac{2 (M - 1)}{M \log_{2} M}, ρ = η \sqrt{2 \times \frac{6 \log_{2} M}{M^{2} - 1}}, σ^{2} = σ_{0}^{2} + σ_{1}^{2}, ϵ \frac{ε^{2}}{A_{0}^{ε^{2}}} .

Moreover, we also assume that the pointing and tracking (APT) equipment is used in FSO communication systems for precise pointing, then there is ϵ = 1 [29]. So the Eq. (16) can be written as

\begin{array}{l} \bar{P_{B E R}} = ξ \int_{0}^{\infty} \frac{2 {(α β)}^{(α + β) / 2}}{Γ (α) Γ (β)} {〈 I 〉}^{(α + β) / 2 - 1} K_{α - β} (2 \sqrt{α β 〈 I 〉}) \times \\ Q (\frac{ρ 〈 I 〉}{\sqrt{2 σ^{2}}}) d 〈 I 〉 \\ = ξ \frac{2 {(α β)}^{(α + β) / 2}}{Γ (α) Γ (β)} G_{5, 2}^{2, 4} [{{(\frac{2}{α β})}^{2} 〈 u_{S N R} 〉 |}_{0, \frac{1}{2}}^{\frac{1 - α}{2}, \frac{2 - α}{2} \frac{1 - β}{2} \frac{2 - β}{2}, 1}] . \end{array}

Where 〈u_SNR〉 = ρ²I²/(2σ²E(u_SER)) is the normalized signal-noise ratio (SNR) and G(·) is the Meijer G function, see [30, Sec. (2.24.1) and (2.24.1.3)]. The Eq. (22) is about the function of SNR, which is of great significance for evaluating signal quality. As shown in Fig. 9, the ABER begins to fall below the limit 3.8 × 10⁻³ when turbulence is below

C_{n}^{2} = 3 \times 10^{- 15} m^{- 2 / 3}

and the SNR is greater than 20dB. When ABER is 1.0 × 10⁻⁸ m^−2/3, the SNR under the condition of

C_{n}^{2} = 3.0 \times 10^{- 15} m^{- 2 / 3}

need spends 20 dB more than one under the condition of

C_{n}^{2} = 1.0 \times 10^{- 13} m^{- 2 / 3}

. It shows that the 4-PAM is sensitive to the turbulence intensity and is suitable for the communication under weak turbulence conditions. The experimental result was exactly the same as that we obtained in [11], it successfully demonstrated the 64 Gb/s 4-PAM optical signal propagation in the free-space channel, where the BER is less than 10⁻⁹ when the atmospheric turbulence condition is considered to be weak turbulence, i.e.,

C_{n}^{2} < 10^{- 17} m^{- 2 / 3}

.

Fig. 9 ABER of the 4-PAM as a function of SNR for the atmospheric structure constant $C_{n}^{2}$ , where λ = 1550 nm, L = 1500 m, T = 300 K, Symbol rate = 1 GB/s.

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5. Conclusion

In this paper, when employing the APD to detect the optical field, the ABER performance expression of M-PAM optical signal propagation based on the G-G atmospheric turbulence channel was illustrated. The FSO communication systems of M-PAM is greatly affected by the intensity of turbulence and it is more suitable for the communication of short-wavelength infrared under short-distance and weak turbulence conditions. As the symbol transmission rate increases, the noise of the system increases. It is very advantageous for the M-PAM optical signal to improve the transmission rate by increasing the gain of the APD to mitigate the noise obtained on the active area of the detector. Furthermore, it was analyzed that as the temperature of the APD increases, the ABER becomes higher. And the effect of PEs is considered, that is, as the optical beam deviates from the main axis, the communication performance deteriorates. After applying the temperature control system and APT precise pointing device, the model of SNR-ABER was derived. The simulation results demonstrated that the SNR needs to be greater than 20 dB, the ABER is below the 7% FEC of 3.8 × 10⁻³ at this time. All in all, the proposed model provides a reference for the selection of actual M-PAM communication parameters and provides theoretical support for experimental data analysis.

Funding

National Natural Science Foundation of China (61775022 and 61475025); Development Program of Science and Technology of Jilin Province of China (20170521001HJ, 20180519012JH); Postdoctoral Science Foundation of China (2017M621179).

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Performance of M-PAM FSO communication systems in atmospheric turbulence based on APD detector

Abstract

1. Introduction

2. Channel models

3. M-PAM model based APD

4. Numerical analysis

5. Conclusion

Funding

References and links

Cited By

Figures (9)

Equations (22)

Optics Express