performance. To overcome this inconvenience, the PDF in Eq. (2.34) can be approximated by using the first term of the Taylor expansion at i = 0 as fI(i) = aib−1+ O(ib) [60], which can provide a deeper insight on how atmospheric turbulence and pointing errors deteriorate the performance of FSO communication systems. In this way, the PDF in Eq. (2.34) is approximated by a single polynomial term as follows
fI(i) .
= aib−1, i≥ 0 (2.37)
based on the fact that the asymptotic behavior of the system performance is dominated by the behavior of the PDF near the origin, i.e. fI(i) at i → 0 determines high SNR performance [60]. The parameter a is a positive constant, and the parameter b quantifies the order of smoothness of fI(i) at the origin. The high SNR approximations are especially useful for performance analysis of FSO communication systems, where severe fading renders necessary a large SNR for achieving a target in terms of the BER and outage probability.
Hence, we can obtain the following asymptotic expression for the PDF of GG atmospheric turbulence model in Eq. (2.34) as
fI(i) .
= aib−1=
ϕ2(αβ)min(α,β)Γ(|α−β|)
(A0L)min(α,β)Γ(α)Γ(β)(ϕ2−min(α,β))imin(α,β)−1, ϕ2 > min(α, β)
ϕ2(αβ)ϕ2Γ(α−ϕ2)Γ(β−ϕ2)
(A0L)ϕ2Γ(α)Γ(β) iϕ2−1. ϕ2 < min(α, β)
(2.38)
It is noteworthy to mention that the above asymptotic expression is dominated by b− 1, i.e., min(α, β, ϕ2)− 1. Hence, different expressions for a and b are derived in Eq. (2.38) depending on the relation between ϕ2 and min(α, β). In plane wave propagation, it is demonstrated that the relation α > β is always satisfied and, hence, β is lower bounded above 1 as turbulence strength increases [61]. In this case, the asymptotic expression given in Eq. (2.38) is dominated by min(β, ϕ2)− 1. The corresponding asymptotic CDF is obtained as follows
FI(i) = Z i
0
fI(x)dx .
= a
bib, i≥ 0. (2.39)
The asymptotic behavior has been applied to the combined effect of atmospheric turbulence (GG in this case) and pointing errors (I), but this procedure can directly be applied to atmospheric turbulence (Ia) as long as this one can always be expanded into Maclaurin series.
For instance, both GG and EW atmospheric turbulence can be expanded into Maclaurin series, among others. In the case of GG atmospheric turbulence, Ia can be expressed as
fIa(i) .
= aib−1= (αβ)min(α,β)Γ(|α − β|)
Γ(α)Γ(β) imin(α,β)−1, i≥ 0. (2.40)
2.4 Performance of FSO Communication Systems
The measurement of the performance in FSO communication systems has always been a subject of real interest. The SNR is a common measure of system performance, but
there are other metrics, which are able to provide a better performance in communications, such as probability of error, outage probability and channel capacity. The performance of FSO communication systems is usually quantified in terms of the BER. At the same time, outage probability is another performance metric that is defined as the probability that SNR falls below a certain specified threshold. Finally, the performance of FSO communication systems is measured in terms of ergodic capacity, which basically provides information about the limiting error-free information rate that can be achieved. Unlike outage probability and ergodic capacity, the BER is the only metric that depends on the modulation scheme employed by the FSO communications system.
The main difficulties when evaluating these performance metrics lie on the fact that the PDF of the irradiance might not be known in closed-form when a generalized pointing errors model is assumed. In this way, there are no available closed-form expressions for the BER, outage probability and ergodic capacity. The performance analysis is restricted to the numerical evaluation.
In this section, the performance of an FSO link over GG atmospheric turbulence channels with zero boresight pointing errors is analyzed in terms of the BER, outage probability and ergodic capacity in order to establish the baseline performance.
2.4.1 Bit Error-Rate (BER) Performance Analysis
The BER of IM/DD systems with OOK modulation in the presence of AWGN and assuming perfect channel state information (CSI) at the receiver is given by
Pb(e|i) = Pb(0)Pb(e|0) + Pb(1)Pb(e|1), (2.41) where Pb(0) and Pb(1) are the probabilities of sending 0 and 1 bits, respectively, and Pb(e|0) and Pb(e|1) denote the conditional bit error probability when the transmitted bit is 0 and 1, respectively. Assuming each bit is equally likely, the conditional BER at the receiver is given by
where Q(·) is the Gaussian Q-function defined as Q(x) = (1/2π)R∞
x exp −t2/2
dt, and γ = Pt2Tb/N0 is the received electrical SNR in absence of turbulence. Hence, the average BER Pb can be obtained by averaging Pb(e|i) over the combined PDF fI(i) as follows
Pb =
where fI(i) is given by Eq. (2.37). To evaluate the integral in Eq. (2.43), we can use that the Q-function is related to the complementary error function erfc(·) by erfc(x) = 2Q(√
2x) [62,
2.4. PERFORMANCE OF FSO COMMUNICATION SYSTEMS 29 eqn. (6.287)] and, then, we can use [62, eqn. (6.281)] (See Appendix A.9), obtaining an asymptotic closed-form expression for a generic average BER as follows
Pb .
where the parameters a and b depend on the atmospheric turbulence and pointing errors.
Interestingly, it is straightforward to show that the average BER behaves asymptotically as Pb .
= (Gcγ)−Gd, (2.45)
where Gd and Gc denote diversity order and coding gain, respectively [60]. At high SNR, the diversity order determines the slope of the BER versus average SNR curve in a log-log scale, and the coding gain (in decibels) determines the shift of the curve in SNR.
It can be observed that the diversity order is independent of pointing errors when the re-lation ϕ2> min(α, β) holds, i.e., atmospheric turbulence is the dominant effect in relation to pointing errors. In other words, the diversity order only depends on atmospheric turbu-lence when larger amounts of misalignment are not assumed. It can be shown that most practical terrestrial FSO systems operate under the condition of atmospheric turbulence is the dominant effect. It must also be mentioned that a much higher diversity order can be achieved under this condition and, hence, a much better BER performance is obtained. As a result, the adoption of the transmitter with accurate control of their beam width is espe-cially important here to satisfy this desired FSO scenario in order to maximize the diversity order.
2.4.2 Outage Performance Analysis
The outage probability, Pout, can be defined as the probability that the instantaneous com-bined SNR, γT, falls below a certain specified threshold, γth, that is
Pout := P (γT ≤ γth) = Z γth
0
fγT(i)di, (2.46)
where γT is the resulting received electrical SNR given by γT(i) = 1
By using Eq. (2.46), the outage probability can be written as Pout = P (4γi2 ≤ γth) =
where Pout represents the exact closed-form solution for the outage probability. In the case of GG atmospheric turbulence, FI
qγ
th
4γ
is expressed according to Eq. (2.36). Similar to
BER, Pout can also be expressed in terms of its asymptotic behavior by using Eq. (2.37) as
In this way, the outage probability also behaves asymptotically as Pout .
= (Ocγ)−Od, (2.50)
where Od and Oc denote outage diversity and coding gain, respectively [60]. As in BER performance, the outage diversity determines at high SNR the slope of the outage probability versus average SNR curve in a log-log scale and the coding gain (in decibels) determines the shift of the curve in SNR. Note that the same conclusions drawn from the BER performance about diversity order can be applied to outage diversity.
2.4.3 Ergodic Capacity Analysis
Assuming instantaneous CSI at the receiver, the ergodic capacity of FSO links in bps is given by Shannon’s well-known expression [63]
C = B
where B is the channel bandwidth, ln(·) is the natural logarithm [62, eqn. (1.511)], and fI(i) is the combined PDF of atmopsheric turbulence and pointing errors. It should be noted that the factor 1/2 in Eq. (2.51) is because the transmitter is assumed to operate in half-duplex mode and, hence, we consider a transmission in one direction at a time. By other hand, this analysis can be extended to full-duplex transmission, i.e., both directions simultaneously. Now, we solve the integral in Eq. (2.51) for GG atmospheric turbulence as given in Eq. (2.34). This integral can be solved using [59, eqn. (8.4.6.5)] in order to express the natural logarithm in terms of the Meijer’s G-function (See Appendix A.3.1) and, then, we can use [58, eqn. (07.34.21.0013.01)]. Hence, the closed-form expression for the ergodic capacity of an FSO link in bits/s/Hz is given by
C/B = ϕ22α+β−4 An asymptotic analysis can be performed in order to obtain a simpler closed-form expression for the ergodic capacity of FSO links. An asymptotic expression at high SNR is easily and accurately lower-bounded due to the fact that ln(1 + z)≈ ln(z) when z → ∞ as follows
C/B .
2.5. SUMMARY 31