## 2. System and channel model

We adopt a multiple-input/multiple-output array based on L laser sources, assumed to be intensity-modulated only and all pointed towards a distant array of M photodetectors, assumed to be ideal noncoherent (direct-detection) receivers. The transmit and receive apertures are physically situated so that all transmitters are simultaneously observed by each receiver. The use of infrared technologies based on IM/DD links is considered, where the instantaneous current ylm(t) in the mth receiving photodetector corresponding to the information signal transmitted from the lth laser can be written as

$ylm(t) = ηilm(t)x(t) + zm(t)$
where η is the detector responsivity, assumed hereinafter to be the unity, Xx(t) represents the optical power supplied by the lth source and Ilmilm(t) the equivalent real-valued fading gain (irradiance) through the optical channel between the lth transmit and the mth receive aperture. Additionaly, the fading experienced between source-detector pairs Ilm is assumed to be statistically independent. Zmzm(t) is assumed to include any front-end receiver thermal noise as well as shot noise caused by ambient light much stronger than the desired signal at the mth detector. In this case, the noise can usually be modeled to high accuracy as AWGN with zero mean and variance $σm2 = N0/2M$, i.e. ZmN(0, N 0/2M), independent of the on/off state of the received bit. The factor M is consequence of the fact that the sum of the M receive aperture areas is the same as the aperture area of a system with no receive diversity, wherein the noise is usually modelled as AWGN with zero mean and variance σ 2 = N 0/2. This allows the systems to be compared fairly [6]. Since the transmitted signal is an intensity, X must satisfy ∀t x(t) ≥ 0. Due to eye and skin safety regulations, the average optical power is limited and, hence, the average amplitude of X is limited. The received electrical signal Ylmylm(t), however, can assume negative amplitude values. We use Ylm, X, Ilm and Zm to denote random variables and ylm(t), x(t), ilm(t) and zm(t) their corresponding realizations.

The irradiance is susceptible to either strong atmospheric turbulence conditions and pointing error effects. In this case, it is considered to be a product of two independent random variables, i.e. $Ilm = Ilm(a) Ilm(p)$, representing $Ilm(a)$ and $Ilm(p)$ the attenuation due to atmospheric turbulence and the attenuation due to geometric spread and pointing errors, respectively, between lth transmitter and mth receiver. Although the effects of turbulence and pointing are not strictly independent, for smaller jitter values they can be approximated as independent [19]. Considering a limiting case of strong turbulence conditions [4, 22, 23], the turbulence-induced fading is modelled as a multiplicative random process which follows the negative exponential distribution, whose probability density function (PDF) is given by

$fIlm(a) (i) = e−i, i ≥ 0.$
This PDF has also been adopted in different works [8, 9, 24] to describe turbulence-induced fading, leading to an easier mathematical treatment. Regarding to the impact of pointing errors, we use the general model of misalignment fading given in [16] by Farid and Hranilovic, wherein the effect of beam width, detector size and jitter variance is considered. Assuming a Gaussian spatial intensity profile of beam waist radius, ωz, on the receiver plane at distance z from the transmitter and a circular receive aperture of radius r, the PDF of $Ilm(p)$ is given by
$fIlm(p) (i) = φ2A0φ2iφ2−1, 0 ≤ i ≤ A0$
where φ = ωzeq/2σs is the ratio between the equivalent beam radius at the receiver and the pointing error displacement standard deviation (jitter) at the receiver, $ωzeq2 = ωz2πerf(v)/2vexp(−v2)$, $v = πr/2ωz$, A 0 = [erf(v)]2 and erf(·) is the error function [25, eqn. (7.1.1)]. Here, independent identical Gaussian distributions for the elevation and the horizontal displacement (sway) are considered, being $σs2$ the jitter variance at the receiver. Using the previous PDFs for turbulence and misalignment fading, a closed-form expression of the combined PDF of Ilm was derived in [18] as
$fIlm(i) = φ2A0φ2iφ2−1Γ(1 − φ2, iA0), i ≥ 0$
where Γ(·,·) is the upper incomplete Gamma function [25, eqn. (6.5.3)]. Here, it must be commented that the pointing error model in Eq. (3) is not general enough to be directly applied to FSO systems with more than one photodetector (i.e. M > 1) since it depends on locations of the transmitter and receiver, requiring to take into account the relative position for each receive aperture regarding initial pointing in undisturbed position. This represents a fixed pointing error called boresight. As previously indicated, we adopt a MIMO array wherein the transmit and receive apertures are physically situated so that all transmitters are simultaneously observed by each receiver. For the sake of simplicity, it is here assumed that the receivers are placed very closely together and beam radius is large enough so that the relative displacement for each receiver regarding the initial pointing can be considered negligible, and hence the pointing error model in Eq. (3) can be approximately applied. This assumption can be reinforced by the fact that strong turbulence channels are more robust to boresight [19]. Additionally, as shown in the appendix, we consider OOK formats with any pulse shape and reduced duty cycle, allowing the increase of the PAOPR parameter [12, 23].

## 3. Outage performance analysis

In this section, the outage probability as a performance measure is evaluated for different MIMO FSO configurations over strong atmospheric turbulence channels with pointing errors. Together with the average error rate, outage probability, P out, is another standard performance criterion of communications systems operating over fading channels [26]. It is defined as the probability that the instantaneous combined SNR, γT, falls below a certain specified threshold, γth, which represents a protection value of the SNR above which the quality of the channel is satisfactory, i.e.,

$Pout = Prob(γT ≤ γth) = ∫0γthfγT (γT)dγT = FγT (γth)$
where fγT (γT) and FγT (γT) are the PDF and the cumulative distribution function (CDF) of γT, respectively. Next, the outage performance of a SISO FSO system is firstly evaluated in order to be considered as a benchmark in the analysis of the remaining MIMO configurations, corresponding to a multiple-input/single-output (MISO) system with a L × 1 array, a single-input/multiple-output (SIMO) system with a 1 × M array and, finally, a MIMO FSO system with a L × M array.

#### 3.1. SISO FSO system

In this subsection, the outage probability for the FSO system model previously presented with L = M = 1 is evaluated. According to Eq. (1) and the OOK signaling described in the appendix, a constellation of two equiprobable points in a one-dimensional space with an Euclidean distance of d, the statistical channel model corresponding to the SISO configuration can be written as

$Y = XI + Z, X ∈ {0,d}, Z ∼ N(0,N0/2).$
The received electrical SNR can be defined, as in [16], as
$γTSISO (i) = 12 d2N0/2i2 = 4Popt2TbξN0i2 = γξi2$
where γ represents the received electrical SNR in absence of turbulence when the classical rectangular pulse shape is adopted for OOK formats, Tb parameter is the bit period, P opt is the average optical power transmitted and ξ represents the square of the increment in Euclidean distance due to the use of a pulse shape of high PAOPR, as explained in a greater detail in the appendix. Using Eq. (5), the outage probability for a SISO FSO system can be written as
$Pout = Prob(γξi2 ≤ γth) = Prob(i2 ≤ γthξγ) = FI(1ξγthγ)$
where the CDF of I, FI (i), can be easily derived from Eq. (4) as
$FI(i) = 1 − (iA0)φ2 φ2Γ (−φ2, iA0), i ≥ 0$
by applying the differential relation in [25, eqn. (6.5.26)] and the fact that the series expansion corresponding to the upper incomplete Gamma function can be simplified by $Γ(a,z) ∝ Γ(a) − (za/a)(1 − az1+a + O(z2))$ [27, eqn. (8.354.2)], where Γ(·) is the well-known Gamma function. The results corresponding to this FSO scenario are illustrated in Fig. 1, where rectangular pulse shapes with ξ = 1 are used, assuming different values of normalized beamwidth, ωz/r, and normalized jitter, σs/r (i.e. ωz/r = 5 with σs/r = {1,3,4,5} and ωz/r = 10 with σs/r = {1,4,7,11}). Outage simulation results are furthermore included as a reference, demonstrating an excellent agreement with the analytical result in Eq. (8) for different misalignment fading conditions.

Fig. 1 Probability of outage versus normalized average SNR in FSO IM/DD links over the exponential atmospheric turbulence channel with pointing errors, assuming different values of normalized beamwidth, ωz/r, and normalized jitter, σs/r.

Additionally, we can take advantage of this series expansion in order to quantify the outage probability at high SNR, showing that the asymptotic performance of this metric as a function of the average SNR is characterized by two parameters: the diversity and coding gains. Therefore, for large enough SNR (γ → ∞), the asymptotic outage performance can be expressed after some algebraic manipulations as

$Pout ≐ (φ4(1 − φ2)2A02ξ γthγ)1/2 + (Γ (1 − φ2)−2φ2A02ξγthγ)φ2/2.$
Here, an even greater simplification can be deduced, leading to different asympotic expressions depending on the value of φ, as follows
$Pout ≐ (φ4(1 − φ2)2A02ξ γthγ)1/2, φ > 1$
$Pout ≐ (Γ (1 −φ2)2/φ2A02ξ γthγ)φ2/2, φ < 1$
Results corresponding to this asymptotic analysis are also illustrated in the Fig. 1 for different cases in Eq. (11) . It is straightforward to show that the outage probability behaves asymptotically as (Oc(γ/γth))Od, where Od and Oc denote outage diversity and coding gain, respectively [26, 28]. At high SNR, if asymptotically the outage probability behaves as (Oc(γ/γth))Od, the outage diversity Od determines the slope of the outage performance versus average SNR curve in a log-log scale and the coding gain Oc (in decibels) determines the shift of the curve in SNR. Additionally, as previously reported by the authors [12, 13, 23], a relevant improvement in performance must be noted as a consequence of the pulse shape used, providing an increment in the average SNR of 10 log10 ξ decibels.

At this point, it can be convenient to compare with the outage performance obtained in a similar context when misalignment fading is not present. In this case, knowing that the CDF corresponding to the negative exponential distribution in Eq. (2) is given by F I(a) (i) = 1 − exp(−i) and the fact that the series expansion for the exponential function can be simplified by exp(z) ∝ 1 + z + O(z 2), an asymptotic expression can be easily derived as $Poutexp ≐ (γth/γξ))1/2$. From these results, it can be deduced that φ is a key parameter in order to optimize the outage diversity. So, it is shown that the diversity order is independent of the pointing errors when the equivalent beam radius at the receiver is at least twice the value of the pointing error displacement standard deviation at the receiver, i.e. φ > 1. Once this condition is satisfied, taking into account the coding gain in Eq. (11a), the impact of the pointing error effects translates into a coding gain disadvantage, D[dB], relative to strong atmospheric turbulence without misalignment fading given by

$D[dB] ≜ 20 log10 (φ2A0 (φ2 − 1)).$
According to this expression, it can be observed in Fig. 1 that coding gain disadvantages of 23.7, 34.4 and 42.7 decibels are achieved for values of (ωz/r,σs/r) = (5, 1), (ωz/r,σs/r) = (10,1) and (ωz/r,σs/r) = (10,4), respectively.

#### 3.2. MISO FSO system with a L ×1 array

In this subsection, the outage probability for the FSO system model previously presented where the information signal is transmitted via L apertures and received by only one aperture (i.e. M = 1) is evaluated. Here, two different transmit diversity strategies are considered: repetition coding (RC) and transmit laser selection (TLS).

### 3.2.1. Transmit diversity using repetition coding

It is assumed that the same symbol is transmitted from the L transmitters at a given bit period [6, 10, 29]. According to Eq. (1), the statistical channel model corresponding to this MISO configuration can be written as

$Y = X1L ∑l=1LIl + Z, X ∈ {0,d}, Z ∼ N(0,N0/2)$
where Il represents the equivalent irradiance through the optical channel between the lth transmit aperture and the photodetector. Here, the division by L is considered so as to maintain the average optical power in the air at a constant level of P opt, being transmitted by each laser an average optical power of P opt/L. In this way, the total transmit power is the same as in a FSO system with no transmit diversity. The resulting received electrical SNR can be defined as
$γTMISO-RC = γξ 1L2 (∑l=1Lil)2 = γξ 1L2iT2$
and, hence, the outage probability for a MISO FSO system with repetition coding can be written as
$Pout = FIT (L1ξγthγ)$
Next, in order to address the cumbersome statistical problem related to the CDF of the sum of L variates, the PDF in Eq. (4) is approximated by a single polynomial term as fIlm (i) ≈ aib, 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. fIlm (i) at i → 0 determines high SNR performance [28]. Using the series expansion corresponding to the upper incomplete Gamma function [27, eqn. (8.354.2)], different asympotic expressions for Eq. (4), depending on the value of φ, can be written as
$fIlm (i) ≈ φ2A0 (φ2 − 1), φ > 1$
$fIlm (i) ≈ φ2Γ(1 − φ2)A0φ2iφ2−1, φ < 1$
Since the variates are independent, knowing that the resulting PDF of their sum is the convolution of their corresponding PDFs, obtained via single-sided Laplace and its inverse transforms, approximate expressions for the CDF, FIT (i), of the combined variate can be easily derived as
$FIT (i) ≈ 1Γ(L + 1) (φ2A0 (φ2 − 1))L iL, φ > 1$
$FIT (i) ≈ Γ (1 + φ2)LΓ ( 1 + Lφ2) (Γ ( 1 − φ2)A0φ2)L iLφ2, φ < 1$
and, then, the corresponding approximate expressions for the outage probability for a MISO FSO system with repetition coding can be obtained from Eq. (15) as follows
$Pout ≐ (L2Γ(L + 1)2/L φ4A02 ( 1 − φ2)2ξ γthγ )L/2, φ > 1$
$Pout ≐ (L2Γ ( 1 + φ2)2/φ2Γ ( 1 + Lφ2)2/Lφ2 Γ ( 1 − φ2)2/φ2A02 ξ γthγ)Lφ2/2, φ < 1$
Monte Carlo simulation results for Eq. (15) together with the asymptotic expressions in Eq. (18) are illustrated in Fig. 2, where rectangular pulse shapes with ξ = 1 are used, assuming a normalized beamwidth of ωz/r = 5 and different values of normalized jitter of σs/r = 1 and σs/r = 4.

Fig. 2 Probability of outage versus normalized average SNR in MISO and SIMO FSO IM/DD links over the exponential atmospheric turbulence channel with pointing errors, assuming a normalized beamwidth of ωz/r = 5 and values of normalized jitter of (a) σs/r = 1 and (b) σs/r = 4.

### 3.2.2. Transmit diversity using laser selection

When channel side information is not only available at the receiver but also on the transmitter, an alternative transmit diversity scheme can be adopted. Following the transmit laser selection (TLS) scheme based on the selection of the optical path with a greater value of fading gain (irradiance) [12], our MISO system model can be considered as an equivalent SISO system model where the channel irradiance corresponding to the TLS scheme, Imax, can be written as Imax = maxl =1,2,… L Il. In this way, having a SISO system as a reference, the statistical channel model corresponding to this MISO configuration can be written as

$Y = XImax + Z, X ∈ {0,d}, Z ∼ N(0,N0/2)$
where, in order to maintain the average optical power transmitted at the same constant level P out, the division by L is not included in Eq. (19) because of not more than one laser is simultaneously used. According to order statistics [30], the CDF, FImax (i), of the resulting channel irradiance, Imax, is given by FImax (i) = [FI (i)]L and, hence, the outage probability can be expressed as
$Pout = FImax (1ξ γthγ) = (FI (1ξ γthγ))L$
In the same way, its corresponding asympotic expressions can be easily derived from Eq. (11) as
$Pout ≐ (φ4( 1 − φ2)2 A02ξ γthγ)L/2, φ > 1$
$Pout ≐ (Γ ( 1 − φ2)2/φ2A02ξ γthγ)Lφ2/2, φ < 1$
Results for the exact outage probability in Eq. (20) together with the corresponding asymptotic expressions in Eq. (21) are also illustrated in Fig. 2. Monte Carlo simulation results are furthermore included as a reference, demonstrating an excellent agreement with the analytical result in Eq. (20). Comparing both transmit diversity techniques and their corresponding asymptotic expressions in Eq. (18) and Eq. (21) , respectively, it can be stated that the superiority of the TLS scheme translates into a coding gain advantage, ATLS[dB], relative to the RC scheme given by
$ATLS[dB] ≜ 10 log10 (L2Γ(L + 1)2/L), φ > 1$
$ATLS[dB] ≜ 10 log10 (L2Γ ( 1 + φ2)2/φ2Γ ( 1 + Lφ2)2/Lφ2), φ < 1$
As observed in Fig. 2, it can be deduced from expressions in Eq. (22) that the greater the number of transmit apertures, the better performance in terms of coding gain advantage. Additionally, this superiority is even more significant when φ < 1 and the value of φ is decreasing.

#### 3.3. SIMO FSO system with a 1 × M array

In this subsection, the outage probability for the FSO system model previously presented is evaluated when the information signal is transmitted via only one aperture (i.e. L = 1) and received by M apertures. Here, two different receive diversity strategies are considered: equal gain combining (EGC) and selection combining (SC).

### 3.3.1. Receive diversity using equal gain combining

Following this combining technique, the receiver adds the information corresponding to M photodetectors and, hence, according to Eq. (1), the statistical channel model for this SIMO configuration can be written as

$Y = X1M ∑m=1MIm + ZEGC,X∈{0,d},ZEGC ∼ N(0, N0/2)$
where Im represents the equivalent irradiance through the optical channel between the transmit aperture and the mth photodetector. Here, the division by M is considered to ensure that the area of the receive aperture in SISO links has the same size as in the sum of M receive aperture areas of SIMO links [6]. It can be noted that the resulting expression is equivalent to the expression in Eq. (13), obtained for MISO FSO links assuming RC at the transmitter side [10]. Therefore, we can interchange L by M in Eq. (15) and Eq. (18) to obtain the outage probability and its corresponding asympotic expressions for SIMO FSO links using equal gain combining.

### 3.3.2. Receive diversity using selection combining

This type of combining is based on the selection of the receive aperture corresponding to the optical path with a greater value of fading gain (irradiance) [6, 10]. In this way, in a similar way as previously presented for MISO FSO links with transmit laser selection, our SIMO system model can be considered as an equivalent SISO system model where the channel irradiance corresponding to the SC scheme, Imax, can be written as Imax = maxm =1,2,··· M Im and, hence, the statistical channel model can be written as

$Y = X1MImax + ZSC,X ∈ {0,d},ZSC ∼ N(0, N0/2M)$
where M represents the same role as previously explained when EGC is adopted. It can be observed that the noise variance is given by N 0/2M since only one of the M receive apertures is used. Taking into account Eq. (7), the resulting received electrical SNR can be defined as
$γTSIMO-SC = γξ1MImax2$
and, hence, the outage probability for a SIMO FSO system with selection combining can be written as
$Pout = FImax (Mξ γthγ) = (FI (Mξ γthγ))M.$
In the same way, its corresponding asympotic expressions, depending on the value of φ, can be easily derived from Eq. (11) as
$Pout ≐ (Mφ4(1 − φ2)2 A02ξ γthγ)M/2, φ > 1$
$Pout ≐ (MΓ( 1 − φ2)2/φ2A02 ξ γthγ)Mφ2/2, φ < 1$
Results for the exact outage probability in Eq. (26) together with the corresponding asymptotic expressions in Eq. (27) are also illustrated in Fig. 2. Monte Carlo simulation results are furthermore included as a reference, demonstrating an excellent agreement with the analytical result in Eq. (26). Comparing both receive diversity techniques and their corresponding asymptotic expressions, a different relative performance is deduced depending on the value of φ, corroborating the superiority of EGC scheme when φ > 1 and the superiority of both SC scheme or EGC scheme when φ < 1, depending on the values of M and φ. This improvement in performance of the EGC scheme when φ > 1 translates into a coding gain advantage, AEGC[dB], relative to the SC scheme as follows
$AEGC[dB] ≜ 10 log10 (Γ(M + 1)2/MM), φ > 1$
corroborating the fact that the greater the number of apertures, the higher coding gain advantage. Nonetheless, it can be noted that there is no difference between both diversity techniques for a value of M = 2. In a similar way, the superiority of SC scheme when φ < 1 translates into a coding gain advantage, ASC[dB], relative to the EGC scheme as follows
$ASC[dB] ≜ 10 log10 (MΓ ( 1 + φ2)2/φ2Γ ( 1 + Mφ2)2/Mφ2), φ < 1$
Nonetheless, it can be concluded from this expression that the superiority of the SC scheme is only valid for low values of φ and when a small number of receive apertures is considered, being only justified the adoption of this receive diversity technique in the context of very severe pointing errors. As depicted in Fig. 3 for a normalized beamwidth of ωz/r = 10, better performance for the EGC scheme is again achieved as the number of receive apertures increases and the value of the normalized jitter, σs/r, is lower and lower. So, the value of φ becomes higher and higher, and hence it is closer to the unity.

Fig. 3 Coding gain advantage for the SC scheme relative to the EGC scheme, assuming a normalized beamwidth of ωz/r = 10 and values of normalized jitter of 6, 7, 8 and 9, corresponding to values of φ of 0.83, 0.71, 0.62 and 0.55, respectively.

#### 3.4. MIMO FSO system with a L ×M array

According to results in previous subsections, the outage probability for MIMO FSO IM/DD links is here evaluated when transmit laser selection and equal gain combining are the diversity strategies adopted in the transmit and receive side, respectively. The statistical channel model for this MIMO configuration can be written as

$Y = X 1M ∑m=1MIm + ZEGC,X ∈ {0,d},ZEGC ∼ N(0,N0/2)$
where Im represents the equivalent irradiance through the optical channels between the L transmit apertures and the mth photodetector, based on the selection of the optical path with a greater value of irradiance, i.e. Im = maxl =1,2,··· L Ilm. The resulting received electrical SNR can be defined as
$γTMIMO = γξ 1M2 (∑m=1Mim)2 = γξ 1M2iT2$
and, hence, the outage probability is given by
$Pout = FIT (M1ξ γthγ)$
where IT represents the sum of M variates following the Lth-order statistics distribution for L observations from the distribution in Eq. (4). As in previous subsections, in order to address the cumbersome statistical problem related to the CDF of the sum of these M variates and making use of Eq. (16) , we use an approximated expression for the PDF of Im, which is given by
$fIm (i) ≈ L (φ2(φ2 − 1)A0)L iL−1, φ > 1$
$fIm (i) ≈ Lφ2 (Γ( 1 − φ2)A0φ2)L iLφ2−1, φ < 1$
according to order statistics [30]. Since the variates are independent, knowing that the resulting PDF of their sum is the convolution of their corresponding PDFs, obtained via single-sided Laplace and its inverse transforms, approximate expressions for the CDF, FIT (i), of the combined variate can be easily derived as
$FIT (i) ≈ (LΓ(L))MΓ(LM + 1) (φ2(φ2 − 1)A0)LM iLM, φ > 1$
$FIT (i) ≈ Γ (Lφ2 + 1)MΓ (LMφ2 + 1) (Γ ( 1 − φ2)A0φ2)LM iLMφ2, φ < 1$
and, then, the corresponding approximate expressions for the outage probability for a MIMO FSO system with transmit laser selection and equal gain combining can be obtained from Eq. (32) as follows
$Pout ≐ ((M2Γ(L + 1)2/LΓ (LM + 1)2/LM φ4(φ2 − 1)2 A02ξ γthγ)LM2, φ > 1$
$Pout ≐ (M2Γ (Lφ2 + 1)2/Lφ2Γ (LMφ2 + 1)2/LMφ2 Γ ( 1 − φ2)2/φ2A02ξ γthγ)LM φ2/2, φ < 1$
As expected, these expressions can be reduced to those corresponding to MISO and SIMO FSO scenarios in the special case of M = 1 and L = 1, respectively. In this way, it can be noted that expressions in Eq. (35) are equivalent to the expressions in Eq. (21) when a MISO FSO system with transmit laser selection is assumed. In the same way, expressions in Eq. (35) can also be reduced to those corresponding to SIMO FSO system with equal gain combining, being equivalent to the expressions in Eq. (18) when L is interchanged by M. As previously commented in section 3.3.1, expressions corresponding to a SIMO FSO system using equal gain combining are equivalent to those corresponding to a MISO FSO system assuming RC at the transmitter side. Monte Carlo simulation results for Eq. (32) together with the asymptotic expressions in Eq. (35) are illustrated in Fig. 4, where rectangular pulse shapes with ξ = 1 are used, assuming a normalized beamwidth of ωz/r = 5 with a normalized jitter of σs/r = 1 and a normalized beamwidth of ωz/r = 10 with a normalized jitter of σs/r = 7 and values of L = {2, 4} and M = {2, 4}.

Fig. 4 Probability of outage versus normalized average SNR in MIMO FSO IM/DD links over the exponential atmospheric turbulence channel with pointing errors, assuming a normalized beamwidth of ωz/r = 5 with a normalized jitter of σs/r = 1 and a normalized beamwidth of ωz/r = 10 with a normalized jitter of σs/r = 7, corresponding to values of φ of 2.55 and 0.71, respectively. Additionally, results assuming the optimum beamwidth corresponding to values of jitter of σs/r = 1 and σs/r = 7 are also included for L = 4 and M = 2.

## 4. Outage optimization

From obtained results, it can be concluded that the outage diversity is independent of the pointing error effects when the equivalent beam radius value at the receiver is at least twice the value of the pointing error displacement standard deviation at the receiver, i.e. φ > 1, being φ a main parameter to consider in order to optimize the outage performance. Once this condition is satisfied, comparing different diversity techniques and their corresponding asymptotic expressions, it can be noted that better performance is achieved when increasing the number of transmit apertures, i.e. L, instead of the number of receive apertures, i.e. M, in order to guarantee a same diversity order, as shown in Fig. 4 when assuming a normalized beamwidth of ωz/r = 5 with a normalized jitter of σs/r = 1 and a diversity order of 8 with values of {L,M} = {4,2} and {L,M} = {2,4}. Taking into account expressions in Eq. (35a) and Eq. (21a), this can be quantified as a coding gain disadvantage, DMIMO[dB], for the MIMO FSO system relative to the MISO FSO system using laser selection as follows

$DMIMO[dB] ≜ 20log10 (MΓ (Od/M + 1)M/OdΓ(Od + 1)1/Od).$
where Od and M denote outage diversity and the number of receive apertures, respectively. In agreement with this expression, it can be seen in Fig. 4 that a difference of 2.13 decibels is achieved for a diversity order of 8 and values of M = 2 and M = 4 when a normalized beamwidth of ωz/r = 5 with a normalized jitter of σs/r = 1 is assumed. Lastly, since proper FSO transmission requires transmitters with accurate control of their beamwidth, the optimization procedure is finished by finding the optimum beamwidth, ωz/r, that gives the minimum outage [16, 17, 31]. From Eq. (35a), it can be observed that this is equivalent to minimize the expression in Eq. (12) corresponding to the SISO configuration, being independent of the MIMO configuration adopted. In this way, the optimum beamwidth can be achieved using numerical optimization methods for different values of normalized jitter, σs/r [32]. Numerical results for the optimum beamwidth are illustrated in Fig. 5 for a range of values in the normalized jitter from σs/r = 1 to σs/r = 10 in discrete steps of 0.5 when the stochastic function minimizer simulated annealing is used. From this figure, it can be deduced that the outage optimization provides numerical results following a linear performance. This leads to easily obtain a first-degree polynomial as follows
$ωz/roptimum ≈ 2.85(σs/r − 1) + 2.6$
considering the values of optimum beamwidth corresponding to normalized jitters of σs/r = 1 and σs/r = 10, respectively. As can be seen in this figure, it is clearly depicted that the approximative analytical expression remains very accurate to numerical results. The use of this expression is shown in Fig. 4, where results assuming the optimum beamwidth corresponding to values of jitter of σs/r = 1 and σs/r = 7 are also included for L = 4 and M = 2. As expected, a greater improvement in performance is achieved when σs/r = 7 compared to the improvement in performance corresponding to the normalized jitter of σs/r = 1 since not only coding gain but also diversity gain is obtained, making outage diversity performance independent of the impact of misalignment fading.

Fig. 5 Optimum normalized beamwidth versus normalized jitter, σs/r in FSO IM/DD links over the exponential atmospheric turbulence channel with pointing errors.

## Appendix

We consider OOK formats with any pulse shape and reduced duty cycle, allowing the increase of the PAOPR parameter [12, 23]. A new basis function ϕ (t) is defined as $ϕ (t) = g(t)/Eg$ where g(t) represents any normalized pulse shape satisfying the non-negativity constraint, with 0 ≤ g(t) ≤ 1 in the bit period and 0 otherwise, and $Eg = ∫−∞∞g2 (t)dt$ is the electrical energy. In this way, an expression for the optical intensity can be written as

$x(t) = ∑k=−∞∞ak2TbPoptG(f =0)g(t − kTb)$
where G(f = 0) represents the Fourier transform of g(t) evaluated at frequency f = 0, i.e. the area of the employed pulse shape, and Tb parameter is the bit period. The random variable (RV) ak follows a Bernoulli distribution with parameter 1/2, taking the values of 0 for the bit “0” (off pulse) and 1 for the bit “1” (on pulse). From this expression, it is easy to deduce that the average optical power transmitted is P opt, defining a constellation of two equiprobable points (x 0 = 0 and x 1 = d) in a one-dimensional space with an Euclidean distance of $d = 2PoptTbξ$ where ξ = Tb Eg/G 2(f = 0) represents the square of the increment in Euclidean distance due to the use of a pulse shape of high PAOPR, alternative to the classical rectangular pulse.

## Acknowledgments

The authors would like to thank the anonymous reviewers for their useful comments that helped to improve the presentation of the paper. The authors are grateful for financial support from the Junta de Andalucía (research group “ Communications Engineering ( TIC-0102)”).

1. J. M. Kahn and J. R. Barry, “Wireless infrared communications,” Proc. IEEE 85, 265–298 (1997). [CrossRef]

2. L. B. Stotts, L. C. Andrews, P. C. Cherry, J. J. Foshee, P. J. Kolodzy, W. K. McIntire, M. Northcott, R. L. Phillips, H. A. Pike, B. Stadler, and D. W. Young, “Hybrid optical RF airborne communications,” Proc. IEEE 97(6), 1109–1127 (2009). [CrossRef]

3. W. Lim, C. Yun, and K. Kim, “BER performance analysis of radio over free-space optical systems considering laser phase noise under gamma-gamma turbulence channels,” Opt. Express 17(6), 4479–4484 (2009). [CrossRef]   [PubMed]

4. L. Andrews, R. Phillips, and C. Hopen, Laser Beam Scintillation with Applications (SPIE Press, 2001). [CrossRef]

5. X. Zhu and J. M. Kahn, “Free-space optical communication through atmospheric turbulence channels,” IEEE Trans. Commun. 50(8), 1293–1300 (2002). [CrossRef]

6. E. J. Lee and V. W. S. Chan, “Part 1: optical communication over the clear turbulent atmospheric channel using diversity,” IEEE J. Sel. Areas Commun. 22(9), 1896–1906 (2004). [CrossRef]

7. I. B. Djordjevic, S. Denic, J. Anguita, B. Vasic, and M. Neifeld, “LDPC-coded MIMO optical communication over the atmospheric turbulence channel,” J. Lightwave Technol. 26(5), 478–487 (2008). [CrossRef]

8. M. Simon and V. Vilnrotter, “Alamouti-type space-time coding for free-space optical communication with direct detection,” IEEE Trans. Wireless Commun. 4(1), 35–39 (2005). [CrossRef]

9. C. Abou-Rjeily and W. Fawaz, “Space-time codes for MIMO ultra-wideband communications and MIMO free-space optical communications with PPM,” IEEE J. Sel. Areas Commun. 26(6), 938–947 (2008). [CrossRef]

10. T. A. Tsiftsis, H. G. Sandalidis, G. K. Karagiannidis, and M. Uysal, “Optical wireless links with spatial diversity over strong atmospheric turbulence channels,” IEEE Trans. Wireless Commun. 8(2), 951–957 (2009). [CrossRef]

11. E. Bayaki and R. Schober, “On space-time coding for free-space optical systems,” IEEE Trans. Commun. 58(1), 58–62 (2010). [CrossRef]

12. A. Garcia-Zambrana, C. Castillo-Vazquez, B. Castillo-Vazquez, and A. Hiniesta-Gomez, “Selection transmit diversity for FSO links over strong atmospheric turbulence channels,” IEEE Photon. Technol. Lett. 21(14), 1017–1019 (2009). [CrossRef]

13. A. García-Zambrana, C. Castillo-Vázquez, and B. Castillo-Vázquez, “Space-time trellis coding with transmit laser selection for FSO links over strong atmospheric turbulence channels,” Opt. Express 18(6), 5356–5366 (2010). [CrossRef]   [PubMed]

14. S. Arnon, “Optimization of urban optical wireless communication systems,” IEEE Trans. Wireless Commun. 2(4), 626–629 (2003). [CrossRef]

15. S. Arnon, “Effects of atmospheric turbulence and building sway on optical wireless-communication systems,” Opt. Lett. 28(2), 129–131 (2003). [CrossRef]   [PubMed]

16. A. A. Farid and S. Hranilovic, “Outage capacity optimization for free-space optical links with pointing errors,” J. Lightwave Technol. 25(7), 1702–1710 (2007). [CrossRef]

17. H. G. Sandalidis, T. A. Tsiftsis, G. K. Karagiannidis, and M. Uysal, “BER performance of FSO links over strong atmospheric turbulence channels with pointing errors,” IEEE Commun. Lett. 12(1), 44–46 (2008). [CrossRef]

18. H. G. Sandalidis, “Coded free-space optical links over strong turbulence and misalignment fading channels,” IEEE Trans. Commun. 59(3), 669–674 (2011). [CrossRef]

19. D. K. Borah and D. G. Voelz, “Pointing error effects on free-space optical communication links in the presence of atmospheric turbulence,” J. Lightwave Technol. 27(18), 3965–3973 (2009). [CrossRef]

20. W. Gappmair, S. Hranilovic, and E. Leitgeb, “Performance of PPM on terrestrial FSO links with turbulence and pointing errors,” IEEE Commun. Lett. 14(5), 468–470 (2010). [CrossRef]

21. A. A. Farid and S. Hranilovic, “Diversity gains for MIMO wireless optical intensity channels with atmospheric fading and misalignment,” in Proc. IEEE GLOBECOM Workshops (GC Wkshps, 2010), pp. 1015–1019. [CrossRef]

22. M. A. Al-Habash, L. C. Andrews, and R. L. Phillips, “Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media,” Opt. Eng. 40, 1554–1562 (2001). [CrossRef]

23. A. García-Zambrana, C. Castillo-Vázquez, and B. Castillo-Vázquez, “Rate-adaptive FSO links over atmospheric turbulence channels by jointly using repetition coding and silence periods,” Opt. Express 18(24), 25422–25440 (2010). [CrossRef]   [PubMed]

24. B. Castillo-Vazquez, A. Garcia-Zambrana, and C. Castillo-Vazquez, “Closed-Form BER expression for FSO links with transmit laser selection over exponential atmospheric turbulence channels,” Electron. Lett. 45(23), 1185–1187 (2009). [CrossRef]

25. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th ed. (Dover, 1970).

26. M. K. Simon and M.-S. Alouini, Digital Communications over Fading Channels, 2nd ed. (Wiley-IEEE Press, 2005).

27. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 7th ed. (Academic Press Inc., 2007).

28. Z. Wang and G. B. Giannakis, “A simple and general parameterization quantifying performance in fading channels,” IEEE Trans. Commun. 51(8), 1389–1398 (2003). [CrossRef]

29. S. M. Navidpour, M. Uysal, and M. Kavehrad, “BER performance of free-space optical transmission with spatial diversity,” IEEE Trans. Wireless Commun. 6(8), 2813–2819 (2007). [CrossRef]

30. H. A. David and H. N. Nagaraja, Order Statistics, 3rd ed. (John Wiley and Sons Inc., 2003). [CrossRef]

31. H. G. Sandalidis, “Optimization models for misalignment fading mitigation in optical wireless links,” IEEE Commun. Lett. 12(5), 395–397 (2008). [CrossRef]

32. Wolfram Research Inc., Mathematica, version 8.0.1. ed. (Wolfram Research, Inc., Champaign, Illinois, 2011).

### References

• View by:
• |
• |
• |

1. J. M. Kahn and J. R. Barry, “Wireless infrared communications,” Proc. IEEE 85, 265–298 (1997).
[Crossref]
2. L. B. Stotts, L. C. Andrews, P. C. Cherry, J. J. Foshee, P. J. Kolodzy, W. K. McIntire, M. Northcott, R. L. Phillips, H. A. Pike, B. Stadler, and D. W. Young, “Hybrid optical RF airborne communications,” Proc. IEEE 97(6), 1109–1127 (2009).
[Crossref]
3. W. Lim, C. Yun, and K. Kim, “BER performance analysis of radio over free-space optical systems considering laser phase noise under gamma-gamma turbulence channels,” Opt. Express 17(6), 4479–4484 (2009).
[Crossref] [PubMed]
4. L. Andrews, R. Phillips, and C. Hopen, Laser Beam Scintillation with Applications (SPIE Press, 2001).
[Crossref]
5. X. Zhu and J. M. Kahn, “Free-space optical communication through atmospheric turbulence channels,” IEEE Trans. Commun. 50(8), 1293–1300 (2002).
[Crossref]
6. E. J. Lee and V. W. S. Chan, “Part 1: optical communication over the clear turbulent atmospheric channel using diversity,” IEEE J. Sel. Areas Commun. 22(9), 1896–1906 (2004).
[Crossref]
7. I. B. Djordjevic, S. Denic, J. Anguita, B. Vasic, and M. Neifeld, “LDPC-coded MIMO optical communication over the atmospheric turbulence channel,” J. Lightwave Technol. 26(5), 478–487 (2008).
[Crossref]
8. M. Simon and V. Vilnrotter, “Alamouti-type space-time coding for free-space optical communication with direct detection,” IEEE Trans. Wireless Commun. 4(1), 35–39 (2005).
[Crossref]
9. C. Abou-Rjeily and W. Fawaz, “Space-time codes for MIMO ultra-wideband communications and MIMO free-space optical communications with PPM,” IEEE J. Sel. Areas Commun. 26(6), 938–947 (2008).
[Crossref]
10. T. A. Tsiftsis, H. G. Sandalidis, G. K. Karagiannidis, and M. Uysal, “Optical wireless links with spatial diversity over strong atmospheric turbulence channels,” IEEE Trans. Wireless Commun. 8(2), 951–957 (2009).
[Crossref]
11. E. Bayaki and R. Schober, “On space-time coding for free-space optical systems,” IEEE Trans. Commun. 58(1), 58–62 (2010).
[Crossref]
12. A. Garcia-Zambrana, C. Castillo-Vazquez, B. Castillo-Vazquez, and A. Hiniesta-Gomez, “Selection transmit diversity for FSO links over strong atmospheric turbulence channels,” IEEE Photon. Technol. Lett. 21(14), 1017–1019 (2009).
[Crossref]
13. A. García-Zambrana, C. Castillo-Vázquez, and B. Castillo-Vázquez, “Space-time trellis coding with transmit laser selection for FSO links over strong atmospheric turbulence channels,” Opt. Express 18(6), 5356–5366 (2010).
[Crossref] [PubMed]
14. S. Arnon, “Optimization of urban optical wireless communication systems,” IEEE Trans. Wireless Commun. 2(4), 626–629 (2003).
[Crossref]
15. S. Arnon, “Effects of atmospheric turbulence and building sway on optical wireless-communication systems,” Opt. Lett. 28(2), 129–131 (2003).
[Crossref] [PubMed]
16. A. A. Farid and S. Hranilovic, “Outage capacity optimization for free-space optical links with pointing errors,” J. Lightwave Technol. 25(7), 1702–1710 (2007).
[Crossref]
17. H. G. Sandalidis, T. A. Tsiftsis, G. K. Karagiannidis, and M. Uysal, “BER performance of FSO links over strong atmospheric turbulence channels with pointing errors,” IEEE Commun. Lett. 12(1), 44–46 (2008).
[Crossref]
18. H. G. Sandalidis, “Coded free-space optical links over strong turbulence and misalignment fading channels,” IEEE Trans. Commun. 59(3), 669–674 (2011).
[Crossref]
19. D. K. Borah and D. G. Voelz, “Pointing error effects on free-space optical communication links in the presence of atmospheric turbulence,” J. Lightwave Technol. 27(18), 3965–3973 (2009).
[Crossref]
20. W. Gappmair, S. Hranilovic, and E. Leitgeb, “Performance of PPM on terrestrial FSO links with turbulence and pointing errors,” IEEE Commun. Lett. 14(5), 468–470 (2010).
[Crossref]
21. A. A. Farid and S. Hranilovic, “Diversity gains for MIMO wireless optical intensity channels with atmospheric fading and misalignment,” in Proc. IEEE GLOBECOM Workshops (GC Wkshps, 2010), pp. 1015–1019.
[Crossref]
22. M. A. Al-Habash, L. C. Andrews, and R. L. Phillips, “Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media,” Opt. Eng. 40, 1554–1562 (2001).
[Crossref]
23. A. García-Zambrana, C. Castillo-Vázquez, and B. Castillo-Vázquez, “Rate-adaptive FSO links over atmospheric turbulence channels by jointly using repetition coding and silence periods,” Opt. Express 18(24), 25422–25440 (2010).
[Crossref] [PubMed]
24. B. Castillo-Vazquez, A. Garcia-Zambrana, and C. Castillo-Vazquez, “Closed-Form BER expression for FSO links with transmit laser selection over exponential atmospheric turbulence channels,” Electron. Lett. 45(23), 1185–1187 (2009).
[Crossref]
25. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th ed. (Dover, 1970).
26. M. K. Simon and M.-S. Alouini, Digital Communications over Fading Channels, 2nd ed. (Wiley-IEEE Press, 2005).
27. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 7th ed. (Academic Press Inc., 2007).
28. Z. Wang and G. B. Giannakis, “A simple and general parameterization quantifying performance in fading channels,” IEEE Trans. Commun. 51(8), 1389–1398 (2003).
[Crossref]
29. S. M. Navidpour, M. Uysal, and M. Kavehrad, “BER performance of free-space optical transmission with spatial diversity,” IEEE Trans. Wireless Commun. 6(8), 2813–2819 (2007).
[Crossref]
30. H. A. David and H. N. Nagaraja, Order Statistics, 3rd ed. (John Wiley and Sons Inc., 2003).
[Crossref]
31. H. G. Sandalidis, “Optimization models for misalignment fading mitigation in optical wireless links,” IEEE Commun. Lett. 12(5), 395–397 (2008).
[Crossref]
32. Wolfram Research Inc., Mathematica, version 8.0.1. ed. (Wolfram Research, Inc., Champaign, Illinois, 2011).

#### 2011 (1)

H. G. Sandalidis, “Coded free-space optical links over strong turbulence and misalignment fading channels,” IEEE Trans. Commun. 59(3), 669–674 (2011).
[Crossref]

#### 2010 (4)

E. Bayaki and R. Schober, “On space-time coding for free-space optical systems,” IEEE Trans. Commun. 58(1), 58–62 (2010).
[Crossref]

W. Gappmair, S. Hranilovic, and E. Leitgeb, “Performance of PPM on terrestrial FSO links with turbulence and pointing errors,” IEEE Commun. Lett. 14(5), 468–470 (2010).
[Crossref]

#### 2009 (6)

B. Castillo-Vazquez, A. Garcia-Zambrana, and C. Castillo-Vazquez, “Closed-Form BER expression for FSO links with transmit laser selection over exponential atmospheric turbulence channels,” Electron. Lett. 45(23), 1185–1187 (2009).
[Crossref]

A. Garcia-Zambrana, C. Castillo-Vazquez, B. Castillo-Vazquez, and A. Hiniesta-Gomez, “Selection transmit diversity for FSO links over strong atmospheric turbulence channels,” IEEE Photon. Technol. Lett. 21(14), 1017–1019 (2009).
[Crossref]

L. B. Stotts, L. C. Andrews, P. C. Cherry, J. J. Foshee, P. J. Kolodzy, W. K. McIntire, M. Northcott, R. L. Phillips, H. A. Pike, B. Stadler, and D. W. Young, “Hybrid optical RF airborne communications,” Proc. IEEE 97(6), 1109–1127 (2009).
[Crossref]

T. A. Tsiftsis, H. G. Sandalidis, G. K. Karagiannidis, and M. Uysal, “Optical wireless links with spatial diversity over strong atmospheric turbulence channels,” IEEE Trans. Wireless Commun. 8(2), 951–957 (2009).
[Crossref]

#### 2008 (4)

C. Abou-Rjeily and W. Fawaz, “Space-time codes for MIMO ultra-wideband communications and MIMO free-space optical communications with PPM,” IEEE J. Sel. Areas Commun. 26(6), 938–947 (2008).
[Crossref]

H. G. Sandalidis, T. A. Tsiftsis, G. K. Karagiannidis, and M. Uysal, “BER performance of FSO links over strong atmospheric turbulence channels with pointing errors,” IEEE Commun. Lett. 12(1), 44–46 (2008).
[Crossref]

H. G. Sandalidis, “Optimization models for misalignment fading mitigation in optical wireless links,” IEEE Commun. Lett. 12(5), 395–397 (2008).
[Crossref]

#### 2007 (2)

S. M. Navidpour, M. Uysal, and M. Kavehrad, “BER performance of free-space optical transmission with spatial diversity,” IEEE Trans. Wireless Commun. 6(8), 2813–2819 (2007).
[Crossref]

#### 2005 (1)

M. Simon and V. Vilnrotter, “Alamouti-type space-time coding for free-space optical communication with direct detection,” IEEE Trans. Wireless Commun. 4(1), 35–39 (2005).
[Crossref]

#### 2004 (1)

E. J. Lee and V. W. S. Chan, “Part 1: optical communication over the clear turbulent atmospheric channel using diversity,” IEEE J. Sel. Areas Commun. 22(9), 1896–1906 (2004).
[Crossref]

#### 2003 (3)

S. Arnon, “Optimization of urban optical wireless communication systems,” IEEE Trans. Wireless Commun. 2(4), 626–629 (2003).
[Crossref]

Z. Wang and G. B. Giannakis, “A simple and general parameterization quantifying performance in fading channels,” IEEE Trans. Commun. 51(8), 1389–1398 (2003).
[Crossref]

#### 2002 (1)

X. Zhu and J. M. Kahn, “Free-space optical communication through atmospheric turbulence channels,” IEEE Trans. Commun. 50(8), 1293–1300 (2002).
[Crossref]

#### 2001 (1)

M. A. Al-Habash, L. C. Andrews, and R. L. Phillips, “Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media,” Opt. Eng. 40, 1554–1562 (2001).
[Crossref]

#### 1997 (1)

J. M. Kahn and J. R. Barry, “Wireless infrared communications,” Proc. IEEE 85, 265–298 (1997).
[Crossref]

#### Abou-Rjeily, C.

C. Abou-Rjeily and W. Fawaz, “Space-time codes for MIMO ultra-wideband communications and MIMO free-space optical communications with PPM,” IEEE J. Sel. Areas Commun. 26(6), 938–947 (2008).
[Crossref]

#### Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th ed. (Dover, 1970).

#### Al-Habash, M. A.

M. A. Al-Habash, L. C. Andrews, and R. L. Phillips, “Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media,” Opt. Eng. 40, 1554–1562 (2001).
[Crossref]

#### Alouini, M.-S.

M. K. Simon and M.-S. Alouini, Digital Communications over Fading Channels, 2nd ed. (Wiley-IEEE Press, 2005).

#### Andrews, L.

L. Andrews, R. Phillips, and C. Hopen, Laser Beam Scintillation with Applications (SPIE Press, 2001).
[Crossref]

#### Andrews, L. C.

L. B. Stotts, L. C. Andrews, P. C. Cherry, J. J. Foshee, P. J. Kolodzy, W. K. McIntire, M. Northcott, R. L. Phillips, H. A. Pike, B. Stadler, and D. W. Young, “Hybrid optical RF airborne communications,” Proc. IEEE 97(6), 1109–1127 (2009).
[Crossref]

M. A. Al-Habash, L. C. Andrews, and R. L. Phillips, “Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media,” Opt. Eng. 40, 1554–1562 (2001).
[Crossref]

#### Arnon, S.

S. Arnon, “Optimization of urban optical wireless communication systems,” IEEE Trans. Wireless Commun. 2(4), 626–629 (2003).
[Crossref]

#### Barry, J. R.

J. M. Kahn and J. R. Barry, “Wireless infrared communications,” Proc. IEEE 85, 265–298 (1997).
[Crossref]

#### Bayaki, E.

E. Bayaki and R. Schober, “On space-time coding for free-space optical systems,” IEEE Trans. Commun. 58(1), 58–62 (2010).
[Crossref]

#### Castillo-Vazquez, B.

B. Castillo-Vazquez, A. Garcia-Zambrana, and C. Castillo-Vazquez, “Closed-Form BER expression for FSO links with transmit laser selection over exponential atmospheric turbulence channels,” Electron. Lett. 45(23), 1185–1187 (2009).
[Crossref]

A. Garcia-Zambrana, C. Castillo-Vazquez, B. Castillo-Vazquez, and A. Hiniesta-Gomez, “Selection transmit diversity for FSO links over strong atmospheric turbulence channels,” IEEE Photon. Technol. Lett. 21(14), 1017–1019 (2009).
[Crossref]

#### Castillo-Vazquez, C.

A. Garcia-Zambrana, C. Castillo-Vazquez, B. Castillo-Vazquez, and A. Hiniesta-Gomez, “Selection transmit diversity for FSO links over strong atmospheric turbulence channels,” IEEE Photon. Technol. Lett. 21(14), 1017–1019 (2009).
[Crossref]

B. Castillo-Vazquez, A. Garcia-Zambrana, and C. Castillo-Vazquez, “Closed-Form BER expression for FSO links with transmit laser selection over exponential atmospheric turbulence channels,” Electron. Lett. 45(23), 1185–1187 (2009).
[Crossref]

#### Chan, V. W. S.

E. J. Lee and V. W. S. Chan, “Part 1: optical communication over the clear turbulent atmospheric channel using diversity,” IEEE J. Sel. Areas Commun. 22(9), 1896–1906 (2004).
[Crossref]

#### Cherry, P. C.

L. B. Stotts, L. C. Andrews, P. C. Cherry, J. J. Foshee, P. J. Kolodzy, W. K. McIntire, M. Northcott, R. L. Phillips, H. A. Pike, B. Stadler, and D. W. Young, “Hybrid optical RF airborne communications,” Proc. IEEE 97(6), 1109–1127 (2009).
[Crossref]

#### David, H. A.

H. A. David and H. N. Nagaraja, Order Statistics, 3rd ed. (John Wiley and Sons Inc., 2003).
[Crossref]

#### Farid, A. A.

A. A. Farid and S. Hranilovic, “Diversity gains for MIMO wireless optical intensity channels with atmospheric fading and misalignment,” in Proc. IEEE GLOBECOM Workshops (GC Wkshps, 2010), pp. 1015–1019.
[Crossref]

#### Fawaz, W.

C. Abou-Rjeily and W. Fawaz, “Space-time codes for MIMO ultra-wideband communications and MIMO free-space optical communications with PPM,” IEEE J. Sel. Areas Commun. 26(6), 938–947 (2008).
[Crossref]

#### Foshee, J. J.

L. B. Stotts, L. C. Andrews, P. C. Cherry, J. J. Foshee, P. J. Kolodzy, W. K. McIntire, M. Northcott, R. L. Phillips, H. A. Pike, B. Stadler, and D. W. Young, “Hybrid optical RF airborne communications,” Proc. IEEE 97(6), 1109–1127 (2009).
[Crossref]

#### Gappmair, W.

W. Gappmair, S. Hranilovic, and E. Leitgeb, “Performance of PPM on terrestrial FSO links with turbulence and pointing errors,” IEEE Commun. Lett. 14(5), 468–470 (2010).
[Crossref]

#### Garcia-Zambrana, A.

B. Castillo-Vazquez, A. Garcia-Zambrana, and C. Castillo-Vazquez, “Closed-Form BER expression for FSO links with transmit laser selection over exponential atmospheric turbulence channels,” Electron. Lett. 45(23), 1185–1187 (2009).
[Crossref]

A. Garcia-Zambrana, C. Castillo-Vazquez, B. Castillo-Vazquez, and A. Hiniesta-Gomez, “Selection transmit diversity for FSO links over strong atmospheric turbulence channels,” IEEE Photon. Technol. Lett. 21(14), 1017–1019 (2009).
[Crossref]

#### Giannakis, G. B.

Z. Wang and G. B. Giannakis, “A simple and general parameterization quantifying performance in fading channels,” IEEE Trans. Commun. 51(8), 1389–1398 (2003).
[Crossref]

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 7th ed. (Academic Press Inc., 2007).

#### Hiniesta-Gomez, A.

A. Garcia-Zambrana, C. Castillo-Vazquez, B. Castillo-Vazquez, and A. Hiniesta-Gomez, “Selection transmit diversity for FSO links over strong atmospheric turbulence channels,” IEEE Photon. Technol. Lett. 21(14), 1017–1019 (2009).
[Crossref]

#### Hopen, C.

L. Andrews, R. Phillips, and C. Hopen, Laser Beam Scintillation with Applications (SPIE Press, 2001).
[Crossref]

#### Hranilovic, S.

W. Gappmair, S. Hranilovic, and E. Leitgeb, “Performance of PPM on terrestrial FSO links with turbulence and pointing errors,” IEEE Commun. Lett. 14(5), 468–470 (2010).
[Crossref]

A. A. Farid and S. Hranilovic, “Diversity gains for MIMO wireless optical intensity channels with atmospheric fading and misalignment,” in Proc. IEEE GLOBECOM Workshops (GC Wkshps, 2010), pp. 1015–1019.
[Crossref]

#### Kahn, J. M.

X. Zhu and J. M. Kahn, “Free-space optical communication through atmospheric turbulence channels,” IEEE Trans. Commun. 50(8), 1293–1300 (2002).
[Crossref]

J. M. Kahn and J. R. Barry, “Wireless infrared communications,” Proc. IEEE 85, 265–298 (1997).
[Crossref]

#### Karagiannidis, G. K.

T. A. Tsiftsis, H. G. Sandalidis, G. K. Karagiannidis, and M. Uysal, “Optical wireless links with spatial diversity over strong atmospheric turbulence channels,” IEEE Trans. Wireless Commun. 8(2), 951–957 (2009).
[Crossref]

H. G. Sandalidis, T. A. Tsiftsis, G. K. Karagiannidis, and M. Uysal, “BER performance of FSO links over strong atmospheric turbulence channels with pointing errors,” IEEE Commun. Lett. 12(1), 44–46 (2008).
[Crossref]

S. M. Navidpour, M. Uysal, and M. Kavehrad, “BER performance of free-space optical transmission with spatial diversity,” IEEE Trans. Wireless Commun. 6(8), 2813–2819 (2007).
[Crossref]

#### Kolodzy, P. J.

L. B. Stotts, L. C. Andrews, P. C. Cherry, J. J. Foshee, P. J. Kolodzy, W. K. McIntire, M. Northcott, R. L. Phillips, H. A. Pike, B. Stadler, and D. W. Young, “Hybrid optical RF airborne communications,” Proc. IEEE 97(6), 1109–1127 (2009).
[Crossref]

#### Lee, E. J.

E. J. Lee and V. W. S. Chan, “Part 1: optical communication over the clear turbulent atmospheric channel using diversity,” IEEE J. Sel. Areas Commun. 22(9), 1896–1906 (2004).
[Crossref]

#### Leitgeb, E.

W. Gappmair, S. Hranilovic, and E. Leitgeb, “Performance of PPM on terrestrial FSO links with turbulence and pointing errors,” IEEE Commun. Lett. 14(5), 468–470 (2010).
[Crossref]

#### McIntire, W. K.

L. B. Stotts, L. C. Andrews, P. C. Cherry, J. J. Foshee, P. J. Kolodzy, W. K. McIntire, M. Northcott, R. L. Phillips, H. A. Pike, B. Stadler, and D. W. Young, “Hybrid optical RF airborne communications,” Proc. IEEE 97(6), 1109–1127 (2009).
[Crossref]

#### Nagaraja, H. N.

H. A. David and H. N. Nagaraja, Order Statistics, 3rd ed. (John Wiley and Sons Inc., 2003).
[Crossref]

#### Navidpour, S. M.

S. M. Navidpour, M. Uysal, and M. Kavehrad, “BER performance of free-space optical transmission with spatial diversity,” IEEE Trans. Wireless Commun. 6(8), 2813–2819 (2007).
[Crossref]

#### Northcott, M.

L. B. Stotts, L. C. Andrews, P. C. Cherry, J. J. Foshee, P. J. Kolodzy, W. K. McIntire, M. Northcott, R. L. Phillips, H. A. Pike, B. Stadler, and D. W. Young, “Hybrid optical RF airborne communications,” Proc. IEEE 97(6), 1109–1127 (2009).
[Crossref]

#### Phillips, R.

L. Andrews, R. Phillips, and C. Hopen, Laser Beam Scintillation with Applications (SPIE Press, 2001).
[Crossref]

#### Phillips, R. L.

L. B. Stotts, L. C. Andrews, P. C. Cherry, J. J. Foshee, P. J. Kolodzy, W. K. McIntire, M. Northcott, R. L. Phillips, H. A. Pike, B. Stadler, and D. W. Young, “Hybrid optical RF airborne communications,” Proc. IEEE 97(6), 1109–1127 (2009).
[Crossref]

M. A. Al-Habash, L. C. Andrews, and R. L. Phillips, “Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media,” Opt. Eng. 40, 1554–1562 (2001).
[Crossref]

#### Pike, H. A.

L. B. Stotts, L. C. Andrews, P. C. Cherry, J. J. Foshee, P. J. Kolodzy, W. K. McIntire, M. Northcott, R. L. Phillips, H. A. Pike, B. Stadler, and D. W. Young, “Hybrid optical RF airborne communications,” Proc. IEEE 97(6), 1109–1127 (2009).
[Crossref]

#### Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 7th ed. (Academic Press Inc., 2007).

#### Sandalidis, H. G.

H. G. Sandalidis, “Coded free-space optical links over strong turbulence and misalignment fading channels,” IEEE Trans. Commun. 59(3), 669–674 (2011).
[Crossref]

T. A. Tsiftsis, H. G. Sandalidis, G. K. Karagiannidis, and M. Uysal, “Optical wireless links with spatial diversity over strong atmospheric turbulence channels,” IEEE Trans. Wireless Commun. 8(2), 951–957 (2009).
[Crossref]

H. G. Sandalidis, T. A. Tsiftsis, G. K. Karagiannidis, and M. Uysal, “BER performance of FSO links over strong atmospheric turbulence channels with pointing errors,” IEEE Commun. Lett. 12(1), 44–46 (2008).
[Crossref]

H. G. Sandalidis, “Optimization models for misalignment fading mitigation in optical wireless links,” IEEE Commun. Lett. 12(5), 395–397 (2008).
[Crossref]

#### Schober, R.

E. Bayaki and R. Schober, “On space-time coding for free-space optical systems,” IEEE Trans. Commun. 58(1), 58–62 (2010).
[Crossref]

#### Simon, M.

M. Simon and V. Vilnrotter, “Alamouti-type space-time coding for free-space optical communication with direct detection,” IEEE Trans. Wireless Commun. 4(1), 35–39 (2005).
[Crossref]

#### Simon, M. K.

M. K. Simon and M.-S. Alouini, Digital Communications over Fading Channels, 2nd ed. (Wiley-IEEE Press, 2005).

L. B. Stotts, L. C. Andrews, P. C. Cherry, J. J. Foshee, P. J. Kolodzy, W. K. McIntire, M. Northcott, R. L. Phillips, H. A. Pike, B. Stadler, and D. W. Young, “Hybrid optical RF airborne communications,” Proc. IEEE 97(6), 1109–1127 (2009).
[Crossref]

#### Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th ed. (Dover, 1970).

#### Stotts, L. B.

L. B. Stotts, L. C. Andrews, P. C. Cherry, J. J. Foshee, P. J. Kolodzy, W. K. McIntire, M. Northcott, R. L. Phillips, H. A. Pike, B. Stadler, and D. W. Young, “Hybrid optical RF airborne communications,” Proc. IEEE 97(6), 1109–1127 (2009).
[Crossref]

#### Tsiftsis, T. A.

T. A. Tsiftsis, H. G. Sandalidis, G. K. Karagiannidis, and M. Uysal, “Optical wireless links with spatial diversity over strong atmospheric turbulence channels,” IEEE Trans. Wireless Commun. 8(2), 951–957 (2009).
[Crossref]

H. G. Sandalidis, T. A. Tsiftsis, G. K. Karagiannidis, and M. Uysal, “BER performance of FSO links over strong atmospheric turbulence channels with pointing errors,” IEEE Commun. Lett. 12(1), 44–46 (2008).
[Crossref]

#### Uysal, M.

T. A. Tsiftsis, H. G. Sandalidis, G. K. Karagiannidis, and M. Uysal, “Optical wireless links with spatial diversity over strong atmospheric turbulence channels,” IEEE Trans. Wireless Commun. 8(2), 951–957 (2009).
[Crossref]

H. G. Sandalidis, T. A. Tsiftsis, G. K. Karagiannidis, and M. Uysal, “BER performance of FSO links over strong atmospheric turbulence channels with pointing errors,” IEEE Commun. Lett. 12(1), 44–46 (2008).
[Crossref]

S. M. Navidpour, M. Uysal, and M. Kavehrad, “BER performance of free-space optical transmission with spatial diversity,” IEEE Trans. Wireless Commun. 6(8), 2813–2819 (2007).
[Crossref]

#### Vilnrotter, V.

M. Simon and V. Vilnrotter, “Alamouti-type space-time coding for free-space optical communication with direct detection,” IEEE Trans. Wireless Commun. 4(1), 35–39 (2005).
[Crossref]

#### Wang, Z.

Z. Wang and G. B. Giannakis, “A simple and general parameterization quantifying performance in fading channels,” IEEE Trans. Commun. 51(8), 1389–1398 (2003).
[Crossref]

#### Young, D. W.

L. B. Stotts, L. C. Andrews, P. C. Cherry, J. J. Foshee, P. J. Kolodzy, W. K. McIntire, M. Northcott, R. L. Phillips, H. A. Pike, B. Stadler, and D. W. Young, “Hybrid optical RF airborne communications,” Proc. IEEE 97(6), 1109–1127 (2009).
[Crossref]

#### Zhu, X.

X. Zhu and J. M. Kahn, “Free-space optical communication through atmospheric turbulence channels,” IEEE Trans. Commun. 50(8), 1293–1300 (2002).
[Crossref]

#### Electron. Lett. (1)

B. Castillo-Vazquez, A. Garcia-Zambrana, and C. Castillo-Vazquez, “Closed-Form BER expression for FSO links with transmit laser selection over exponential atmospheric turbulence channels,” Electron. Lett. 45(23), 1185–1187 (2009).
[Crossref]

#### IEEE Commun. Lett. (3)

H. G. Sandalidis, T. A. Tsiftsis, G. K. Karagiannidis, and M. Uysal, “BER performance of FSO links over strong atmospheric turbulence channels with pointing errors,” IEEE Commun. Lett. 12(1), 44–46 (2008).
[Crossref]

W. Gappmair, S. Hranilovic, and E. Leitgeb, “Performance of PPM on terrestrial FSO links with turbulence and pointing errors,” IEEE Commun. Lett. 14(5), 468–470 (2010).
[Crossref]

H. G. Sandalidis, “Optimization models for misalignment fading mitigation in optical wireless links,” IEEE Commun. Lett. 12(5), 395–397 (2008).
[Crossref]

#### IEEE J. Sel. Areas Commun. (2)

E. J. Lee and V. W. S. Chan, “Part 1: optical communication over the clear turbulent atmospheric channel using diversity,” IEEE J. Sel. Areas Commun. 22(9), 1896–1906 (2004).
[Crossref]

C. Abou-Rjeily and W. Fawaz, “Space-time codes for MIMO ultra-wideband communications and MIMO free-space optical communications with PPM,” IEEE J. Sel. Areas Commun. 26(6), 938–947 (2008).
[Crossref]

#### IEEE Photon. Technol. Lett. (1)

A. Garcia-Zambrana, C. Castillo-Vazquez, B. Castillo-Vazquez, and A. Hiniesta-Gomez, “Selection transmit diversity for FSO links over strong atmospheric turbulence channels,” IEEE Photon. Technol. Lett. 21(14), 1017–1019 (2009).
[Crossref]

#### IEEE Trans. Commun. (4)

E. Bayaki and R. Schober, “On space-time coding for free-space optical systems,” IEEE Trans. Commun. 58(1), 58–62 (2010).
[Crossref]

X. Zhu and J. M. Kahn, “Free-space optical communication through atmospheric turbulence channels,” IEEE Trans. Commun. 50(8), 1293–1300 (2002).
[Crossref]

Z. Wang and G. B. Giannakis, “A simple and general parameterization quantifying performance in fading channels,” IEEE Trans. Commun. 51(8), 1389–1398 (2003).
[Crossref]

H. G. Sandalidis, “Coded free-space optical links over strong turbulence and misalignment fading channels,” IEEE Trans. Commun. 59(3), 669–674 (2011).
[Crossref]

#### IEEE Trans. Wireless Commun. (4)

S. M. Navidpour, M. Uysal, and M. Kavehrad, “BER performance of free-space optical transmission with spatial diversity,” IEEE Trans. Wireless Commun. 6(8), 2813–2819 (2007).
[Crossref]

M. Simon and V. Vilnrotter, “Alamouti-type space-time coding for free-space optical communication with direct detection,” IEEE Trans. Wireless Commun. 4(1), 35–39 (2005).
[Crossref]

S. Arnon, “Optimization of urban optical wireless communication systems,” IEEE Trans. Wireless Commun. 2(4), 626–629 (2003).
[Crossref]

T. A. Tsiftsis, H. G. Sandalidis, G. K. Karagiannidis, and M. Uysal, “Optical wireless links with spatial diversity over strong atmospheric turbulence channels,” IEEE Trans. Wireless Commun. 8(2), 951–957 (2009).
[Crossref]

#### Opt. Eng. (1)

M. A. Al-Habash, L. C. Andrews, and R. L. Phillips, “Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media,” Opt. Eng. 40, 1554–1562 (2001).
[Crossref]

#### Proc. IEEE (2)

J. M. Kahn and J. R. Barry, “Wireless infrared communications,” Proc. IEEE 85, 265–298 (1997).
[Crossref]

L. B. Stotts, L. C. Andrews, P. C. Cherry, J. J. Foshee, P. J. Kolodzy, W. K. McIntire, M. Northcott, R. L. Phillips, H. A. Pike, B. Stadler, and D. W. Young, “Hybrid optical RF airborne communications,” Proc. IEEE 97(6), 1109–1127 (2009).
[Crossref]

#### Other (7)

L. Andrews, R. Phillips, and C. Hopen, Laser Beam Scintillation with Applications (SPIE Press, 2001).
[Crossref]

H. A. David and H. N. Nagaraja, Order Statistics, 3rd ed. (John Wiley and Sons Inc., 2003).
[Crossref]

Wolfram Research Inc., Mathematica, version 8.0.1. ed. (Wolfram Research, Inc., Champaign, Illinois, 2011).

A. A. Farid and S. Hranilovic, “Diversity gains for MIMO wireless optical intensity channels with atmospheric fading and misalignment,” in Proc. IEEE GLOBECOM Workshops (GC Wkshps, 2010), pp. 1015–1019.
[Crossref]

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th ed. (Dover, 1970).

M. K. Simon and M.-S. Alouini, Digital Communications over Fading Channels, 2nd ed. (Wiley-IEEE Press, 2005).

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 7th ed. (Academic Press Inc., 2007).

### Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

### Figures (5)

Fig. 1

Probability of outage versus normalized average SNR in FSO IM/DD links over the exponential atmospheric turbulence channel with pointing errors, assuming different values of normalized beamwidth, ωz /r, and normalized jitter, σs /r.

Fig. 2

Probability of outage versus normalized average SNR in MISO and SIMO FSO IM/DD links over the exponential atmospheric turbulence channel with pointing errors, assuming a normalized beamwidth of ωz /r = 5 and values of normalized jitter of (a) σs /r = 1 and (b) σs /r = 4.

Fig. 3

Coding gain advantage for the SC scheme relative to the EGC scheme, assuming a normalized beamwidth of ωz /r = 10 and values of normalized jitter of 6, 7, 8 and 9, corresponding to values of φ of 0.83, 0.71, 0.62 and 0.55, respectively.

Fig. 4

Probability of outage versus normalized average SNR in MIMO FSO IM/DD links over the exponential atmospheric turbulence channel with pointing errors, assuming a normalized beamwidth of ωz /r = 5 with a normalized jitter of σs /r = 1 and a normalized beamwidth of ωz /r = 10 with a normalized jitter of σs /r = 7, corresponding to values of φ of 2.55 and 0.71, respectively. Additionally, results assuming the optimum beamwidth corresponding to values of jitter of σs /r = 1 and σs /r = 7 are also included for L = 4 and M = 2.

Fig. 5

Optimum normalized beamwidth versus normalized jitter, σs /r in FSO IM/DD links over the exponential atmospheric turbulence channel with pointing errors.

### Equations (48)

$y lm ( t ) = η i lm ( t ) x ( t ) + z m ( t )$
$f I lm ( a ) ( i ) = e − i , i ≥ 0.$
$f I lm ( p ) ( i ) = φ 2 A 0 φ 2 i φ 2 − 1 , 0 ≤ i ≤ A 0$
$f I lm ( i ) = φ 2 A 0 φ 2 i φ 2 − 1 Γ ( 1 − φ 2 , i A 0 ) , i ≥ 0$
$P out = Prob ( γ T ≤ γ th ) = ∫ 0 γ th f γ T ( γ T ) d γ T = F γ T ( γ th )$
$Y = XI + Z , X ∈ { 0 , d } , Z ∼ N ( 0 , N 0 / 2 ) .$
$γ T SISO ( i ) = 1 2 d 2 N 0 / 2 i 2 = 4 P opt 2 T b ξ N 0 i 2 = γ ξ i 2$
$P out = Prob ( γ ξ i 2 ≤ γ th ) = Prob ( i 2 ≤ γ th ξ γ ) = F I ( 1 ξ γ th γ )$
$F I ( i ) = 1 − ( i A 0 ) φ 2 φ 2 Γ ( − φ 2 , i A 0 ) , i ≥ 0$
$P out ≐ ( φ 4 ( 1 − φ 2 ) 2 A 0 2 ξ γ th γ ) 1 / 2 + ( Γ ( 1 − φ 2 ) − 2 φ 2 A 0 2 ξ γ th γ ) φ 2 / 2 .$
$P out ≐ ( φ 4 ( 1 − φ 2 ) 2 A 0 2 ξ γ th γ ) 1 / 2 , φ > 1$
$P out ≐ ( Γ ( 1 − φ 2 ) 2 / φ 2 A 0 2 ξ γ th γ ) φ 2 / 2 , φ < 1$
$D [ dB ] ≜ 20 log 10 ( φ 2 A 0 ( φ 2 − 1 ) ) .$
$Y = X 1 L ∑ l = 1 L I l + Z , X ∈ { 0 , d } , Z ∼ N ( 0 , N 0 / 2 )$
$γ T MISO-RC = γ ξ 1 L 2 ( ∑ l = 1 L i l ) 2 = γ ξ 1 L 2 i T 2$
$P out = F I T ( L 1 ξ γ th γ )$
$f I lm ( i ) ≈ φ 2 A 0 ( φ 2 − 1 ) , φ > 1$
$f I lm ( i ) ≈ φ 2 Γ ( 1 − φ 2 ) A 0 φ 2 i φ 2 − 1 , φ < 1$
$F I T ( i ) ≈ 1 Γ ( L + 1 ) ( φ 2 A 0 ( φ 2 − 1 ) ) L i L , φ > 1$
$F I T ( i ) ≈ Γ ( 1 + φ 2 ) L Γ ( 1 + L φ 2 ) ( Γ ( 1 − φ 2 ) A 0 φ 2 ) L i L φ 2 , φ < 1$
$P out ≐ ( L 2 Γ ( L + 1 ) 2 / L φ 4 A 0 2 ( 1 − φ 2 ) 2 ξ γ th γ ) L / 2 , φ > 1$
$P out ≐ ( L 2 Γ ( 1 + φ 2 ) 2 / φ 2 Γ ( 1 + L φ 2 ) 2 / L φ 2 Γ ( 1 − φ 2 ) 2 / φ 2 A 0 2 ξ γ th γ ) L φ 2 / 2 , φ < 1$
$Y = X I max + Z , X ∈ { 0 , d } , Z ∼ N ( 0 , N 0 / 2 )$
$P out = F I max ( 1 ξ γ th γ ) = ( F I ( 1 ξ γ th γ ) ) L$
$P out ≐ ( φ 4 ( 1 − φ 2 ) 2 A 0 2 ξ γ th γ ) L / 2 , φ > 1$
$P out ≐ ( Γ ( 1 − φ 2 ) 2 / φ 2 A 0 2 ξ γ th γ ) L φ 2 / 2 , φ < 1$
$A TLS [ dB ] ≜ 10 log 10 ( L 2 Γ ( L + 1 ) 2 / L ) , φ > 1$
$A TLS [ dB ] ≜ 10 log 10 ( L 2 Γ ( 1 + φ 2 ) 2 / φ 2 Γ ( 1 + L φ 2 ) 2 / L φ 2 ) , φ < 1$
$Y = X 1 M ∑ m = 1 M I m + Z EGC , X ∈ { 0 , d } , Z EGC ∼ N ( 0 , N 0 / 2 )$
$Y = X 1 M I max + Z SC , X ∈ { 0 , d } , Z SC ∼ N ( 0 , N 0 / 2 M )$
$γ T SIMO-SC = γ ξ 1 M I max 2$
$P out = F I max ( M ξ γ th γ ) = ( F I ( M ξ γ th γ ) ) M .$
$P out ≐ ( M φ 4 ( 1 − φ 2 ) 2 A 0 2 ξ γ th γ ) M / 2 , φ > 1$
$P out ≐ ( M Γ ( 1 − φ 2 ) 2 / φ 2 A 0 2 ξ γ th γ ) M φ 2 / 2 , φ < 1$
$A EGC [ dB ] ≜ 10 log 10 ( Γ ( M + 1 ) 2 / M M ) , φ > 1$
$A SC [ dB ] ≜ 10 log 10 ( M Γ ( 1 + φ 2 ) 2 / φ 2 Γ ( 1 + M φ 2 ) 2 / M φ 2 ) , φ < 1$
$Y = X 1 M ∑ m = 1 M I m + Z EGC , X ∈ { 0 , d } , Z EGC ∼ N ( 0 , N 0 / 2 )$
$γ T MIMO = γ ξ 1 M 2 ( ∑ m = 1 M i m ) 2 = γ ξ 1 M 2 i T 2$
$P out = F I T ( M 1 ξ γ th γ )$
$f I m ( i ) ≈ L ( φ 2 ( φ 2 − 1 ) A 0 ) L i L − 1 , φ > 1$
$f I m ( i ) ≈ L φ 2 ( Γ ( 1 − φ 2 ) A 0 φ 2 ) L i L φ 2 − 1 , φ < 1$
$F I T ( i ) ≈ ( L Γ ( L ) ) M Γ ( LM + 1 ) ( φ 2 ( φ 2 − 1 ) A 0 ) LM i LM , φ > 1$
$F I T ( i ) ≈ Γ ( L φ 2 + 1 ) M Γ ( LM φ 2 + 1 ) ( Γ ( 1 − φ 2 ) A 0 φ 2 ) LM i LM φ 2 , φ < 1$
$P out ≐ ( ( M 2 Γ ( L + 1 ) 2 / L Γ ( LM + 1 ) 2 / LM φ 4 ( φ 2 − 1 ) 2 A 0 2 ξ γ th γ ) LM 2 , φ > 1$
$P out ≐ ( M 2 Γ ( L φ 2 + 1 ) 2 / L φ 2 Γ ( LM φ 2 + 1 ) 2 / LM φ 2 Γ ( 1 − φ 2 ) 2 / φ 2 A 0 2 ξ γ th γ ) LM φ 2 / 2 , φ < 1$
$D MIMO [ dB ] ≜ 20 log 10 ( M Γ ( O d / M + 1 ) M / O d Γ ( O d + 1 ) 1 / O d ) .$
$ω z / r optimum ≈ 2.85 ( σ s / r − 1 ) + 2.6$
$x ( t ) = ∑ k = − ∞ ∞ a k 2 T b P opt G ( f = 0 ) g ( t − k T b )$