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Coherent Wireless Optical Communications With Predetection and Postdetection EGC Over Gamma–Gamma Atmospheric Turbulence Channels Mingbo Niu, Josh Schlenker, Julian Cheng, Jonathan F. Holzman, and Robert Schober
Abstract—Wireless optical communication systems with coherent detection are analyzed for Gamma–Gamma distributed turbulence channels. In addition to the shot noise, we consider the impacts of both turbulence amplitude fluctuations and phase fluctuations on the error performance. Error rate analyses of predetection and postdetection equal gain combining (EGC) are carried out. We derive the exact error rate expressions for predetection and postdetection EGC using a characteristic function method. In the case of predetection EGC, we also study the impact of phase noise compensation error on the error rate performance. It is shown that the error rate performance of predetection EGC is sensitive to phase noise compensation errors for both weak and strong turbulence conditions. In order to alleviate the impact of phase noise, postdetection EGC with differential phase-shift keying is introduced and analyzed. In addition, postdetection EGC is compared with predetection EGC in the presence of phase noise compensation errors, and it is found to be an effective alternative to predetection EGC with low complexity implementation. Index Terms—Atmospheric turbulence; Coherent detection; Diversity; Optical communications; Phase noise.
I. I NTRODUCTION ireless optical communication, also known as free-space optical (FSO) communication, is a cost-effective, highly secure, and license-free wireless technology [1–5]. FSO communication is particularly attractive in dense urban areas with no pre-existing fiber optic infrastructure. There are, however, challenges associated with atmospheric amplitude and phase distortion. When an optical wave propagates through an atmospheric turbulence channel, both the amplitude and the phase of the electrical field suffer from random fluctuations. Such fluctuations are caused by the variation of the refractive index due to temperature and pressure changes in the wireless channels. The received optical signals can suffer
W
Manuscript received April 1, 2011; revised July 21, 2011; accepted October 7, 2011; published October 27, 2011 (Doc. ID 145234). Mingbo Niu, Julian Cheng (e-mail:
[email protected]), and Jonathan F. Holzman are with the School of Engineering, University of British Columbia, Kelowna, British Columbia V1V 1V7, Canada. Josh Schlenker and Robert Schober are with the Department of Electrical and Computer Engineering, University of British Columbia, Vancouver, British Columbia V6T 1Z4, Canada. Digital Object Identifier 10.1364/JOCN.3.000860
1943-0620/11/110860-10/$15.00
from this turbulence, resulting in severe system performance degradation. Recently, it has been shown that the impact of turbulenceinduced fading in irradiance modulation with direct detection (IM/DD) can be mitigated through the use of diversity reception [5–7]. In the literature, IM/DD with on–off keying (OOK) often assumes adaptive detection thresholds. Such systems, while feasible, can be costly and subject to channel state estimation errors, leading to incorrect detection thresholds [8,9]. As an alternative, IM/DD OOK-based FSO systems with fixed detection thresholds are easier to implement, compared to those with adaptive thresholds, but they suffer from irreducible error floors [9]. High transmitted powers are required in strong turbulence, and this can lead to high operational costs. To overcome the shortcomings of IM/DD-based FSO systems, coherent FSO communication with homodyne/heterodyne detection1 is an attractive alternative technology to mitigate the adverse effects of turbulence channels and improve the overall system performance [11–17]. These coherent FSO systems, with or without phase compensation, ultimately offer excellent background noise rejection, higher sensitivity, and improved spectral efficiency with provisions for a variety of modulation formats. More importantly, coherent FSO systems do not use adaptive detection thresholds and, unlike IM/DD OOK with a fixed threshold, do not have irreducible error floors when the phase noise is perfectly compensated. The benefits in power efficiency and background noise rejection of coherent FSO communications can lead to a better system performance compared to IM/DD-based FSO systems in turbulent FSO channels. Coherent FSO communication systems must accommodate the presence of amplitude distortion (fading) of the atmospheric channels. The evaluation of the error rate performance in such a system will therefore require a statistical model of turbulence-induced fading. The most popular statistical models in the FSO literature to describe this atmospheric turbulence are lognormal, K, and Gamma–Gamma distributions [18]. Weak turbulence is typically modeled with lognormal distribution models for propagation distances within several hundred meters in a clear sky environment [5,6,19–21], while strong turbulence conditions are described by K distribution models [7,12,22,23], which are particularly suitable for FSO 1 More details on the implementation and tradeoffs of coherent FSO systems can be found in [10].
© 2011 Optical Society of America
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communications over longer (km) distances. Beyond these two specific models, the Gamma–Gamma distribution is an important turbulence model, because it applies to a much wider range of turbulence conditions (from weak to strong) [24–27]. It can be shown that the K distribution is simply a special case of the Gamma–Gamma distribution. In this paper, we focus on the error rate analysis of coherent FSO systems in Gamma–Gamma turbulence channels and discuss methods for overcoming the impairment of atmospheric turbulence-induced fading as well as phase aberration. While IM/DD FSO systems have received considerable attention [5–7,19–22,25], there exist fewer analyses of coherent FSO systems. With the benefits of coherent FSO communications in mind [13], exact bit error rate (BER) expressions were developed for differential phase-shift keying (DPSK) over K-distributed turbulence by Kiasaleh [12] and over Gamma–Gamma-distributed turbulence by Tsiftsis [26]. A heterodyne FSO system with pointing errors was also studied by Sandalidis et al. [14], and Belmonte and Kahn studied the performance of coherent FSO links using modal phase compensation in lognormal turbulence channels [15–17]. In [28] and [29], the error rate performance of coherent FSO systems with maximum ratio combining (MRC), predetection equal gain combining (EGC), and selection diversity reception was studied for K-distributed atmospheric turbulence. In this work, we extend the work in [29] to the Gamma–Gamma turbulence model for predetection EGC with phase noise compensation error as well as postdetection EGC. It is shown that uncompensated phase noise errors are an important consideration in coherent FSO systems. The paper is organized as follows. Section II describes the coherent FSO system model. Section III introduces the Gamma–Gamma turbulence model and derives the characteristic function (CHF) for the square root of the Gamma–Gamma-distributed statistics. In Section IV, an exact BER expression is derived for binary phase-shift keying (BPSK) with EGC reception using a phase-locked loop (PLL) at the receiver for phase noise compensation. In Section V, DPSK with postdetection EGC is introduced for coherent FSO links to mitigate the impact of phase noise errors. We derive the exact error rate expression for such schemes in Gamma–Gamma turbulence channels. Finally, Section VI presents numerical results, and Section VII draws some important conclusions.
laser phase noise. When a laser source with a narrow linewidth (on the order of 10 kHz) is used, both of these phase noise mechanisms have variations on millisecond timescales [12,13,31], and tracking can be applied for phase compensation. For the L-branch coherent FSO system of interest here, with L identical local oscillators,3 multiple PLL phase noise compensation mechanisms are assumed at the receivers to accomplish independent phase tracking on each branch. In a practical coherent FSO system, the local oscillator power is made sufficiently large such that shot noise is the dominant noise source. Thermal noise and dark photocurrents are much smaller than the DC local oscillator photocurrent and thus can be neglected. The shot noise can be modeled as zero-mean additive white Gaussian noise (AWGN) with high accuracy [22,32]. Under these conditions, the optical power incident on the lth photodetector can be written as [28] q P l (t) = P s,l + P LO + 2 P s,l P LO g(t)
× cos(ω IF t + φs + φn,l ),
In this section, we present a coherent FSO system with L-branch diversity reception over turbulence-induced fading. Heterodyne coherent detection employing BPSK is considered. A collimated optical signal beam propagates from the transmitter [1]. The optical signal beam and a local oscillator beam are then mixed coherently at the receiver. The polarization state of the local oscillator and optical signal beams are matched with polarization controllers at the receiver. The phase error that is present during the superposition of these beams can be attributed to both atmospheric turbulence2 phase noise and 2 We assume that the correlation time is on a millisecond timescale [12]. Since the correlation time is inversely proportional to the scintillation cutoff frequency [3,30], the cutoff frequency for the temporal spectrum of irradiance fluctuations is on the order of kilohertz. Recent results on the temporal spectrum of irradiance fluctuations for FSO communications can be found in [30].
l = 1, 2, . . . , L,
(1)
where P s,l is the instantaneous incident optical signal power on the lth branch beamsplitter, P LO denotes the local oscillator power, φs ∈ {0, π} is the phase information, φn,l denotes the phase noise for the lth branch, and ω IF = ω0 − ωLO is the intermediate frequency, where ω0 and ωLO denote the carrier frequency and local oscillator frequency, respectively. In Eq. (1), g(t) is the signal pulse, defined as r 1 , g(t) = T 0,
0≤t≤T
(2)
else,
where T denotes the bit duration. In obtaining Eq. (1), we have assumed that the received optical beam and the local oscillator beam are mixed in perfect spatial coherence over a sufficiently small photodetector area. The product of the responsivity R and the incident optical power gives the photocurrent i l (t) = RP l (t) = i dc,l + i ac,l (t) + n l (t),
l = 1, 2, . . . , L,
(3)
where i dc,l = R(P s,l + P LO ) and i ac,l (t) = 2R
II. S YSTEM M ODEL
861
q
P s,l P LO g(t)
× cos(ω IF t + φs + φn,l ),
l = 1, 2, . . . , L
(4)
represent, respectively, the DC and AC terms at the receiver, and n l (t) is an AWGN process with equal variance σ2 for all branches. In practice, a coherent FSO system is operated in the regime P LO À P s,l , and the DC term in Eq. (3) can be approximated by the dominant term RP LO . The variance of the shot noise process n l (t) is then given by [32] σ2 = 2qRP LO ∆ f ,
(5)
where q is the electronic charge and ∆ f is the noise-equivalent bandwidth of the photodetector. The DC term can be removed using an appropriate bandpass filter. 3 For simplicity, we assume that the local oscillator beams used for phase tracking have identical powers and frequencies.
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By defining c l , 2R current as
q
P s,l P LO , we can express the AC
£ i ac,l (t) = c l g(t) cos φs cos(ω IF t + φn,l ) ¤ − sin φs sin(ω IF t + φn,l ) .
p © 2 c l g(t)[cos φs cos(ω IF t + φn,l ) ª − sin φs sin(ω IF t + φn,l )] cos(ω IF t)
(6)
(7)
and p © ys,l (t) = − 2 c l g(t)[cos φs cos(ω IF t + φn,l ) ª − sin φs sin(ω IF t + φn,l )] sin(ω IF t).
(8)
After a lowpass filter, we obtain the equivalent baseband signal i˜ac,l (t). With the relationship P s,l = AI s,l , where A denotes the photodetector area, the equivalent baseband signal of i l (t) can be expressed as
Andrews et al. proposed the modified Rytov theory, which defines the optical field as a function of perturbations due to large-scale and small-scale atmospheric effects [33]. This leads to the Gamma–Gamma turbulence model [24]. In this paper, we model the signal irradiance I s,l using the Gamma–Gamma model having a probability density function (PDF) f I s (I s,l ) =
p i˜l (t) = i˜ac,l (t) + n˜ l (t) = 2R
+ n˜ l (t),
AP LO g(t)
q
I s,l e jφs e jφn,l
l = 1, 2, . . . , L,
(9)
where n˜ l (t) is the complex envelope of the real white Gaussian noise process with power spectral density (PSD) 4qRP LO u(ω + ω IF ) and u(·) is the unit step function. In Eq. (9), I s,l denotes the lth optical signal irradiance incident on the beamsplitter and is independent and identically distributed (i.i.d.). Sufficiently large receiver separations, being larger than the centimeter-sized coherence lengths of the turbulence [5], are used to create this condition. After correlation and sampling, assuming perfect bit synchronization, we obtain
i˜l =
Z Tp q 2AR I s,l P LO g(t)e jφn,l e j φs g(t)dt
(11)
E[I 2s,l ] (E[I s,l ])2
−1 =
1 α
+
1 β
+
1 αβ
.
(12)
In weak turbulence regimes, the scintillation index is proportional to the Rytov variance, defined as [24] 7
11
σ2R = 1.23C 2n k 6 L p6 ,
(13)
where C 2n stands for the index of refraction structure parameter,4 k is the optical wave number, and L p is the propagation path length between the transmitter and receiver. When the optical turbulence strength extends to the moderate-to-strong irradiance fluctuation regime, by increasing either C 2n and/or the path length L p , the scintillation index for a plane wave and that for a spherical wave are related to the Rytov variance through Eqs. (33) and (43) in [33]. Parameters α and β can be related to physical parameters [3,27]. Estimation of the α and β shaping parameters was recently considered in [35].
0
Z T
+ =
α+β α+β −1 2 (αβ) 2 I s,l2 Γ(α)Γ(β) ´ ³ q × K α−β 2 αβ I s,l , I s,l > 0,
where Γ(·) denotes the Gamma function; K α−β (·) is the modified Bessel function of the second kind of order α − β; the positive parameters α and β, respectively, represent the effective number of large-scale and small-scale cells of the scattering process in the atmosphere; and I s,l is the lth signal irradiance incident on the beamsplitter with E[I s,l ] = 1, where E[·] denotes the expectation operation. The parameters α and β in Eq. (11) are related to the scintillation index σ2si via [33] σ2si ,
q
AND
A. Gamma–Gamma Turbulence Model
Two real filters are then used to implement the complex filtering in the down-conversion process. The real and imaginary parts of the baseband signal are, respectively, obtained as yc,l (t) =
III. G AMMA –G AMMA T URBULENCE M ODEL S TATISTICS
0
B. Gamma–Gamma Turbulence Statistics
n˜ l (t)g(t)dt
q p 2AR I s,l P LO e j φn,l e jφs + n˜ l ,
l = 1, 2, . . . , L, (10)
where n˜ l is a zero-mean complex Gaussian random variable (RV), and its real and imaginary parts are Gaussian RVs with equal variance σ2 . Subsequent combiner-based processing in the receiver will be described in Section IV for predetection EGC with phase noise compensation. To facilitate the performance analyses, we first provide a brief introduction to the Gamma–Gamma atmospheric turbulence model in the following section.
To facilitate the performance analysis of coherent FSO systems with diversity reception in Gamma–Gamma turbulence, we study in this subsection the statistical properties of the optical irradiance I s following a Gamma–Gamma distribution. To derive the error rate expressions for predetection/ postdetection EGC, we require knowledge of the momentgenerating function (MGF) of I s and the CHF of the square root of I s . Both will be shown to be effective tools for the 4 The most commonly used model to describe C 2 is the Hufnagel–Valley n model [3], and typical values of C 2n for weak to strong turbulence conditions can be found in [34].
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error rate performance analysis of coherent FSO transmission in Gamma–Gamma turbulent channels. First of all, with Eqs. 6.643(3) and 9.220(4) from [36], the MGF of I s 5 can be directly obtained from its definition: M I s (s) = =
Z ∞ 0
α+β−1 2
αβ
e− 2s
µ ¶ Γ(β − α) αβ M k1 ,k2 − Γ(β) s µ ¶¸ Γ(α − β) αβ + , M k 1 ,− k 2 − Γ(α) s
2
f y (y) =
µ ¶ µ ¶ α−β+1 αβ 2 αβ αβ M k1 ,k2 − = − e− 2s s s µ ¶ αβ ×1 F1 α, α − β + 1, − s
Φ z (ω ) = (14)
(15)
and µ
2ββ y2β−1 e−β y , Γ(β)
y > 0,
µ ¶ β−α+1 αβ 2 αβ αβ e− 2s − = − s s µ ¶ αβ ×1 F1 β, β − α + 1, − . s
Z ∞ 0
e j ω z f z (z)dz = E[e jω z ],
! Ã Ã ! − jωx −ω2 x2 21−β Γ(2β) D −2β p exp , = Γ(β) 8β 2β
α(α I x )α−1 e−α I x
Γ(α)
,
Ix > 0
(23)
where D ρ (·) is the parabolic cylinder function of the ρ th order. The parabolic cylinder function in Eq. (23) can be further expanded into the confluent hypergeometric function [36, Eq. 9.240]. With this in mind, we can rewrite Eq. (23) as ³ ´ ! Ã ! Γ β + 21 1 ω2 x 2 ωx φ z| x (ω) = 1 F1 β, , − +j p 2 4β Γ(β) β ! Ã 1 3 ω2 x 2 . ×1 F1 β + , , − 2 2 β Ã
(16)
(17)
(18)
(24)
Averaging the CHF of z conditioned on x gives the desired CHF as Φ z (ω) = R{Φ z (ω)} + j I{Φ z (ω)}6 [39], where 1 ω2 R{Φ z (ω)} = 2 F1 β, α, , − 2 4αβ
To obtain the CHF of z, a direct calculation using the definition of the CHF can be difficult. Alternatively, we may exploit the property of I s = I x I y [24], where I x and I y are two independent RVs with Gamma PDFs, respectively, defined as
f I x (I x ) =
(22)
φ z| x (ω) = E z| x [e j ω z ]
Ã
α+β 4 (αβ) 2 zα+β−1 Γ(α)Γ(β) ³ p ´ × K α−β 2 αβ z , z > 0.
ω ∈ R.
Using Eq. 3.462(1) of [36], we can obtain the CHF of z conditioned on x as
¶
Next, we seek the CHF for the square root of the irradiance pI s. We denote the square root of the irradiance by z, i.e., z , I s , and the PDF of z can be shown to be f z (z) =
(21)
The CHF of z is defined as
·
α+β−1 (− s) 2
where M·,· (·) denotes the Whittaker function [36], k 1 = 1 − α − β/2, and k 2 = α − β/2. Equation (14) can be further expressed in terms of the confluent hypergeometric function 1 F1 (·, ·, ·) by way of [36, Eqs. 9.220(2), 9.220(3)]
M k 1 ,− k 2
and
respectively. The PDFs in Eqs. (20) and (21) are Nakagami PDFs if α Ê 1/2 and β Ê 1/2.
e sI s f I s (I s )dI s
(αβ)
863
!
(25)
and
I{Φ z (ω)} =
³ ´ ³ ´ Γ α + 12 Γ β + 12 ω ¡
¢− 1 αβ 2
Γ(α)Γ(β) Ã ! 1 3 ω2 1 × 2 F1 β + , α + , , − . 2 2 2 4αβ
(26)
Here, R{·} and I{·} denote the real and imaginary operations, respectively, and 2 F1 (·, ·, ·; ·) is the Gaussian hypergeometric function.
and
f I y (I y ) =
β(β I y )β−1 e−β I y
Γ(β)
,
I y > 0.
(19)
p p We let z = x y with x , I x and y , I y , and x and y can be shown to be two independent RVs, with their PDFs given by 2
f x (x) =
2αα x2α−1 e−α x , Γ(α)
x>0
(20)
5 We note that a different form of the MGF of Gamma–Gamma-distributed RVs has been documented in a technical report [37].
IV. P REDETECTION EGC W ITH P HASE N OISE C OMPENSATION E RROR In this section, we employ the CHF of z derived in Section III to study the impact of phase estimation error on the error rate performance of predetection EGC coherent FSO systems in Gamma–Gamma turbulence. As shown in Fig. 1, the receiver compensates for the random phase noise in the optical links on all diversity branches by multiplying the received signals with 6 We remark that the MGF of the product of two independent Nakagami RVs was derived independently in [38] in terms of the Meijer G-function.
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Beam combiner
Ps ,1 (t )
P1 (t ) Photodetector
i1 (t )
Match filter 1
Downconverter
Sampling
PLO Local oscillator
Ps ,2 (t )
e
P2 (t ) Photo- i2 (t ) Downdetector
Match filter 2
converter
PLO
e
PL (t ) Photo- iL (t ) Downdetector converter
i
Sampling
Local oscillator
Ps , L (t )
− jφˆn ,1
− jφˆn ,2
Match filter L
Since BPSK is assumed here, the real part of Eq. (27) can be used to express the decision variable as
Sampling
PLO
e
Local oscillator
ˆ Decision φs device
both turbulence and laser phase noise, which is justified as follows. The transmitter laser phase noise is correlated on each branch, and can be made negligible with a sufficiently stable and narrow linewidth laser [45]. This can be achieved with either a single transmitter optical injection PLL or an ultrastable laser [45–47]. The remaining laser phase noise due to the L separate receiver local oscillators is i.i.d. At the same time, turbulence phase noise can be made i.i.d. by ensuring that the receiver apertures of the L branches are separated by distances beyond the centimeter-sized turbulence coherence length [5]. We also assume the irradiance I s,l to be independent of ∆φl .
− jφˆn ,L
D=
q q L p X 2R AP LO cos φs I s,l cos ∆φl + R{ν} l =1
Fig. 1. Block diagram for coherent wireless optical receiver with predetection EGC reception.
= cos φs
L X
S l + νR ,
(29)
l =1
the complex conjugate of the phase noise estimates from the respective channels. The output of the combiner can then be found as i˜ =
=
l =1
L p q X e j∆φl 2R I s,l AP LO e jφs + ν,
q p p 2R AP LO I s,l cos ∆φl , and νR = R{ν} is a
real-valued zero-mean Gaussian noise RV with variance σ2νR = L σ2 .
q L L X X ˆ p ˆ e− j φn,l 2R P s,l P LO e jφn,l e jφs + e− j φn,l n˜ l l =1
where S l =
(27)
Based on the decision variable at the output of the combiner, we can find the signal-to-noise ratio (SNR) for EGC reception with phase noise compensation error as
l =1
where φˆ n,l is the estimate of φn,l at the lth branch, ∆φl = φn,l − φˆ n,l denotes the phase noise compensation error, and P ˆ ν= L e− jφn,l n˜ l is the complex noise term at the output of l =1 the combiner. The real and imaginary parts of the noise term ν are Gaussian RVs with equal variance Lσ2 . PLL phase noise compensation is applied at the receiver to accomplish independent phase tracking on each branch. For the PLL circuit, we require the loop bandwidth, describing the response rate of the PLL [40], to be at least kilohertz to compensate for the millisecond-duration phase noise originating from the turbulence channels and laser source. We note that a PLL phase detector can work successfully up to 10 MHz [41]. Phase noise estimates are derived from an unmodulated carrier using a first-order PLL, and only Gaussian noise is present in the PLL circuit [42,43]. In this case, the PDF of the phase noise compensation error ∆φ is given by7 [43] µ
exp f ∆φ (∆φl ) =
cos(∆φl ) σ2∆φ
2π I 0
µ
1 σ2∆φ
¶
¶ ,
|∆φ| ≤ π,
(28)
where I 0 (·) denotes the zeroth-order modified Bessel function of the first kind, and σ∆φ is the standard deviation of the phase noise compensation error ∆φl for l = 1, . . . , L. It is assumed that ∆φ1 , ∆φ2 , . . . , ∆φL are i.i.d. RVs associated to 7 When the loop signal-to-noise ratio is large, the Tikhonov distribution is also a good statistical model for describing the phase noise compensation error for a second-order PLL [44].
γ˜ EGC =
=
à L q 2R 2 AP LO X
L σ2 Ã L q γ X
L l =1
!2
I s,l cos ∆φl
l =1
!2
I s,l cos ∆φl
,
(30)
where γ = R A/(q∆ f ) denotes the average SNR per branch for a given coherent FSO system. Of importance to the ongoing investigation is the fact that the SNR in Eq. (30) is related to q I s,l and cos ∆φl , but it is independent of the local oscillator
power. When the phase noise is tracked and compensated perfectly, the SNR expression at the output of the combiner reduces to
γEGC =
³P ´2 q γ L I s,l l =1
L
.
(31)
From Eq. (31), we make an important observation that, unlike the predetection EGC for IM/DD-based FSO systems [7], where the combining output SNR γEGC is related to the sum over all I s,l values, γEGC for predetection coherent FSO systems is q associated with the sum of all I s,l values. This difference can impact the BER performance of these systems. From the expression of the decision variable in Eq. (29), we now derive the average BER for EGC with phase noise compensation error through a CHF approach. Without loss of generality, we assume that φs = 0. We define the cumulative distribution function (CDF) of the decision variable as FD (ξ|φs = 0) = Pr{D < ξ|φs = 0}.
(32)
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The average BER can thus be written as P e = FD (0|φs = 0) when Eq. (32) is evaluated at ξ = 0. To find FD (·|φs = 0), we write the conditional CHF of D as
ΦD (ω|φs = 0) = ΦνR (ω)
L Y l =1
Beam combiner
Ps ,1 (t )
P1 (t ) Photo- i1 (t ) detector
PLO
~ Match Vk ,1 filter 1 and sampling
Local oscillator
ΦS l (ω) = ΦνR (ω)[ΦS1 (ω)]L (33)
~
Delay
Ps ,2 (t )
P2 (t ) Photo- i2 (t ) Downdetector
for i.i.d. RVs S l ’s (l = 1, . . . , L), where ΦS l (ω) is the CHF of S l
for l = 1, 2, . . . , L, and ΦνR (ω) = exp(−Lσ2 ω2 /2) is the CHF of
converter
Averaging Eq. (34) over ∆φ1 gives the CHF of S 1 as ΦS1 (ω) = p E ∆φ1 [Φ z (ω 2AP LO R cos ∆φ1 )]. Then, the CHF of D can be found as
ΦD (ω|φs = 0) = [ΦS1 (ω)]L ΦνR (ω) ´i´L ³ h ³ q = E ∆φ1 Φ z ω 2AP LO R cos ∆φ1 × ΦνR (ω).
Z 1 1 ∞ I{ΦD (ω|φs = 0)e− j ωξ } − d ω. 2 π 0 ω
Z 1 1 ∞ I{ΦD (ω|φs = 0)} − d ω. 2 π 0 ω
(36)
(37)
Substituting Eq. (35) into Eq. (37) gives the BER expression for EGC reception with phase compensation error. Two integrations are required to evaluate the BER performance of an L-branch predetection EGC system using Eqs. (35)–(37). In the absence of phase noise compensation error, it can be shown that the average BER expression is given by [39] Ãp !#L γ tan θ 1 2 ∗ Pe = R sec θ Φ z p 2π − π2 2L ) z(tan θ ) × dθ, 2 Z
π 2
(
PLO
~
Delay
à ! µ ¶ 2 3 ω2 j − ω4 F 1, , − + 1 − e 1 1 2 4 ω π
1
PL (t ) Photo- iL (t ) Downdetector
converter
Match filter L and sampling
Vk −1, L
~
Vk , L
PLO Local oscillator
Fig. 2. Block diagram for coherent wireless optical receiver with postdetection EGC reception.
Up to this point, we have discussed the error rate performance of predetection EGC with phase compensation error. Since the characteristics of atmospheric turbulence are time variant, the phase-tracking device may not recover the signal phase perfectly. This can lead to system performance losses and may reduce the diversity order. In this section, we present an efficient technique using DPSK for coherent FSO communications over atmospheric turbulence channels. Since DPSK does not require phase noise estimation,8 it is a useful alternative to coherent PSK. We consider a coherent FSO system employing postdetection EGC to mitigate amplitude fading. Postdetection EGC does not require carrier phase estimation and is therefore well suited for differential coherent detection. The receiver is shown in Fig. 2. The FSO system is implemented with L-branch wireless optical links through Gamma–Gamma turbulence. From Eq. (9), the received complex envelope at the lth branch in the kth bit interval can be written as i˜k,l (t) = i˜ac,k,l (t) + n˜ k,l (t) q p q = 2R AP LO g(t) I s,l e j φs,k e j φn,l + n˜ k,l (t), (40)
"
(38)
where (·)∗ denotes complex conjugation. Here,
z(ω) = p
Δφˆs , k
(35)
Finally, the BER can be obtained as
Pe =
D
V. P OSTDETECTION EGC W ITH DPSK
The CDF FD (ξ|φs = 0) can be evaluated by the Gil-Pelaez formula [48,49]:
FD (ξ|φs = 0) =
~
{}
Local oscillator
Ps , L (t )
(34)
Vk −1,2
Match V~ filter 2 and k ,2 sampling
Gaussian RV νR . The CHF of S 1 conditioned on ∆φ1 can be found to be ³ q ´ ΦS1 |∆φ1 (ω) = Φ z ω 2AP LO R cos ∆φ1 .
~
Vk −1,1
Delay Downconverter
865
(39)
is the Fourier transform of the complementary error function, and ω denotes the angular frequency. Equation (38) can be easily evaluated numerically as it involves only a single definite integration.
where φs,k = φs,k−1 + ∆φs,k is the differentially coded phase. Here, ∆φs,k ∈ {0, π} denotes the differential carrier phase, and the encoded phase differences are assumed to be equally likely transmitted. Due to the high data rates (gigabits per second), compared to the kilohertz variation rates of atmospheric turbulence [12] and the kilohertz linewidth9 of the laser sources [31], one can assume a frozen phase model in which the phase characteristics remain constant over at least two successive symbol intervals. At the same branch, the signal in 8 An optical PLL at the transmitter is not needed, which further reduces the system complexity. 9 The kilohertz linewidth ensures that the laser source phase noise changes slowly.
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the (k − 1)th bit interval can, therefore, be similarly expressed as
10 10
i˜k−1,l (t) = i˜ac,k−1,l (t) + n˜ k−1,l (t) p q = 2R AP LO g(t) q × I s,l e j φs,k−1 e j φn,l + n˜ k−1,l (t).
(41)
The shot noise processes n˜ k,l (t) and n˜ k−1,l (t) are i.i.d. complex Gaussian random processes with PSD 4qRP LO u(ω + ω IF ). In postdetection EGC reception, the outputs of the correlator at the lth branch, after normalizing the noise terms, are given by
BER
10 10 10 10 10
0
–1
–2
–3
–4
Single branch, L=1 MRC, L=2 MRC, L=3 MRC, L=4 EGC, L=2 EGC, L=3 EGC, L=4
–5
–6
p
R A q I s,l e jφs,k e jφn,l + µ˜ k V˜ k,l = p qR ∆ f p q = γ¯ I s,l e j φs,k e j φn,l + µ˜ k
10
(43)
where µ˜ k and µ˜ k−1 are i.i.d. Gaussian RVs with unit variance. Therefore, the decision variable D˜ at the output of the postdetection combiner is obtained as L X
U˜ l =
l =1
l =1
R{V˜ k∗−1,l V˜ k,l },
(44)
where U˜ l = R{V˜ k∗−1,l V˜ k,l }. Using Eq. (B.5) from [43], the CHF of U˜ l conditioned on I s,l can be found as
ΦU˜ | I (ω|∆φs,k = 0) = s,l
1 ω2 + 1
Ã
exp −γ¯
ω2 − j ω ω2 + 1
!
I s,l .
(45)
With the help of Eq. 6.643(3) from [36], averaging Eq. (45) over I s,l gives the CHF of U˜ l as
ΦU˜ (ω|∆φs,k = 0) =
1 ω2 + 1
Ã
M I s −γ¯
ω2 − j ω
!
ω2 + 1
,
(46)
where M I s (·) is given by Eq. (14). For i.i.d. Gamma–Gamma turbulence, we can express the CHF of D˜ as
ΦD˜ (ω|∆φs,k = 0) = ΦL˜ (ω|∆φs,k = 0) U
=
1 (ω2 + 1)L
"
Ã
M I s −γ¯
5 10 Average SNR per branch (dB)
15
Fig. 3. BER of BPSK for MRC and predetection EGC reception (assuming perfect channel state information) operating over L strongly turbulent Gamma–Gamma channels with channel parameters α = 2.23, β = 1.70.
p R A q I s,l e jφs,k−1 e jφn,l + µ˜ k−1 V˜ k−1,l = p qR ∆ f p q = γ¯ I s,l e j φs,k−1 e j φn,l + µ˜ k−1 ,
L X
0
(42)
and
D˜ =
–7
ω2 − j ω
!#L
ω2 + 1
.
(47)
With the Gil-Pelaez formula, we can obtain the average BER for DPSK with postdetection EGC as P e = Pr{D˜ < 0|∆φs,k = 0} ½h ³ ´iL ¾ ω2 − j ω Z ∞ I M I s −γ¯ 2 ω +1 1 1 = − d ω, 2 π 0 ω(ω2 + 1)L
(48)
where only one integral is needed for the DPSK BER calculation.
VI. N UMERICAL R ESULTS In this section, we evaluate the average BER of BPSK for predetection EGC diversity reception and DPSK for postdetection EGC diversity reception. We define the average SNR per branch as γ = R A/(q∆ f ). The turbulent channel parameters α and β are determined by the value of the Rytov variance σ2R [24]. The corresponding scintillation index can also be found from Eq. (13) or in [33]. It is assumed that the atmospheric channels have equal turbulence severity parameters, i.e., α and β are identical for all diversity branches. The average BER of Gamma–Gamma turbulence-induced fading in L independent atmospheric channels with predetection EGC and MRC benchmark reception is plotted in Fig. 3 for BPSK when σ2R = 2 (with the corresponding channel parameters α = 2.23, β = 1.70). Such parameters imply strong turbulence conditions. In Fig. 3, the channel states are assumed to be known at the EGC/MRC receiver. As expected, the BER performance improves as the number of diversity branches increases. It is readily seen that even a small diversity order can significantly mitigate the effects of atmospheric turbulence. Furthermore, the error rate performance of an FSO system employing EGC is close to the MRC benchmark case. For example, it is apparent from Fig. 3 that there is less than 1 dB SNR difference between four-branch MRC and EGC reception at a BER of 2 × 10−6 . Though it is not shown in Fig. 3, as well as in the following figures, all the numerical results have been verified by computer simulations. As a next step, we study the impact of phase noise on the error performance of predetection EGC systems over Gamma–Gamma turbulence in both strong and weak turbulence regimes. Figure 4 shows the impact of phase
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0
10
0
–1
10
10
–2
–2 °
°
10 10 10
–3
σΔΦ
=10 ° ,
σΔΦ =0 , L=2
L=2
BER
BER
σΔΦ =0 , L=2
10
σΔΦ =20 ° , L=2
–4
–5
σΔΦ
=30 ° ,
σΔΦ
=0 °,
10
σΔΦ =10 ° , L=2
–4
° σΔΦ =20 , L=2
σΔΦ =30 ° , L=2
L=2
10
L=3
–6
°
σΔΦ =0 , L=3 σΔΦ =10 ° , L=3
σΔΦ =10 ° , L=3 σΔΦ =20 ° , L=3
–6
10
σΔΦ =20 ° , L=3
–8
σΔΦ =30 ° , L=3
σΔΦ =30 ° , L=3
10
867
–7
0
5
10
15
Average SNR per branch (dB)
Fig. 4. BER of BPSK for predetection EGC reception with phase noise compensation error operating over L strongly turbulent Gamma–Gamma channels with channel parameters α = 2.23, β = 1.70.
noise compensation errors on dual-branch and three-branch predetection EGC reception in strongly turbulent wireless optical links when σ2R = 2 (with the corresponding channel parameters α = 2.23, β = 1.70). One can see that there is a performance loss due to the phase noise compensation error. The performance degradation, as expected, worsens when the standard deviation of the phase noise compensation error increases. A notable point in Fig. 4 is that there is relatively little difference in the error performance between the system with perfect phase noise compensation and those with phase noise compensation errors of σ∆φ = 10◦ or σ∆φ = 20◦ . In particular, only small performance losses (less than 1 dB SNR loss) are observed with a phase noise compensation error of σ∆φ = 20◦ for both dual-branch and three-branch reception. As the standard deviation of the phase noise compensation error increases, however, the wireless optical systems suffer from a larger performance degradation. For instance, for an SNR per branch of 15 dB, the BER is 2.8 × 10−5 for σ∆φ = 20◦ , but it degrades to 1.6 × 10−3 for σ∆φ = 30◦ . The BER performance for strong turbulence conditions shown in Figs. 3 and 4 is compared next to the performance under weak turbulence conditions. Figure 5 presents dualbranch and three-branch predetection EGC reception in weak wireless optical turbulence links when σ2R = 0.3 (with the corresponding channel parameters α = 6.52, β = 6.92). Again, we observe that the performance degradation becomes larger when the standard deviation of the phase noise compensation error increases. We also find that, comparing Figs. 4 and 5, the error rate performance in weak turbulence is much better than that in strong turbulence, as expected. For example, at an average SNR per branch of 10 dB, the BER with three-branch predetection EGC reception is on the order of 10−3 in strong turbulence, but the BER is on the order of 10−5 in weak turbulence. From Figs. 4 and 5, we also conclude that the standard deviation of the phase noise compensation error is a crucial parameter in evaluating the error rate performance of a wireless optical communication system with predetection EGC reception. We find that predetection EGC is a viable choice
0
5
10
15
Average SNR per branch (dB)
Fig. 5. BER of BPSK for predetection EGC reception with phase noise compensation error operating over L weakly turbulent Gamma–Gamma channels with channel parameters α = 6.52, β = 6.92.
with a relatively simple implementation when σ∆φ is less than 10◦ . When σ∆φ is much larger than this, predetection EGC suffers from great performance losses. For optical wireless systems with large phase noise compensation errors, there exists a viable alternative by way of DPSK with postdetection EGC reception. For postdetection EGC, there is no need to estimate the phase noise. Figure 6 demonstrates this fact for postdetection EGC with DPSK under strong turbulence conditions when σ2R = 2 (with the corresponding channel parameters α = 2.23, β = 1.70). We find that postdetection EGC enjoys the full benefits of the diversity order. For instance, at an average SNR per branch of 15 dB, a BER of 1.8 × 10−5 is achieved. This point is further evident from Fig. 7, which gives a performance comparison between BPSK with predetection EGC and DPSK with postdetection EGC. Both employ dual-branch reception. We observe that predetection EGC outperforms postdetection EGC when the phase noise has been compensated perfectly or the standard deviation of the phase noise compensation error is relatively small. However, DPSK with postdetection EGC outperforms BPSK with predetection EGC when σ∆φ = 30◦ and the SNR per branch is greater than approximately 10 dB.10 A complete comparison between the performance of DPSK with postdetection EGC and BPSK with predetection EGC is given in Fig. 8, which shows the BER versus standard deviation of phase noise compensation error σ∆φ from 0◦ to 30◦ . For low levels of phase noise, the displayed curves are largely independent of σ∆φ . When σ∆φ reaches 20◦ , however, the performance of predetection BPSK worsens. At approximately 25◦ , the BPSK performance degrades below that of postdetection DPSK at an average SNR per branch of 15 dB. These observations agree with those in Fig. 7. We conclude that DPSK with postdetection EGC is an excellent alternative to BPSK with predetection EGC in coherent FSO communication systems with large phase noise compensation errors. 10 We would expect that M-ary PSK will be even more sensitive to these phase noise compensation errors with M greater than two.
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10
10
0
10
–1
10
BER
10
10
10
DPSK, L=1 DPSK, L=2 DPSK, L=3 DPSK, L=4
–4
10
BER
10
0
5
10
15
0
–4
BPSK, Average SNR per branch=10 dB DPSK, Average SNR per branch=10 dB BPSK, Average SNR per branch=15 dB DPSK, Average SNR per branch=15 dB
0
5
10
15 σΔΦ (°)
20
25
30
Fig. 8. BER versus standard deviation of phase noise compensation error for BPSK with predetection EGC reception and DPSK with postdetection EGC reception operating over dual-branch strongly turbulent Gamma–Gamma channels with channel parameters α = 2.23, β = 1.70.
as a solution for coherent FSO systems operating with large phase compensation errors, DPSK with postdetection EGC was proposed and shown to be a viable alternative in wireless optical communications.
–1
–2
A CKNOWLEDGMENT
BPSK, σΔΦ =0°
10
–3
–5
Fig. 6. BER of DPSK for postdetection EGC reception operating over L strongly turbulent Gamma–Gamma channels with channel parameters α = 2.23, β = 1.70.
10
10
–3
Average SNR per branch (dB)
10
–2
–2
BER
10
–1
This work is supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC).
BPSK, σΔΦ =10 °
–3
BPSK, σΔΦ =20 ° BPSK, σΔΦ =30 °
R EFERENCES
DPSK
10
–4
0
5 10 Average SNR per branch (dB)
15
Fig. 7. BER of BPSK with predetection EGC reception and DPSK with postdetection EGC reception operating over dual-branch strongly turbulent Gamma–Gamma channels with channel parameters α = 2.23, β = 1.70.
VII. C ONCLUSION In this paper, we have analyzed the error rate performance of coherent wireless optical communication systems and computed the BER of BPSK with predetection EGC reception in Gamma–Gamma turbulent environments. Error rates for predetection EGC reception with perfect phase noise compensation have been obtained, and coherent FSO systems employing predetection EGC were shown to have a greatly improved system performance (compared to a single-branch FSO link). Furthermore, we have studied the impact of phase noise on the performance of predetection EGC reception. It was found that an FSO system with a relatively small phase noise compensation error (with a standard deviation of 20◦ for strong turbulence and with a standard deviation of 10◦ for weak turbulence) offers comparable error rate performance to that of the perfect phase compensation case. Furthermore,
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