Impact of Pointing Errors on the Error Performance of ... - OSA Publishing

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Impact of Pointing Errors on the Error Performance of Intersatellite Laser Communications Tianyu Song, Student Member, IEEE, Qian Wang, Student Member, IEEE, Ming-Wei Wu, Member, IEEE, Tomoaki Ohtsuki, Senior Member, IEEE, Mohan Gurusamy, Senior Member, IEEE, and Pooi-Yuen Kam, Fellow, IEEE

Abstract—The effect of pointing errors on the average bit error probability (ABEP) of an intersatellite laser communication link is studied. A closed-form expression in terms of the Marcum Q-function is given for calculating the instantaneous channel gain. This expression provides great potential in further performance analysis and system optimization. In addition, tight, closed-form upper and lower bounds on the ABEP are derived, by using bounds on the Gaussian Q-function. These bounds are in simpler form than existing results and are much more efficient in numerical evaluations. Further simplifications of these bounds are performed and invertible ABEP expressions are given. Via numerical comparisons, these invertible ABEP expressions are shown to be accurate approximations within a wide range of transmit power of interest and thus provide efficient error prediction methods. The diversity gain is easily obtained using the invertible ABEP expressions, which is related to the ratio of the equivalent beam radius to the pointing error displacement standard jitter at the receiver. Moreover, we show explicitly how the ABEP depends on different system parameters. Index Terms—Average bit error probability (ABEP), beam waist, diversity gain, inter-satellite laser communications, invertible expressions, pointing errors, the Gaussian Q-function, the Marcum Q-function, tight bounds.

I. INTRODUCTION ONG distance inter-satellite laser communication links are highly vulnerable due to the degrading effect of pointing errors [1]–[5]. The pointing errors are due to platform vibrations, which cause vibrations of the transmitter telescope and,

L

Manuscript received August 17, 2016; revised March 27, 2017; accepted April 30, 2017. Date of publication May 16, 2017; date of current version June 19, 2017. This work was supported in part by the Singapore Ministry of Education (AcRF Tier 2 under Grant MOE2013-T2-2-135), in part by the National Natural Science Foundation of China under Grants 61302112 and 61571316, in part by the Ministry of Education of the PRC (SRF for the ROCS, the 47th batch), and in part by the Qianjiang Talent Project (QJD1402023). (Corresponding author: Qian Wang.) T. Song, Q. Wang, M. Gurusamy, and P.-Y. Kam are with the Department of Electrical and Computer Engineering, National University of Singapore, Singapore 117576, and also with the National University of Singapore Suzhou Research Institute, Suzhou 215123, China (e-mail: [email protected]; [email protected]; [email protected]; [email protected]). M.-W. Wu is with School of Information and Electronic Engineering, Zhejiang University of Science and Technology, Hangzhou 310023, China (e-mail: [email protected]). T. Ohtsuki is with Department of Information and Computer Science, Keio University, Yokohama 223-8522, Japan (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/JLT.2017.2705132

therefore, misalignment between the transmitter and the receiver [1], [3]. Various statistical models have been proposed over the years to describe the pointing errors [1], [2]. In these works, the effects of misalignment on the error performance have been investigated. However, the existing results for the average bit error probability (ABEP) involve multiple integration [1], or higher-order transcendental functions [3], which do not facilitate further analysis. No simple, closed-form expressions for the ABEP are given so far, and the diversity gain cannot be easily derived. Here, we analyze the effect of pointing errors on the error performance of an inter-satellite laser link. Our approach is to obtain tight, algebraic-form, upper and lower bounds on the ABEP, by using the tight bounds on the Gaussian Q-function derived in [6]–[9]. Moreover, the bounds derived by using the approaches in [6], [7] can be made arbitrarily tight. For large transmit power, all the bounds can simplify to invertible expressions, which turn out to be accurate approximations corresponding to the ABEP values of 10−3 and lower, which are of practical interest. More importantly, the diversity gain is straightforwardly obtained, which is related to the ratio of the equivalent beam radius to the pointing error displacement standard jitter at the receiver. We will show the explicit insights into how the channel parameters affect the ABEP via simulations. This paper is organized as follows. Section 2 gives the integral form expression of the exact ABEP. The double integral form of the exact instantaneous channel gain hp is briefly reviewed and a new closed-form expression in terms of the Marcum Q-function is derived accordingly. In section 3, tight and closed-form upper and lower bounds on the ABEP are derived and the tightness is analyzed via numerical comparisons. Invertible approximate expressions of the ABEP are given in section 4, and the system diversity order is analyzed using these invertible expressions. Our final concluding remarks are given in section 5. II. EXPRESSIONS OF THE EXACT ABEP The system ABEP can be obtained by averaging the instantaneous BEP over all possible pointing error angle θ values, i.e.,  ∞ P (e|θ)pθ (θ)dθ. (1) P (e) = 0

Here, pθ (θ) is the probability density function (pdf) of θ. Assuming that binary phase-shift keying (BPSK) with coherent detection is adopted, the instantaneous BEP conditioned on a

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SONG et al.: IMPACT OF POINTING ERRORS ON THE ERROR PERFORMANCE OF INTERSATELLITE LASER COMMUNICATIONS

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given value of the pointing error angle θ is [10, eq. (10.4.10)]   γb (θ) (2) P (e|θ) = QG where QG (x) is the Gaussian Q-function given by  2  ∞ 1 u G √ exp − Q (x) = du. 2 2π x

(3)

Here, γb (θ) is the instantaneous SNR of BPSK and depends on the coherent detection techniques: homodyne detection or heterodyne detection. The use of coherent detection allows one to use a local oscillator to amplify the received signal, and thus the shot-noise limit can be achieved even for p − i − n receivers whose performance is generally limited by thermal noise [10, P. 482]. For appropriately large local-oscillator power, the instantaneous SNR of BPSK with homodyne detection dominated by shot noise is given by [10, eq. (10.1.13)] γb (θ) ≈

4hp (θ, ωz , z, a)Pt R . eRdata

(4)

Here, hp (θ, ωz , z, a) is the instantaneous channel gain, R is the photodetector responsivity, Pt is the transmit power, Rdata is the data rate, and e = 1.6 × 10−19 C is the elementary charge. In the heterodyne case, the SNR is lower by a factor of 1/2. For homodyne detection, (2) is rewritten as   G hp (θ, ωz , z, a)γ (5) P (e|θ) = Q where the quantity γ is γ=

4Pt R . eRdata

(6)

The channel gain hp (θ, ωz , z, a) due to pointing errors is the fraction of the power collected by the receiver aperture. In this section, we give two methods to calculate the exact value of hp . A. Expression of hp Based on a Double Integral The calculation of hp using a double integral has been introduced in detail in [2]. In this paper, to keep our manuscript self-contained, we briefly introduce it in this subsection. The channel gain hp can be obtained by integrating the normalized power spatial distribution density function at the distance z, which is denoted by Ib eam (ρ2 ; ωz ), over the aperture surface C, whose radius is a. Mathematically, it is expressed as  Ib eam (x2 + y 2 ; ωz )dxdy, (7) hp (θ, a, ω0 , z) = C

where Ib eam (ρ2 ; ωz ) is given by [11]   2 2ρ2 2 exp − Ib eam (ρ ; ωz ) = . πωz2 ωz2

Fig. 1. Misalignment with radial displacement d between receiver aperture C with radius a and Gaussian beam footprint in red.

beam waist ω0 by [12, P. 665] ωz = ω0

1+



zλ πω02

2 ,

(9)

where λ is the laser wavelength. The validation of the Gaussian beam model requires ω0 > 2λ/π [12, P. 630]. In (7), the integration region is related to θ, a and z. Specifically, the radial displacement between the aperture center and the beam center is d = z · tan θ and can be approximated by d = z · θ with very high accuracy since θ is very small (< 10−4 rad). Referring to Fig. 1, a is the radius of the circular aperture C, which determines the aperture size. Though the double integral approach given in (7) provides an accurate method to evaluate the instantaneous channel gain, it is too complex and numerically intensive. Further analysis based on it is thus intractable. Few research work has been done using this exact form, and assumptions are made in existing literature to simplify this calculation. For instance, [13]–[15] assume negligible detector aperture size with respect to the beam width at the receiver due to the large link distance, resulting in that the optical intensity variation over the detector surface is negligible; and [16] directly assumes that the intensity distribution within the beam width is uniform. Under both assumptions, the total power collected, hp , is simply the product of the intensity in (8) and the detector area, which inevitably results in some biases from using the double integral in (7). In the next subsection, we will give a mathematically tractable approach, which avoids the double integral non-closed form and involves the Marcum Q-function. B. Expression of hp Based on the Marcum Q-Function

(8)

In (8), ρ is the radial vector from the beam center, and ωz is the beam radius at which the intensity drops to e−2 of the axial value at the distance z. The beam radius ωz is related to the

Let X and Y denote two independent and identically distributed Gaussian random variables with mean zero and variance σ 2 = ωz2 /4 . If we let ρ = (X, Y ), the joint pdf of X and Y is exactly the same as (8). Hence, the exact channel gain hp (θ, a, ω0 , z) can be interpreted as the probability of a random point (X, Y ) falling onto the aperture C. We use d to denote the radial displacement vector, which can also be interpreted as the

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coordinates of the aperture center. Thus, we have d = d and hp (θ, a, ω0 , z) = Pr ((X, Y )−d ≤ a). Notation  ·  here denotes the Euclidean norm. Due to the circular symmetry of the beam cross-sectional shape and the aperture area, hp (θ, a, ω0 , z) depends only on the radial displacement. Hence, without loss of generality, we consider the case where the center of aperture C is located at d = (−d, 0) for simplicity,  and hp (θ, a, ω0 , z) is interpreted as the probability of event

(X + d)2 + Y 2 ≤ a, i.e.,

hp (θ, a, ω0 , z) = Pr ((X, Y )−d ≤ a)   2 2 (X + d) + Y ≤ a . = Pr

(10)

Using the geometric interpretation of the first-order Marcum M ar (, ) can be used to Q-function QM ar (, ), given in [17],  Q

represent the probability of event (X + d)2 + Y 2 > a, i.e.,     d a 2 M ar 2 , (X+d) + Y > a = Q Pr . (11) σ σ Here, QM ar (, ) is given by  2   ∞ r + x2 M ar Q (x, y) = r exp − I0 (xr) dr, 2 y

(12)

where I0 () is the modified Bessel  function of the first kind of order zero. Since events (X + d)2 + Y 2 ≤ a and  (X + d)2 + Y 2 > a are mutually exclusive, we have the channel gain hp (d, ωz , a) expressed as   2 2 hp (θ, a, ω0 , z) = Pr (X + d) + Y ≤ a   2 2 (X + d) + Y > a = 1 − Pr  = 1 − QM ar

2d 2a , ωz ωz

 .

(13)

It should be emphasized that in deriving (13), no approximation is made. The calculation results of hp (θ, a, ω0 , z) using (13) and (7) are exactly the same. Our result (13) is a simple, closedform expression. Since the Marcum Q-function QM ar (, ) is built in by some commercial softwares, e.g., MATLAB, the evaluation of (13) is much faster than that of the double integral given in (7), and thus (13) facilitates later numerical study of error performance. The numerical results in Fig. 2 computed from (13) using MATLAB show how the fraction of the power collected by the receiver aperture decreases as the pointing error increases. For a given pointing error angle θ, the radial displacement d increases as z increases, which thus leads to a smaller received power. Besides facilitating numerical studies, (13) gives a new idea for calculating the channel gain using the Marcum Q-function. This enables the possibility of analyzing the system performance using the existing knowledge about the Marcum Q-function, e.g., [17]–[21]. For instance, in the current literature, there is no closed-form solution for calculating the optimum beam waist

Fig. 2. Channel gain h p as a function of pointing error angle θ with parameters in Table I and ω 0=3 mm.

ω0 for achieving the minimum ABEP. Based on the expressions given in (1), (2), and (13), one can obtain an efficient method to calculate the exact or approximate optimum value of ω0 . Another idea is inspired by the fact that the pointing error angle θ can be measured at the transmitter side [14], [22] and that the laser beam waist ω0 can be adjusted sufficiently fast [23], [24]. Using these pointing error angle measurement and beam waist adjustment technologies, one can derive a method to find the optimum dynamic ωz according to the instantaneous measured θ value. This problem can be mathematically formulated as:   2zθ 2a , (14) max hp (θ, a, ω0 , z) = 1 − QM ar ω0 ωz ωz s.t. ω0 > 2λ/π, which can be further reduced to:   2zθ 2a M ar , min Q ω0 ωz ωz

(15)

s.t. ω0 > 2λ/π. We have proposed a suboptimal method to continuously adjust the beam waist ω0 in [25], where we use a square to approximate the circular detector area and express the instantaneous channel gain in terms of the Gaussian Q-function. This method has been shown effective for inter-satellite communications. However, since an approximation is made in calculating the instantaneous channel gain, the method is not mathematically optimum and the performance can be further improved by solving (15), which gives the real optimum solution. The work of finding the optimum beam waist is beyond the scope of this paper and these two examples given in the above two paragraphs are to highlight the importance and usefulness of (13). III. APPROXIMATE ABEP USING BOUNDS ON THE GAUSSIAN Q-FUNCTION A. An Alternative Expression of the ABEP For the calculation of the ABEP, besides averaging the instantaneous BEP over all possible pointing error angle θ values,

SONG et al.: IMPACT OF POINTING ERRORS ON THE ERROR PERFORMANCE OF INTERSATELLITE LASER COMMUNICATIONS

i.e., using equation set (1), (2), and (7), or (1), (2), and (13), we can also average the instantaneous BEP over all possible hp values. Specifically from (5), the conditional BEP of BPSK for a given value of hp is given as  P (e|hp = h) = QG ( hγ). (16) Thus, the ABEP P (e) can be obtained by  ∞ P (e|hp = h)ph p (h)dh. P (e) =

(17)

0

Here, ph p (h) is the pdf of hp . Several references, e.g., [1], [2], study the pointing error effects on the channel gain hp , and thus on the ABEP. However, it should be emphasized that the exact closed-form expression of ph p (h) is not available in the literature at this point. In this paper, we use the model given in [2], which is obtained by approximating the circular detector using a square of the same area and assuming θ to be Rayleigh distributed with the scale parameter σθ . The pdf of hp is given as [2, eq. (11)] ph p (h) =

s2 s 2 −1 , 0 ≤ h ≤ A0 . 2 h As0

(18)

Here, A0 is the fraction of the collected power when no pointing error occurs, and s = ωz e q /(2σd ) is the ratio between the equivalent beam radius ωz e q at the receiver and the point[2]. ing error displacement standard jitter √ σd at the receiver 2 We have A0 = [erf(v)]2 , ωz2e q = ωz2 πerf(v)/(2ve−v ), and √ √ v = πa/( 2ωz ), where erf(.) is the error function. Here, σθ is called the standard pointing jitter, and we have σd = z · σθ . In the literature, there are abundant analytical results based on (18), e.g., [3], [26], [27]. However, these results are given in terms of very complicated hypergeometric functions or Meijer’s G-function, which are too difficult to facilitate further analytical studies. In the next subsection, we will provide efficient upper and lower bounds on the error probability given in (17), which involves (18). These bounds can give more insights into the system error performance. B. Efficient Bounds In this subsection, using some existing upper and lower bounds on the Gaussian Q-function, we derive several tight upper and lower bounds on the ABEP of (17), which involve simple closed-form expressions. In [6] and [7], we have developed an upper bound on the Gaussian Q-function [6, eq. (9)] QG (x) ≤ QU B 1 (x) =

q

ak k =0

x

exp(−bk x2 )

and a lower bound on the Gaussian Q-function [7, eq. (12)] QG (x) ≥ QL B 1 (x) =

q −1

k =1

ck x exp(−dk x2 ),

where the adjustable coefficients ak , bk , ck and dk are given, respectively, as ⎧ 1 ⎪ k=0 ⎪ ⎨ √2π , , ak = ⎪  − k −1 ⎪ ⎩ − √k , k≥1 2πk k −1 ⎧1 ⎪ k=0 ⎨ , bk = 2 , ⎪ ⎩ k k −1 , k ≥ 1 2 k − k −1 ck = √ , k ≥ 1, (21) 2π and 2k + 2k −1 + k k −1 , k ≥ 1. 6 In (21), k ’s are chosen such that 1 = 0 < 1 < . . . < q < . . .. In [6] and [7], it has been shown that these bounds can be made arbitrarily tight as the number of k ’s, i.e., q, increases and the differences between adjacent k ’s, i.e., k − k −1 , reduces. In this paper, we use equally-spaced k ’s for numerical illustration, i.e., the values of (k − k −1 ) for any k are equal, the accuracy of which has been validated in [28]. Using the upper bound (19) on (16) in conjunction with (18), we have an upper bound on (17) given as dk =

P (e) ≤ PU B 1 (e) q  A0

a s2 2 √ k exp(−bk γh) s 2 hs −1 dh. = hγ A0 k =0 0 After some operations, the above bound PU B 1 (e) becomes PU B 1 (e) =

q

k =0



(20)

1 ak s2 2 (bk γ) 2 −s s2 √ A0 γ

A 0 bk γ

exp(−bk γh)(bk γh)s

2

− 32

d(bk γh). (22)

0

The integral form in (24) can be expressed in terms of the lower incomplete gamma function, which is defined as [29, eq.(8.350)]  x e−t tα −1 dt, [Re α > 0]. (23) Γ(α, x) = 0

Thus, the closed-form upper bound on (17) is P (e) ≤ PU B 1 (e) =

q (1/2−s

ak s2 b k

k =0

(19)

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A0 s

2

2

)

  1 2 γ −s Γ s2 − , bk γA0 . (24) 2

According to the definition of the lower incomplete gamma function, which is given in (25), the real part of (s2 − 12 ) in (26) is required to be non-negative. Thus, PU B 1 (e) in (26) works for s > 0.71. However, the case of s < 1, caused by very large standard pointing jitter σθ , occurs seldom in practice. All the other bounds we introduce later do not have this limitation, i.e., they work for any value of s.

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Similarly, we can use the lower bound (20) on the Gaussian Q-function to obtain a lower bound on the ABEP given in (17), which is

TABLE I PARAMETER VALUES FOR NUMERICAL RESULTS

P (e) ≥ PL B 1 (e) =

q −1 −(1/2+s

ck s2 d k

A0 s

k =1

2

2

)

γ

−s 2



1 Γ s + , dk γA0 2



2

. (25)

Both bounds PU B 1 (e) and PL B 1 (e) in (26) and (27) are explicit, closed-form expressions, which provide explicit insights into how the performance depends on the system parameters. They can be arbitrarily tight by adjusting the values of q and (k − k −1 ). We will show later that these bounds are extremely tight and mathematically tractable for any system parameters. Next, for simplicity, we also apply pure exponential upper and lower bounds on QG (x) to obtain the bounds on the ABEP with fixed non-adjustable parameters. The tightness of these pure exponential bounds on QG (x) has been demonstrated in [8], [9], which guarantees the tightness of the following bounds on the ABEP. A well-known pure exponential approximation on QG (x) is given by [8, eq.(14)] QG (x) ≈

 2   x 2x2 1 1 exp − + exp − , 12 2 4 3

(26)

which is verified in [8] to be a tight upper bound for x > 0.71. Thus, in this paper, we use QU B 2 (x) to denote the right side of (26), i.e., QU B 2 (x) =

 2   x 2x2 1 1 exp − + exp − . 12 2 4 3

(27)

This approximation in conjunction with (17) and (18) gives a simple approximation on P (e), that is,   2 s2 2s −s 2 2 γA0 γ Γ , s 2 2 12As0  −s 2   2 −s 2 2 2γA0 γ Γ s , . 3 3

P (e) ≈ PU B 2 (e) = +

s2 2 4As0

(28)

In most cases, we have P (e) ≤ PU B 2 (e).

(29)

In addition, a simple, pure exponential lower bound on QG (x) is given by [9, eq.(9)]  √   √  2 −2 − 3x2 3x 1 1 QG (x) ≥ QL B 2 (x) = exp + exp . 6 π 6 π (30)

This bound (30) has been shown to be tight in [9] and can lead to a tight lower bound on P (e) as  √ −s 2 3 s2 2 γ −s P (e) ≥ PL B 2 (e) = 2 π 6As0      √ √ 3γA 3γA 2 2 0 0 + Γ s2 , . × 2−s Γ s2 , π π (31) As we will show, PU B 2 (e) and PL B 2 (e) in (29) and (31) are tight enough. But they cannot be arbitrarily tight as the bounds PU B 1 (e) and PL B 1 (e) in (24) and (25), since they have fewer fixed terms. We can see that all of these bounds have similar forms. C. Numerical Results For numerical illustration throughout this paper, we use (1) in conjunction with (2) and (13) as the exact ABEP result. We regard θ as a Rayleigh distributed random variable with the scale parameter σθ , as in [1]. Since the accuracy of the satellite tracking and pointing system allows θ to be less than 100 μrad [31], we thus set σθ ∈ [10, 60] μrad. The lengths of inter-satellite links are commonly between 1,000 km and 80,000 km and we choose z = 8000, 104 , 2 × 104 km here. The receiver aperture diameter is set to be 2a = 0.25 m; and the carrier wavelength is λ = 1.064 μm. The data rate is set to be Rdata = 1 Gbps and the transmit power Pt has been limited up to 37 dBm (5 Watts). The parameters are also summarized in Table I. We set q = 10 and (k − k −1 ) = 0.1 throughout this paper, which are sufficient for the accuracy of the bounds PU B 1 (e) and PL B 1 (e). The comparison with the exact ABEP in (1) in Fig. 3 shows that for beam waist ω0 = 3 and 5 mm, the bounds (26), (27), (31) and (33) are extremely tight in the whole transmit power region of interest. Note that (26) and (27) can be made arbitrarily tight by adjusting the parameters q and k in (21), i.e., by increasing the number of sub-ranges, q, or shifting the partition points k to make the values of (k − k −1 ) smaller. Fig. 3 shows that for the power region of Pt > 21 dBm, the performance of ω0 = 3 mm is much better, and the ABEP drops most rapidly as Pt increases. For Pt ∈ [15, 21] dBm, ω0 = 5 mm performs better. The crosspoint in Fig. 3 implies that to achieve minimum ABEP, we need to optimize the beam waist ω0 case by case for different values of Pt .

SONG et al.: IMPACT OF POINTING ERRORS ON THE ERROR PERFORMANCE OF INTERSATELLITE LASER COMMUNICATIONS

reduces to P (e) ≤ PU B 1 (e) ≈

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  q

ak s2 Γ s2 − 12 A0

k =0

s2

1

bk2

−s 2

γ −s , 2

(32)

which is shown to be a good approximation later. Therefore, we show that for a large value of Pt , i.e., large γ, PU B 1 (e) in (24) can reduce to an invertible expression, given in the form: (33) P (e) ≈ Cγ −s = DPt−s .  −s 2 . It thus Here, C is a constant, and we have D = C eR4R data follows that we obtain C from (32) as   q

ak s2 Γ s2 − 12 12 −s 2 bk . (34) C = CU B 1 = 2 A0 s k =0 2

Fig. 3. Performance with different beam waist ω 0 values; σ θ = 20 μrad, z = 2 × 104 km.

2

For a given value of P (e), one can easily obtain the required value of Pt :  s −2 D Pt ≈ . (35) P (e) It is shown later that (33) turns out to be an accurate approximation to the exact ABEP (17). Actually, all of the above bounds can reduce to the invertible expression given in (33). Next, we show the values of C in (33), respectively, for different bounds. As γ increases, the lower bound in (25) can be well approximated as (33) with C given by   q −1

ck s2 Γ s2 + 12 − 12 −s 2 C = CL B 1 = dk . (36) 2 A0 s k =1

Fig. 4. Performance with different pointing error jitter σ θ values; ω 0=5.5 mm, z = 2 × 104 km.

Moreover, Fig. 4 gives the comparison of performance with different levels of pointing errors, where the exact results obtained from (1) and results of our bounds (26), (27), (31) and (33) are all given. We can see that our bounds are very tight in the three cases where σθ varies. Inevitably, as σθ increases, the system error performance degrades. With the ABEP value of 10−6 , compared to the case of σθ = 10 μrad, the power penalties are about 4 dB and 12.5 dB, respectively, for the cases of σθ = 15 and 20 μrad. IV. INVERTIBLE APPROXIMATIONS AND DIVERSITY GAIN A. Invertible Expressions and Diversity Analysis Although all the bounds on the ABEP are expressed in closed form, it is still not straightforward to get the required value of Pt or γ for a given value of P (e), since γ is involved in the lower incomplete gamma function Γ(·, ·). Therefore, we will derive invertible approximations to the ABEP. First, for the upper  bound PU B 1 (e)  in (24),  as γ increases, Γ s2 − 12 , bk γA0 can reduce to Γ s2 − 12 , since Γ(.) is the ∞ gamma function defined as Γ(α) = 0 e−t tα −1 dt. Thus, (24)

For large γ, the bounds (31) and (33) can simplify to (35) with C, respectively, given as    2 s2 Γ s2 2s s2 + 1.5 (37) C = CU B 2 = 2 3 4As0 and C = CL B 2

   √ −s 2   s2 Γ s2 3 −s 2 2 = + 1 . 2 π 6As0

(38)

We can see that all the values of C above are constants related to A0 and s. The validity of this invertible expression (35) with different values of C is demonstrated later. Moreover, the invertible expression (35) provides direct insight on obtaining the diversity gain, i.e., diversity order. Normally, the diversity gain is defined as limγ →∞ (− log P (e)/ log γ) given by [32, eq. 3]. However, since the transmit power Pt is one of the most important system parameters for the inter-satellite laser link in practice, we will consider the diversity gain with respective to Pt here. Thus, the diversity gain Gd with respective to Pt , is given as − log P (e) P t →∞ log Pt

Gd = lim

(39)

It thus follows from (35) that the diversity order Gd is Gd = s2 .

(40)

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Fig. 5.

JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 35, NO. 14, JULY 15, 2017

ω z and ω z e q versus ω 0 ; λ = 1.064 μ m, z = 2 × 104 km. Fig. 7. Invertible ABEP as a function of ω z for different (σ θ , P t ) combinations at z = 2 × 104 km.

Fig. 6. Invertible ABEP comparison as a function of P t for different z at ω 0 = 3 mm, σ θ = 20 μrad.

Note that Gd depends on the ratio s = ωz e q /(2σd ), and it obviously decreases as σd increases. More importantly, we cannot easily get the value of Gd from the exact ABEP expression (17). B. Numerical Results For numerical illustration, we will use PU B 1−in v (e), PL B 1−in v (e), PU B 2−in v (e) and PL B 2−in v (e) to denote the invertible expression (35) with CU B 1 , CL B 1 , CU B 2 and CL B 2 , respectively. It should be noted that the diversity order Gd increases as ωz increases, since ωz has a positive correlation with ωz e q as Fig. 5 shows. Besides, we can see from Fig. 5 that for z = 2 × 104 km, the beam waist ω0 varies inversely with respect to ωz . Therefore, we conclude that Gd decreases as the small values of ω0 increases, which is validated by Fig. 3 where ω0 varies from 3 mm to 5 mm. Fig. 6 shows that for σθ = 20 μrad, all the invertible approximations (35) with different values of C perform almost the same as the exact ABEP (1), in the region of Pt corresponding to the ABEP values of 10−4 and lower which are of practical interest. Moreover, as the link distance z increases, the radial displacement d increases, which results in a worse error performance. As Fig. 6 shows, for the ABEP value of 10−6 , the case of z = 2 × 104 km has a 8 dB power penalty compared to that of z = 8000 km.

Fig. 8. ABEP comparison between OOK and BPSK as a function of P t for ω 0 = 5.5 mm, σ θ = 15 μrad and z = 2 × 104 km.

Furthermore, we explicitly show the effect of the beam radius ωz on the ABEP. We find that the transmit power Pt and standard jitter σd jointly determine the optimum value of ωz , the adjustment of which can be done by adjusting ω0 according to (9). As shown in Fig. 7, the optimum values of ωz are 1935 m, 2185 m and 2509 m, respectively, for (σθ , Pt ) combinations of (15 μrad, 25 dBm), (20 μrad, 25 dBm) and (20 μrad, 27 dBm). In some literatures, e.g., [1], there are wrong impressions that the ratio of ωz and σd , i.e., (ωz /σd ), can be optimized to a fixed value given a certain value of Pt . We hope to emphasize that this is not true from our observation. As given in our example for z = 2 × 104 km, at Pt = 25 dBm, the corresponding optimum values of (ωz /σd ) are 6.45 and 5.46, respectively, for σd = σθ · z = 300 m and 400 m, which are obviously nonidentical. We emphasize that our approach can be generalized to any modulation format, for which the BEP result in pure AWGN can be expressed in terms of a linear combination of single Gaussian Q-functions. Here, we will compare the performance between coherent BPSK and on-off keying (OOK) with direct detection. Using the same approach, the ABEP results of OOK are given

SONG et al.: IMPACT OF POINTING ERRORS ON THE ERROR PERFORMANCE OF INTERSATELLITE LASER COMMUNICATIONS

TABLE II INTERMEDIATE VARIABLES IN DERIVING (41) AND (44)

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Using the upper bound (19) on (41) in conjunction with (18), we have an upper bound on (17) given as P (e) ≤ PU B 1 (e) =

q (1−s

ak s2 b

2A0 s

k =0

in the Appendix. As Fig. 8 shows, at the ABEP value of 10−5 , BPSK has about 15 dB of power gain over OOK. This justifies the benefit of optically coherent receivers.

)/2



2

− s2

γo

2

Γ

s2 − 1 , bk γo A0 2 2

 . (43)

Similarly, we can use the lower bound (20) on the Gaussian Q-function to obtain a lower bound on the ABEP, which is P (e) ≥ PL B 1 (e)

V. CONCLUDING REMARKS In this paper, we have developed a closed-form expression to calculate the exact instantaneous channel gain for inter-satellite links. This expression is simple and thus facilitates later numerical study, and due to the involvement of the Marcum Q-function, it is potentially helpful in further analytical studies. Tight, simple, upper and lower bounds on the ABEP are derived, which lead to invertible ABEP expressions that reveal the achievable diversity order and the dependence on system parameters. Our approach can be further used to analyze the combined effect of atmospheric turbulence and pointing errors for terrestrial laser links, and to optimize parameters to achieve the minimum ABEP. These will be given in a future report. APPENDIX Here, we present the closed-form bounds and invertible results of OOK with direct detection limited by thermal noise. For OOK, the instantaneous BEP conditioned on a given value of the pointing error angle θ is  √  P (e|θ) = QG hp (θ, ωz , z, a) γ o ,

2

k

(41)

=

q −1 −(1+s

ck s2 d

2

)/2

k

2A0 s

k =1

2



2

− s2

γo

Γ

s2 + 1 , dk γo A0 2 2

 . (44)

Next, for simplicity, we also apply pure exponential upper and lower bounds (26) and (30) on QG (x) to obtain the bounds on the ABEP with fixed non-adjustable parameters. The upper bound is  2  2 s γo A0 2 s2 2s /2 − s22 , Γ P (e) ≈ PU B 2 (e) = 2 γo 2 2 24As0  −s 2 /2  2  2 2 s 2γo A0 2 s2 − s2 , γ Γ + . (45) o 2 3 2 3 8As0 The lower bound is  √ − s22 2 3 s2 −s γo 2 P (e) ≥ PL B 2 (e) = 2 s π 12A0      √ √ √ −s 2 3γo A0 2 s2 2 3γo A0 2 s2 , , × ( 2) Γ +Γ . 2 π 2 π (46)

with γo given by γo =

2(Pt R)2 Rload , N0 Rdata

(42)

where Rload is the receiver circuit resistance, and N0 is the receiver one-sided thermal noise power spectral density. All the parameters are given in Table I. The derivation of (41) and (44) is as follows. We use Pr = hp Pt to denote the average received optical power. Thus for OOK, the instantaneous received power for ON symbol is 2Pr , and for OFF symbol is 0. Table II sequentially shows the optical currents (noiseless), the voltages through the receiver resistor, and the samples after the matched filter (MF) for both ON and OFF symbols. The MF here is u(t) = √1T , 0 < t < Ts . Apparently, we can obtain the minis √ mum Euclidean distance of OOK as dm in = 2hp Pt RRload Ts . This direct-detection system is thermal noise-limited, and the noise term after the MF is Gaussian distributed with mean zero and variance σ 2 = N20 Rload . The BEP of OOK follows the form QG ( d m iσn /2 ), which can be easily shown equivalent to that expressed in (41).

It is obvious that for a large value of Pt , i.e., large γo , all the bounds above can reduce to an invertible expression 2

− s2

P (e) ≈ Co γo

= Do Pt−s . 2

(47)

 2 − s22 Rlo a d Here, Co is a constant, and we have Do = Co 2R . N0 Rd ata It thus follows that we obtain Co from (43), (44), (45) and (46), respectively, as  2  q a s2 Γ s −1

k 1 −s 2 2 bk 2 (48) Co = Co−U B 1 = s2 2A0 k =0  2  q −1 c s2 Γ s +1 2

k 2 − 1 +2s d (49) Co = Co−L B 1 = 2 k 2A0 s k =1  2  √ 2 s2 Γ s2 √ ( 2)s s2 + ( 1.5) (50) Co = Co−U B 2 = 2 3 8As0

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and Co = Co−L B 2 =

s2 Γ



 2

s 2

12As0

2

√  3 π

2 − s2

 √ 2 ( 2)−s + 1 . (51)

The closed-form bounds and invertible results can be easily shown to be very tight and accurate. REFERENCES [1] M. Toyoshima, T. Jono, K. Nakagawa, and A. Yamamoto, “Optimum divergence angle of a Gaussian beam wave in the presence of random jitter in free-space laser communication systems,” J. Opt. Soc. Amer. A, vol. 19, no. 3, pp. 567–571, Mar. 2002. [2] A. Farid and S. Hranilovic, “Outage capacity optimization for free-space optical links with pointing errors,” J. Lightw. Technol, vol. 25, no. 7, pp. 1702–1710, Jul. 2007. [3] H. Sandalidis, T. Tsiftsis, G. Karagiannidis, and M. Uysal, “BER performance of FSO links over strong atmospheric turbulence channels with pointing errors,” IEEE Commun. Lett., vol. 12, no. 1, pp. 44–46, Jan. 2008. [4] J. Ma, Y. Jiang, L. Tan, S. Yu, and W. Du, “Influence of beam wander on bit-error rate in a ground-to-satellite laser uplink communication system,” Opt. Lett., vol. 33, no. 22, pp. 2611–2613, Nov. 2008. [5] H. Guo, B. Luo, Y. Ren, S. Zhao, and A. Dang, “Influence of beam wander on uplink of ground-to-satellite laser communication and optimization for transmitter beam radius,” Opt. Lett., vol. 35, no. 12, pp. 1977–1979, Jun. 2010. [6] H. Fu, M. Wu, and P. Y. Kam, “Explicit, closed-form performance analysis in fading via new bound on Gaussian Q-function,” in Proc. IEEE Int. Conf. Commun., Jun. 2013, pp. 5819–5823. [7] H. Fu, M. Wu, and P. Y. Kam, “Lower bound on averages of the product of L Gaussian Q-functions over Nakagami-m fading,” in Proc. IEEE 77th Veh. Technol. Conf., Dresden, Germany, Jun. 2013, pp. 1–5. [8] M. Chiani, D. Dardari, and M. K. Simon, “New exponential bounds and approximations for the computation of error probability in fading channels,” IEEE Trans. Wireless Commun., vol. 2, no. 4, pp. 840–845, Jul. 2003. [9] M. Wu, X. Lin, and P.-Y. Kam, “New exponential lower bounds on the Gaussian Q-function via Jensen’s inequality,” in Proc. IEEE 73rd Veh. Technol. Conf., Yokohama, Japan, May 2011, pp. 1–5. [10] P. A. Govind, Fiber-Optic Communication Systems, 3rd ed. New York, NY, USA: Wiley, 2002. [11] M. A. Bandres and J. Guti´errez-Vega, “Ince gaussian beams,” Opt. Lett., vol. 29, no. 2, pp. 144–146, 2004. [12] A. Siegman, Lasers. Sausalito, CA, USA: University Science, 1986. [13] C.-C. Chen and C. S. Gardner, “Impact of random pointing and tracking errors on the design of coherent and incoherent optical intersatellite communication links,” IEEE Trans. Commun., vol. 37, no. 3, pp. 252–260, Mar. 1989. [14] M. Toyoshima, Y. Takayama, H. Kunimori, T. Jono, and S. Yamakawa, “In-orbit measurements of spacecraft microvibrations for satellite laser communication links,” Opt. Eng., vol. 49, no. 8, pp. 083 604–083 604, Aug. 2010. [15] H. G. Sandalidis, “Performance analysis of a laser ground-station-tosatellite link with modulated gamma-distributed irradiance fluctuations,” J. Opt. Commun. Netw., vol. 2, no. 11, pp. 938–943, Nov. 2010. [16] Q. Ran, Z. Yang, J. Ma, L. Tan, H. Liao, and Q. Liu, “Weighted adaptive threshold estimating method and its application to satellite-to-ground optical communications,” Opt. Laser Technol., vol. 45, pp. 639–646, Feb. 2013. [17] P. Y. Kam and R. Li, “Computing and bounding the first-order Marcum Q-function: A geometric approach,” IEEE Trans. Commun., vol. 56, no. 7, pp. 1101–1110, Jul. 2008. [18] P. Y. Kam and R. Li, “A new geometric view of the first-order Marcum Q-function and some simple tight erfc-bounds,” in Proc. IEEE 63rd Veh. Technol. Conf., 2006, vol. 5, pp. 2553–2557. [19] P. Y. Kam and R. Li, “Simple tight exponential bounds on the first-order Marcum Q-function via the geometric approach,” in Proc. IEEE Int. Symp. Inform. Theory, 2006, pp. 1085–1089. [20] R. Li and P. Y. Kam, “Generic exponential bounds on the generalized Marcum Q-function via the geometric approach,” in Proc. IEEE Global Telecommun. Conf., Washington, DC, USA, 2007, pp. 1754–1758.

[21] R. Li and P.-Y. Kam, “Computing and bounding the generalized Marcum Q-function via a geometric approach,” in Proc. IEEE Int. Symp. Inf. Theory, Seattle, WA, USA, 2006, pp. 1090–1094. [22] K. Schultz, D. Kocher, J. Daley, J. Theriault, J. Spinks, and S. Fisher, “Satellite vibration measurements with an autodyne CO2 laser radar,” Appl. Opt., vol. 33, no. 12, pp. 2349–2355, Apr. 1994. [23] K. Inagaki and Y. Karasawa, “Ultra high speed optical beam steering by optical phased array antenna,” in Proc. SPIE, Free-Space Laser Commun. Technol. VIII, San Jose, CA, Jan. 1996, vol. 2699, pp. 210–217. [24] W. M. Neubert, K. H. Kudielka, W. R. Leeb, and A. L. Scholtz, “Experimental demonstration of an optical phased array antenna for laser space communications,” Appl. Opt., vol. 33, no. 18, pp. 3820–3830, Jun. 1994. [25] T. Song, Q. Wang, M.-W. Wu, and P.-Y. Kam, “Performance of laser inter-satellite links with dynamic beam waist adjustment,” Opt. Express, vol. 24, no. 11, pp. 11 950–11 960, 2016. [26] P. Sharma, A. Bansal, P. Garg, T. Tsiftsis, and R. Barrios, “Performance of FSO links under exponentiated Weibull turbulence fading with misalignment errors,” in Proc. IEEE Int. Conf. Commun., Jun. 2015, pp. 5110– 5114. [27] R. Boluda-Ruiz, A. Garc´ıa-Zambrana, B. Castillo-V´azquez, and C. Castillo-V´azquez, “Impact of relay placement on diversity order in adaptive selective DF relay-assisted FSO communications,” Opt. Express, vol. 23, no. 3, pp. 2600–2617, Feb. 2015. [28] Q. Wang, H. Lin, and P. Y. Kam, “Tight bounds and invertible average error probability expressions over composite fading channels,” J. Commun. Netw., vol. 18, no. 2, pp. 182–189, 2016. [29] I. S. Gradshteyn, Table of Integrals, Series, and Products. New York, NY, USA: Academic, 2007. [30] S. Dolinar, D. Divsalar, J. Hamkins, and F. Pollara, “Capacity of pulseposition modulation (PPM) on gaussian and webb channels,” JPL TMO Progress Rep., vol. 42, no. 142, pp. 1–31, 2000. [31] S. Arnon and N. Kopeika, “Laser satellite communication network vibration effect and possible solutions,” Proc. IEEE, vol. 85, no. 10, pp. 1646–1661, Oct. 1997. [32] L. Zheng and D. N. Tse, “Diversity and multiplexing: A fundamental tradeoff in multiple-antenna channels,” IEEE Trans. Inf. Theory, vol. 49, no. 5, pp. 1073–1096, May 2003.

Tianyu Song (S’13) was born in Wuchang, Heilongjiang Province, China in 1989. He received the B.Eng. degree from Honors School of Harbin Institute of Technology, Harbin, China, in 2011, and the Ph.D. degree from National University of Singapore, Singapore, in January 2016, supervised by Prof. P.-Y. Kam. From September 2009 to June 2010, he was an exchange student in the Department of Electrical Engineering, Korea Advanced Institute of Science and Technology, Daejon, South Korea. His research interests include Free Space Optical communications, optimal receiver design, stochastic processes and algorithms. He received the Best Paper Award at the IEEE/CIC ICCC2015.

Qian Wang (S’14) was born in Songyuan, Jilin, China, in 1990. She received the B.Eng. degree from the College of Information and Communication Engineering, Harbin Engineering University, Harbin, China, in 2012, the Ph.D. degree from National University of Singapore, Singapore, in January 2017, supervised by Prof. P.-Y. Kam under who she is currently working as a Research Fellow. Her research interests include performance analysis and receiver design for wireless and optical communications, constellation design and optimization, and information theory and algorithms.

Ming-Wei Wu received the B.E. (with First Class Hons.), M.E., and Ph.D. degrees in electrical engineering from the National University of Singapore, Singapore, in 2000, 2003, and 2011, respectively. From 2002 to 2004, she was as a Research Engineer in the Institute for Infocomm Research, Singapore, and worked on Ethernet passive optical networks standardization and implementation. In 2004, she joined the School of Information and Electronic Engineering, Zhejiang University of Science and Technology, China, as a Lecturer. Her research interests include wireless communication, detection and estimation theory, and performance analysis. She is a Technical Program Committee Member for international communications conferences including the IEEE ICC2012, the IEEE ICC2011, the IEEE VTC2013-Fall, the IEEE VTC2013-Spring, and the IEEE ICCC2012. She received the Best Paper Award at the IEEE ICC2011, Kyoto, Japan.

SONG et al.: IMPACT OF POINTING ERRORS ON THE ERROR PERFORMANCE OF INTERSATELLITE LASER COMMUNICATIONS

Tomoaki Ohtsuki (SM’01) received the B.E., M.E., and Ph.D. degrees from Keio University, Yokohama, Japan, in 1990, 1992, and 1994, respectively, all in electrical engineering. From 1994 to 1995, he was a Postdoctoral Fellow and a Visiting Researcher of electrical engineering in Keio University. From 1995 to 2005, he was with Tokyo University of Science, Tokyo, Japan. From 1998 to 1999, he was with the Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, CA, USA. He is currently an Associate Professor at Keio University. His research interests include wireless communications, optical communications, signal processing, and information theory. He was a Special Researcher with Fellowships with the Japan Society for the Promotion of Science for Japanese Junior Scientists from 1993 to 1995. He received the 1997 Inoue Research Award for Young Scientists, the 1997 Hiroshi Ando Memorial Young Engineering Award, the 2000 Ericsson Young Scientist Award, the 2001 IEEE First Asia-Pacific Young Researcher Award, and the 2002 Funai Information and Science Award. He is a member of the Symposium on Information Theory and Its Applications.

Mohan Gurusamy (M’00–SM’07) received the Ph.D. degree in computer science and engineering from the Indian Institute of Technology, Madras, Chennai, India, in 2000. He joined the National University of Singapore, Singapore, in June 2000, where he is currently an Associate Professor in the Department of Electrical and Computer Engineering. He has about 150 publications to his credit, including two books and three book chapters in the area of optical networks. His research interests include the areas of cloud data center networks, software defined networks, green networks and optical networks. He is currently an Editor for the IEEE TRANSACTIONS ON CLOUD COMPUTING, Computer Networks (Elsevier), and Photonic Network Communications (Springer). He was the Lead Guest Editor for two special issues of the IEEE COMMUNICATIONS MAGAZINE (OCS), August 2005 and November 2005, and a Co-Guest Editor for a special issue of the Optical Switching and Networking (Elsevier), November 2008. He was also a TPC Co-Chair for several conferences including IEEE ICC 2008 (ONS). He was the organizer and Lead Chair for CreateNet GOSP workshops colocated with Broadnets conference, October 2005 and October 2006, and September 2008.

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Pooi-Yuen Kam (F’10) was born in Ipoh, Malaysia. He received the S.B., S.M., and Ph.D. degrees in electrical engineering from Massachusetts Institute of Technology, Cambridge, MA, USA, in 1972, 1973, and 1976, respectively. From 1976 to 1978, he was a Member of the Technical Staff in the Bell Telephone Laboratories, Holmdel, NJ, USA, where he was engaged in packet network studies. Since 1978, he has been with the Department of Electrical and Computer Engineering, National University of Singapore, Singapore, where he is currently a Professor. He was the Deputy Dean of Engineering and the Vice Dean for Academic Affairs, Faculty of Engineering of the National University of Singapore, from 2000 to 2003. His research interests are in the communication sciences and information theory, and their applications to wireless and optical communications. He spent the sabbatical year 1987 to 1988 in the Tokyo Institute of Technology, Tokyo, Japan, under the sponsorship of the Hitachi Scholarship Foundation. In year 2006, he was invited to the School of Engineering Science, Simon Fraser University, Burnaby, B.C., Canada, as the David Bested Fellow. He was appointed a Distinguished Guest Professor (Global) in the Graduate School of Science and Technology of Keio University, Tokyo, Japan, for the academic years April 2015 to March 2017. Dr. Kam is a member of Eta Kappa Nu, Tau Beta Pi, and Sigma Xi. Since September 2011, he has been a Senior Editor of the IEEE WIRELESS COMMUNICATIONS LETTERS. From 1996 to 2011, he was the Editor of Modulation and Detection for Wireless Systems of the IEEE TRANSACTIONS ON COMMUNICATIONS. He also served on the editorial board of PHYCOM, the Journal of Physical Communications of Elsevier, from 2007 to 2012. He was also a Co-Chair of the Communication Theory Symposium of IEEE Globecom 2014. He was elected a Fellow of the IEEE for his contributions to receiver design and performance analysis for wireless communications. He received the Best Paper Award at the IEEE VTC2004-Fall, at the IEEE VTC2011-Spring, at the IEEE ICC2011, and at the IEEE/CIC ICCC2015.