Optics & Laser Technology 45 (2013) 705–707

Contents lists available at SciVerse ScienceDirect

Optics & Laser Technology

0030-39

http://d

n Corr

E-m

journal homepage: www.elsevier.com/locate/optlastec

Optimum signal input distribution design in the presence of random pointingjitter for intersatellite optical communications

Xin Li n, Jing Ma, Siyuan Yu, Liying Tan, Tao Shen

National Key Laboratory of Tunable Laser Technology, Institute of Opto-electronics, Harbin Institute of Technology, Harbin 150001, China

a r t i c l e i n f o

Article history:

Received 2 February 2012

Received in revised form

7 May 2012

Accepted 8 May 2012Available online 29 May 2012

Keywords:

Input distribution

Channel capacity

Satellite optical communications

92/$ - see front matter & 2012 Elsevier Ltd. A

x.doi.org/10.1016/j.optlastec.2012.05.007

esponding author.

ail address: cammilee@163.com (X. Li).

a b s t r a c t

Channel capacity is widely investigated for free space optical links to approach high-speed data-rate

communication. Instead of traditional equiprobable binary symbol input distribution, an optimum

input distribution is proposed with respect to channel capacity by maximizing mutual information for

intersatellite optical communications in the presence of random pointing jitter. It is shown that the

optimum input distribution varies with the variance of pointing jitter s and laser beam divergence

angle w0 and the normalized intensity threshold IT. For traditional normalized intensity threshold

IT¼0.5, the optimum input distribution ranges from about p(x¼0)¼0.52 for weak pointing jitter to

about p(x¼0)¼0.24 for strong pointing jitter given the same laser beam divergence angle. The results

obtained in this paper will be useful for intersatellite optical communication system design.

& 2012 Elsevier Ltd. All rights reserved.

1. Introduction

With increasing demands for real-time communicationthroughout the whole world, intersatellite optical communica-tions have attracted considerable attention for potential high-speed data-rate and global coverage access [1–5]. Channelcapacity, which describes the maximum achievable code rateallowing reliable communication between transmitters and recei-vers, is the key parameter for optical links and has beenresearched by various studies [6–10]. The report by Borosonshows the channel capacity limits for free space optical links [7].And the impacts of pointing and tracking error on channelcapacity are also investigated for intersatellite optical commu-nications [8,9]. The effects of misalignment and atmosphericturbulence on terrestrial free space optical channel capacityhave been studied by Farid and Hranilovic [10]. However in allcase it is assumed that the equiprobable binary symbols inputdistribution is applicable to the atmospheric or non-atmo-spheric free space optical channels. However, equiprobableinput distribution only serves symmetric channels [6]. In factthe channel for intersatellite optical communications is subjectto random pointing jitter, and the channel is asymmetric [11].As a result the optimum input distribution is no longerp(x¼0)¼p(x¼1)¼1/2. The input probability is considered max-imizing mutual information in order to find the capacityboundary for a photon-counting Poisson channel by Wyner[12]. And the optimum input distribution is also given for fiber

ll rights reserved.

optical channel numerically [13]. However, an analytical opti-mum input distribution for intersatellite optical channel hasnot been presented as far as the authors know.

The optimum signal input distribution with respect to thechannel capacity for intersatellite optical communications in thepresence of random pointing jitter is the issue of concern in thispaper. First the channel matrix is modeled based on the statisticcharacteristic of normalized optical intensity in the presence ofrandom pointing jitter. Then the optimum input distribution isanalytically derived by maximizing the mutual information.Moreover, the channel capacity introducing optimum input dis-tribution is also presented.

2. Channel matrix modeling

A typical intersatellite optical link is sketched in Fig. 1.The transmitter modulates data onto the instantaneous intensityof an optical beam, and it is de-modulated at the receiver. Thechannel for intersatellite link is non-atmospheric free space. Forintersatellite optical communications the channel state is subjectto random pointing jitter, which induces optical intensity fluctua-tion at the receiver and has a significant impact on the linkcommunication performance.

In the presence of random pointing jitter, the probabilitydensity function (PDF) of the received optical intensity follows aBeta distribution expressed as [14]

f ðIÞ ¼w2

0

4s2Iw2

0=4s2�1

ð1Þ

Transmitter

Receiver

Random point jitter angle

Free space channel

Fig. 1. Sketch of intersatellite optical link.

X. Li et al. / Optics & Laser Technology 45 (2013) 705–707706

where IA[0,1] is the normalized intensity, w0 is the divergenceangle of Gaussian laser beam, and s2 is the variance of randompointing jitter.

According to the Shannon capacity theory, the channel matrixmodel must be developed to investigate the channel capacity. In thispaper, we consider the intensity-modulated/direct-detection (IM/DD)with on–off keying (OOK) system, which is widely employed inpractical optical terminals [15,16]. With OOK, the transmitted signalis xA{0,1} and received signal is yA{0,1}. In the presence of randompointing jitter the received signal demonstrates different statisticsdepending on whether ‘0’ or ‘1’ is transmitted.

With the development of photodetector, the detector noise canbe well constricted and in practical system design, threshold ishigher than the detector dark noise level. Hence, when the trans-mitted signal is 0 the impact of detector noise can be neglected byan appropriate threshold. In this case, we can assume that thereceived signal can be identified as ‘0’ when the transmitted signal is‘0’, which can be expressed by the conditional probability

pðy¼ 09x¼ 0Þ ¼ 1 ð2Þ

Furthermore, when the transmitted signal is ‘1’, generally thedetector shot noise raises the received signal intensity level. Thus,the impact of detector noise can be also neglected. However therandom pointing jitter results in optical intensity fluctuationwhich would decrease the intensity level at the receiver, whichis the main reason we investigate the influence of randompointing jitter on channel communication performances in thispaper. In this case, the probability of falsely identifying thereceived signal as ‘0’ when the transmitted signal is ‘1’ isdetermined by threshold level, which can be derived by theconditional probability

pðy¼ 09x¼ 1Þ ¼

Z IT

0f ðIÞ dI¼ I

w20=4s2

T ð3Þ

where IT is the normalized threshold of optical intensity at thereceiver.

Generally the channel matrix is described as

P¼pðy¼ 09x¼ 0Þ pðy¼ 19x¼ 0Þ

pðy¼ 09x¼ 1Þ pðy¼ 19x¼ 1Þ

" #ð4Þ

Therefore, introducing Eqs. (2) and (3) into Eq. (4), the channelmatrix for intersatellite optical links in the presence of randompointing jitter can be obtained as

P¼1 0

Iw2

0=4s2

T 1�Iw2

0=4s2

T

" #ð5Þ

3. Optimum signal input distribution

Channel capacity is defined as the maximum of the mutualinformation MI(x;y) between the channel input and output [6,16].

The maximization is done over the input distribution p(x)

C ¼maxp xð Þ

MIðx; yÞ ð6Þ

MIðx; yÞ ¼ �X2

j ¼ 1

pðyjÞlog2 pðyjÞþX2

i ¼ 1

X2

j ¼ 1

pðxiÞpðyj9xiÞlog2 pðyj9xiÞ

ð7Þ

X2

i ¼ 1

pðxiÞ ¼ 1 ð8Þ

where C denotes the Shannon capacity, the range of which is [0,1]for OOK system, Eq. (7) is the expression of mutual information,and Eq. (8) is the constraint of Eq. (6).

The maximum mutual information MI(x;y) can be foundthrough Lagrange multiplier method, which is widely used inmathematical optimization and provides a strategy for finding themaximum or minimal of a function subject to constraints.Introducing a new parameter l called the Lagrange multiplier,we define the Lagrange function as

L¼MIðX;YÞ�lX2

i ¼ 1

pðxiÞ�1

" #ð9Þ

The critical values of L occur where its gradient is zero. And thepartial derivations are

@L@p xið Þ

¼�X2

j ¼ 1

½pðyj9xiÞlog2 pðyjÞþpðyj9xiÞlog2 e�

þX2

j ¼ 1

pðyj9xiÞlog2 pðyj9xiÞ�l¼ 0 ð10Þ

@L@l¼�

X2

i ¼ 1

pðxiÞþ1¼ 0 ð11Þ

Solving these equations yields the values of p(x), which

maximize the mutual information. Defining e¼ Iw2

0=4s2

T for simpli-

fication, the optimum p(x) obtained from Eqs. (10) and (11) can bedescribed as

pðx¼ 0Þ ¼1�e1=ð1�eÞ

1þð1�eÞee=ð1�eÞð12Þ

pðx¼ 1Þ ¼ee=ð1�eÞ

1þð1�eÞee=ð1�eÞð13Þ

It should be noted that eA[0,1] since ITA[0,1], therefore theprobabilities we got here are non-negative. And it is obvious thatthe optimum signal input distribution satisfies Eq. (8). Conse-quently, the results obtained here are validated.

Eqs. (12) and (13) show the relation between optimum inputdistribution and channel state analytically. It is also indicated thatthe optimum input distribution is dependent on normalizedthreshold IT, divergence angle w0, and especially the standardvariance of random pointing jitter s. Introducing the systemdesign parameter b¼w0/s, which is defined as the ratio ofdivergence angle to the standard variance of pointing jitter, weassess the optimum signal input distribution for various channeland system conditions based on Eqs. (12) and (13).

Fig. 2 shows the optimum signal input probability of x¼0 as afunction of normalized intensity threshold IT for various systemdesign parameters b. It is observed that optimum probability of x¼0is different with traditional 1/2 for various channel and systemconditions. In the presence of random pointing jitter, the channel isasymmetric, and the input distribution that maximizes the mutualinformation is no longer p(x¼0)¼p(x¼1)¼1/2. It is also observedthat the optimum input distribution varies with the jitter strength

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

0.1

0.2

0.3

0.4

0.5

0.6

0.7

IT

Prob

abili

ty o

f x=0

β = 1β = 2β = 5

Fig. 2. Probability of x¼0 as a function of normalized intensity threshold IT for

various system design parameters b.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

IT

Cha

nnel

Cap

acity

(bits

/sym

bol)

β = 1β = 2β = 5

Fig. 3. Channel capacity as a function of normalized intensity threshold IT for

various system design parameters b.

X. Li et al. / Optics & Laser Technology 45 (2013) 705–707 707

and system parameter. Given the same divergence angle w0, thestronger the random pointing jitter, the smaller the b. For strongrandom pointing jitter b¼1, the optimum input probability of x¼0decreases with increasing IT. And for weak random pointing jitterb¼2,5, the optimum input probability of x¼0 first increases slowlyand then decreases fast with increasing IT. It is implied that thestrength of random pointing jitter has a dramatic impact onoptimum input distribution. For traditional normalized intensitythreshold IT¼0.5, the optimum input distribution ranges from aboutp(x¼0)¼0.52 for weak pointing jitter to about p(x¼0)¼0.24 forstrong pointing jitter with the same laser beam divergence angle.

Substituting the optimum input distribution Eqs. (11) and (12)into mutual information Eq. (6), we can obtain the channel capacity,which is the maximum mutual information, expressed as

C ¼ log2 1þð1�eÞee=1�eh i

ð14Þ

Eq. (13) shows the dependence of channel capacity on channelstate. According to Eq. (13), Fig. 3 shows the channel capacity as afunction of normalized intensity threshold IT for various systemdesign parameters b. It is observed that the lower the normalizedintensity threshold is, the higher the channel capacity. This isbecause we assume that detector noise is neglectable. In fact thedetector noise exists, hence the normalized intensity threshold

cannot be so low to approach channel capacity of 1. It is alsoobserved from Fig. 3 that the higher the system design parameter b,the better the channel capacity. It results from the fact that for agiven laser beam divergence angle, the higher the b, the lower thevariance of pointing jitter, in which case the channel is more stable.

4. Conclusion

The optimum signal input distribution is investigated in thispaper with respect to channel capacity for intersatellite opticalcommunication in the presence of random pointing jitter. First thechannel matrix is obtained in the presence of random pointing jitter.And then the optimum input distribution is derived as a function ofnormalized intensity threshold system IT, laser beam divergenceangle and variance of pointing jitter by maximizing the mutualinformation. Numerical simulation shows that the optimum inputdistribution varies with the jitter strength and system parameter.For traditional normalized intensity threshold IT¼0.5, the optimuminput distribution ranges from about p(x¼0)¼0.52 for weak point-ing jitter to about p(x¼0)¼0.24 for strong pointing jitter with thesame laser beam divergence angle. Finally the channel capacity ispresented introducing the optimum input distribution. The resultsobtained in this paper will be useful for intersatellite opticalcommunication system design.

Acknowledgments

This work is supported by the National Natural Science Foundationof China (NSFC) financially with Project no. 10904026.

References

[1] Kaufmann JE, Chan VWS. Coherent optical intersatellite crosslink systems.Proceedings of the IEEE 1988;2:0533–40.

[2] Mahdieh MH, Pournoury M. Atmospheric turbulence and numerical evalua-tion of bit error rate (BER) in free space communication. Optical LaserTechnology 2010;42:55–60.

[3] Chan VWS. Optical satellite networks. IEEE: Journal of Lightwave Technology2003;21(11):2811–27.

[4] Toyoshima M. Trends in satellite communications and the role of optical free-space communications. Journal of Optical Networking 2005;6587:300–11.

[5] Cesarone RJ, Abraham DS, Deutsch L. Prospects for a next-generation deepspace network. Proceedings of the IEEE 2007;95(10):1902–15.

[6] Shannon CE. A mathematical theory of communication. Bell System TechnicalJournal 1948;27:379–423.

[7] Boroson DM. Channel capacity limits for free space optical links. Proceedingsof the SPIE 2008;6951 69510A-1-11.

[8] Anguita JA, Djordjevic IB, Neifeld MA, Vasic BV. Shannon capacities and error-correction codes for optical atmospheric turbulent channels. Journal ofOptical Networking 2005;4(9):586–601.

[9] Barron RJ, Boroson DM. Analysis of capacity and probability of outage forfree-space optical channels with fading due to pointing and tracking error.Proceedings of the SPIE 2006;6105 61050B-1-12.

[10] Farid AA, Hranilovic S. Outage capacity optimization for free space opticallinks with pointing errors. Journal of Lightwave Technology 2007;25:1702–10.

[11] Anguita JA, Djordjevic IB, Neifeld MA, Vasic V. Shannon capacities and error-correction codes for optical atmospheric turbulent channels. Journal ofOptical Networking 2005;4(9):586–601.

[12] Wyner AD. Capacity and error exponent for the direct detection photonchannel. IEEE Transactions: Information Theory 1988;34:1449–71.

[13] Ho KP. Exact evaluation of the capacity for intensity modulated directdetection channels with optical amplifier noises. IEEE Photonics TechnologyLetters 2005;17:858–60.

[14] Toyoshima M, Jono T, Nakagawa K, Yamamoto A. Optimum divergence angleof a Gaussian beam wave in the presence of random jitter in free-space lasercommunication systems. Journal of the Optical Society of America A2002;19(3):567–71.

[15] Li J, Uysal M. Optical wireless communications: system model, capacity andcoding. Proceedings of the IEEE 2003;1:168–72.

[16] Khalighi MA, Schwarts N, Aitamer N, Bourennane S. Fading reduction byaperture averaging and spatial diversity in optical wireless systems. Journalof Optical Communications and Networking 2009;1(6):580–93.