UBC Faculty Research and Publications

Free-space optical communications using on-off keying and source information transformation Yang, Luanxia; Zhu, Bingcheng; Cheng, Julian; Holzman, Jonathan F. Jun 30, 2016

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0Free-space Optical Communications Using On-offKeying and Source Information TransformationLuanxia Yang, Bingcheng Zhu, Julian Cheng, Senior Member, IEEE, Jonathan F. Holzman, Member, IEEEAbstract—Free-space optical communication using on-off key-ing (OOK) and source information transformation is proposed.It is shown that source information transformation allows theproposed system to detect the OOK signal without requiringthe knowledge of instantaneous channel state information andthe probability density function (pdf) of the turbulence model.Analytical expressions are derived for the pdf of the detectionthreshold, and an upper bound is obtained on the average bit er-ror rate (BER). Numerical studies show that the proposed systemcan achieve comparable performance to the idealized adaptivedetection system, with a greatly reduced level of implementationcomplexity and a signal-to-noise ratio performance loss of only):0 dB at a BER of ))(−9 for a lognormal turbulence channelwith  5 (:25.I. INTRODUCTIONFree-space optical (FSO) communication systems have im-portant advantages over their radio frequency counterparts.For example, FSO systems can offer broadband operation,high link security, and freedom from spectral license regu-lations. Optical signals that are transmitted over free-spaceare subject to amplitude and phase distortion. Due to transientinhomogeneities of atmospheric temperature and pressure [1],[2], however, and the resulting scintillation or fading is amajor cause of performance degradation for FSO systems.The performance degradation is especially pronounced forFSO systems using irradiance modulation and direct detec-tion (IM/DD) with on-off keying (OOK) and fixed detectionthresholds, which are non-adaptive and unoptimized [3], [4].Such systems can produce irreducible error floors if operationis extended to high signal-to-noise ratios (SNRs) [5].Attempts to overcome the irreducible error floors of OOKIM/DD systems have focused on the application of adaptivedetection thresholds. The standard approach proposed foradaptive detection applies perfect knowledge of the instanta-neous channel state information (CSI), as it uses the instanta-neous SNR to detect each data symbol [2], [6]. This approachhas obvious practical concerns for OOK IM/DD operationwith nanosecond data symbol durations (i.e., Gbps rates) andmillisecond turbulence coherence times, as rapid detectionthreshold adjustments are needed on the timescale of themillisecond turbulence coherence times [2], [7]-[9]. To accom-modate these practical concerns, the electrical-SNR-optimizedLuanxia Yang, Julian Cheng and Jonathan F. Holzman are withSchool of Engineering, The University of British Columbia, Kelowna,BC, Canada V1V 1V7 (e-mail: yangluanxia@hotmail.com, {julian.cheng,jonathan.holzman}@ubc.ca).Bingcheng Zhu is with School of Information Science and Engi-neering, Southeast University, Nanjing, Jiangsu, China 210096 (e-mail:230109122@seu.edu.cn)detection system was proposed in [10], [11]. Electrical-SNR-optimized detection thresholds only need to change over theespecially long timescales, of seconds or minutes, over whicha stationary turbulence channel assumption applies [12]. How-ever, this method requires knowledge of the probability densityfunction (pdf) of the turbulence channel, which increases thecomputation complexity.In [13], the authors introduced pilot-symbol (PS) assistedmodulation (PSAM) to mitigate the turbulence fading andimprove the system performance. The PS provided the receiverwith explicit turbulence fading references for detection, andthis can be used to mitigate the effects of fading. However,PSAM can cause delays in the receiver as it is necessary tostore the whole frame before decoding[14, Chap. 5]. In [15], itwas demonstrated that an OOK IM/DD system could use twolaser wavelengths at the transmitter and two photodetectorsat the receiver to work in a differential mode, and achieveexcellent bit error rate (BER) performance with a detectionthreshold fixed at zero. Unfortunately, this scheme suffers fromlow throughput, as two lasers are used to transmit the sameinformation in each symbol duration.In this work, we propose an alternative scheme that usestwo (or more) laser transmitters. Such a scheme can improvethe throughput beyond that of [15]. Moreover, the proposedscheme does not require knowledge of the instantaneous CSIand pdf of the turbulence channel, as the receivers haveexplicit turbulence fading references for detection. It is shownthat such a system provides good BER performance withoutirreducible error floors.The remainder of this work is organized as follows. SectionII describes the system and channel models. Section III derivesthe pdf of the detection threshold. Section IV obtains an upperbound on the average BER. Numerical results and discussionsare presented in Section V. Finally, Section VI makes someconcluding remarks.II. SYSTEM AND CHANNEL MODELSWe consider an IM/DD system with M laser source trans-mitters and M photodetectors operating through atmosphericturbulence channels. The operation of the proposed schemeis as follows. At the transmitter, which is shown in Fig.1, there are M distinct optical wavelengths, ); 2; M M M ; M ,assigned to the M laser transmitters. Each wavelength is usedto transmit an independent information sequence, with sourceinformation transformation used to ensure that one or morelasers transmit bit \1: during each symbol duration. WhenM = 2, for example, the proposed system can almost double1Fig. 1. Block diagram of the transmitter for the system using sourceinformation transformation.Fig. 2. Block diagram of the source information transformation. A binaryinformation sequence of length Q is converted to a (2M−))-nary informationsequence of length J , and this (2M −))-nary information sequence of lengthJ is mapped to a binary information sequence of length JM .the multiplexing gain achieved in the system of [15] usingdouble-laser differential signaling.For source information transformation, we first convert abinary information sequence of length L to a (2M − 1)-naryinformation sequence of length J as shown in Fig. 2. Thismapping can be written asg) : {0; 1}L → {0; M M M ; 2M − 1}J M (1)Then we map each element of the (2M − 1)-nary sequenceinto an M -bit binary sequence that does not contain the all-zero binary sequence. The resulting M -bit binary sequenceafter the serial-to-parallel conversion determines, among theM transmitted lasers, which link transmits bit \0: and whichlink transmits bit \1:. For example, when M = 3, we mapthe seven elements of the 7-nary sequence (0; 1; 2; 3; 4; 5 ; 6)to the binary sequence {001; 010; 011; 100; 101; 110; 111}.(Note here that the all-zero binary sequence is avoided.) Thismapping can be written asg2 : {0; M M M ; 2M − 1}J → {0; 1}JM M (2)The mapping described in (1) and (2), which we call sourceinformation transformation, will ensure that the M receivedsignals have an explicit turbulence fading reference for de-tection in each symbol duration, meaning that at least onelaser is on (ie., at least one bit \1: is transmitted). Each bit\1: suffers turbulence distortion and therefore be used as aturbulence fading reference.It is desirable to select values of L and J that make themapping of g) be a one-to-one mapping, i.e.,2L = (2M − 1)J M (3)Fig. 3. Block diagram of the receiver for the system using source informationtransformation.However, the above equality is difficult to achieve in practicefor arbitrary values of L and J . Thus, to approximate the idealcase of (3), we considermin(LNJ)[(2M − 1)J − 2L] (4)Subject to 2L ≤ (2M − 1)J MSince there might be more than one pair (L; J) that satisfies(4), we will choose the smallest pair (L; J) for our system,i.e., the value of L + J that is the smallest among the pairssatisfying (4). This is done to minimize the system delay. Forexample, when M = 3, we select L = 14 and J = 5 by usinga computer search.At the receiver, as shown in Fig. 3, diffractive opticalelements and/or narrowband optical filters are used to separatethe wavelengths for detection of the M transmitted signals. InFig. 3, we use the acronym PD to represent the photodetector.After the M parallel photodetectors and the parallel-to-serialconversion, in each symbol duration, a value of one-half ofthe largest received signal is used to define the detectionthreshold for theM received signals. If all of theM -bit binarysequence are demodulated as bit \0:, which may happen dueto the noise, this is an incorrect decision (since an all-zerobinary sequence is not transmitted for our system), and wewill assume the source transmits 00 M M M 01. The demodulatedJM -bit binary sequence will be mapped to a (2M − 1)-narysequence of length J , and then this (2M − 1)-nary sequenceof length J will ultimately be converted back to a binaryinformation sequence of length L.At themth transmitter, the baseband signal to be transmittedcan be expressed assm(t) =∑itiNmgm(t− igp); m = 1; 2; M M M ;M (5)where tiNm ∈ {0; 1} is the ith data in the mth transmitter,and gp is the symbol duration. In (5), pulse shaping in themth transmitter is defined as gm(t) = 1 for 0 < t < gp,and gm(t) = 0 otherwise. These M signals are transmittedthrough atmospheric turbulence channels and are distorted bya multiplicative process I(u; t). We have assumed that thechannel fading is the same for all the wavelengths in eachsymbol duration. This assumption can be achieved by ensuringthat the transmitter wavelengths are sufficiently close to each2other (being separated only by tens of nanometers). Thisensures that the transmitter beams are spatially overlapped andexperience the same atmospheric turbulence distortion [15].At the mth receiver, the received signal after the photode-tection can be written asrm(t) = eI(u; t)∑itiNmgm(t− igp) + nm(t);m = 1; 2; M M M ;MM(6)Without loss of generality, the photodetector responsivity eis assumed to be unity. In (6), I(u; t) is assumed to be astationary random process for signal scintillation caused byatmospheric turbulence, and nm(t) is additive white Gaussiannoise (AWGN) due to thermal noise and/or ambient shotnoise in mth receiver. Using a p-i-n photodiode and following[10], the shot noise is assumed to be dominated by ambientshot noise. (Both ambient shot noise and thermal noise arestatistically independent of the desired signal.) The total noisepower is .2g = .2s+.2g , where .2s and .2g denote the respectivevariances of the ambient shot noise power and the thermalnoise power.The mth received signal is sampled at time gp. The sampleI(u; t = gp) is a RV I , and the sample nm(t = gp) is anAWGN RV nm having zero mean and variance .2g . Whenbit \0: is transmitted, s( becomes true and the laser is off.The demodulation sample is rm|s0 = nmM When bit \1:is transmitted, s) becomes true and the laser is on. Thedemodulation sample is rm|s1 = I + nm.III. THE PROBABILITY DENSITY FUNCTION OF THEDETECTION THRESHOLDWith perfect knowledge of the instantaneous CSI, the mini-mum error probability is provided by the maximum-likelihood-based decision threshold which can be expressed by [11]gth =.((I( + I)) + .)I(.( + .)(7)where .) and .( are the standard deviations of the noisecurrents for bits \1: and \0:, respectively, and I) and I( areaverages of the generated currents at the receiver for bits \1:and \0:. For simplicity, we assume .( = .) = .g , I( = 0 andI) = I . The maximum-likelihood-based detection threshold isgth = IP2, which is an adaptive detection threshold, as itvaries with the fading coefficient. Note that this approach iscomplex to realize in practice, as it requires perfect knowledgeof the instantaneous CSI for each symbol detection. However,when the average SNR (denoted by ) approaches infinity, orfor a noiseless system, we havelim→∞max{r); r2; M M M ; rM} = IM (8)Thus, we can intuitively set the detection threshold for thesystem to begth =max{r); r2; M M M ; rM}2M (9)The most important feature of the detection threshold proposedin (9) is that it only depends on the received signal, and unlikean ideal optimized OOK detection scheme, an estimate of theCSI is not required.We now derive the pdf of the detection threshold gth in(9). In a symbol duration, we first consider the case for whichk branches transmit bit \1:, where k = 1; 2; M M M ;M , and therest of the M − k branches transmit bit \0:. Without loss ofgenerality, we assume the first k branches transmit bit \1:,and the rest of the M − k branches transmit bit \0:. Theconditional pdf of gth can be written asfgth(tth|I; k)= 2k(2tth − I.g)k−)(2tth.g)M−kfa (2tth − I)+ 2(M − k)(2tth − I.g)k(2tth.g)M−k−)fa (2tth)(10)where (x) =∫ x−∞)√2,ex“(− r22)dr is the cumulativedistribution function of a standard Gaussian RV, and fa (x) =)√2,.gex“(− x22.2g)denotes the noise pdf. The detailed deriva-tions of (10) are given in the Appendix. The pdf of gthconditioned on I can be obtained asfgth(tth|I) =M∑k5)fgth(tth|I; k)p(k)=fa (2tth − I)2M−){M[(2tth.g)+(2tth − I.g)]M−)+((2tth.g))M−)}+fa (2tth)2M−)×{M((2tth.g)+(2tth − I.g))M−)+((2tth.g))M−2 [(M − 1)(2tth − I.g)−M(2tth.g)]}(11)where p(k) =(Mk)P2M is the probability that there are kbranches transmitting bit \1: values. Averaging (11) withrespect to the fading coefficient I , one can obtain the pdf ofgth asfgth(tth) = EI [fgth(tth|I)] (12)where EI [·] represents the statistical expectation with respectto I .IV. THE UPPER BOUND ON THE AVERAGE BERAs it is challenging to find the exact BER expression forour proposed system, we will find an upper bound on theaverage BER. For expository purposes, we first analyze theerror caused in the detection process. The error in the detectionprocess, before the (2M − 1)-nary sequence conversion atthe receiver, is the turbulence induced error when the binarysequence is transmitted through the turbulence channel. Wethen analyze the average BER of the output binary sequence.3A. Error in the Detection Process at the ReceiverWithout loss of generality, it is assumed that the first kbranches transmit bit \1:, and the rest of the M −k branchestransmit bit \0:. The detection threshold becomes”gth =max{I + n); M M M ; I + nk; nk+); M M M ; nM}2M (13)We define N = [n[1] MMM n[M ]]g as the noise vector, andNk⊙ =[n[1] MMM n[k − 1] n[k + 1] MMM n[M ]]g asthe noise vector without the kth noise component nk. Theprobability of having incorrect detection in one or more linkscan be written asP (x2|k)=1M{EN1⊙[EI[P(I + n) < ”gth∣∣∣N)⊙ ; I)]]+ M M M+ ENk⊙[EI[P(I + nk < ”gth∣∣∣Nk⊙ ; I)]]+ ENk+1⊙[EI[P(nk+) > ”gth∣∣∣Nk+)⊙ ; I)]]+ M M M+ ENM⊙[EI[P(nM > ”gth∣∣∣NM⊙ ; I)]]} M(14)Since all components of the noise vector N are independentand identically distributed (i.i.d.), for k) ̸= k2, where k); k2 ∈{1; 2; M M M ;M}, we haveENk1⊙[EI[P(I + nk1 <”gth∣∣∣Nk1⊙ ; I)]]= ENk2⊙[EI[P(I + nk2 <”gth∣∣∣Nk2⊙ ; I)]] (15)andENk1⊙[EI[P(nk1 >”gth∣∣∣Nk1⊙ ; I)]]= ENk2⊙[EI[P(nk2 >”gth∣∣∣Nk2⊙ ; I)]] M (16)Thus, the probability of having incorrect detection in one ormore links can be written asP (x2|k) = kMEN1⊙[EI[P(I + n) < ”gth∣∣∣N)⊙ ; I)]]+M − kMENM⊙[EI[P(nM > ”gth∣∣∣NM⊙ ; I)]] M(17)The first term in (17) can be upper-bounded as (18) on thetop of the next page. The second term in (17) can be upper-bounded as (19) on the next page. Subtituting (18) and (19)into (17), we haveP (x2|k) < kM{EI [P (n) < −I)] + (k − 1)× En2[EI[P(n) <n2 − I2∣∣∣∣n2; I)]]+ (M − k)EnM[EI[P(n) <nM2− I∣∣∣nM ; I)]]}+M − kMEn1[EI[P(nM >I + n)2∣∣∣∣n); I)]] M(20)The upper bound on the average BER for the binary sequencetransmitted through the turbulence channel with M transmitlasers is obtained as (21) on the next page. It is difficult tofind a closed-form expression of (21), as it contains a doubleintegral, however, this integral can be evaluated numericallywith high accuracy.B. Average BER of the SystemAt the transmitter, a binary sequence tL M M M t2t) is con-verted into a (2M − 1)-nary sequence hJhJ−) M M M h2h). This(2M −1)-nary sequence of length J is mapped to a binary se-quence of length JM . At the receiver, after the demodulation,we will map a binary sequence of length JM to a (2M − 1)-nary sequence h^J h^J−) M M M h^2h^). This (2M −1)-nary sequenceh^J h^J−) M M M h^2h^) is converted to a binary sequence, denotedas t^L+)t^L M M M t^2t^). We comment that when a (2M − 1)-narysequence of length J will be converted to a binary sequenceof length L + 1; however, at the transmitter, we convert abinary information sequence of length L to a (2M − 1)-naryinformation sequence of length J . Thus, at the receiver, wewill ignore t^L+) and use t^L M M M t^2t^) as our binary informationsequence.To calculate the probability of having an incorrect decisionin the binary sequence t^L M M M t^2t^), for expository purposes,we consider M = 3. When M = 3, we will convert L = 14binary information bits t)4 M M M t2t) to a 7-nary sequence witha length of J = 5 (h5h4h+h2h)). Then we map the 7-narysequence of length of J = 5 into a binary sequence of lengthJM = 15. At the receiver, after the demodulation, we mapeach sequence of M = 3 binary bits to an element of the7-nary sequence. The mapping can be seen as follows:000→ 0; 001→ 0; 010→ 1; 011→ 2;100→ 3; 101→ 4; 110→ 5; 111→ 6M (22)Tabel I shows the conditional probability of the received (2M−1)-nary number h^l given the transmitted (2M−1)-nary numberhl. The conditional probability in Tabel I can be calculated byusing the bit error probability for the binary bit transmittedthrough the turbulence channel, Px2 . For example, P (0|0) =P (000|001)+P (001|001) = (1−Px2)2Px2+(1−Px2)+. TabelII shows the conditional probability of the received (2M −1)-nary number h^l given the transmitted (2M − 1)-nary numberhl, written in terms of Px2 .Since all elements of h^5 M M M h^2h^) and h5 M M M h2h) are inde-pendent, the conditional probability for the received (2M−1)-nary sequence h^5 M M M h^2h^) given the transmitted (2M−1)-narysequence h5 M M M h2h) isP (h^5 M M M h^2h^)|h5 M M M h2h))=5∏j5)P (h^j |h5 M M M h2h)) =5∏j5)P (h^j |hj)M(23)The conditional probability of the received binary sequencet^L M M M t^2t^) given the transmitted binary sequence tL M M M t2t)can be written asP (t^)4 M M M t^2t^)|tL M M M t2t))= P (h^5 M M M h^2h^)|h5 M M M h2h)) =5∏j5)P (h^j |hj)M (24)If the transmitted binary sequence is t)4 M M M t2t), and thereceived binary sequence is t^)4 M M M t^2t^), the conditional errorprobability of this system given t)4 M M M t2t) and t^)4 M M M t^2t^)isP (x|t)4 M M M t2t); t^)4 M M M t^2t^)) =∑)4l5) tl ⊕ t^l14(25)4P(I + n1 < ~Tth∣∣∣N1⊙ ; I) 5 P ( I + n1 < max{I + n1; : : : ; I + nk; nk+1; : : : ; nM}2∣∣∣∣N1⊙ ; I)5 P({I + n1 <I + n12}∪ : : : ∪{I + n1 <I + nk2}∪{I + n1 <nk+12}∪ : : : ∪{I + n1 <nM2}∣∣∣N1⊙ ; I)≤ P(I + n1 <I + n12∣∣∣∣ I)+ P ( I + n1 < I + n22∣∣∣∣n2; I)+ : : :+ P(I + n1 <I + nk2∣∣∣∣nk; I)+ P ( I + n1 < nk+12∣∣∣nk+1; I)+ : : :+ P(I + n1 <nM2∣∣∣nM ; I)5 P (n1 < −I| I) + P(n1 <n2 − I2∣∣∣∣n2; I)+ : : :+ P (n1 < nk − I2∣∣∣∣nk; I)+ P(n1 <nk+12− I∣∣∣nk+1; I)+ : : :+ P (n1 < nM2− I∣∣∣nM ; I)5 P (n1 < −I) + (p − ))P(n1 <n2 − I2∣∣∣∣n2; I)+ (M − p)P (n1 < nM2 − I∣∣∣nM ; I) :(18)P(nM > ~Tth∣∣∣NM⊙ ; I) 5 P (nM > max{I + n1; : : : ; I + nk; nk+1; : : : ; nM}2∣∣∣∣NM⊙ ; I)5 P({nM >I + n12}∩ : : : ∩{nM >I + nk2}∩{nM >nk+12}∩ : : : ∩{nM >nM2}∣∣∣NM⊙ ; I)≤ P(nM >I + n12∣∣∣∣nM ; I) :(19)Pe2 5M∑k21P (e2|p)p(p)<M∑k21(Mk)2M − ){pM[JI [P (n1 < −I)] + (p − ))Jn2[JI[P(n1 <n2 − I2∣∣∣∣n2; I)]]+ (M − p)JnM[JI[P(n1 <nM2− I∣∣∣nM ; I)]] + M − pMJn1[JI[P(nM >I + n12∣∣∣∣n1; I)]]}5M∑k21(Mk)2M − ){pM(JI[Q(Ig)]+ (p − ))Jn2[JI[Q(I − n22g)]]+(M − p)JnM[JI[Q(2I − nM2g)]])+M − pMJn1[JI[Q(I + n12g)]]}:(21)TABLE ICONDITIONAL PROBABILITY OF THE RECEIVED (2M − ))-NARY NUMBER h^l GIVEN THE TRANSMITTED (2M − ))-NARY NUMBER hl .h^l 0 1 2 3 4 5 6P (h^ljhl 5 () P ((j() P ()j() P (2j() P (3j() P (4j() P (5j() P (6j()P (h^ljhl 5 )) P ((j)) P ()j)) P (2j)) P (3j)) P (4j)) P (5j)) P (6j))P (h^ljhl 5 2) P ((j2) P ()j2) P (2j2) P (3j2) P (4j2) P (5j2) P (6j2)P (h^ljhl 5 3) P ((j3) P ()j3) P (2j3) P (3j3) P (4j3) P (5j3) P (6j3)P (h^ljhl 5 4) P ((j4) P ()j4) P (2j4) P (3j4) P (4j4) P (5j4) P (6j4)P (h^ljhl 5 5) P ((j5) P ()j5) P (2j5) P (3j5) P (4j5) P (5j5) P (6j5)P (h^ljhl 5 6) P ((j6) P ()j6) P (2j6) P (3j6) P (4j6) P (5j6) P (6j6)5TABLE IITHE CONDITIONAL PROBABILITY OF THE RECEIVED (2M − ))-NARY NUMBER h^l GIVEN THE TRANSMITTED (2M − ))-NARY NUMBER hl .h^l 0 1 2 3 4 5 6P (h^l|hl 5 () ()− Pe2 )2Pe2 ()− Pe2 )P 2e2 ()− Pe2 )P 2e2 ()− Pe2 )P 2e2 ()− Pe2 )P 2e2 ()− Pe2 )P 2e2 ()− Pe2 )P 2e2+()− Pe2 )3P (h^l|hl 5 )) ()− Pe2 )2Pe2 ()− Pe2 )3 ()− Pe2 )P 2e2 ()− Pe2 )P 2e2 P 3e2 ()− Pe2 )2Pe2 ()− Pe2 )P 2e2+()− Pe2 )P 2e2P (h^l|hl 5 2) ()− Pe2 )2Pe2 ()− Pe2 )2Pe2 ()− Pe2 )3 P 3e2 ()− Pe2 )P 2e2 ()− Pe2 )P 2e2 ()− Pe2 )2Pe2+()− Pe2 )P 2e2P (h^l|hl 5 +) ()− Pe2 )2Pe2 ()− Pe2 )P 2e2 P 3e2 ()− Pe2 )3 ()− Pe2 )2Pe2 ()− Pe2 )2Pe2 ()− Pe2 )P 2e2+()− Pe2 )P 2e2P (h^l|hl 5 4) ()− Pe2 )2Pe2 P 3e2 ()− Pe2 )P 2e2 ()− Pe2 )2Pe2 ()− Pe2 )3 ()− Pe2 )P 2e2 ()− Pe2 )P 2e2+()− Pe2 )P 2e2P (h^l|hl 5 5) P 3e2 + ()− Pe2 )P 2e2 ()− Pe2 )2Pe2 ()− Pe2 )P 2e2 ()− Pe2 )2Pe2 ()− Pe2 )P 2e2 ()− Pe2 )3 ()− Pe2 )P 2e2P (h^l|hl 5 6) P 3e2 + ()− Pe2 )P 2e2 ()− Pe2 )P 2e2 ()− Pe2 )2Pe2 ()− Pe2 )P 2e2 ()− Pe2 )P 2e2 ()− Pe2 )P 2e2 ()− Pe2 )3where ⊕ implements an exclusive OR. Thus, the BER of thissystem with M = 3 can be written asP (x) =∑t14MMMt2t1Nt^14MMMt^2t^1P (x|t)4 M M M t2t); t^)4 M M M t^2t^))× P (t)4 M M M t2t); t^)4 M M M t^2t^))=∑t14MMMt2t1Nt^14MMMt^2t^1∑)4l5) tl ⊕ t^l14× P (t)4 M M M t2t); t^)4 M M M t^2t^))=∑t14MMMt2t1Nt^14MMMt^2t^1∑)4l5) tl ⊕ t^l14× P (t^)4 M M M t^2t^)|t)4 M M M t2t))P (t)4 M M M t2t))M(26)In (26), the elements of t)4 M M M t2t) are independent, so wehave P (t)4 M M M t2t)) = )214 . Substituting (24) into (26), wehaveP (x) =∑t14MMMt2t1Nt^14MMMt^2t^1∑)4l5) tl ⊕ t^l14×5∏j5)P (h^j |hj)P (t)4 M M M t2t))=12)4∑t14MMMt2t1Nt^14MMMt^2t^1∑)4l5) tl ⊕ t^l145∏j5)P (h^j |hj)M(27)In general, the BER for the proposed system with Mtransmitted lasers can be written asP (x) =12L∑tLMMMt2t1Nt^LMMMt^2t^1∑Ll5) tl ⊕ t^lLJ∏j5)P (h^j |hj)M(28)V. NUMERICAL RESULTSIn this section, the pdf of the detection threshold is firstverified, and the BER performance of the proposed system isnumerically studied.In Fig. 4, the derived pdf of the detection threshold gth iscompared with the simulated pdf. For expository purposes, we0 0.2 0.4 0.6 0.8 110−510−410−310−210−1100101Variable TthProbability Density Function  Simulated pdf, M=2, σ=0.25Derived pdf, M=2,σ=0.25Simulated pdf, M=3,σ=0.25Derived pdf, M=3,σ=0.25Simulated pdf, M=3,σ=0.2Derived pdf, M=3,σ=0.2Fig. 4. Comparison of the derived and simulated pdfs for the detectionthreshold Tth over a lognormal fading channel with (;M) 5 ((:25; +),((:2; +) and ((:25; 2).let the parameters (.;M) = (0M25; 3), (0M2; 3) and (0M25; 2).The simulated pdf is obtained by using Monte Carlo computersimulations with 104 trials. The derived pdf shows excellentagreement with the simulated pdf.In Fig. 5, we plot the BER versus electrical SNR whenthe OOK IM/DD system uses a fixed detection threshold ofgth = 0M5. Note that an error floor appears in the large SNRregime. The system using source information transformationcan eliminate the error floor, although its BER performanceis worse than that of the OOK IM/DD system using fixeddetection thresholds in low SNR regimes. This is due to thefact that a value of one-half of the largest received signalis used to define the detection threshold for the M receivedsignals in each symbol duration. This detection threshold isonly optimum when the electrical SNR approaches infinityand/or there is no noise. In the low SNR regimes, the detectionthreshold is not an optimum detection threshold for our62 4 6 8 10 12 14 16 18 20 22 2410−1010−810−610−410−2Average SNR (dB)Bit Error Rate  BER upper boundIdealized adaptive detectionSimulated BERFixed threshold detection, Tth=0.5σ=0.25σ=0.5Fig. 5. The simulated BERs and BER upper bounds of the system usingidealized adaptive detection thresholds, source information transformation,and fixed detection thresholds over lognormal turbulence channels (with 5 (:25,  5 (:5 and M 5 +).proposed system, due to the noise influence, and the BERof our proposed system becomes worse than that of the OOKIM/DD system using a fixed detection threshold.In Fig. 5, we also plot the upper bounds on the average BERfor the proposed system over lognormal fading channels withdifferent turbulence conditions. Simulated BER curves are alsoused to verify the analytical BER upper-bound solutions. Theupper bound is tight when M = 3. However, as we have usedthe union upper bound technique, it can be shown that theupper bound becomes loose with increasingM . It is seen fromFig. 5 that the OOK modulated system using idealized adaptivedetection thresholds with a lognormal turbulence model having. = 0M25 requires an SNR of 24M8 dB to attain a BERof 1 × 10−1, while the proposed system requires an SNRof 26M6 dB to achieve the same BER performance. Thus,the corresponding SNR penalty factor for the system usingOOK and source information transformation in a lognormalturbulence channel with . = 0M25 is only 1M8 dB at BER of1 × 10−1. This performance difference can be factored intothe ultimate FSO system design to offset the complexity ofimplementing systems with adaptive detection thresholds (andtheir need for knowledge of the instantaneous CSI).VI. CONCLUSIONFSO communication systems using OOK and source infor-mation transformation have been proposed. It was shown thatsuch systems can achieve good BER performance without theneed for knowledge of the instantaneous CSI and pdf of theturbulence model. We have derived an analytical expressionfor the pdf of the detection threshold and developed a tightupper bound on the average BER. Numerical studies ultimatelyshowed that the system using coded wavelength multiplexingachieves comparable performance to the idealized adaptive de-tection system, with a greatly reduced level of implementationcomplexity (exhibiting an SNR penalty factor of only 1M8 dBat a BER of 1 × 10−1, for a lognormal turbulence channelwith . = 0M25).APPENDIXIn a symbol duration, we first consider the case for whichk branches transmit bit \1:, where k = 1; 2; M M M ;M , and therest of the M − k branches transmit bit \0:. Without loss ofgenerality, we assume the first k branches transmit bit \1:, andthe rest of theM−k branches transmit bit \0:. The conditionalcumulative distribution function of gth can be written as (29)on the top of the next page. Since all the noise componentsn); n2; M M M ; nM are assumed to be i.i.d., we have (30) on thetop of the next page. It follows thatFgth(tth|I; k)=[P (n) < 2tth − I)]k[P (nM < 2tth)]M−k=[(2tth − I)]k[(2tth)]M−kM(31)The pdf of gth conditioned on k branches transmitting bits\1:s and I can be written asfgth(tth|I; k) =ddtthFgth(tth|I; k)= 2k(2tth − I.g)k−)(2tth.g)M−kfa (2tth − I)+ 2(M − k)(2tth − I.g)k(2tth.g)M−k−)fa (2tth)M(32)REFERENCES[1] V. W. S. Chan, “Free-space optical communications,” IEEE/OSA J.Lightwave Technol., vol. 24, pp. 4750-4762, Dec. 2006.[2] J. H. Shapiro and R. C. Harney, “Burst-mode atmospheric optical com-munication,” in Proc.1980 Nat. Telecommun. Conf., 1980, pp. 27.5.1-27.5.7.[3] N. Chand, J. J. Loriz, A. J. Hunton, and B. M. Eteson, “Performancecomparison of NRZ and RZ modulations with and without forward errorcorrections for free-space optical communication,” in Proc. SPIE, vol.5892, pp. 58920U-1-58920U-8, Sept. 2005.[4] N. Liu, W. D. Zhong, Y. He, K. H. Heng, and T. H. Cheng, “Comparisonof NRZ and RZ modulations in laser intersatellite communicationsystems,” in Proc. 2008 Int. Conf. Advanced Infocomm Tech., Shenzhen,2008, pp. 29-32.[5] J. Li, J. Q. Liu, and D. P. Taylor, “Optical communication usingsubcarrier PSK intensity modulation through atmospheric turbulencechannels,” IEEE Trans. Commun., vol. 55, pp. 1598-1606, Aug. 2007.[6] M. Tycz, M. W. Fitzmaurice, and D. A. Premo, “Optical communicationsystem performance with tracking error induced signal fading,” IEEETrans. Commun., vol. 21, pp. 1069-1072, Sept. 1973.[7] M. L. B. Riediger, R. Schober, and L. Lampe, “Fast multiple-symbol de-tection for free-space optical communications,” IEEE Trans. Commun.,vol. 57, pp. 1119-1128, Apr. 2009.[8] M. L. B. Riediger, R. Schober, and L. Lampe, “Blind detection of on-off keying for free-space optical communications.” in Proc. CCECE,Niagara Falls, Canada, 2008, pp. 1361-1364.[9] H. R. Burris, et al., “Laboratory implementation of an adaptive thresh-olding system for free-space optical communication receivers with signaldependent noise,” in Proc. SPIE, 2005, vol. 5892, pp. 1-20.[10] X. Zhu and J. M. Kahn, “Free-space optical communication throughatmospheric turbulence channels,” IEEE Trans. Commun., vol. 50, pp.1293-1300, Oct. 2002.7FTth (tth|I; p) 5 P(max{I + n1; : : : ; I + nk; nk+1; : : : ; nM}2< tth)5 P(I + n12< tth; : : : ;I + nk2< tth;nk+12< tth; : : : ;nM2< tth)5 P (n1 < 2tth − I; : : : ; nk < 2tth − I; nk+1 < 2tth; : : : ; nM < 2tth):(29)P (n1 < 2tth − I; : : : ; nk < 2tth − I; nk+1 < 2tth; : : : ; nM < 2tth)5 P (n1 < 2tth − I) : : : P (nk < 2tth − I)P (nk+1 < 2tth) : : : P (nM < 2tth)5 [P (n1 < 2tth − I)]k[P (nM < 2tth)]M−k:(30)[11] H. Moradi, H. H. Refai, and P. G. LoPresti, “Thresholding-based optimaldetection of wireless optical signals,” J. Opt. Comm. Net., vol. 2, pp.689-700, Sept. 2010.[12] D. A. DeWolf, “Are strong irradiance fluctuations log normal orRayleigh distributed,” J. Opt. Soc. Amer., vol. 57, pp. 787-797, Jun.1967.[13] X. Zhu and J. M. Kahn, “Pilot-symbol assisted modulation for correlatedturbulent free-space optical channels.” in Proc. SPIE, 2002, vol. 4489,pp. 138C145.[14] J. G. Proakis, Digital Communications. New York: McGraw-Hill,5th ed., 2008.[15] M. Khalighi, F. Xu, Y. Jaafar, and S. Bourennane, “Double-laserdifferential signaling for reducing the effect of background radiationin free-space optical systems,” IEEE/OSA J. Opt. Commun. Netw., vol.3, pp. 145-154, Feb. 2011.

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