Xu MA Khalighi S Bourennane Ecole Centrale Marseille Institut Fresnel UMR CNRS 6133 Marseille France FangXufresnelfr AliKhalighifresnelfr SalahBourennanefresnelfr Abstract For terrestrial free space optical FSO sys tems we investigate the use of pu ID: 26744 Download Pdf

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Xu MA Khalighi S Bourennane Ecole Centrale Marseille Institut Fresnel UMR CNRS 6133 Marseille France FangXufresnelfr AliKhalighifresnelfr SalahBourennanefresnelfr Abstract For terrestrial free space optical FSO sys tems we investigate the use of pu

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Pulse Position Modulation for FSO Systems: Capacity and Channel Coding F. Xu, M.A. Khalighi, S. Bourennane Ecole Centrale Marseille, Institut Fresnel, UMR CNRS 6133, Marseille, France Fang.Xu@fresnel.fr, Ali.Khalighi@fresnel.fr, Salah.Bourennane@fresnel.fr Abstract — For terrestrial free space optical (FSO) sys- tems, we investigate the use of pulse position modulation (PPM) which has the interesting property of being average- energy efﬁcient. We ﬁrst discuss the upper bound on the information transmission rate for a Gaussian channel. Next, we consider the more

practical aspect of channel coding and look for a suitable solution for the case of -ary PPM. Instead of using a non-binary channel code, we suggest to use a simple binary convolutional code and to perform iterative soft demodulation (demapping) and channel decoding at the receiver. We show that the proposed scheme is quite efﬁcient against demodulation errors due to the receiver noise. Moreover, we propose a simple soft-demapping method of low complexity for the general case of -ary PPM. The receiver complexity remains then reasonable in view of implementation in a terrestrial FSO

system. I. I NTRODUCTION Free Space Optics (FSO) is a promising solution for very high data rate point-to-point communication [1], [2], [3]. In practice, several factors such as pointing errors and propagation loss associated with visibility can degrade the performance of an FSO system. This paper assumes clear atmosphere conditions and that the transmitter and the receiver are perfectly aligned. Even under these con- ditions, the system performance can be limited due to atmospheric turbulence. The resulting channel fading, i.e., random ﬂuctuations in both the amplitude and the phase of

the received signal, can deteriorate considerably the quality of data transmission [4]. To reduce the destructive effect of turbulence, channel coding could be used under weak turbulence conditions [5], [6]. However, in the cases of moderate to strong turbulence, channel coding alone is not sufﬁcient to mitigate fading efﬁciently and diversity techniques should be employed [6]. The performance of the FSO link can also be improved by employing an appropriate modulation scheme that makes a good compromise between complex- ity and performance. In this view, we consider in this paper

the pulse position modulation (PPM) which has the interesting advantage of being average-energy efﬁcient. We ﬁrst study the upper bound on the information transmission rate for PPM modulation in the absence of turbulence. Then, we consider the channel coding adapted to PPM. As a matter of fact, for non-binary PPM, we should use a non-binary code to correct efﬁciently the demodulation errors. The disadvantage of such a method is that it necessitates computationally complex decoding at the receiver. An important point is hence to use a channel coding technique adapted to PPM

and of reasonable complexity. We propose to use a classical binary convolutional code and to perform iterative soft signal demodulation and channel decoding at the receiver. We study the performance of this scheme that we call BCID (standing for Binary Convolutional encoding with Iterative Detection), by presenting some simulation re- sults. In this paper we use interchangeably the terms of demodulation (demodulator) and demapping (demapper). We assume that we do not have any source of diversity available: we use a monochromatic laser with a single beam at the transmitter and a single lens of

very small size at the receiver, usually referred to as a point receiver This assumption of the absence of any source of diversity allows us to focus on the impact of signal modulation. We study the system performance in the absence of turbulence, as well as in the weak turbulence regime. Note that this latter case is equivalent to the case of moderate or strong turbulence when a relatively large lens is used at the receiver for aperture averaging [4]. The remainder of this paper is organized as follows. In Section II, we present our system model and general assumptions. Next, a brief state of

the art on PPM and the proposed coding schemes for this modulation are pre- sented in Section III. Also, the information transmission bounds are studied for a non-random PPM channel, i.e., in the absence of turbulence. Iterative detection, including PPM and soft demodulation, is then described in Section IV. Some numerical results are presented in Section V, and Section VI concludes the paper. II. S YSTEM MODEL AND ASSUMPTIONS At the transmitter, the encoded information bits are transformed into symbols according to the PPM modu- lation that is explained in the next section. The encoder is a

classical binary convolutional code. We denote by the transmitted light intensity and by the corresponding received intensity in an ON PPM slot. Denoting the chan- nel fading coefﬁcient by , we have in fact, hI .The received signal after optical/electrical conversion is: ηhI n, (1) where is the optical/electrical conversion efﬁciency assumed here to be unity for simplicity. Also, is the receiver noise, assumed to be dominated by thermal noise, 10th International Conference on Telecommunications - ConTEL 2009 ISBN: 978-953-184-131-3, June 8-10, 2009, Zagreb, Croatia 31

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01 10 11 00 TT TT TT TT Figure 1. Examples of PPM symbols. stands for the transmitted light intensity in an ON slot. and modelled by a zero-mean Gaussian additive random process. Note that for an OFF PPM slot, we have We perform soft demodulation on the received signal based on the maximum a posteriori (MAP) criterion, followed by soft channel decoding. Soft information at the demodulator or decoder outputs is considered in the form of logarithmic likelihood ratio (LLR). We assume that we do not have inter-slot interference (ISI). That is, we assume perfect time

synchronization and that the signal corresponding to each time slot is independent of those of the other slots. A. Turbulence modeling We consider the Gamma-Gamma ( ) model for the fade statistics [4]. According to this model, the probability density function (PDF) of the channel fading is: )= 2( Γ( )Γ( 2( αβh ,h> (2) Parameters and are the effective numbers of large- and small-scale eddies of the scattering environment, respectively, and is the modiﬁed Bessel function of the second kind and order . Let us consider the classical plane wave propagation model, as well

as turbulent eddies of zero inner scale. By these assumptions, we have [4]: exp 49 1+1 11 12 (3) exp 51 1+0 69 12 (4) where is the Rytov variance. Channel time variations are considered according to the theoretical quasi-static model, also called the frozen channel model. By this model, channel fading is considered to be constant over the duration of a frame of symbols, changing to a new independent value from one frame to next. III. P ULSE POSITION MODULATION Due to implementation complexity issues, most current FSO systems use intensity modulation with direct detec- tion (IM/DD). The on-off

keying (OOK) modulation is commonly used due to its simplicity where the presence of the transmitted light intensity represents a symbol one, and its absence, a symbol zero. An interesting alternative to OOK is PPM. For the -ary PPM modulation, a symbol corresponds to log bits. The symbol duration is cut into time slots (or chips) of duration /Q , and an optical pulse is sent in one of slots. Figure 1 shows an example of quaternary PPM. Since -ary PPM contains one pulse per slots, it has a duty cycle of /Q and a peak-to- average power ratio (PAPR) of . We can vary to make a ﬂexible

compromise between power efﬁciency and bandwidth efﬁciency. For notational simplicity, we denote the -ary PPM by PPM when we want to specify the modulation order. We will denote the binary PPM, by BPPM. The important advantage of PPM over OOK is that it is more average- energy efﬁcient. In fact, to achieve a given bit error rate (BER), the required average power by OOK is more than that of PPM for Q> . But at the same time, it has some disadvantages. For the same data transmission rate, more bandwidth is in general required than for OOK. The bandwidth requirement, however,

does not cause any problem because a very large bandwidth is available in FSO systems. However, in practice, the real problem with a larger is the increased required switching speed for electronic circuits. Moreover, the receiver synchronization becomes more difﬁcult [7]. Another disadvantage of PPM is that for a given average transmission power, by increas- ing , the PAPR increases. A. Channel coding for PPM An important issue is to use an appropriate channel coding scheme in the case of using PPM. A number of coding techniques including convolutional codes, turbo codes (TC), and

Reed-Solomon (RS) codes, have been proposed so far. The classical binary convolutional codes have been applied to PPM in [8] and [9]. Convolutional coded interleaved PPM has been proposed in [10]. More powerful codes such as turbo-codes have also been considered for PPM [11], [12]. However, the main draw- back of these schemes is that binary codes are not suitable for using with -ary symbols in the sense that they are not efﬁcient for correcting demodulator output errors. Non-binary codes are more appropriate [13]. However, the problem with non-binary convolutional or turbo-codes is

their decoding complexity that can be prohibitively large for a practical implementation in a Gbps-rate FSO system. RS codes are appropriate from this point of view: an n,k RS code is naturally matched to PPM by choosing [14], [15]. However, the performance improvement by RS coding is not considerable due to the fact that, usually hard RS decoding is performed at the receiver which has the advantage of low complexity. Soft RS decoding, on the other hand, is computationally too complex and is rarely implemented. A concatenation of convolutional and RS coding has also been proposed in [16]. This

scheme has a rather limited performance as hard decision Viterbi decoding is done for the former. In this work, we explain how to adapt a simple binary convolutional code to the case of PPM. As we will see, in order to efﬁciently correct demodulation errors, we perform iterative demodulation and channel decoding, hence, beneﬁt from the channel coding gain in PPM demodulation. The receiver complexity remains reason- able, as compared to the case of non-binary convolutional coding. F. Xu, M.A. Khalighi, S. Bourennane ConTEL 2009, ISBN: 978-953-184-131-3 32

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B.

Channel capacity It is interesting to compare the upper bounds on the information transmission rate for OOK and PPM mod- ulations. Although these bounds cannot really be called capacity in the sense of Shannon (because of restricting the distribution of the transmitted signal according to the underlying modulation), for simplicity we will call them capacity of OOK or PPM. The capacity of PPM channel has been studied in [17], [18] for the case of deep-space communication using a photon counting receiver, where the Poisson channel model is considered. We consider in this paper the case of

terrestrial FSO systems used over ranges up to several kilometers. In such systems, photon counting is not feasible in practice. In fact, the received photon ﬂux is important and we can detect the received signal based on the beam intensity directly. Here, we consider the channel capacity for different modulation schemes for intensity- based signal detection. According to the assumptions of Section II, the receiver noise has a Gaussian distribution. The channel capacity is the maximum of the mutual information between the channel input and output , with respect to the input distribution

= max (5) is in fact the probability mass function (PMF) of . Let us denote by the distribution of and by the conditional distribution of to Variable takes the values 0 or 1, corresponding to an OFF or ON slot, respectively. Then, it can easily be shown that, =0 ) log dy (6) Inspired by [17], we evaluate in the following the channel capacity for the case of an additive white Gaussian noise (AWGN) channel. That is to say, we consider the absence of atmospheric turbulence and set the channel fading coefﬁcient in (1) to one. For the sake of completeness, we also consider the case of OOK

modulation. 1) Case of OOK: Given the Gaussian assumption for the receiver noise, the conditional distributions are: =0)= exp =1)= exp 1) (7) where is the receiver noise variance. Note that we take (0) = (1) = 0 and do not optimize the capacity with respect to . In other words, we consider equally likely symbols. 2) Case of PPM: Let us use bold-face characters for denoting channel input and output corresponding to a PPM symbol. Hence, a symbol corresponds to slot- values that we denote by =1 . As previously mentioned, we assume the absence of ISI, that is, the signal corresponding to a time

slot is independent of those 10 15 20 25 30 35 40 45 50 /N (dB) C (bits/slot) BPPM 4PPM 8PPM 16PPM 32PPM 64PPM 128PPM 256PPM Figure 2. Capacity in units of bits per slot versus SNR for OOK and PPM, =2 16 32 64 128 256 of the other slots. The conditional distribution is then, )= =1 (8) Since we assumed equiprobable symbols, without loss of generality, let us assume that the ﬁrst slot is ON and the others are OFF . We denote this symbol by . In addition, let us denote by a PPM symbol with the th slot ON Then the channel capacity in (6) turns to: )log =1 (9) where, is the set of real

numbers, and for instance, )= =1 ,i (10) Distributions and are given in (7). Instead of calculating the -dimensional integral in (9) numer- ically, which becomes computationally too complex for large , we use Monte Carlo simulations to evaluate [17]. We have shown in Fig. 2 the capacities of OOK and PPM modulations, versus the receiver signal-to- noise ratio (SNR). Throughout the paper we consider the electrical SNR in the form of /N , where is the averaged received energy per information bit and the noise unilateral power spectral density. To set the SNR, we ﬁx the channel bandwidth

for different modulations, or equivalently, we ﬁx the slot duration for any .The capacity is represented in units of bits per symbol. The capacity in units of bits per slot can be obtained by dividing these values by . Note that the capacities of OOK and BPPM are identical. For SNR the asymptotic capacity equals log bits per symbol, or equivalently, log Q/Q bits per slot. Pulse Position Modulation for FSO Systems: Capacity and Channel Coding ConTEL 2009, ISBN: 978-953-184-131-3 33

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De−interleaver SISO Demapper Channel Decoder SISO Interleaver LLR ext DEM LLR ext

DEC Figure 3. Block diagram of the iterative receiver. LLR ext DEM and LLR ext DEC denote the extrinsic soft-values at the demapper and the channel decoder, respectively. IV. I TERATIVE DETECTION OF PPM SYMBOLS In this section, we describe the proposed iterative detection at the receiver. We also provide details on PPM bit-symbol mapping and soft-demodulation. A. Iterative demodulation and decoding As explained previously, we use here a classical binary convolutional code which has a far less decoding com- putational complexity than a non-binary code. In order to correct the demodulator errors

efﬁciently, we perform the demodulation and channel decoding iteratively: By providing the decoder soft outputs for the demodulator, it can reﬁne symbol demodulation by beneﬁting from the coding gain. We were inspired by the idea that is used in turbo-equalization or turbo-detection, where respectively, channel equalization or signal detection is performed in an iterative manner together with channel decoding. The block diagram of the proposed receiver is shown in Fig. 3. We perform soft-input soft-output (SISO) demod- ulation and channel decoding. The former is based on

the MAP criterion and the latter on the maximum likelihood (ML) criterion using the soft-output Viterbi algorithm (SOVA) [19]. The demodulator provides bit-level LLRs on the transmitted (coded) bits using the received signal samples (see (1)). The interleaved demapper output LLRs are then fed to the SISO decoder that provides at its output LLRs on the coded bits. Extrinsic soft-values are exchanged between the demodulator and the decoder to ensure proper convergence of the receiver. After a few iterations, we practically obtain no improvement, i.e., we attain the convergence. Decision making

on the transmitted bits can then be done using the a posteriori LLRs at the decoder output. The interleaver (or equivalently, the de-interleaver) in Fig. 3 has the task of decorrelating the demapper outputs in order to improve the performance of the SOVA decoder. For this purpose, equivalently, at the transmitter, the encoded bits are ﬁrst interleaved prior to modulation. The receiver performance after convergence depends on the interleaver design. Here, we consider pseudo-random interleaving of large enough size. We found out that a somehow similar scheme has been proposed by Moision

and Hamkins in [20], [21], [22]. The proposed scheme, called serially concatenated pulse-position modulation (SCPPM) is proposed for use in deep space communication with data rates on the order of Mbps. Also, [23] considered the use of low-density- parity-check (LDPC) codes instead of a binary convolu- tional code. Later, single-parity-check (SPC) codes have been applied to SCPPM in [24]. The main difference with our BCID scheme is that in SCPPM, interleaved coded bits are passed through the combination of a bit- accumulator and a PPM modulator, called accumulated- PPM (APPM). The use of APPM

scheme improves the performance of the iterative receiver but at the same time, increases its complexity. Having a lower receiver complexity, our scheme can be considered as to be more suitable for use in a Gbps-rate FSO system. Moreover, we propose later a simple and low computational complexity formulation for the soft demodulator for the general case of PPM. B. PPM mapping At the transmitter, the information bits are ﬁrst encoded by a binary convolutional code and then interleaved. Each block of interleaved coded bits is then mapped into a PPM symbol of slots: =( ,x ,...,x . We will

refer to as a slot-word . As we assumed an ISI-free channel, the way of mapping the bits to symbols has no importance. So, we consider here the simple natural mapping by which, the position of the optical pulse in is determined by =1 +1 . An example of natural mapping for 4PPM is shown in Fig. 1. C. Soft demapping PPM symbols When performing hard signal detection, a decision is made based on the slot with the maximum signal level [13]. Here, we perform MAP-based soft signal demodu- lation. According to (1), at the receiver, corresponding to a slot-word , we receive =( ,r ,...,r , where ,i =1

,...,Q. (11) equals 0 or 1, depending on the mapped PPM symbol. Signals are statistically independent according to the assumptions of Section II. For the sake of demonstration simplicity, let us again consider the case of 4PPM with the bit-symbol mapping illustrated in Fig. 1. 1) Demodulator soft-value calculation, basic formula- tion: The likelihood ratio (LR) on the bit is given by: LR )= =1 ,b =0)+ =1 ,b =1) =0 ,b =0)+ =0 ,b =1) )+ )+ (12) Given the independence of and the AWGN assumption for ,wehave: LR )= exp +exp exp +exp (13) The LLR on is obtained by taking the logarithm of (13). We

use the approximation log( max( m,n for simplifying the calculation of LLRs. The performance loss due to this approximation is negligible at relatively high SNRs. We obtain, LLR max max (14) F. Xu, M.A. Khalighi, S. Bourennane ConTEL 2009, ISBN: 978-953-184-131-3 34

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Similarly, the LLR on the bit is calculated as follows. LLR max max (15) 2) Taking into account a priori information: To use in the iterative scheme of Fig. 1, we should modify the formulation of LLR calculation in order to, ﬁrstly, make use of apriori information at the demodulator input, and secondly, to

provide extrinsic LLRs at its output. Let us denote by and the apriori LLRs on the bits and . These are, in fact, the extrinsic LLRs at the decoder output from the previous iteration. Let us use the superscripts apost and ext to indicate a posteriori and extrinsic information, respectively. Then, (13) becomes: LR apost )= exp +exp exp +exp (16) Then, the a posteriori LLR on is: LLR apost max max (17) The extrinsic LLR on is: LLR ext )= LLR apost = max max (18) Similarly, the extrinsic LLR on is calculated as follows. LLR ext ) = max max (19) Checking (18) and (19), we verify that the extrinsic

LLR on a bit does not depend on its apriori LLR. Notice that for the case of BPPM modulation, as the demodulation is done bit-wise, the iterative demodulation and decoding cannot be done. However, a binary code is well adapted to this case. We have presented in the Appendix a simple formu- lation of the soft demodulator for the general case of PPM. V. N UMERICAL RESULTS Here we present some simulation results to study the performance of our proposed BCID scheme. The system performance is evaluated in terms of average BER as a function of /N . For channel coding, we consider the rate 1/2

recursive systematic convolutional (RSC) code (1 7) of constraint length =3 , where the numbers 5 and 7 represent the code polynomial generators in octal. Also, we consider two cases of additive white Gaus- sian noise (AWGN) channel, i.e., without atmospheric turbulence, and weak-turbulence channel. For the latter case, we set the Rytov variance to =0 04 , resulting in =51 and =49 in the model of (2). Normalized channel is considered, i.e., E =1 , and the channel fading is kept constant for each frame of symbols. A. Comparison of different modulations To do a fair comparison of the performance

of different modulation schemes, we ﬁx for all modulation schemes the information transmission rate that we denote by ,as well as the average transmitted optical power, denoted by av . Fixing this latter parameter is very important because it turns to the total energy consumption for the transmission of a given volume of data. On the other hand, as we consider RSC coding with coding rate =1 the (encoded data) bit rate equals R/R . Since for all modulations, channel coding and are the same, ﬁxing turns to ﬁxing Taking into account ﬁxed and av , we set the signal

intensity (in ON slots) for a given modulation. We also set the receiver noise variance according to the occupied bandwidth that we consider as /T , where is the slot duration. We specify below the calculation of and for the different modulations that we will consider. For the simple OOK modulation, we have /R and hence we set: =2 av av (20) For PPM, we have log /Q log .Weset: QP av QR log QP av (21) Note that in (20) and (21), we have taken the channel coding rate into account. B. Classical convolutional coding Before studying the proposed BCID scheme, let us consider the simple binary RSC

coding for PPM mod- ulations of different orders. Figure 4 compares the BER performance for a Gaussian channel when using the RSC (1 7) code. Although we do not perform iterative detection, interleaving coded bits permits to decorrelate the demodulator output errors and to improve slightly the BER. We have set the interleaver size to about 2000, 2000, 4000, 8000, and 16000 for BPPM, 4PPM, 8PPM, 16PPM, and 32PPM, respectively. Remember that the role of the (de-)interleaver is to decorrelate the LLRs input to the SISO decoder. The larger the interleaver size, the less correlated the LLRs. By

increased number of bits per symbol, we should increase the interleaver size, because the outputs of bit-level demapper are more correlated. Negligible improvement is obtained for larger interleavers than those speciﬁed above. Note that a larger interleaver implies a larger latency in data detection. We see from Fig. 4 that the performance of OOK is the same as BPPM. It is seen that, as expected, for PPM, the performance is improved by increasing . For example, at BER =10 compared to BPPM, we have an SNR gain of 2.1 dB, 3.2 dB, 3.85 dB, and 4.25 dB, by using 4PPM, 8PPM, 16PPM, and

32PPM, respectively. Pulse Position Modulation for FSO Systems: Capacity and Channel Coding ConTEL 2009, ISBN: 978-953-184-131-3 35

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10 10 −6 10 −5 10 −4 10 −3 10 −2 10 −1 10 /N (dB) BER BPPM 4PPM 8PPM 16PPM 32PPM Figure 4. Performance of classical receiver for different PPM modu- lations, Gaussian channel, RSC code (1 7) code. 10 12 14 10 −6 10 −5 10 −4 10 −3 10 −2 10 −1 10 /N (dB) BER BPPM 4PPM 8PPM 16PPM 32PPM Figure 5. Performance of classical receiver for different PPM modu- lations, Weak turbulence

channel with =0 04 , RSC code (1 7) code. We have also compared the BER performances for the case of a weak turbulence channel in Fig. 5. Compared to the Gaussian channel case, the performance improvement by increasing the modulation order is less signiﬁcant, but still considerable. For example, at BER =10 compared to BPPM, we have an SNR gain of 1.75 dB, 2.6 dB, 2.95 dB, and 3.2 dB, by using 4PPM, 8PPM, 16PPM, and 32PPM, respectively. C. Performance of BCID Consider now our proposed BCID scheme for PPM, Q> . The interleaver size for different is as we spec- iﬁed in the previous

subsection. Negligible improvement is obtained for larger interleavers. Remember that by in- creased number of bits per symbol, or equivalently ,we should increase the interleaver size, because the outputs of bit-level PPM demapper will be more correlated. 10 −6 10 −5 10 −4 10 −3 10 −2 10 −1 10 /N (dB) BER 4PPM, IT1 4PPM, IT2 4PPM, IT5 8PPM, IT1 8PPM, IT2 8PPM, IT5 16PPM, IT5 Error floor 16PPM Figure 6. Performance of the iterative receiver for Gaussian channel, PPM with =4 16 ,RSC (1 7) code, IT denotes th iteration. 1) Performance over the Gaussian

channel: We have presented in Fig. 6 curves of BER versus SNR corre- sponding to the ﬁrst, second, and ﬁfth iterations, for the cases of 4PPM and 8PPM, as well as the BER corresponding to the ﬁfth iteration for the case of 16PPM. Note that, curves corresponding to the ﬁrst iteration are equivalent to those shown in Fig. 4. Practically, the full convergence is attained after about ﬁve iterations. The interesting point is that, it is sufﬁcient to process only two iterations at high enough SNR. As expected, the gain obtained by the iterative method is

specially considerable for 8PPM and 16PPM modulations. For instance, for BER =10 , the SNR gain after full convergence is about 0.88 dB, 1.61 dB, and 2.25 dB for the cases of 4PPM, 8PPM, and 16PPM, respectively. For the sake of completeness, we have also shown in Fig. 6 for the case of 16PPM the BER ﬂoor, that is obtained by feeding the demapper with perfect apriori LLRs. 2) Performance for weak turbulence regime: Consider now the case of weak turbulence channel. We have presented the performance curves in Fig. 7 for the three previous modulation schemes. Other simulation parame- ters

are the same as in Fig. 6. The SNR gain is again signiﬁcant: it is about 1.05 dB, 1.35 dB, and 1.4 dB for the cases of 4PPM, 8PPM, and 16PPM, respectively, at BER =10 and after ﬁve receiver iterations. Again, the major part of the gain is obtained after only two iterations. From the results of Fig. 7, we deduce that, in a practical channel submitted to turbulence, there is no interest to increase beyond 8 and the 8PPM seems to make a good compromise between complexity and performance. VI. C ONCLUSIONS We considered in this paper a channel coding scheme adapted to PPM, based on a

simple binary convolutional code. At the receiver, we proposed to perform iterative soft demodulation and channel decoding. We showed that a signiﬁcant performance improvement is obtained F. Xu, M.A. Khalighi, S. Bourennane ConTEL 2009, ISBN: 978-953-184-131-3 36

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10 11 12 10 −6 10 −5 10 −4 10 −3 10 −2 10 −1 10 /N (dB) BER 4PPM, IT1 4PPM, IT2 4PPM, IT5 8PPM, IT1 8PPM, IT2 8PPM, IT5 16PPM, IT5 Figure 7. Performance of the iterative receiver for the weak turbulence channel, PPM with =4 16 ,RSC (1 7) code, IT denotes th iteration. by

using the proposed scheme, especially for increased modulation order. However, in the presence of turbulence, there is practically no interest to increase beyond 8. We proposed a method for LLR calculation in PPM de- modulator for the general case of PPM. The complexity of the proposed scheme returns in major part to that of the soft channel decoder. This latter remains reasonable, as we use a simple code of short constraint length, =3 . The interesting point is that, at large enough SNR, corresponding to practical BERs, most of the gain is obtained after only two iterations. So, by processing

only two iterations, we would have a reasonable overall receiver complexity for use in commercial FSO systems. PPENDIX For the case of natural mapped -ary PPM, we provide here a simple trick to obtain the LLR expressions on bits. Remember that we consider natural mapping, that is, the “on” slot position is given by =1 +1 , with the LSB and the MSB. Let us remind the corresponding expression for for the case of 4PPM: LLR ext 4PPM ) = max max (22) For the case of 8PPM, the LLR expression for is as follows: LLR ext 8PPM )= max max (23) Comparing (23) and (22), we see that, in each max( term in

(23), we should keep the previous terms in (22), and add two other terms. For these two additional terms, the indices of are obtained by adding 8/2=4 to those of the two previous terms; they also contain the apriori LLR on the third bit, . The same rule applies to LLR ext that can be obtained from (19) directly: LLR ext 8PPM )= max max (24) The LLR on the third bit (MSB) is obtained by mod- ifying (24): in the ﬁrst max( term we should set the indices from 5 to 8 (corresponding to =1 ), and in the second max( term, we should set the indices from 1 to 4. Finally, we change to LLR ext 8PPM

)= max max (25) Similarly, for 16PPM, the LLR on , and can be obtained by modifying the corresponding expressions for 8PPM. For example, the expressions for the LLRs on the bits and (the MSB), denoted respectively by LLR ext 16PPM and LLR ext 16PPM are provided at the bottom of this page. LLR ext 16PPM )= max 10 12 14 16 max 11 13 15 LLR ext 16PPM )= max 10 11 12 13 14 15 16 max Pulse Position Modulation for FSO Systems: Capacity and Channel Coding ConTEL 2009, ISBN: 978-953-184-131-3 37

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CKNOWLEDGMENT The authors wish to thank Dr. Steve Hranilovic from McMaster University,

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