Modulation and coding for throughput-efficient optical systems

zyxwvutsrqp zyxwvuts 1313 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 40, NO. 5, SEPTEMBER 1994 Modulation and Coding for Throughput-Efficient Optical Systems Costas N. Georghiades, Senior Member, IEEE Abstract-Optical direct-detection systems are currently being considered for some high-speed intersatellite links, where data rates of a few hundred megabits per second are envisioned under power and pulsewidth constraints. In this paper we investigate the capacity, cutoff rate, and error-probability performance of uncoded and trellis-coded systems for various modulation schemes and under throughput and power constraints. Modulation schemes considered are on-off keying, pulse-position modulation (PPM), overlapping PPM, and multipulse (combinatorial) PPM. Index Terms-Optical systems, capacity, cutoff rate, error probability, modulation, coding, throughput efficiency. I. INTRODUCTION D compare the efficiency of the various modulation schemes in terms of their throughput in nats/slot, i.e., under the constraint that they have the same pulsewidth. The throughput limitations of PPM provide a motivation for investigating the use of other modulation schemes for high-rate systems which, hopefully, do not have its limitations. One such scheme is overlapping PPM (OPPM), which was originally studied in [6] and later in [7]-[101. OPPM is a generalization of PPM that allows more than one pulse-position per pulsewidth and preserves some of the desirable properties of PPM, such as equal energy signals and low duty-cycle. If Q is the number of nonoverlapping pulse-positions in a T-second symbol interval (i.e., Q = T/T,) and N (referred to as the index of ouerlap) is the number of pulse-positions per pulsewidth, then the total number of OPPM symbols J is J = N ( Q - 1) + 1. For N = 1, OPPM reduces to PPM. If we constrain Q to be an integer, then to obtain a desirable number of modulation signals J , the index of overlap must be N = ( J - l)/(Q - l), which is a rational number. For r the rate in nats/s, the following is true for OPPM: zyxwv zyxwv zyxwvuts UE T O their small size and relatively high-power efficiency, optical direct-detection systems have long been considered by NASA for deep-space communication [11. For these low-power, low-data-rate applications (a few tens of kilobits/per second), good performance for little power is paramount; under these constraints, pulse-position modulation (PPM) was shown to be a well-suited modulation scheme [2], [3]. PPM has also been the modulation of choice for NASA's direct-detection intersatellite link (ISL) applications for which quaternary PPM (QPPM) has been much studied [4], [5] for data rates of a few hundred megabits per second. In the future, even higher data rates are envisioned for which PPM may not be well suited due to its inherent throughput limitations: with PPM, the only way throughput can increase is by reducing the pulsewidth* Thus, if is the 'PM size, T, the slot duration (pulsewidth), and r the rate in nats per second, we have the following severe bound on the PPM throughput: rT, In [ N ( Q - 1) = Q + 11 In (N I 2 + 1) nats/slot , ( 2 ) zyxwvuts \ rT, In<Q> ln(3) = -< - Q - 3 nats/slot , (') which suggests ternary 'PM can yield the largest throughput for a fixed pulsewidth T,. In the following, we will I. Manuscript received August 16, 1993; revised March 18, 1994. This work was supported by the NASA Lewis Research Center under Grant NAG 3-1353. This paper was presented in part at the 1993 Global Telecommunications Conference (Globecom '93), Houston, TX, December 1993. The author is with the Electrical Engineering Department, Texas A & M University, College Station, TX 77843-3128. IEEE Log Number 9405284. which (at least in theory) can be made as large as desired by increasing N. Clearly, there is a penalty to be paid when N is increased, both in error probability and in more stringent synchronization requirements,' which must be taken into account in an ultimate comparison of OPPM - to other modulation schemes. Fig. 1 illustrates OPPM for N = 3 and Q = 2, which results in J = 4 signals. Another modulation scheme that has been considered recently in the literature is multipulse or combinatorial PPM (MPPM) [ll], [12]. As with OPPM, MPPM is yet another generalization of PPM that allows more than one pulse per symbol interval. Thus, if the number of pulse. ' zyxw ( allowed is p , the number of MPPM symbols is M = f), which increases monotonically for 1 5 p 5 lQ/2]. 'see Of P are in the Fig. 2. C1ear1y7 the interesting interval 1 I p I [ Q / 2 ] . Like PPM, MPPM iS an equal energy signaling scheme, and like OPPM it increases the number of available signals for the same pulsewidth compared to PPM. The data rate achieved by MPPM relates zyxwvutsrqp 0018-9448/94$04.00 0 1994 IEEE . 1114 zyxwvutsrqponmlkjihg IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 40, NO. 5, SEPTEMBER 1994 Before moving on to the next section, we note that hybrid modulation schemes can be designed that allow overlap and multiple pulses per baud interval [141. We will not pursue these schemes here for space reasons and also because they are invariably too complicated to implement. In the next section, we derive capacity, cutoff rate, and error-probability expressions for OPPM and MPPM. Section I11 compares the various modulation schemes in terms of peak-power requirements and throughput efficiency. Section IV gives a flavor of the coding problem over OPPM and MPPM symbols, and Section V concludes. zyxwvutsrqp zyxwvutsrq zyxwvutsrqpo zyxwvutsrqp zyxwvut zyxwvutsrqp zyxwvuts zyx Fig. 1. An example of OPPM for Q = 2,n = 3. 11. ERROR PROBABILITY, CUTOFF RATE, AND CAPACITY 1100 1010 1001 01 10 0101 0011 - Fig. 2. A n example of MPPM for Q = 4, p to Q and p according to We derive expressions for error probability, cutoff rate, and capacity for MPPM and OPPM, assuming a direct-detection (Poisson) channel. For derivations of the capacity of direct-detection Poisson channels under peak and average power constraints, the reader may wish to read [lS], [16], and for binary inputs with a constraint on the interval between level transitions, [171. A. Multipulse PPM An expression for the optimum receiver for MPPM is easily derived in Appendix A. The resulting log-likelihood function is zyxwvut = 2. P Ik Nwkz7 i=l rT, nats/slot . ( 3 ) where w k = { w k l , w k 2 ; . . , w k Yis} the set of p integers taking values in the set {1,2,..-,Q} indicating the positions of the p ones in the kth symbol. In other words, the In view of some well-known inequalities (see, for example, receiver accumulates the number of photons in each Cover and Thomas [13, page 284]), we have the following pulsed slot for each possible transmitted symbol and debounds on the rate in (3), clares the symbol corresponding to the one with the largest accumulated counts as the transmitted symbol. This receiver is equivalent to one that decides for the MPPM symbol whose pulses are in the slots containing the p largest number of photons. where h ( x ) = - x In ( x ) - (1 - x ) In (1 - x) is the binary 1) Error Probability: In this section we derive an exact entropy function. The ratio p / Q can be identified as the expression for the symbol error probability for a probability of a pulsed-slot in a sequence of MPPM quantum-limited channel’ and an upper bound when symbols. As Q + CO, rT, approaches h ( p / Q ) , which is the background noise is present. largest amount of information that can be produced by Quantum-limited channel: We assume that all M symany binary source with prior symbol probabilities p / Q bols are equiprobable, in which case and (1 - p / Q ) [13]. Thus, at least asymptotically, MPPM is a throughput-efficient scheme. For comparison, QPPM 1 M has a probability of a pulsed-slot of and a throughput of In (2)/2 = 0.346 nats/slot. For the same probability of $, MPPM can achieve (for large Q ) close to h ( + ) = 0.562 nats/slot, a potential throughput increase of The second equality is due to the symmetry of MPPM that implies P ( Z I di) is the same for all transmitted more than 60%. symbols d i . Since for the quantum-limited channel errors 1 . Since equal to apparent stringent = Q the smallest time distance between two pulse-positions is now T, = T”, (the interval T, referred to as a chip), it seems that the timing requirements have become N times more compared to PPM. ’An expression for the error probability derived in [ll] ignores the possibility of making a right decision even when one or more pulses are erased. zyxwvu zyxwvutsrq zyxwvutsrqponmlk zyxwvutsrqp zyxwvutsrq zyxwv zyxwv GEORGHIADES: MODULATION AND CODING FOR THROUGHPUT-EFFICIENT OPTICAL SYSTEMS occur only when one or more pulses are erased, we can write P P(g) = c P ( 8 I k erasures)P(k erasures). 1315 partitioned into groups of u k terms each, for each of which the Chernoff bound in (11) equals exp ( - kd2). Under these observations, (10) and (11) combine to give (7) k= 1 When k pulses are erased, a random decision must be made among the Ak = symbols that have ( k ) pulses at the ( p - k ) positions where pulses were detected. Since the probability of exactly k erased pulses is ( ; ) ~ ‘ ( l - CY-’, we have -’ + where E = c p A y T s is the pulse-erasure probability. An excellent approximation to (8) for E less than approximately l o p 2 is The general observation to be made from the above derivation is that what determines the performance of MPPM signals (at least. for large signal levels when the bound above is tight) is the minimum Hamming distance between symbols. We will use this observation in Section IV in determining the performance of trellis-coded MPPM signals. 2) Cutoff Rate: We make use of the following general expression for the cutoff rate derived in [ 3 ] , valid for optical channels with observations modeled by conditional Poisson processes and both for quantum-limited3 and noisy channels: (9) Background noise channel: For the noisy channel, we have P(8)5P IM U U, I 1,) I d , j=2 M I 1 c P(1, s l j I d l ) , j= 2 (9,) i = 1 j = 1 where M is the number of MPPM signals, qi is the prior probability for the ith signal, and (10) zyxwvutsr where the first inequality is obtained by assuming pessimistically that whenever equal counts occur an error is made, and the second follows from the union bound. Focusing on the probability inside the sum, we can show using the Chernoff bound (refer to Appendix B) that where In (161, [ s i ( t )+ A,] in photons/sec is the mean rate (intensity) of the observed Poisson process when the ith signal is sent, s,(t) is the signal intensity due to the optical beam impinging on the photodetector, and A, is the noise intensity as defined previously. For MPPM signals, we can express (16) as d; = d2d,(di, d j ) , where d 2 and d,(d,, d j ) were defined above. Letting Ymin be the minimum value the double summation in (15) can attain, we have (12) M and d,(d,, d;) is the Hamming distance between d , and dj. To proceed further, we need to find the Hamming distance profile for the MPPM signals. Towards this end, we first note that the possible values that d,(d,, d,) can take are 2,4;-., 2p. Further study using counting arguments shows that if a k , k = l , 2 , . . . , p , is the number of symbols at distance 2 k from d,, then P Q-P “=(k)( k ) M Using Lagrange multipliers, we obtain the following necessary (and in view of the strict convexity of the objective function, sufficient) conditions for a minimum: i= 1 (13) In fact, it can be shown that because of the symmetry of the MPPM signals noted above, we have the same distance profile when any signal di is sent (not just when d , is sent). It can be easily verified that C,Y=,ak = ( M - 1). Thus, the ( M - 1) terms in the sum in (10) can be for some constant ZF’. Summing over all n on both sides of (18) and observing that for M P P M the sum 3Although expressions for the cutoff rate and capacity for MPPM were derived in [18] for quantum-limited channels (A, = O), we found that these expressions significantly underestimate the cutoff rate and capacity of the channel, because of the pessimistic way erasures were defined. 1316 zyxwvutsrqponmlkji zyxwv zyxwvutsrqp zyxwvutsr zyx zyxwvu zyxwvutsr IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 40, NO. 5 , SEPTEMBER 1994 E =: expi- id2dH(d,, d i l l is not a function of i, we obtain (for all i) 1100 1010 1100 1001 (19) where we have used the fact that dH(d,, d,) = d,(d,, d,). Comparing (19) with (181, we conclude that the uniform distribution, 4, = 1/M, i = 1,2;.., M is the minimizing distribution. Multiplying (18) by 4, and summing over all n, we obtain Pmi = e. Further, since as seen above, there are a, = terms in the sum in (19) that have distance d,(d,, d , ) = 2k for k = O,l;..,p, we finally obtain b -0 1010 1001 01 10 0110 0101 0101 001 1 001 1 zyxwvut 1000 0100 (E]( ‘ i ’ ) 0010 Fig. 3. DMC model for (4,2)-MPPM. number of binary codewords of length Q and weight at most p : zyxwvutsrqp It is easy to verify that for p = 1, the cutoff-rate expression for MPPM above yields the cutoff rate of Q-ary PPM, as expected. The above derivation and results can be summarized in the following general theorem: Theorem: For an optical direct-detection channel utilizing M signals with distances d;j satisfying E: exp( - i d ; ) = %5’ (not a function of i), the cutoff rate is given by R, = -In( L= q). k=O (9 Of the L possible outputs, correspond to the input symbols and the rest to words containing one or more erasures. An example of a DMC model for MPPM with Q = 4 and p = 2 is shown in Fig. 3. The capacity of the channel is given by g), It can be seen that P ( y , Ix,) = 0 when y, is not an “erased” version of x, (i.e., y, contains one or more 1’s at and is achieved by a uniform distribution. An immediate consequence of the theorem is that, if positions where x, does not), and P ( y , I xr)= e k ( l the rows of the matrix with entries {exp(- i d : ) } are E ) P - , when y, is an “erased” version of x, containing k permutations of each other, then a uniform distribution erasures (i.e., y, has 0’s in k positions in all of which x, achieves the cutoff rate. Another consequence of the contains 1’s). For each input x,,corresponding to the ith theorem, since d,, = d,,, is that the cutoff rate for any row of the transition matrix, there are 2P possible “erased” binary optical system is achieved with a uniform distribu- outputs with corresponding transition probabilities (which tion and it equals R , = - I n [ i ( l + exp(- i p ’ ) ) ] , where are the same for all x,’s) given by the binomial distribup 2 is the distance between the two signals as determined tion above. Therefore, for each x, there are ( L - 2p) remaining elements of the ith row of the transition matrix by (16). that are zero, again independently of i. Clearly then, the If the necessary condition above is not satisfied, then rows of the transition matrix are permutations of one obtaining the optimizing prior probabilities and Pm,, can be done by solving the following system of ( M 1) linear another. Turning to the columns of the transition matrix, we can partition the set of columns into ( p + 1) subsets equations: + 9 , of ( f ) columns each corresponding to outputs with k = 0,1,2;.., p erasures (and corresponding weights p, ( p - l),..., 0). Looking at any column in a subset Y k , we see that it contains exactly A , = nonzero en- zyxwvutsrqpo zyxwvuts (z) M 4,,-$f2d,(d,,,d,) -Pm,,, = 0, n = 1,2;.-, M , r=l M C4;= 1. i= 1 3) Capacity: The MPPM direct-detection channel can be modeled as a discrete memoryless channel (DMC) with M = inputs and L outputs. For a quantum-limited channel where the only degradation occurs when 1,2;.., p pulses are erased, the number of outputs equals the -1 - ’ + (21) ( k ) tries (since when k erasures occur, there are A , inputs that could have resulted in the output), all equal to e k ( l - e Y k . We conclude from the above arguments that the channel is symmetric (see Gallagher [201), and thus the capacity-achieving prior distribution is uniform, P(x,) = 1/M, i = 1,2;.., M . Further, the inside sum in (22) is the same for each x, and it equals capacity; the outputs within each subset Ykall have the same probabil- zyxwvu zyxwvutsrq zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGF zyxwvutsrq zyxwvutsrqp zyxwvuts GEORGHIADES: MODULATION AND CODING FOR THROUGHPUT-EFFICIENT OPTICAL SYSTEMS ity given by (l/M)EEl P ( y I xi>.Putting everything together, we finally obtain 1317 in the range from 1 to 4 and arbitrary J . In all of these cases it can be shown (we skip the derivations here as they are somewhat tedious but otherwise straightforward) that + 1 In (( Q -:‘)) + (23) The expression is easily seen to simplify to that of PPM for p = 1. An upper bound on the capacity, very tight for small E , can be obtained by using Jensen’s inequality on the first expression in (23) to yield where J = N ( Q - 1) 1 is the OPPM alphabet size. Computation for larger values of N is easy but becomes progressively more tedious. Based on the results for the values of N considered, we conjecture that the above expression is true for all N. Background noise channel: When background noise is present, the following upper bound can be derived using the results in Appendix B: l N POp,,(8) I bkepkd2/N, 2J k = l where bk= A further bound can be obtained by rewriting the sum in (24) as ~ ( p - ~ ) C k p =( ,f ) ~ Q - ,( lE), and recognizing the summation as the probability that a binomially distributed random variable is less than or equal to p . This probability can be bounded using the Chernoff bound. Skipping the details of the derivation, and combining the Chernoff bound with the bound in (24), we finally obtain + p In (1 - E ) . (25) is valid provided p / Q + E 5 1, and becomes Cmppm I Qh(p/Q) (28) ( 2(J - k ) , ( J -N)(J - N + l), k k = = 1,2;-., ( N - l), N, (29) zyx zyxw and d 2 is as defined in (12). The bound is slightly tighter than the one derived in [8]. 2) Cutoff-Rate: Here again we make use of (15) and (16). Using (16) we can write d ‘12 = 2d2 -6 “1’ (30) where d 2 is as defined in (12) and The bound tight for small E and in the limit of large Q for a fixed P/Q. A tighter bound can be obtained by noting that the capacity of MPPM in nats per slot is upper-bounded by the average mutual information of the binary erasure channel with prior “one” probability equal to p / Q and erasure probability E. The bound, which is tight for large Q for a fixed p / Q , is due to the fact that MPPM imposes a block constraint that tends to decrease capacity. Thus, B. Overlapping PPM Note that for i # j, aijtakes values in {1,2;--, N I ; if we = k , then consider the number of pairs i, j for which we can show that it is equal to b, given in (29). A brief check indicates that the conditions of the above theorem are not satisfied, in which case the uniform distribution may not be the optimizing one. In order to (at least empirically) assess the difference between the cutoff rates when the optimum prior probabilities and a uniform distribution are used, we look at an example using the Maple symbolic package to solve the system in (22): Let y = e-d2/N and consider Q = 2, N = 3, and therefore J = 4 OPPM. The optimum prior probabilities are given by [1/2(2 - 71, ( y - 1>/2(y - 2), ( y - 1)/2(y - 2),1/2(2 - 71, and the cutoff rate is zyxwvutsrq In this section we present expressions for the error probability, cutoff rate, and capacity of OPPM. The optimum receiver for OPPM was derived in [8] and (as expected) consists of finding the slot with the largest number of observed photons. 1) Error Probability Quantum-limited channel: An exact expression4 for the error probability has been derived in this case for N Computed with a uniform distribution, the cutoff rate is 1 - 4 3 + -8y + 1 -y2 4 + -8y 3 4An expression derived in [8] for the error probability is exact only for N = 2, and an approximation otherwise. 1318 zyxwvutsrqponmlkji zyxwv zyxwv zyxwvuts zyxwvut IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 40, NO. 5, SEPTEMBER 1994 = ln(4) 3 - -y 2 + -81y 2 + O ( y 3 ) . limited channel and is given here (slightly rewritten) for reference: N Note that the optimum prior probabilities quickly become uniform as d 2 increases (and thus y decreases). On the other hand, as the signal level decreases ( y + 11, only the Q signals corresponding to orthogonal pulse-positions are sent with a nonzero probability ( l / Q each), which suggests that the advantages of OPPM are attained at the higher signal levels. The difference between the optimum cutoff rate and that attained by a uniform distribution was seen to be small, as indicated by the example above. Since it is much easier to obtain general closed-form expressions for a uniform distribution, we will assume it in what follows. In this case we obtain = {i+ -In (J-N)(J-N+ - 2 y(l + -J - YN J2 = -1n ’(’ (1 - y Y -- i!, + Q-2 -E’ (32) k-1 where c(k,j) = 1 -min(k,N-j,J+N-k-j). J (35) C. On-Off Keying Finally, for comparison purposes, we present the cutoff rate and error probability of OOK, which was not in the past considered for free-space applications. One reason for not considering OOK in such applications is the possibility of getting long sequences of 1’s or 0’s that degrade synchronization performance, and in the former case require the laser to be on for a long time. Both problems are solvable, if one is willing to sacrifice some throughput through the use of appropriate line coding, but we will not pursue this in this paper. The well-known expression for the cutoff rate of OOK, achieved with equiprobable signaling, is 2 (36) and the error probability is 2(1 - E’) (Q - l)d2 12-1 + c(k,j)ln[c(k,j)l i = 2 j=i-l “)zyxw ]I. [ lNI: For the noiseless case, i.e., A, = 0, (32) holds by substituting d 2 = h,T,, which implies that y = e-hsTs/N is the erasure probability for signaled chips of duration T,/N. Even though more general than the expression obtained in [71 (valid only when A, = O), (32) is much simpler to compute. In the limit as N + m, (32) reduces to R,, JtN-j-l zyxwvutsrq yN-1) 1 - y + -J22 1) N-1 coppm = 2E’ - ( Q - 1I2d2 1, 2(1 (Q - E’) I lI2d4 1 (37) 2 The capacity of OOK is attained by a nonuniform prior distribution and is given by POOk(k?)= Cook= In --E. + 11. (38) 111. PERFORMANCE COMPARISONS In this section we use the expressions derived above to compare the performances of the various modulation schemes in terms of power efficiency for a given throughput, capacity and cutoff rate, and peak-power requirements. We address coded error-probability performance in the section that follows. A. Power Eficiency For coded systems, the capacity is a fundamental limit (33) on the rates for reliable communication, whereas it was argued by Wozencraft and Kennedy [21]and subsequently by Massey [22]that the cutoff rate of a system, which is where 6 ’ = e-d2. The approximation in (33) is valid for upper-bounded by capacity, is a practical limit. In this large values of d 2 and was also derived in [7] for quan- section we will use both the cutoff rate and capacity as indicators of the achievable rates in order to investigate tum-limited channels. 3) Capacig: The capacity of OPPM (assuming the throughput efficiencies of OOK, PPM, OPPM, and equiprobable signaling) was derived in [71 for a quantum- MPPM. =In[ (Q - l)d2 Q>L zyxwvu zyxwvut zy GEORGHIADES: MODULATION AND CODING FOR THROUGHPUT-EFFICIENT OPTICAL SYSTEMS Following [23], [24], we let r be the desired throughput in nats per second, and T, the desired pulsewidth. Both constraints stem from practical considerations, where a certain throughput is required but the pulsewidth cannot be reduced beyond some limit. Fixing the throughput and the pulsewidth implies the following constraining equation (since T = QT, for PPM, MPPM, and OPPM): rT, Ro = - Q nats/slot. (39) 1319 prior probabilities (not the case with the other schemes). Letting q be the probability of a 1 (laser on), we have zyx 1- YTS 2q(l Roph = e-rT, - - < 1 - q , (44) q) q 2 - (1 - qI2 with equality in the limit rT, -+ 0. The above expression can now be optimized with respect to q to obtain the best possible efficiency. Alternatively, one can assume for simplicity equal probabilities, in which case zyxwvutsr zyxwvuts For a fixed average noise photons per symbol, Zn,and a fixed overlap N for OPPM or a fixed p for MPPM, (39) can be satisfied by varying the average number of signal photons/symbol q ,where for PPM and OPPM, q = h,T,, and for MPPM, 3 =ph,T,. If we let %(Q,rT,,h) (where h = N for OPPM and h - p for MPPM) be the value of satisfying (39), then the throughput efficiency in nats per photon is (40) In the following, we assume a quantum-limited channel for simplicity and also because, at the high data rates envisioned for ISL systems, only a small fraction of a noise photon is expected per slot (assuming the sun is not in the field of view) [19]. In general, explicit analytical solution of (39) for Z ( Q , rT,, h) is not possible, except for some special cases, such as for PPM, p = 2 MPPM, and N = 2 OPPM, for which: ROph = rTsQ l n < Q - 1) - In [Qexp(-rT,Q) - 11 ' PPM, (41) ROph The denominator of the above equations for ROphcorresponds to the average symbol energy that satisfies (39). For values of p and N other than 2, numerical solutions are easily obtained. The functions Roph(as well as the corresponding capacity expressions below) were seen to be concave with respect to Q for PPM, OPPM, and MPPM, and with respect to q for OOK. A comparison between PPM, OPPM, MPPM, and OOK as a function of the required nats/slot is given in Table I. The values of Q in the table are optimum for the corresponding rT,. Both MPPM and OPPM outperform PPM, especially at high throughputs. N = 2 OPPM is uniformly better than p = 2 MPPM but becomes worse than p = 4 MPPM (not shown in the table) as the throughput increases. For OPPM, we list the two extreme overlap cases: N = 2 and N = 00. Most of the gain in allowing overlap is obtained for N = 2, with progressively less incremental improvement as N is increased. OOK does not perform as well for small required nats/slot, but becomes significantly better at high-rates. For a practical comparison, let us consider the efficiencies of each modulation scheme at the rate of ln(2)/2 = 0.35 nats/slot, which is the rate at which the currently developed QPPM system operates. At rT, Q = 2 l n ( Q - 3) - 21n , [ Q < Q - 3) exp (-rT,Q) + 21 MPPM,p=2, -2 (42) ROph = 2In(2Q - 3) - 21n rT, Q (4Q2 - lOQ + 5 ) (2Q - 1I2(2Q - 3) exp (-rT,Q) 2(Q - 1) 2(Q - 1) , - For OOK, which, unlike the other modulation schemes, does not have signals with equal energies, better performance can be obtained if one leaves the prior probabilities as parameters to be varied in order to maximize R,,, directly, instead of using the prior probabilities that achieve the cutoff rate (or capacity). This is so for OOK since the average energy expended is a function of the ll OPPM, N = 2 . zyx (43) this rate, PPM has an efficiency of 0.284, OOK 0.482 for N = 2 OPPM 0.528, the optimal q and 0.392 for q = p = 2 MPPM 0.482, and p = 4 MPPM 0.533. Clearly, all modulation schemes can do better than QPPM, whose performance is upper-bounded by 0.284 nats/photon (since this is the value obtained by the optimum value of Q = 3 for PPM). 4, 1320 zyxwvutsrqponmlkji zyxwv IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 40,NO. 5, SEPTEMBER 1994 zyxwvutsrqpo TABLE I CUTOFF RATE IN nats/photon. THE NUMBERS I N PARENTHESES FOR PPM, MPPM, AND OPPM CORRESPOND TO THE OPTIMUM VALUESOF Q, AND FOR OOK, TO THE OPTIMUM4. zyxwvut zyxwvu TABLE I1 CAPACITY ’ IN nats/photon. THE NUMBERS IN PARENTHESES FOR MPPM AND OPPM CORRESPOND TO THE OPTIMUM VALUESOF Q, AND FOR OOK, TO THE OPTIMUM4. Results similar to the above using the capacity instead of the cutoff rate are shown in Table 11. For PPM, N = 2 OPPM, and p = 2 MPPM, these results were obtained by using equations which parallel those in (41)-(43). We present below the equations for PPM and p = 2 MPPM and skip the one for N = 2 OPPM, as it is rather long. For OOK, no closed-form expression is available: Cph = In r zyxwvut zyxwvutsr zyxwvut -,.r n ‘A’X In ( Q ) - rT’Q 1’ PPM, (46) In general, the capacity results are qualitatively similar to those using the cutoff rate, with some exceptions. For example, whereas N = 2 OPPM is uniformly superior to p = 2 MPPM in Table I, Table I1 indicates that the latter is sightly better than the former for the smaller values of rT,. Also, the capacity results indicate that OOK with the optimum prior distribution is uniformly superior to the other schemes for the parameters studied, and not just for the higher throughputs as suggested by the cutoff rate. Larger values of N and p are needed for MPPM and T OPPM to compete with OOK. The optimizing values of Q, also shown in the table, are closely similar to those obtained using the cutoff rate. Table I11 summarizes the ultimate limits in throughput (nats/slot) for each modulation scheme. As can be seen, only OPPM can provide throughputs greater than In (2) nats/slot, but it requires indexes of overlap above N = 4 to do so (which will make synchronization significantly more difficult). B. Peak-Power Requirements Here we investigate the peak-power requirements of PPM, MPPM, OPPM, and OOK as the data rate r in nats per second increases and uncoded error probability is kept fixed. We assume a quantum-limited channel. To make the comparison fair between the various modulation schemes, we compare the peak power needed to convey a sequence of bits through the channel at the same-sequence error probability for each modulation scheme. zy I zyxwvutsrqponmlkjih OOK II rT, 5 ln(2) I I/ PPM MPPM rT, 5 h ( p / Q )- equality for Q = 3 I = - In T, Substituting T, A, _ - - r ~ [ = zyxwv rT, 5 1 equality for Q = 2 I = 03 TABLE IV PEAK-POWER PERFORMANCE IN photons/nat zyxwvu zyxwvut P(Q ( Q - p + 1)P(8) ] photons/s. (48) In ( M ) / Q r , we obtain [ OPPM 5 rT, 5 h ( p / Q ) equality for Q MPPM: Remembering that MPPM has p times the average energy per symbol compared to PPM and OPPM (for the same A,T,) and using the approximate expression for error probability in (9), we have A, zyxw zyxwvut zyxwvutsr zyxwv 1321 GEORGHIADES: MODULATION AND CODING FOR THROUGHPUT-EFFICIENT OPTICAL SYSTEMS ’(e+-’) 1)J‘(g)] PQ In ln(M> (Q-p PROBABILITY OF AT AN ERROR -3. MPPM OPPM Q,N,J X,/r Q , p , M X./r 10.96 4 19.10 3,2,5 24.92 5,2,10 31.76 17.34 8 26.06 5.2.10 30.90 , 5.2.10 , , , 31.76 23.92 16 39.49 9,2,17 43.50 7,2,21 34.11 30.70 32 63.49 17,2,33 66.87 12,2,66 42.99 37.62 64 106.06 33,2,65 108.97 12,2,66 42.99 OOK 2L 4 8 16. 32 64 IO L/r PPM Q L/r / I photons/nat. (49) The above expression holds as a special case for PPM by setting p = 1. Clearly, for fixed Q, p , and P ( 8 ) , peak power increases linearly with the data rate, and thus, to reduce the peakpower requirements, we must minimize the slope (which has units of photons/nat) on the right-hand side of (49). Another quantity of interest for practical systems is the peak-to-average power a , which for MPPM is given by a = Q / p (note that 1 / a is the probability of a pulsed slot in a sequence of data). OPPM: An expression parallel to (49) relating the peak-power requirements of OPPM to the throughput for a fixed error probability can be similarly derived and is given by bits/symbol, but becomes significantly better at the higher rates. PPM is uniformly better than OPPM, although the difference becomes smaller at the higher rates. The latter observation may seem surprising at first glance, since whereas throughput in nats/s for PPM can only increase at the expense of a smaller T, (which means a larger peak power A, to maintain the same performance), this is not the case for OPPM. On the other hand, OPPM requires larger peak powers to achieve the same error probability as PPM, which apparently is the reason for it being inferior compared to the latter. C. Capacity and Cutoff-Rate Comparisons We compare the capacities and cutoff rates of the various modulation schemes in nats/slot as a function of the average energy per nat. Fig. 4 compares the cutoff rates in nats/slot for OOK, N ( Q - 1) NQ PPM,-OPPM, and MPPM. For OPPM we consider N = 2 4 _ -and N = 3, which are small enough not to make synchror l n [ N ( Q - 1) + 11 In [ N ( Q - 1) + l I P ( 8 ) nization impractical, and for MPPM we consider p = 2, photons/nat. (50) p = 3, and p = 4. The values of Q for PPM, OPPM, and The peak-to-average power requirements for OPPM are MPPM were chosen to maximize the capacity/cutoff rate the same as those for PPM: a = Q. at large signal levels, namely Q = 3 for PPM, Q = 2 for OOK: For OOK (which, assuming equal prior probabil- OPPM, and Q = 2 p 1 for MPPM. The superiority of ities, has an average energy per symbol equal to A,T,/2), OOK in this comparison is obvious from the figure. OPPM the probability of sequence error for a sequence of L bits is inferior to MPPM for small signal levels but becomes is P ( 8 ) = 1 - (1 - exp [ - (2A,T,/L)l]L. Then, better at the higher levels. For N 2 4, OPPM will perform better than OOK as the average number of signal photons increases, since (see Table 111) OOK saturates at ln(2) nats/slot, whereas OPPM saturates at In(N + 1) nats/slot. The peak-to-average power for OOK is a = 2. A comparison using capacity instead of cutoff rate Table IV compares the quantity A,/r in photons/nat for the various modulation schemes at 2, 3, 4, 5, and 6 yields results qualitatively similar to those in Fig. 4, which bits/symbol and an error probability of lop3. For OPPM are not presented here to save space. and MPPM, the parameter values were chosen to yield IV. CODING the best results for the number of bits per symbol required. N = 2 for OPPM and p = 2 for MPPM were seen In this section we compare the coded performances of to yield the smallest peak powers for a fixed data rate. OPPM and MPPM. In the interest of space, this compariThe table shows that OOK has the best performance. son is not exhaustive and is only meant to illustrate what MPPM is inferior to both PPM and OPPM at 2 and 3 the possibilities are with each modulation scheme. We [ 1 + zyxwvutsr 1322 zyxwvutsrqponmlk zyxwv zy zyxwvutsrqpo zyxwvuts zyxwvutsr zyxwvutsrqpo IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 40, NO. 5, SEPTEMBER 1994 ......-..,.....,-. c $ 0.5 --. 4- 2 6 c $ U+ c 5 zyxwvutsr 0.4 0.3 0.2 0.1 oPPM, Q = 3 D MPPM, p=2, Q = 5 MPPM, p=3, Q = 7 9 MPPM, p=4, Q=9 -. OPPM, N=2, Q=2 -t OPPM, N=3, Q=2 .. -4-PPM - i + OOK 1 * 0=2, N=7, %state Q=2, N=7, 15state + MPPM, &state, Q=9, p=2 2 3 4 5 6 7 9\ 8 4 + 9101112 Signal Photons/dibit, dB Fig. 5. Symbol-error probability for trellis-coded OPPM and MPPM compared to QPPM. focus on trellis-coded modulation (TCM), rather than block coding (and specifically Reed-Solomon coding), which has been previously studied for PPM 111, 1251, [261 and MPPM [12]. Some work on TCM for optical OPPM with Q = 2 was presented in [SI, [27]. Here we present new results for both OPPM and MPPM. A. OPPM We consider two examples of how trellis-coded modulation can be used in conjunction with OPPM to obtain a coding gain and/or increase the throughput. For brevity, we present results only for the quantum-limited channel, but qualitatively similar results were obtained for the background noise channel as well. The first example, which was also studied in [SI, starts with Q = 2 PPM, which yields a throughput of 1/2 bits/slot, and has a peak-to-average power ratio of CY = 2. With an index of overlap N = 7, the number of OPPM signals is J = 8, which results in a threefold increase in the number of bits per slot from 1/2 to 3/2, while a remains the same. Using a rate-2/3 Ungerboeck code [281, we can trade off some throughput for performance. Our reference for comparison is QPPM, which has a throughput efficiency of 1/2 bits/slot. If we let p be the eneqyper information bit, R the rate of the trellis code, and afreethe minimum free distance of the code using the metric in (31), then ( =: means "asymptotically") P ( g ) x exp ( - 2 p ) (52) for QPPM, and for the trellis-coded system. The asymptotic coding gain of the trellis-coded system over QPPM is and the corresponding rate in bits/slot is log ( J ) R,= R bits/slot . ~ Q (55) Eq. (54) indicates that any trellis code that has parallel t r a n s i t i o n s will h a v e a gain b o u n d e d by 10 log,, [ R log (J)/2], since in this case afree5 N . For the 8-state code, afree= 5, and thus the loss of the coded system for high SNR's is about 1.5 dB. Note, however, that the coded system is operating at twice the throughput of QPPM. On the negative side, the coded OPPM system has half the peak-to-average power of QPPM and requires a relatively large index of overlap N = 7, which implies more stringent synchronization requirements. Fig. 5 shows simulation results for the 8- and 16-state Ungerboeck codes with the trellises populated by OPPM instead of phase-shift-keying (PSK) signals [28]. The simulations verify the analytical results obtained above, and indicate that the 16-state code is only slightly more than 1 dB worse than QPPM at a symbol-error probability of As a second example, we consider OPPM with Q = 4. With an index of overlap N = 3, we obtain J = 10 signals, two of which can be discarded to yield eight modulation signals. However, a more efficient scheme is to use a fractional index of overlap N = 7/3, which also yields eight modulation signals. The throughput efficiency of this scheme is 3/4 bits/slot and a rate-2/3 Ungerboeck code can be used to reduce the rate to 1/2 bits/slot (same as for QPPM) in exchange for an asymptotic 3.3 dB coding gain for the 8-state code. Fig. 5 shows the performance of the 8- and 16-state Ungerboeck codes using this signal set. It can be seen that the %state and 16-state codes are about 3.0 dB and 3.5 dB better than QPPM, respectively, at an error probability of lo-', for the same throughput and peak-to-average power as the latter. The small discrepancy between the asymptotic gains and the simulations is expected to diminish further at smaller error probabilities. If both a coding gain and a throughput gain over QPPM are desired, this can be achieved with Q = 3 and zyxwvu zyxwvutsrqp GEORGHIADES: MODULATION AND CODING FOR THROUGHPUT-EFFICIENT OPTICAL SYSTEMS z, 1323 N = which again yield J = 8 signals. Using the rate-2/3 8-state code (with = 51, a coding gain of about 1.5 dB can be achieved at a rate of 2/3 bits/slot, a 33% throughput improvement over QPPM. Clearly, even more powerful codes can be designed using this approach that not only give a coding gain but possibly a throughput gain as well at the expense of more complexity. B. MPPM Here we are interested in coding for MPPM signals at throughputs in bits/slot close to those of QPPM. Also, of interest is the peak-to-average power parameter a , which for MPPM is a = Q/p. As indicated above, the minimum Hamming distance for uncoded MPPM signals is 2, which is also that for uncoded QPPM. However, whereas QPPM conveys only 2 bits/symbol, MPPM can convey more. Let us compare the performance of uncoded QPPM with that of uncoded MPPM for the same energy per bit p photons/bit. For MPPM (using the approximate expression in (9)) 0 0.5 1 1.5 2 zyxwvutsr zyxwvu zyxwvutsr zyxwvuts (56) Normalized Rate Fig. 6. Coding gain of MPPM over QPPM as a function of the normalized code rate. All shaded points are achievable for sufficiently large Q. TABLE V VARIOUS CODING RATESFOR MPPM SIGNAL SETS Wl-rH M I N I M U M HAMMING DISTANCE OF 4 (i.e., HAVING A 3-dB CODING GAIN Q, P , a Q=24, p = 6 , a = 4 Q = 2 8 , p = 7 , CY=^ Number of Symbols 4316 37202 Code Rate (bitslslot) 112 15/28 Thus, the gain of MPPM over QPPM is as a function of the rate in bits/slot (normalized by that (57) of QPPM). As an example of what can be practically achieved, where the bound is in view of (4) and is achieved in the consider Q = 16, p = 4 MPPM, resulting in 1820 signals; limit Q =. It is easy to see that G increases monotoni- this is the example studied in [12]. Deleting enough sigcally as p / Q decreases. However, as p / Q decreases, so nals to obtain 1024 signals (10 bits/symbol), it is easy to does the rate in bits/slot: rT, = log (:)]/Q I h(p/Q). see that the minimum distance of the 1024-signal constellation is still 2, the same as the original constellation; thus If we constrain the rate in bits/slot to be at least 1/2 (56) still holds by replacing M by 1024. The resulting (that of QPPM) and the peak-to-average power to be at coding gain over QPPM is 0.969 dB, which is what was least 4 (that of QPPM), then for large Q it must be reported in [12] using simulations. At the same time the 1/4 2 p / Q 2 K1(1/2). Evidently, the largest gain is obachieved throughput is 5/8 bits/slot, a 25% increase over tained for p / Q = h-'(1/2) and is QPPM. Next we investigate the use of coding over MPPM G i -1Olog,, [4hP1(1/2)] =: 3.56 dB, (58) signals. It is clearly possible to use only a subset of the indicating that MPPM can provide some gain even with- MPPM signals for a given Q and p, for which the miniout coding, while satisfying the same constraints as QPPM. mum distance is greater than 2, at the expense of a The 3.56-dB gain is the maximum that can be obtained throughput reduction. In particular, we are interested in a without the use of further coding, and can only be achieved 3-dB gain over QPPM, by insisting that the minimum at very large (theoretically infinite) values of Q. Both a Hamming distance for the subset of MPPM symbols be at coding gain and a throughput gain can be achieved over least 4. Table V shows the results of a computer search QPPM by setting p / Q = 1/4 to satisfy the peak-to-aver- for such codes for different values of Q and p . The table age power constraint. In this case (again for large Q), gives the number of MPPM symbols whose distance is at G = 10log[2h(i)l = 2.10 dB, but the throughput is least 4, and the rate in bits/slot of a practical code h(1/4) = 0.8113 bits/slot, a 62% increase over QPPM. A obtained by deleting additional symbols (codewords) in throughput gain can be traded off for coding gain and vice order to obtain a number which is a power of 2. This versa through the use of coding. Fig. 6 plots the power deletion of symbols can be made intelligently in order to gain that is achieved with (uncoded) MPPM over QPPM facilitate, for example, synchronization and to relax the -j [ zyxwvutsrq zyxwvu 1324 zyxwvutsrqponm zyxwvutsrqponmlkj zyx IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 40, NO. 5, SEPTEMBER 1994 D’o D2 D4 D1 D7 D5 D6 D4 D2 D7 D1 D3 D4 D6 DO D5 D3 D1 - D2 DO D6 / D3 D5 D7 Fig. 7. Trellis for the 8-state code used with the 32 MPPM signals and the subsets resulting from set-partitioning: Q p = 2. = 9, zyxw zyxwvutsrqp strain on the laser. In the interest of space, we do not list the codewords for the codes listed in the table. As can be seen from the table, it is possible to obtain codes with a 3-dB gain over QPPM at rates of 1/2 bits/slot or better and for a = 4. The problem is that these are nonlinear codes and efficient techniques for decoding must be found before they can become practical. If the laser can support smaller values of a , then smaller codes can be designed with a 3-dB gain over QPPM and the same rate. For example, for a = 3.2, a code with a 3-dB gain over 256 MPPM symbols and rate 1/2 can be designed. Finally, for a = 2, a rate-9/16 code can be designed with Q = 16 that has a 3-dB gain over QPPM. For more powerful codes that can also be practically implemented, a concatenated coding scheme where a block or trellis code is used over a set of MPPM symbols (which can be thought of as the inner code) may be necessary. Such an approach was employed in C121 where a Reed-Solomon code was used in conjunction with 1024 MPPM symbols. Here, instead, we investigate briefly the use of trellis coding over MPPM symbols. For the example here, we use the 8-state Ungerboeck trellis shown in Fig. 7 [29]. With Q = 9, p = 2 MPPM, we have M = 36 signals, four of which can be deleted to yield a set of 32 modulation signals. Two of those deleted signals could be those having pulses at the first two and last two slots, to avoid the possibility of the laser being on over four consecutive slots. The other two signals to be deleted can be chosen based on other criteria, such as to help synchronization. The rate of the trellis code is 4/5, which when multiplied by 5/9, the rate in bits/slot of the MPPM signal set, gives an overall rate of 4/9 bits/slot. The peak-to-average power ratio is a = 4.5. The next step is to partition the 32 MPPM symbols into eight subsets of four signals each whose Hamming distance (since, as noted above, it is the Hamming distance that determines performance) is greater than the minimum of the 32-MPPM constellation. Since for p = 2 the two possible signal distances are 2 and 4, this means that the distance between signals in the same subset (which correspond to parallel transitions) must be 4. This further implies that the maximum free distance of the code is 4. Since signals leaving and entering a state have a Hamming distance of at least 2, this implies that the minimum free distance of the code is 4, which, combined with the above observation, means that the code has a free distance of 4. Thus, for the same energy per bit, the code has a 3-dB asymptotic coding gain over QPPM. This gain was verified through simulations, as shown in Fig. 5. Fig. 7 shows the partitioning (done by hand in this case) of the 32 MPPM signals into subsets DO, Dl;.., 0 7 . In describing the subsets, an MPPM signal is represented by p numbers ( p = 2 here) enclosed in parentheses, that indicate the bit positions where the pulses are located. Let us now briefly compare the complexities of two coding schemes both having a 3-dB coding gain over QPPM. One uses a small signal constellation combined with trellis coding and Viterbi decoding, and another that (9 MPPM for a given Q and p expels enough of the symbols to achieve a 3-dB coding gain. As the trellis-coding example above indicates, this can be done with 32 MPPM symbols and an 8-state Viterbi decoder at about the same rate as QPPM (slightly worse) and at a sightly better peak-to-average power a than the latter. The alternative code, obtained from Table V, achieves the 3-dB coding gain for the same throughput and a as QPPM with 4096 MPPM signals. Since there is as yet no efficient way to decode the 4096 signals, whereas the Viterbi algorithm can be used to decode the trellis code, it seems clear that trellis coding over comparatively smaller constellations, compared to block coding, offers the better complexity/performance trade-offs. This conclusion, although based on a single comparison, is expected to hold more generally true, unless an efficient way to decode large numbers of MPPM symbols is found. zyx zyxwvu GEORGHIADES: MODULATION AND CODING FOR THROUGHPUT-EFFICIENT OPTICAL SYSTEMS More work on the topic needs to be done to obtain high-rate codes (greater than 1/2) that provide good coding gains while satisfying the duty cycle and peak- and average-power constraints imposed by the laser. 1325 where A, and A, are the signal and noise intensities in photons/s, respectively, and T, is the slot duration. A maximum-likelihood (ML) receiver then performs Pr ( X = N I d k ) . Assuming Poisson statistics for the mad, observed counts, we have V. CONCLUSION We have studied various aspects of modulation and coding for high-rate optical links, by analyzing and comparing the performance of various modulation schemes under different criteria. No modulation scheme considered was seen to be uniformly superior to all others under all constraints and all parameter values. OPPM seems to perform better than MPPM in terms of throughput at small values of Q (Q = 2 is best) whereas MPPM's power is evident at the larger values of Q, that result in large signal constellations. With small Q values, OPPM requires large indices of overlap for large throughput, which will make synchronization more difficult in a practical implementation. OOK performed very well in most comparisons, especially at the higher rates. Its drawback is that it is not an equal energy signaling scheme (which means that an estimate of the received power is needed by the receiver before decisions are made), and it does not guarantee that long streams of zeros or ones will not occur. PPM was seen to be largely inferior to the other modulation schemes, especially at high rates, while satisfying all necessary constraints. All modulation schemes studied in this paper were obtained by imposing block constraints on binary sequences, i.e., sequences of these modulation symbols are subsets of the set of all possible OOK sequences. It is thus entirely possible (if not certain) that one can start with OOK and by more judiciously imposing constraints produce schemes that operate at higher rates than the modulation schemes studied here. zyxwvutsrq zyxwvutsrq zyxwvutsrqp i= 1 N,.! (60) where c is not a function of the data. Taking logarithms and dropping unnecessary terms, we obtain the desired log-likelihood function. zyxwvu APPENDIX B DERIVATION OF THE CHERNOFF BOUND We first derive a general bound for any binary direct-detection system and then apply it to MPPM. Let {Ao(t): 0 I t 5 T} and {A,(t): 0 I t I T } be the two intensities of the observed Poisson process A"= {Nr:0 I tI T } corresponding to each of the two transmitted symbols. Then using the log-likelihood function [30], the likelihood-ratio test for the symbol detection problem is (assuming equal prior probabilities) zyxwvutsrq zyxwvutsrqpo zyxwvu APPENDIX A OPTIMUM RECEIVER FOR MPPM Let 9 = {dk; k = 1,2;.., M } be the set of all binary sequences of length Q having weight (number of ones) equal to p . Clearly, there is a one-to-one correspondence between binary sequences in .9and MPPM signals, the position of ones in the binary sequence indicating the position of the pulsed slots in a (Q,p)-MPPM signal. Further, for the kth MPPM signal let w k = { w k l , w k 2 ; ~ ~ ,be w kthe P } set of p integers taking values in the set {I, 2;.., Q} indicating the position of the p ones. Finally, let X = (xl, X , ; . . , X Q )and N = (N,, N,;.., N,) be the random vector of photons detected in each of the Q slots, and a particular realization of it, respectively. We can show that for the rectangular pulses assumed here, slot photon counts constitute a suficient statistic for the symbol detection problem. The mean number of photons, A,, in the ith slot takes one of two values, depending on whether that slot is pulsed or not: = ( A , + &IT,, (h,T,, if slot is pulsed, otherwise, (59) 1326 zyxwvutsr zyxwvuts zyxwvutsrqpo zyxwv zyxwvu IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 40, NO. 5, SEPTEMBER 1994 The second equality above is due to the fact [30] that, given H,,, the unordered times t, when photons were detected are independent and identically distributed with the common distribution A&)//: A,(t) dt, and the last equality follows from the Poisson distribution of the total number of observed photons j%. The equation above was also derived in [15] using a different approach. Then the upper bound becomes I. Bar-David and G. Kaplan, “Information rates of photon limited overlapping pulse position modulation channels,” IEEE Trans. Inform. Theory, vol. IT-30, pp. 455-464, May 1984. C. N. Georghiades, “Some implications of TCM for optical directdetection channels,” IEEE Trans. Commun., vol. 37, no. 5, pp. 481-487, May 1989. C. N. Georghiades and J. K. Crutcher, “Throughput efficiency considerations for optical OPPM,” in Proc. Global Telecommun. Conf. (Phoenix, AZ),pp. 430-433, Nov. 1991. H. M. H. Shalaby, “Performance of uncoded overlapping PPM under communications constraints,” in Proc. Int. Conf. Commun. (Geneva, Switzerland), pp. 512-516, May 1993. H. Sugiyama and K. Nosu, “MPPM: A method of improving the band-utilization efficiency in optical PPM,” J . Lightwace Technol., vol. 7, no. 3, Mar. 1989. James M. Budinger, Mark Vanderaar, P. Wagner, and Steven Bibyk, “Combinatorial pulse position modulation for power-efficient free-space laser communications,” SPIE Proc., vol. 1866, Jan. 1993. T. M. Cover and J. A. Thomas, Elements of Information Theory. New York: Wiley, 1991. T. Ohtsuki, I. Sasase, and S. Mori, “Overlapping multi-pulse pulse position modulation in optical direct detection channel,” in Proc. Int. Conf. Commun. (Geneva, Switzerland), pp. 1123-1127, May 1993. A. D. Wyner, “Capacity and error exponent for the direct detection photon channel, Parts I and 11,” IEEE Trans. Inform. Theory, vol. 34, pp. 1449-1471, NOV.1988. M. H. Davis, “Capacity and cutoff-rate for Poisson-type channels,” IEEE Trans. Inform. neory, vol. IT-26, pp. 710-715, Nov. 1980. S. Shamai (Shitz), “On the capacity of a direct-detection photon channel with intertransition-constrained binary input,” IEEE Trans. Inform. Theory, vol. 37, pp. 1540-1550, Nov. 1991. T. Ohtsuki, H. Yashima, I. Sasase, and S. Mori, “Cutoff-rate and capacity of MPPM in noiseless photon counting channel,” IEICE Trans., vol. E 74, no. 12, Dec. 1991. J. R. Lesh, W. K. Marshall, and J. Katz, “Simple method for designing or analyzing an optical communication link,” in Proc. Military Commun. Conf. (PLACE), 1986. R. G. Gallagher, Information Theory and Reliable Communication. New York: Wiley, 1968. J. M. Wozencraft and R. S. Kennedy, “Modulation and demodulation for probabilistic coding,” IEEE Trans. Inform. Theory, vol. IT-12, pp. 291-297, July 1966. J. L. Massey, “Coding and modulation in digital communications,” in Proc. Int. Zurich Seminar Digital Commun. (Zurich, Switzerland), Mar. 1974. S. A. Butman, J. Katz, and J. R. Lesh, “Bandwidth limitations on noiseless optical channel capacity,” IEEE Trans. Commun., vol. COM-30, pp. 1262-1264, May 1982. D. L. Snyder and C. N. Georghiades, “Design of coding and modulation for power-efficient use of a band-limited optical channel,” IEEE Trans. Commun., vol. COM-31, pp. 560-565, Apr. 1983. R. J. McEliece, “Practical codes for photon communication,” IEEE Trans. Inform. Theory, vol. IT-27, pp. 393-398, July 1981. J. L. Massey, “Capacity, cutoff-rate and coding for a direct-detection optical channel,” IEEE Trans. Commun., vol. COM-29, pp. 1615-1621, NOV.1981. G. J. Pottie, “Trellis codes for the optical direct-detection channel,” IEEE Trans. Commun., vol. 39, pp. 1182-1183, Aug. 1991. G. Ungerboeck, “Channel coding with multilevel/phase signals,” IEEE Trans. Inform. Theory, vol. IT-28, pp. 55-67, Jan. 1982. G. Ungerboeck, “Trellis-coded modulation with redundant signal sets Part I: Introduction,” IEEE Commun. Mag., vol. 25, no. 2, Feb. 1987. D. L. Snyder and M. I. Miller, Random Processes in Time and Space. New York: Springer-Verlag, 1991. H. V. Poor, An Introduction to Signal Detection and Estimation. New York Springer-Verlag, 1988. zyxwvuts zyxwvutsrqponmlkjih zyxwvutsr which can be tightened by choosing 0 I s I 1 to minimize the right-hand side. The determination of the optimum s is in general analytically intractable. However, it can be shown that if the signals have equal energy and the intensities take only one of two values at any given time, the tightest bound is attained for s = 3 (in all other cases the bound is still valid for s = i, but it may not be the tightest possible). In this case, For MPPM, the intensities Ai(t), i = 0, 1 are binary-valued at any given time, taking values from the set {A,, (A, + As)}. In this case, it is eas;. to see that the general bound above reduces to (11). ACKNOWLEDGMENT Thanks are due James M. Budinger of the NASA Lewis Research Center for some interesting discussions, to Emina Soljanin for providing insight into the shaping gain that can be achieved for OOK, and to an anonymous reviewer who pointed out the bound in (26). REFERENCES [I1 J. R. Lesh, J. Katz, H. H. Tan, and D. Zwillinger, “2.5 bits/detected photon demonstration program: Description, analysis, and phase I results,’’ Jet Propulsion Laboratory, Pasadena, CA, TDA Rep. 42-66, pp. 115-132, Dec. 1981. 121 J. R. Pierce, “Optical channels: Practical limits with photon counting,” IEEE Trans. Commun., vol. COM-26, pp. 1819-1821, Dec. 1978. 131 D. L. Snyder and I. B. Rhodes, “Some implications of the cutoff rate criterion for coded direct-detection optical communications systems,” IEEE Trans. Inform. Theory, vol. IT-26, pp. 327-338, July 1981. 141 X. Sun and F. Davidson, “Direct-detection optical intersatellite link at 220 Mbps using AlGaAs laser diode and silicon APD with 4-ary PPM signaling,” NASA CR-186380, 1990. [51 J. M. Budinger, S. D. Kerslake, L. A. Nagy, M. J. Shalkhauser, N. J. Soni, M. A. Cauley, J. H. Mohamed, J. B. Stover, R. R. Romanofsky, P. J. Lizanich, and D. J. 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