Design of a Coded Modulation for Deep Space Optical Communications

Transcription

1 Deign of a Coded Modulation for Deep Space Optical Communication Bruce Moiion, Jon Hamkin and Michael Cheng Jet Propulion Laboratory, California Intitute of Technology, Paadena, CA 99 USA {bmoiion,hamkin,mkcheng}@jpl.naa.gov Abtract We how how to deign a modulation and error correcting code (ECC) to achieve near-capacity performance on a deep pace optical link. We begiy comparing the capacity of variou modulation, given the power and bandwidth contraint of the channel and how that pule poition modulation (PPM) i near-optimal for a nominal miion cenario to Mar. We preent a coded PPM olution compried of the erial concatenation of a convolutional code, interleaver, accumulator, and PPM. Thi code i amenable to a high-peed, low-complexity hardware implementation and ha performance with about db of capacity and everal db better than competing coded PPM deign. I. INTRODUCTION All of NASA previou deep pace miion have relied upon Radio Frequency (RF) communication for their telemetry link. Such link, particularly at S-band and X- band, are well undertood, with operational flight experience panning everal decade. The additive white Gauian noie channel model for thee RF link i both accurate and one of the implet to analyze, providing a happy convergence of academic reearch and practical communication link deign. Theoretical breakthrough in ECC deign regularly make their way onto deep pace miion. The mot common ignal format ued today by deep pace miion i binary phaehift keying (BPSK) with either a concatenated Reed-Solomon (RS) and convolutional code, or a turbo code. Low denity parity-check code are being tandardized for pace miion a well. RF communication, however, will not alway meet NASA need. Scientit and miion planner continue to demand ever increaing data rate from deep pace, to upport hyperpectral imager, high-reolution tereo camera, and other data-intenive intrument. Optical communication i attractive for thee emerging miion cenario becaue of advantage in tranmitter ma, beam divergence, and bandwidth, compared to RF communication. Thi ha led to the development of the Mar Laer Communication Demontration (MLCD) project, cheduled to launch in 9 and achieve data rate up to 5 Mbp from Mar uing a 5 W laer. NASA trategic redirection ha led to the cancellation of MLCD, but before thi unfortunate new, technique were developed to deign a near-capacity ignal format for deep-pace optical link. Thi paper ummarize the deign proce. The paper i organized a follow. In ection II, we dicu the channel model, power and bandwidth contraint, and how how to elect a modulation. In Section III, we compare RS, turbo, erially concatenated convolutional, and low-denity parity-check code clae. In IV, we dicu the deign of the erially concatenated code, including component code deign, block length, interleaver, and bit-to-ymbol mapping. II. MODULATION In an optical deep pace link, the received ignal, a Poion point proce, i focued by a telecope onto a photodetector that produce a pule in repone to each incident photon. For our purpoe, we aume thi impule tream i efficiently mapped back to the incident Poion proce, o that the entire channel (tranmitter, free-pace channel, photodetector, receiver) i modeled a Poion. We firt addre the choice of modulation for thi channel. The modulation mut atify the following phyical contraint: a maximum average ignal photon tranmiion rate of P av, a maximum peak ignal photon tranmiion rate of P pk, and a minimum pulewidth contraint of T p. Wyner[] howed that under peak and average power contraint there i negligible lo in capacity by retricting the tranmitted ignal to binary value over dicrete time increment (lot). Hence, we retrict ourelve to binary, lotted modulation. Letting be the mean noie photon in a lot and n the mean ignal photon in a puled lot, the channel then reduce to a binary-input, dicrete-output Poion channel, with ignal and noie lot ma function p (y) = (n + ) y e (n + ) y! p (y) = () y e () y! Letting /M be the probability of a pule tranmiion, or duty cycle, we may retate the peak and average power contraint a n P pk n M P av Parameterized by the duty cycle, no modulation for thi channel can exceed the capacity of on-off-keying C OOK = M E Y log p (Y ) p(y ) + M M E Y log p (Y ) bit/lot () p(y )

2 where Y ha ditribution p, Y ha ditribution p, and p(y) = p (y)/m + p (y)(m )/M. It i well known that M-ary pule-poition-modulation (PPM), in which log (M) bit map to an M-lot ymbol with a ingle pule in one lot, with capacity [ ] C = M E ML(Y ) Y,...,Y M log M j= L(Y bit/lot j) where L(y) = p (y)/p (y), Y i ditributed a p and Y j, j i ditributed a p, achieve capacity cloe to that of () at low duty cycle, e.g., [], [3]. Current link budget for deeppace link how that the optimum duty cycle i le than /3 for the entirety of the (in orbit) miion [4], [5]. For thee duty cycle, we ee a negligible lo in chooing PPM. Only when the optimum duty cycle i large, ay greater than /6, doe PPM begin to uffer ignificant loe relative to OOK [6]. In thi region, alternative to PPM hould be purued, e.g., [7]. However, thi occur only for high average power, which are not typical for current deep pace miion. A. Capacity of PPM We decribe a few propertie of the capacity of PPM that will be relevant to the deign. Expanding C in a Taylor erie we have [5] ( ) M n C = + O(n 3 M ln n ) bit/lot b Hence, for mall n, the capacity goe a n /. To ee the behavior a a function of lotwidth T, write the capacity a a function of the incident ignal and noie rate n λ = MT photon/ec λ b = T photon/ec and convert unit to bit/ec yielding ( ) C M(M ) λ T ln bit/ec. Hence the capacity i invariant to the lot width for mall n. Thi i illutrated in Fig., where λ b i held contant and the capacity i plotted a a function of λ with lotwidth varying by power of. λ b Finally, the optimum PPM order in the preence of peak and average power contraint i alway le than or equal to the peak to average power contraint ratio [5]. Hence the peak power contraint tranlate into an order contraint. For the laer in conideration for MLCD, the impoed order contraint wa M 64. B. Partitioning the received ignal power/noie power pace The throughput of the coded modulation i givey Throughput = log M M R bit/ec T where R i the rate of the error correction code (ECC). The received ignal and noie power allow capacitie that range from. Mbp to over 5 Mbp. A ytem deigner can vary M, R and T, to tile thi range of capacitie. How hould one accomplih thi with limited complexity? Each parameter ha phyical contraint and complexity cot. A noted above, we hould chooe M P pk /P av. We are alo bounded by T T p. Finally, the required peed of the hardware for a fixed uer data rate goe a /R. The capacity increae with the invere of the lot width. Hence, an optimum olution i to fix T = T p, chooe the optimum M a a function of λ for the given, and elect a code rate of R = CM/(log (M)T ) (auming a cloeto-capacity achieving code). However, thi would require a wide range of change to the code rate, which i prohibitively complex. Noting that the capacity i roughly invariant to the lotwidth, one can vary the throughput by varying the lotwidth and PPM order, while keeping the rate fixed. A rate of / wa choen to tradeoff power and bandwidth efficiency. Due to fluctuation of the received ignal and noie power and error in the etimation of thee parameter, it would be impractical to have partition of the ignal and noie power pace to a reolution greater than a few tenth of a db. Figure illutrate partition choen for MLCD, optimized under the contraint M 64 and T a multiple of.6 n, and overlaid with a repreentative range of received ignal power. Incident ignal and noie power are caled by photodetector/receiver loe and a ytem margin, o that photon rate at the input to the decoder are λ 8.3 db and λ b 5.3 db. 5 C ( bit/ec ) T = M = 64 =., T =. =., T =. =., T =. =, T = λ b (db photon/n) λ (db photon/n) Fig.. n MT photon/econd Illutrating invariance of PPM capacity to lotwidth Fig.. Incident ignal and noie power and partitioning of the pace. Partition labeled with achievable throughput in Mbp

3 III. CODE SELECTION We now addre the election of an appropriate ECC. We conider four clae of code: Reed-Solomon, parallel concatenated convolutional, erially concatenated convolutional, and low-denity parity-check. A. Serially concatenated convolutional code We were able to obtain the bet performance by treating PPM a an inner code in a erial concatenation with a hortcontraint length outer convolutional code, a tructure illutrated in Fig. 3 (the cyclic-redundancy-check (CRC) i ued for error detection and a a topping rule). Maey [8] firt invetigated the erial concatenation of a convolutional code with PPM with non-iterative decoding. Iteratively decoding the ame concatenation, a done in [9] for convolutional code, provide remarkably good performance a gain of 4.8 db over non-iterative decoding, within le than db from capacity over the wide range of operating point for the deep-pace Mar link with moderate encoding and decoding complexity. We denote thi combination a SCPPM, and will go into detail on the deign of the component in the following ection. Figure 4 illutrate performance of SCPPM and everal conidered alternative for a nominal operating point. We briefly dicu other conidered alternative in the following ection. uer data CRC Fig. 3. B. Reed-Solomon code u SCPPM outer code conv. x a encoder bit-interleaver +D inner code PPM APPM c optical channel Serially concatenated convolutionally coded PPM Reed-Solomon code were firt propoed for ue on the PPM channel by chooing RS ymbol to correpond to PPM ymbol []. The convention of mapping code ymbol to PPM ymbol cae extended by grouping multiple PPM ymbol to form a ingle RS ymbol [], yielding ome improvement in performance. At nominal background rate, RS coded PPM uffer a 3 db lo relative to oft-deciioncode (Thi i for hard-deciion decoding ome of thi lo would be recovered by oft-deciion decoding). C. Parallel concatenated convolutional code With the advent of turbo-code [], everal author invetigated the application of parallel-concatenated-convolutional code to the PPM channel. In [3], [4], the modulation wa not iteratively decoded. In [5], which ue PPM on a dicretetime memoryle Rayleigh-fading channel where a mall duty cycle i alo optimal, PPM i iteratively decoded, providing performance cloe to capacity. However, we find that we have the ame or better performance with lower complexity by uing a lower complexity outer code. y Error Rate 4 Capacity LDPC-PPM SCPPM n /(MT ) (db photon/n) Fig. 4. SCPPM, LDPC-PPM and RS-PPM performance, M = 64, =., T = 3 n D. Low-denity parity-check code Low-denity parity-check (LDPC) code have proven to have performance comparable or better than turbo-like code on the Gauian channel while having a decoder that i more amenable to high-peed implementation. We carried out ome deign [6], uing the approach in [7] to derive a good degree ditribution, and the approach in [8] to implement the degree ditribution a a protograph. We demontrated performance about.5 db wore than that for the SCPPM code. We expect that with additional effort in deign, LDPCcoded PPM performance will match or exceed that of SCPPM. However, for the deep pace channel, the onu on complexity lie with the pace-baed encoder more o than with the ingle, Earth-baed, decoder, and an LDPC encoder may have higher complexity than the SCPPM encoder. At the time of election, performance, maturity of technology, and complexity factor favored the SCPPM code. IV. SCPPM CODE DESIGN In the following ection we decribe the deign of the SCPPM code: component code, block length, interleaver, and bit-to-ymbol mapping. A. Component code election We utilize EXIT chart analyi [9] to elect inner and outer code. We firt examine the chart for PPM. Let L(A), L e (A) be the apriori and extrinic log-likelihood of A and A, X be defined by Fig. 3. We aume that L e (A) i Gauian and ymmetric, i.e, l = log(f Le (l)/f Le ( l)). The initial mutual information i givey ( ( M )) i= I(A; L e (A)) = E log ( + n / ) Yi M/ i= ( + n / ) Yi where Y i ditributed a p and Y j, j =,..., M are ditributed a p. In the limit of large mean L(A), the mean extrinic output of the inner code will approach E[L e (A)] n log E[L(A)] RS-PPM ( + n )

4 and the mutual information approache ( [log I(A; L e (A)) E E[L(X)] + where Y i ditributed a p and Y a p. Hence, the information we can extract from demodulating PPM aturate at a contant independent of the order. The relative gain from demodulation increae with the PPM order, with no gain when M =. By preceding the PPM mapping with a recurive /(+D) binary accumulator, a combination referred to a AP P M, we prevent the aturation contraint. Fig. 5 illutrate EXIT chart for M = 64-ary PPM and APPM with =. and n =.. The figure illutrate the well-known tradeoff of adding a recurive element to the inner code a tranfer function that goe to one at the cot of a lightly lower threhold. Contraining the election of outer code to rate / convolutional code, we found the contraint length 3, (5, 7) (octal tap), convolutional code to be a good fit. The joint EXIT chart with a imulated tranfer function for thi combination i illutrated in Fig. 6. I(A; Le(A)) PPM 64-APPM ( + n ) )] error rate i givey (Y Y ) P w = k c C n/ log (M) h= P [e h ] () where n i the length of x and P [e h ] i the probability of a ditance h event given c wa tranmitted. The pairwie union- Bhattacharrya bound may be tightened below the critical ignal power uing a imilar approach to that ued for the Gauian channel, e.g., []. Let R be the region R = {y : y, x ɛ}, with y, x the inner product, and expand P [e h ] a P [e h ] = P [e h, y / R] + P [e h, y R] The firt term i givey P [y / R] + P [e h, y R] (3) P [y / R] = F S (ɛ ) where F S i a cumulative ditribution and S i Poion with mean λ S = (n/ log (M))(n + ). The econd term may be bounded via an application of the unioound P [e h, y R] P [ɛ U T N] (4) ĉ C h (c) where U i the inner product of y and the pule poition common to x and ˆx o that U, T, N are independent Poion with mean λ T = h (n + ) λ U = λ S λ T I(A; L(A)) Fig. 5. EXIT chart for PPM and APPM, M = 64, n =, =. I(X; L(X)), I(A; Le(A)) Simulated tranfer (5, 7) 64-APPM Fig. 6. I(X;L e (X)), I(A; L(A)) EXIT chart for SCPPM B. Block Length Error Rate Bound Let C be the collection of SCPPM codeword c and C h (c) the collection of codeword at ditance h from c. The word BER contour λ N = h Note that (4) i a function only of h and invariant to c. Hence, although PPM i nonlinear, we may drop the dependence on c and chooe a convenient value to implify analyi. Combining term yield the bound P w n/m h= F S (ɛ ) + ĉ C h (c) P [ɛ U T N] (5) In order to find the optimum ɛ to minimize (3), we retrict the earch to integer ɛ and expand the econd term a n P [ɛ U T N] = f U (u)f N (n)f T (t) u= n=ɛ u t=ɛ u =P [T N]P [U ɛ]+ ɛ n f U (u)f N (n)f T (t) u= n=ɛ u t=ɛ u which lend itelf to efficient numerical computatioy precomputing certain term. Fig. 7 illutrate union-bhattacharrya bound, imulation reult and expurgated verion of (5) [] for the SCPPM code with blocklength of 56, 6 and 5 bit. We ee that block length greater than Kbit have an error floor below 8, with negligible impact for our purpoe.

5 A interleaver length of 5 = wa choen for MLCD a a good tradeoff between threhold and complexity. Thi value i diviible by log M for a large range of M, allowing eay integration with variou PPM order. Pw capacity imulation Π = 5 Π = n /(MT ) (db photon/n) extended bound union-bhatt. bound Π = 6 Fig. 7. Word error rate: imulation, capacity, union-bhattacharyya bound and extended bound, M = 64, =. C. Interleaver Good performance ha been demontrated for long blocklength code with peudo-random pread interleaver. However, peudo-random function may be complex to implement in hardware. In [], quadratic permutation polynomial are propoed for ue a an interleaver and we adopted a permutation polynomial for SCPPM. We oberved no lo in performance relative to pread-interleaver with the polynomial interleaver f(x) = x + x, with invere f (x) = 733x + 777x, which i conveniently alo quadratic. D. Bit-to-ymbol Mapping On a truly memoryle channel, the bit-to-ppm-ymbol mapping ha no impact on performance. However, in practice, finite detector pulewidth and timing jitter may add appreciable inter-lot-interference (ISI). If the detector/receiver pulewidth i on the order of a lotwidth, the effect of ISI will be large. We found the bet performance in the preence of ISI with an anti-gray mapping anti-gray in the ene that we map pair of log M bit at the input to the accumulator with ditance log (M) to PPM ymbol with pule in adjacent lot thoe ymbol cloet in ditance in the preence of ISI. ACKNOWLEDGMENT The deign of the coded modulation for MLCD wa done in collaboration with Richard Barron and Don Boroon of Lincoln Laboratorie. The reearch decribed in thi publication wa carried out at the Jet Propulion Laboratory, California Intitute of Technology, under a contract with the National Aeronautic and Space Adminitration. REFERENCES [] A. D. Wyner, Capacity and error exponent for the direct detection photon channel Part I, IEEE Tranaction on Information Theory, vol. 34, pp , Nov [] J. R. Pierce, Optical channel: Practical limit with photon counting, IEEE Tranaction on Communication, vol. COM-6, pp. 89 8, Dec [3] R. G. Lipe, Pule-poition-modulation coding a near-optimum utilization of photon counting channel with bandwidth and power contraint, DSN Progre Report, vol. 4, pp. 8 3, Apr. 98. [4] A. Biwa and S. Piazzolla, Deep-pace optical communication downlink budget: Mar ytem parameter, IPN Progre Report, July 3. to appear. [5] B. Moiion and J. Hamkin, Deep-pace optical communication downlink budget: Modulation and coding, in IPN Progre Report, vol. 4-54, Aug. 3. [6] J. Hamkin and B. Moiion, Multipule pule poition modulation on dicrete memoryle channel. ubmitted to IEEE Tranaction on Communication, Nov. 5. [7] R. Barron, Binary haping for low duty-cycle communication, in Proceeding of IEEE International Sympoium on Information Theory, p. 53, 4. [8] J. L. Maey, Capacity, cutoff rate, and coding for a direct-dectection optical channel, IEEE Tranaction on Communication, vol. COM-9, pp. 65 6, Nov. 98. [9] S. Benedetto, D. Divalar, G. Montori, and F. Pollara, Serial concatenation of interleaved code: Performance analyi, deign and iterative decoding, Telecommunication and Data Acquiition Progre Report, vol. 4-6, Aug [] R. J. McEliece, Practical code for photon communication, IEEE Tranaction on Information Theory, vol. IT-7, pp , July 98. [] J. Hamkin and B. Moiion, Performance of long blocklength Reed- Solomon code with low-order pule poition modulation. ubmitted to IEEE Tranaction on Communication, Nov. 5. [] C. Berrou and A. Glavieux, Near optimum error correcting coding and decoding: Turbo-code, IEEE Tranaction on Communication, vol. 44, pp. 76 7, Oct [3] K. Kiaaleh, Turbo-coded optical PPM communication ytem, Journal of Lightwave Technology, vol. 6, pp. 8 6, Jan [4] J. Hamkin, Performance of binary turbo coded 56-PPM, TMO Proge Report, vol. 4, pp. 5, Aug [5] M. Peleg and S. Shamai, Efficient communication over the dicretetime memoryle Rayleigh fading channel with turbo coding/decoding, European Tranaction on Telecommunication, vol., pp , September-October. [6] M. Baroum and B. Moiion, An LDPC code for optical pace communication, tech. rep., Jet Propulion Laboratory, Oct. 5. Inter- Office-memorandum. [7] S. ten Brink, G. Kramer, and A. Ahikhmin, Deign of low-denity parity-check code for modulation and detection, IEEE Tranaction on Communication, vol. 5, pp , April 4. [8] J. Thorpe, K. Andrew, and S. Dolinar, Methodologie for deigning LDPC code uing protograph and circulant, in iit, p. 36, 4. [9] S. Brink, Convergence of iterative decoding, Electronic Letter, vol. 35, pp , May 999. [] D. Divalar and E. Biglieri, Upper bound to error probabilitie of coded ytem beyond the cutoff rate, IEEE Tranaction on Communication, vol. 5, pp. 8, December 3. [] B. Moiion and J. Hamkin, Error rate bound for coded PPM on the Poion channel. ubmitted to IEEE Tranaction on Communication, Nov 5. [] J. Sun and O. Y. Takehita, Interleaver for turbo code uing permutation polynomial over integer ring, IEEE Tranaction on Information Theory, vol. 5, pp. 9, Jan. 5.