where L(y) = P Y X (y 0) C PPM = log 2 (M) ( For n b 0 the expression reduces to D. Polar Coding (2)

Transcription

1 Polar-Coded Pulse Position odulation for the Poisson Channel Delcho Donev Institute for Communications Engineering Technical University of unich Georg Böcherer athematical and Algorithmic Sciences La Huawei Technologies France arxiv: v1 [csit] 7 Dec 018 Astract A polar-coded modulation scheme for deep-space optical communication is proposed The photon counting Poisson channel with pulse position modulation PP is considered We use the fact that PP is particularly well suited to e used with multilevel codes to design a polar-coded modulation scheme for the system in consideration The construction of polar codes for the Poisson channel ased on Gaussian approximation is demonstrated to e accurate The proposed scheme uses a cyclic redundancy check outer code and a successive cancellation decoder with list decoding and it is shown that it outperforms the competing schemes I INTRODUCTION This paper designs polar codes for the pulse position modulation PP Poisson channel This channel models a deep-space, direct-detection optical communications link with a photon counting detector [1] It has een shown in [] that for the deep-space Poisson channel, when operating at low signal to noise ratio SNR, PP is near optimal The comination of a inary error-correcting code with a higher-order modulation scheme can e done in several ways, eg, with it-interleaved coded modulation BIC [3], [4] or multilevel coding LC with multi-stage decoding LC-D [5] BIC uses it-metric decoding BD, which calculates it-wise log-likelihood ratios LLR, which are then processed independently However, the LLRs calculated from the same channel output are dependent [6, Sec II-D] This means that BIC cannot always approach the coded modulation C capacity Following [1], we plot in Fig 1 the PP capacity 5 and the achievale BD rate see 31 elow as a function of the average received power per slot in db for n = 0 where n is the average numer of noise photons at the receiver the channel model is explained in Sec II Note that, for a wide range of P av, the BD rate is significantly lower than the PP capacity This has motivated previous works to introduce BIC with iterative demapping BIC- ID, etween the decoder and the PP detector [1] In [1] oth convolutional codes CC and low-density parity-check LDPC codes were studied for the PP Poisson channel Another approach is to use non-inary error correction codes in comination with PP [7] In this paper, we use the fact that PP modulation calls for multilevel codes LC and this is naturally provided y polar-coded modulation PC [6], [8] We use polar codes Rate [its per slot] C PP =64 R BD =64 C PP =16 R BD =16 Op point for PC at BER = Fig 1 PP capacity 4 and BD rate 31 for n = 0 An operating point for the PC scheme at P av = 15 db and BER = is marked with a successive cancellation SC decoder with list decoding LS and a cyclic redundancy check CRC outer codes [9] We use the Gaussian approximation construction to design polar codes for higher-order modulation [6] We show that our proposed scheme improves the it-error rate BER as compared to the scheme proposed in [1] for the lock length 808 its, CRC size of 14 and a dynamic list size with at most entries An operating point of our scheme is shown in Fig 1, which shows that we are operating close to the PP capacity at BER = This paper is organized as follows PC for the PP Poisson channel is explained in Section II Section III descries the polar code construction method for the PP Poisson channel In Section IV the experimental results are presented and Section V concludes the paper A Channel odel II PP POISSON CHANNEL The transmission scheme is depicted in Fig The channel input is a vector X of length n where is the PP order and n is the numer of PP symols in one lock The vector

2 Ũ U 1, U,, U m mn-it vector Encoder C C 1, C,, C m mn-it vector -PP odulator X X 1, X,, X n Length n vector X i Y i Ỹ Poisson Channel Y 1, Y,, Y n Length n vector Used n times serially in N n 0 Fig Communication scheme Ỹ is the channel output We model the channel as memoryless, and we have n PỸ X ỹ x = P Y X y q x q 1 q=1 The channel P Y X y x is modeled as a slotted inary input, discrete output Poisson channel In a given time slot, there is either a pulse, represented y x = 1, or there is no pulse, represented y x = 0 Let n s e the average numer of received photons in a pulsed time slot when no noise is present and let n e the average numer of received noise photons per slot Then the conditional proailities to receive y N 0 = {0, 1,, } photons follow the Poisson distriutions [1]: P Y X y 0 = e n n y y! P Y X y 1 = e n+ns n + n s y 3 y! B Pulse Position odulation PP is a inary slotted modulation scheme where is the numer of slots [] To otain a PP symol, m = log its are input to the PP modulator Each symol consists of 1 slots without energy unpulsed slots and exactly one slot that contains energy pulsed slot [] We model the - PP modulator as an orthogonal inary code with rate m Each code word is of the form x d = 0, 0,, 1,, 0, 0 where the 1 is in the dth position, ie, x d = 1, x p = 0, for p d and p, d {1,,, } Therefore the modulation alphaet is X = {x 1,, x }, with cardinality X = The output alphaet is Y = N 0 with an infinite numer of elements C Achievale Rates The capacity of PP modulation on the Poisson channel is: C PP = 1 I X; Y [its/slot] 4 where I X; Y is the mutual information of the input and the output of the channel, the capacity-achieving input distriution is uniform, ie, P X x = 1 x X [], and P Y X is defined as in and 3 The capacity can e expressed as [1]: C PP = 1 E [ log L Y 1 p=1 L Y p ] [its/slot] 5 where Ly = P Y X y 0 P Y X y 1 is the likelihood ratio of the received value y, Y 1 is distriuted as P Y X 1 whereas Y p, for p 1, is distriuted as P Y X 0 For n > 0 the capacity is: C PP = log 1 E [ log p=1 1 + n ] Yp Y 1 s n [its/slot] 6 For n 0 the expression reduces to C PP = log 1 e n s D Polar Coding [its/slot] 7 A inary polar code of lock length n and dimension k is defined y n k frozen it positions and the polar transform F log n, which denotes the log n-fold Kronecker power of [ ] 1 0 F = Polar encoding can e represented y uf log n = c 9 where the n k frozen positions in u are set to predetermined values and where the unfrozen positions contain k information its The vector c is the code word [10] Successive cancellation SC decoding detects the its u 1 u u n successively, ie, the channel output y and the decisions û 1 û i are used to detect it u i+1 E PP apping and Demapping Fig 3 depicts the canonical polar-coded modulation PC [6], [8] instantiated for PP The encoder input is a vector ũ with length mn that contains the information its and the frozen its The vector ũ is split into m vectors u 1,, u m which are mapped to c i = u i F log n The encoding produces the code word c = c 1,, c m The PP mapper implements a lael function that maps the m its c 1i c mi to the ith transmitted PP symol x i for i {1,, n}, ie, for j {1,, m}, the output c j of the jth polar transformation is mapped to the jth it level of the input of the laeling function We define the PP lael as: i = i1 im := c 1i c mi, i {1,, n} 10 To refer to a lael at a generic time instance we drop the suscript i and write = 1 m [6]

3 u 11 u 1 u 1n u 1 u u n F log n F log n u m1 u m u mn F log n ỹ 11 1 x 1 x x n PP apper 1m 1 PP apper PP apper m Fig 3 Canonical PC [6], [8] PP Demap 1 PP Demap PP Demap m ĉ 1 ĉ ĉ m 1 L 1 L L m n1 n Polar Dec 1 Polar Dec Polar Dec m Fig 4 Receiver side of the PC scheme nm û 1 û û m c 11 c 1 c 1n c 1 c c n c m1 c m c mn The input x i to the PP Poisson channel is the output of the PP mapper defined as where f : {0, 1} m X = {0, 1} 11 d = 1 + x d 1 û j j 1 13 eg, for m = 3, x d010 = For a channel code with lock length mn, n -PP symols are transmitted The demapping and decoding procedure is depicted in Fig 4 The decoding schedule is as follows First demap the first it-level, then decode it Next demap the second it-level, then decode it At the jth step demap the jth itlevel, then decode it Continue until all m it-levels have een demapped and decoded The jth PP demapper uses the channel output Ỹ and the previously detected it-levels Ĉ j 1 = Ĉ1,, Ĉj 1 to calculate a soft-information L j = λ j Ỹ, Ĉ j 1 14 for the jth polar decoder which, in turn, produces a decision Ĉ j In particular, the calculation for each received symol y i at the jth level is L ji = λ j Y i, = log j 1 ˆB i P Y BjB j 1 Y i B ij = 0 P Y BjB j 1 Y i B ij = 1 j 1 ˆB i ˆB j 1 i 15 j 1 where ˆB i = ˆB i1 ˆB ij 1 Example 1: Consider the scenario with = 4 and the PP mapping x 1 = f00, x = f10, x 3 = f01, x 4 = f11 We calculate L 1 y at a generic time instant where y = y 1, y, y 3, y 4 By using Bayes rule and the equiproale distriution of all lael its, L 1 y is calculated as where n P Y B1 y 0 = e n y L 1 y = log P Y B 1 y 0 P Y B1 y n P Y B1 y 1 = e n y1 e n n y4 16 e n n y1 e ns+n n s + n y3 e n s+n n s + n y1 e n n y Similarly, for L y: e n n y3 e n n y e ns+n n s + n y4 e n s+n n s + n y e n n y4 L y = log Suppose ˆ 1 = 0 Then we have n P Y BB 1 y 00 = e n y P Y BB 1 y 0ˆ 1 18 P Y BB 1 y 1ˆ 1 19 e n n y3 e n n y4 e ns+n n s + n y1 0

4 P Y BB 1 y 10 = e n n y1 e n n y3 F Successive Cancellation List Decoding e n n y4 e ns+n n s + n y 1 We use successive cancellation list decoding SCL with list size L N [9] To improve performance, we use an outer CRC code for the information its We found experimentally that the 14-CRC code with polynomial 0x7cf and the 16- CRC code with polynomial 0xd175 are good choices [11] The PP demappers calculate soft information for each active path from the previous level At the end of the decoding process, the most likely path, among the L paths, that passes the CRC is selected as the decoder s decision The complexity of the decoder is OLmn log n [9] III POLAR CODE CONSTRUCTION The polar code construction encompasses first choosing a desired lock length n and a desired rate R = k n for the code and then finding the set of frozen its [10] There are several ways to construct the codes: By onte Carlo simulation By using the Binary Erasure Channel BEC as a surrogate channel By Gaussian approximation The construction of a polar code via onte Carlo C simulations is descried in [10] The C construction method relies on extensive simulations in order to find the est it-channels Polar codes can also e constructed y using the BEC as a surrogate channel, ie y replacing the Poisson channel y a BEC channel with the same capacity Then, y ordering the capacities for each it-channel the "good" channels are found [10] In [1] it is shown that the construction of polar codes can e done efficiently y using the Gaussian approximation GA construction method A Construction of Polar Codes via iawgn Surrogate Channel 1 iawgn Surrogate Channel [6]: The channel is defined as Y = x + σz where x 0 = 1 and x 1 = 1 and Z is zero mean Gaussian noise We define R iawgn σ = I B; x B + σz 3 where B is uniformly distriuted on {0, 1} Gaussian Approximation GA: The GA construction method for polar codes was proposed in [1] The reliaility of the it U i, i {0,, n}, can e quantified y the mutual information I U i ; Y n U i 1 i We can calculate these Is for all i {1,, n} y recursively calculating the Is of the I = I U 1 ; Y 1 Y I + = I U ; Y 1 Y U 1 CER I 1 = I B 1 ; Y 1 Fig 5 Is of the asic polar transform [6] I = I B ; Y I-DGA Const C Const I-DBEC Const Fig 6 Comparison of C, I-DGA and I-BEC construction methods The lock length is 6144, the code rate is 1/ and n = 0 asic polar transform shown in Fig 5 For the iawgn channel the update rule for the asic polar transform is [13] I = 1 J [J 1 1 I 1 ] + [J 1 1 I ] I + = J [J 1 I 1 ] + [J 1 I ] where the J function is J σ = 1 e ξ σ σ 4 5 πσ log 1 + e ξ dξ 6 To calculate the J function and its inverse J 1, we use the approximation H3 J σ 1 H1σH 7 J 1 I 1 1H1 log 1 I 1 H H 3 8 from [14, Eqs 9, 10] where H 1 = 03073, H = and H 3 = I Demapper Gaussian Approximation Construction: For the proposed scheme, we use the construction approach from [6] called I demapper GA I-DGA The idea is to first characterize the it-channels at the output of the PP demappers y mutual information expressions that take into

5 account the dependence of the soft information produced y the PP demappers on all the previous detected its Then we replace the PP demapper it-channels y iawgn surrogate channels with the corresponding I and use the GA construction with the J function defined aove Using the chain rule for mutual information we have IX; Y = IB; Y = IB j ; Y B j 1 = IB 1 ; Y + IB ; Y B IB m ; Y B 1 B m 1 9 Oserve that the jth it-level has the I IB j ; Y B 1 B j 1 Therefore, the I-DGA costruction calculates IB j ; Y B 1 B i 1 for all j {1,,, m} and connects L j with B j y a iawgn channel with noise variance [6] σ j : R iawgn σ j = I Bj ; Y B j 1 30 Then, we use the GA to find the most reliale its in ũ To construct a length mn and rate R = k/n polar code, find the set of nm km most unreliale its in ũ and freeze them Remark 1: Oserve that in 9 on each it-level the conditional I is an achievale rate [15] and the sum-rate of all the it-levels is exactly the capacity of the PP Therefore C PP is an achievale rate with our scheme Note that if in 9 we disregard the conditioning on the previous its of the symol, then we calculate the BD rate [16, Eq 10] IB j ; Y B j 1 IB j ; Y = R BD 31 Fig 1 plots R BD vs P av B Construction of Polar Codes for the PP Poisson Channel via the BEC Surrogate Channel We introduce a I demapper BEC I-DBEC construction for polar codes on the PP Poisson channel We follow the same idea as for the I-DGA Therefore, the I- DBEC construction calculates IB j ; Y B 1 B i 1 for all j {1,,, m} and connects L j with B j y a BEC channel with erasure proaility ɛ j = 1 I B j ; Y B j 1 3 We then use the construction for the BEC from [10] C Comparison of Polar Code Constructions To verify that the results in [6] extend to the PP Poisson channel, we compare the performance of a polar code constructed via the C approach this code is always going to e good with the performance of a polar code otained via I-DGA and I-DBEC Fig 6 plots the codeword error rate CER performance curves for a polar code designed with C simulations, for a polar code designed via I-DBEC, and for a polar code designed with I-DGA The lock length is n = 6144 and the rate is R = 1/ The average numer BER capacity SCPP PC Fig 7 Comparison of the proposed PC scheme with the SCPP scheme in [1] Both codes have a lock length of nm = 808 its and the rate is R = 1/ The maximal allowed list size for the PC code is L = and 14-CRC is used CER 10 0 n = 000 n = 0 10 NB-LDPC PC I-DGA PC I-DBEC n = Fig 8 Comparison of the proposed PC scheme with oth I-DGA and I-DBEC constructions with the NB-LDPC scheme in [7] All codes have a lock length of nm = 808 its, the rate is R = 1/, = 64 and n {000, 0, } The maximal allowed list size for the PC code is L = and 16-CRC is used of noise photons per slot is n = 0 The C, I-DGA and I-DBEC curves virtually coincide We use the I-DGA construction to design the codes However, the I-DBEC construction could also e used IV NUERICAL RESULTS The PC scheme descried aove was implemented and the results are compared with the est scheme in [1] where the rate loss entailed y BD is mitigated y introducing iterations etween the outer decoder and the PP demapper By doing so, a BD-ID scheme is realized where the outer code is a convolutional code serially concatenated with the

6 Tale I DISTRIBUTION OF THE LIST SIZE USED IN SC LIST DECODING L n s PP demapper which is modified y emedding in it a inary accumulator This scheme is referred to as SCPP Fig 7 depicts the simulation results The SCPP code proposed in [1] has a lock length of 808 its and rate R = 1/ orange curve in Fig 7 The PC scheme proposed in this paper red curve in Fig 7 has a lock length of nm = 808 its, with k = 4104 information its An outer 14-CRC code is concatenated to the information its, thus making the polar coding rate R = /808, and the overall rate R = 1/ A dynamic list decoder is used with a maximal allowed list size of L = We define the notion of dynamic list size as follows: Start decoding with list size L = 3 If none of the candidates passes the CRC test, the list size is douled and the decoding is started once again Keep douling the list size either until a code word that passes the CRC valid code word is found or until the limit of is exceeded The performance simulation for oth schemes is done with 64- PP and the ackground noise is n = 0 The parameters are chosen as suitale for a ars-earth downlink [1] The proposed scheme achieves a etter performance than the est code proposed in [1] For the particular parameters, the noninary LDPC NB-LDPC scheme from [7] does not show any improvement over the SCPP scheme, therefore we compare our results only with the SCPP scheme For all of the considered average powers and for all of the simulations presented in this work, the stopping criterion for the simulation was to collect 50 erroneous frames Tale I shows the distriution of the list size needed to find a valid code word For example, at an average power of 149 db, 46 code words were decoded using a list of size L = 3 and = 18 code words needed a list size of L = to e decoded 50 code words were decoded erroneously even at the maximal list size For a low average power, the decoder usually resorts to the maximal list size to find a valid code word and even then very few code words are decoded correctly However, as the power increases, the decoder can find a valid code word with smaller list sizes and at an average power of 147 db a large fraction of code words are decoded with a list size of just 3 However, even though most of the code words are decoded with a list BER L = 3 L = 51 L = 819 Fig 9 Dependence etween the list size and the BER size of 3, having a igger maximal availale list size than will improve the scheme Fig 8 plots the CER of the PC scheme for different choices of n and compares the results with the NB-LDPC scheme from [7] The NB-LDPC scheme performs really well when n is quite small, thus making the Poisson channel ehave like an erasure channel [7] uses EXIT analysis on the erasure channel to construct the NB-LDPC codes However, as n increases the PC scheme starts to gain in performance and, in particular, when n = there is aout 01 db gain in P av with respect to the NB-LDPC scheme A The Impact of the List Size Fig 9 shows that as the list size increases, the performance of the code ecomes etter The codes in Fig 9 have a lock length of 808 its, a 16-CRC outer code with polynomial 0x8d95 [11] and n = 0 Bearing in mind the application of the proposed scheme, the decoding can e done offline, therefore a ig list size is not a major prolem V CONCLUSION In this paper, polar coded modulation PC was applied to the PP Poisson channel The interplay etween the encoder

7 and the modulator was examined and the results in [6] were extended to the PP Poisson channel The results show a slight gain in performance with respect to the state of the art transmission scheme proposed in [1] Additionally, we demonstrated that the existing approaches of designing polar codes for the AWGN channel via iawgn channel surrogates [6] can e extended to design polar codes on the PP Poisson channel We oserved that the list size of the dynamic successive cancellation list decoder has an impact on the performance of the scheme Additionally, the polar code construction for list decoding is an interesting direction for further research [17], [18] as it may lead to improved results for the PC scheme ACKNOWLEDGENT The authors would like to thank Dr Gianluigi Liva for his helpful comments REFERENCES [1] F Barsoum, B E oision, P Fitz, D Divsalar, and J Hamkins, EXIT function aided design of iteratively decodale codes for the poisson PP channel, IEEE Trans Commun, vol 58, no 1, pp , Dec 010 [] H Hemmati, Ed, Deep Space Optical Communications New York, NY: John Wiley & Sons, Inc, 006 [3] E Zehavi, 8-PSK trellis codes for a Rayleigh channel, IEEE Trans Commun, vol 40, no 5, pp , ay 199 [4] G Caire, G Taricco, and E Biglieri, Bit-interleaved coded modulation, IEEE Trans Inf Theory, vol 44, no 3, pp , ay 1998 [5] H Imai and S Hirakawa, A new multilevel coding method using errorcorrecting codes, IEEE Trans Inf Theory, vol 3, no 3, pp , ay 1977 [6] G Böcherer, T Prinz, P Yuan, and F Steiner, Efficient polar code construction for higher-order modulation, IEEE Wireless Commun Netw Conf WCNC, arch 017 [7] B atuz, E Paolini, F Zaini, and G Liva, Non-inary LDPC code design for the Poisson PP channel, IEEE Trans Commun, vol 65, no 11, pp , Nov 017 [8] Seidl, A Schenk, C Stierstorfer, and J B Huer, Polar-coded modulation, IEEE Trans Commun, vol 61, no 10, pp , Oct 013 [9] I Tal and A Vardy, List decoding of polar codes, IEEE Trans Inf Theory, vol 61, no 5, pp 13 6, ay 015 [10] E Arikan, Channel polarization: A method for constructing capacityachieving codes for symmetric inary-input memoryless channels, IEEE Trans Inf Theory, vol 55, no 7, pp , July 009 [11] P Koopman Best CRC Polynomials [Online] Availale: https: //usersececmuedu/~koopman/crc/ [1] P Trifonov, Efficient design and decoding of polar codes, IEEE Trans Commun, vol 60, no 11, pp 31 37, Nov 01 [13] S ten Brink, G Kramer, and A Ashikhmin, Design of low-density parity-check codes for modulation and detection, IEEE Trans Commun, vol 5, no 4, pp , April 004 [14] F Brännström, L K Rasmussen, and A J Grant, Convergence analysis and optimal scheduling for multiple concatenated codes, IEEE Trans Inf Theory, vol 51, no 9, pp , Sep 005 [15] R G Gallager, Information theory and reliale communication John Wiley & Sons, Inc, 1968 [16] A artinez, A Guillén i Fàregas, G Caire, and F J Willems, Bit-interleaved coded modulation revisited: A mismatched decoding perspective, IEEE Trans Inf Theory, vol 55, no 6, pp , June 009 [17] V Bioglio, F Gary, I Land, and J Belfiore, inimum-distance ased construction of multi-kernel polar codes, arxiv preprint, 017 [Online] Availale: [18] P Yuan, T Prinz, and G Böcherer, Polar code construction for list decoding, arxiv preprint, 017 [Online] Availale: http: //arxivorg/as/