Asynchronous Multiple Access in Optical Wireless Scattering Communication: Achievable Transmission Rates and Receiver Design

1 Multi-layer Superimposed T ransmission with Symbol Boundary O ff set for Optical W ireless Scattering Communication Guanchu W ang, Chen Gong, Zhimeng Jiang and Zhengyuan Xu Abstract W e in vestigate the multi-layer superimposed transmission for optical wireless scattering communication where the symbol boundaries on di ff erent signal layers are not necessarily aligned in the time domain. W e characterize the multi-layer transmission based on a hidden markov model. Then, we obtain the achie vable rates of all signal layers and a single layer, and provide a numerical solution. Furthermore, we propose approaches on the channel estimation as well as joint symbol detection and decoding. Finally , both simulations and experiments are conducted to ev aluate the performance of the proposed approaches, and validate the feasibility of the proposed transmission and signal detection approaches. Index T erms NLOS scattering communication, superimposed transmission, hidden markov model, achiev able rate, joint detection and decoding. This work was supported by K ey Program of National Natural Science Foundation of China (Grant No. 61631018) and Ke y Research Program of Frontier Sciences of CAS (Grant No. QYZD Y -SSW -JSC003). The authors are with Ke y Laboratory of Wireless-Optical Communications, Chinese Academy of Sciences, School of Information Science and T echnology , Univ ersity of Science and T echnology of China, Hefei, China. Email: { hegsns, zhimengj } @mail.ustc.edu.cn, { cgong821, xuzy } @ustc.edu.cn. 2 I. I ntr oduction Non-line of sight (NLOS) Ultra-violet (UV) scattering communication serves as a good candidate for the applications where radio-silence is required and the transmitter-recei ver alignment is hard to guarantee due to obstacles or the user mobility . Moreover , it is promising for outdoor communication under strong solar background because of negligible solar radiation in the UV spectrum [1]. Theoretical analysis [2], numerical simulation [3] and real experiments [4], [5] show an extremely large path loss between the transmitter and receiv er , where the recei v ed signal can be detected by photon-counting receiv er and characterized by Poisson distributed number of discrete photoelectrons. The capacity of point-to-point continuous-time Poisson channel has been in vestigated in [6], [7], [8] and the capacity of discrete-time Poisson channel has been deri ved in [9], [10]. Based on the Poisson channel model, se v eral types of channel model such as Poisson fading [11], MIMO [12], interfering [13], broadcast [14] , and multiple access [15], [16], [17] channels ha ve been studied in recent years. Specifically , code-di vision and non-orthogonal multiple transmission has been studied in [18], and random access packet-switched systems was proposed in [19]. Other existing works on NLOS UV scattering communication based on the Poisson and extended channel model are the channel link gain with impulse response [20], [21], channel estimation with inter-symbol interference [22], signal detection with receiv er di versity [23], and the relay protocol [24]. In this work, we characterize multi-layer superimposed transmission in discrete Poisson channel, where the transmitted symbols in various layers are superimposed, and the symbol boundaries on di ff erent signal layers are not necessarily aligned. Specifically , we adopt hidden markov model (HMM) [25], [26] to characterize the superimposed channel. Then, we conceiv e the achie vable transmission rates for all signal layers and a single layer , and obtain the exact and approximated solution [27]. For recei ver -side signal processing, we propose channel estimation based on expectation-maximization (EM) algorithm [28], [29], and adopt V iterbi [30] and Bahl-Cocke-Jelinek-Ravi v (BCJR) [31] algorithms for symbol detection. Furthermore, we propose iterativ e algorithm for maximum-likelihood / maximum a posteriori 3 probability (ML / MAP) joint decoding [32], [33]. Finally , we conduct o ffl ine experiments to ev aluate the performance of the proposed approaches. It is seen that based on the experimental measurements, the proposed approaches perform close to the simulation results with identical channel parameters. The remainder of the paper is organized as follows. In Section II, we characterize the superimposed NLOS scattering communication using HMM. In Section III, we in v estigate the achiev able transmission rates and obtain a numerical solution on the achiev able transmission rate of all signal layers and a single layer . In Section IV , we propose the EM-based channel estimation as well as joint symbol detection and decoding. Numerical and experimental results are giv en in Sections V and VI, respecti vely . Finally , we conclude this paper in Section VII. II. S ystem M odel A. Superimposed T r ansmission based on Discr ete P oisson Asynchr onous Channel W e consider a NLOS scattering communication system adopting on-o ff key (OOK) modulation that outperforms pulse-position modulation (PPM, please refer to Appendix A for more details). The ov erall transmission signal can be split into multiple signal layers which are superimposed possibly in an asynchronous manner , i. e., the symbols in di ff erent layers are not necessarily aligned. As shown in Figure 1, the overall transmission can be split into L signal layers, denoted as layer 1 , 2 , ..., L . Let M denote the number of transmitted symbols in each single layer; T s denote the symbol duration; and ρ 1 , ρ 2 , . . . , ρ L denote the normalized relativ e delay in terms of T s  P L i = 1 ρ i = 1  , where ρ i denotes the normalized delay between layer i and layer i + 1, for 1 ≤ i ≤ L (here layer L + 1 equals Layer 1). In order to characterize the symbol duration o ff set in di ff erent signal layers, we divide the symbols in di ff erent signal layers into chips subjected to symbol boundaries, where the symbol detection is performed based on the receiv ed signal in each chip. The symbol misalignment and relati ve delay are illustrated in Figure 1, where T denotes the number of ov erall chips and T = M L + L − 1. Due to the weak recei ved signal intensity of NLOS scattering communications, the receiv ed signal can be characterized by discrete photoelectrons, whose number satisfies a Poisson distribution. More 4 Fig. 1. Illustration for 3-layer superimposed transmission. specifically , let λ 1 , λ 2 , . . . , λ L denote the mean number of detected photoelectrons in each symbol duration, and z 1 , z 2 , . . . , z L denote the transmitted binary symbols in the L signal layers, where z i = [ z i , 1 , z i , 2 , ..., z i , M ] ∈ { 0 , 1 } M ; z i , m demotes the m th symbol in layer i for 1 ≤ i ≤ L and 1 ≤ m ≤ M ; and the transmitted symbols are independent of each others. The number of detected photoelectrons N t in the t -th chip for 1 ≤ t ≤ T satisfies the following Poisson distribution P ( N t = n ) = τ n t n ! ( λ 0 + Λ T S t ) n e − τ t ( λ 0 + Λ T S t ) , (1) where Λ = [ λ 1 , λ 2 , . . . , λ M ] T ; τ t = ρ ( t − 1 mod M ) + 1 ; S t = [ z 1 , d t L e , z 2 , d t − 1 L e , . . . , z L , d t − L + 1 L e ] T ; z i , 0 = 0 , z i , M + 1 = 0 for 1 ≤ i ≤ L ; and λ 0 denotes the mean number of background radiation photoelectrons in a symbol duration. B. Hidden Markov Model for Asynchr onous Signal Superposition Due to the ov erlap of di ff erent layers, the numbers of detected photoelectrons in adjacent chips are correlated with each other . In the t -th chip, N t depends on S t , which depends on S t − 1 . Consequently , we can adopt HMM to characterize the signal model in the chip lev el. W e denote  T = { S t | 1 ≤ t ≤ T } and N T = [ N 1 , N 2 , · · · , N T ] ∈ N L as the state and observation sequences of the HMM, respecti vely , where S t ∈ B L , and B L denotes the state space of the t -th chip giv en by B L =  L X i = 1 θ i e i | θ i ∈ { 0 , 1 } , 1 ≤ i ≤ L  , (2) where e i denotes the i -th column of L × L identity matrix. 5 An HMM is determined by parameters ( π 1 , A t , B t ), where π 1 , A t and B t denote the initial distribution, state transition matrix and observation emission matrix, respectiv ely . Note that the initial state depends on the first symbol in the first layer , thus π 1 is giv en by π 1 = n q 1 , 1 , 1 − q 1 , 1 , 0 , 0 , . . . , 0 o , (3) where q i , j = P ( z i , j = 1) denotes the prior possibility of symbol z i , j for 1 ≤ i ≤ L and 1 ≤ j ≤ M . The symbols in the same signal layer may hav e di ff erent prior probabilities since they may be allocated to di ff erent users. The state transition matrix is giv en by A t = h a t , i , j | s t , i ∈ B L , s t + 1 , j ∈ B L i , where each element a t , i , j is gi ven by a t , i , j = P ( S t + 1 = s t + 1 , j | S t = s t , i ) = q s t + 1 , j · e k k , d t − k + 2 L e (1 − q k , d t − k + 2 L e ) s t + 1 , j · e k Y r , k  s t + 1 , j · e r    s t , i · e r  , (4) and k = ( t mod L ) + 1, which means A t is cyclical of period L ; s t , i , s t + 1 , j ∈ B L take values among all possible choices of S t and S t + 1 , respectiv ely; Moreov er ,  indicates binary logical XNOR. Pr oof: Please refer to Appendix B. The observ ation emission matrix is gi ven by B t = h b t , i , n + 1 | s t , i ∈ B L , n ∈ N i , where based on Equation (1) each element b t , i , n + 1 is giv en by b t , i , n + 1 = P ( N t = n | S t = s t , i ) = τ n t n ! ( λ 0 + Λ T s t , i ) e − τ t ( λ 0 + Λ T s t , i ) . (5) C. Modeling System W ith Superimposed Communication The superimposed transmission can be applied to multi-user communication. Let K denote the number of users. For K ≤ L , we can assign each signal layer or multiple layers to one user . For K > L , some users have to share a common signal layer . Figure 2 illustrates the scenario with 5 users sharing 2 layers via time-division. 6 Fig. 2. T wo-layer transmission with fiv e users. III. A chiev able T ransmission R a te W e consider the achie v able rates for HMM, and giv e a numerical solution to the achiev able transmission rate of asynchronous signal superposition. A. Achie vable Rates for HMM The achie v able rates can be deriv ed based on the mutual information between hidden states and observ ation sequences for HMM. Let L = { 1 , 2 , . . . , L } denote the entire set of signal layers; U ⊂ L denote a subset of layers; and  U = { z k | k ∈ U } denote the set of transmission symbols in layer set U . Due to the statistical independence of di ff erent transmitted symbols, the entropy and conditional entropy of transmitted symbols are giv en as follows, H(  L ) = L X i = 1 M X j = 1 H ( q i , j ) , H(  U |  L\U ) = X i ∈U M X j = 1 H ( q i , j ) , (6) where H ( x ) = − x log 2 x − (1 − x ) log 2 (1 − x ). The entropy and conditional entropy of the transmitted symbols giv en the observation sequences are gi ven by H(  L | N T ) = − E z ∈ B T n ∈ N T h log 2 P (  L = z | N T = n ) i , H(  U |  L\U , N T ) = − E z ∈ B T n ∈ N T h log 2 P (  U = z U |  L\U = z L\U , N T = n ) i , (7) where N denotes the set of natural number; and Ω T denotes the T -time expansion of set Ω . Note that for ∀U ⊆ L , U , ∅ , the ov erall achiev able rate [34] of the signal layers in set U must 7 satisfy X k ∈U R k ≤ 1 M I(  U ; N T |  L\U ) , (8) where coe ffi cient 1 / M is due to the fact of M symbols in the Markov chain for each signal layer , and R k denotes the achie vable rate of layer k ; and I(  U ; N T |  L\U ) denotes the conditional mutual information gi ven by I(  U ; N T |  L\U ) = H(  U |  L\U ) − H(  U |  L\U , N T ) . (9) Letting U = { k } and U = L , we hav e the following two achiev able rates of the asynchronous signal superposition, R ∗ k = sup R k = 1 M I( Z k ; N T | Z L\ k ) , R ∗ Σ = sup L X k = 1 R k = 1 M I(  L , N T ) , (10) where R ∗ k and R ∗ Σ denote the maximum single-user rate and sum user rate, respecti vely . B. Maximum Achievable T ransmission Rate for a Single Layer W e giv e an algorithm to obtain the maximum achiev able rate of a signal layer R ∗ k for 1 ≤ k ≤ L . According to Equation (10), we hav e R ∗ k = 1 M I( Z k ; N T | Z L\ k ) = 1 M M X i = 1 H ( q k , i ) − 1 M H( Z k | Z L\ k , N T ) . (11) W e hav e the following propositions on H( Z k | Z L\ k , N T ). Proposition 1. The chain rule on the conditional pr obabilities are given as follows P ( Z k | Z L\ k , N T ) = M Y j = 1 P  Z k , j |{ Z i , d t j − i + 1 L e } , { N t j }  , (12) wher e i ∈ L\ k , k + ( j − 1) L ≤ t j ≤ k + j L − 1 ; and P  Z k , j |{ Z i , d t j − i + 1 L e } , { N t j }  is the conditional pr obability of Z k , j given sets { Z i , d t j − i + 1 L e } and { N t j } . 8 Pr oof: Please refer to Appendix C. Proposition 2. The conditional entr opy of a single layer is given by H( Z k | Z L\ k , N T ) = M X j = 1 X Z k , j ∈ B  X Z i , d ( t − i + 1) / L e ∈ B  k + ( j − 1) L ≤ t ≤ k + jL − 1 i ∈L\ k P ( Z k , j ) " Y i ∈L\ k k + jL − 1 Y t = k + ( j − 1) L P ( Z i , d t − i + 1 L e ) # X { N t j }∈ N L " k + jL − 1 Y t = k + ( j − 1) L P ( N t | Z k , j , { Z i , d t − i + 1 L e } ) # log 2 P  Z k , j  Q k + jL − 1 t = k + ( j − 1) L P ( N t | Z k , j , { Z i , d t − i + 1 L e } ) P Z k , j ∈ B P  Z k , j  Q k + jL − 1 t = k + ( j − 1) L P ( N t | Z k , j , { Z i , d t − i + 1 L e } ) , (13) wher e ( P Z i , t ∈ B ) i ∈{ φ 1 ,φ 2 ,... } t ∈{ ω 1 ,ω 2 ,... } is the abbr eviation of P Z φ 1 ,ω 1 ∈ B P Z φ 1 ,ω 2 ∈ B . . . P Z φ 2 ,ω 1 ∈ B P Z φ 2 ,ω 2 ∈ B . . . ; P ( z k , j ) = q z k , j k , j (1 − q k , j ) (1 − z k , j ) ; and P ( N t | Z k , j , { Z i , d t − i + 1 L e } ) = τ N t t N t !  λ 0 + λ k Z k , j + X i ∈L\ k λ i Z i , d t − i + 1 L e  N t e − τ t ( λ 0 + λ k Z k , j + P i ∈L\ k λ i Z i , d t − i + 1 L e ) . (14) Specifically , for single user communication the prior probability of transmitted symbols remains constant, i. e, q i , j = q for 1 ≤ i ≤ L and 1 ≤ j ≤ M , and the entropy of a single layer can be further simplified into H( Z k | Z L\ k , N T ) = M X Z k ∈ B  X Z i , 2 ∈ B X Z i , 3 ∈ B  1 ≤ i < k  X Z i , 1 ∈ B X Z i , 2 ∈ B  k < i ≤ L P ( Z k ) " Y 1 ≤ i < k P ( Z i , 2 ) P ( Z i , 3 ) Y k < i ≤ L P ( Z i , 1 ) P ( Z i , 2 ) # X { N k + L ,..., N k + 2 L − 1 }∈ N L " k + 2 L − 1 Y t = k + L P ( N t | Z k , { Z i , d t − i + 1 L e } ) # log 2 P  Z k  Q k + 2 L − 1 t = k + L P ( N t | Z k , { Z i , d t − i + 1 L e } ) P Z k ∈ B P  Z k  Q k + 2 L − 1 t = k + L P ( N t | Z k , { Z i , d t − i + 1 L e } ) . (15) Pr oof: Please refer to Appendix D. C. Maximum Achievable Sum Rate W e giv e an algorithm to obtain the achie vable sum rate R ∗ Σ . According to Equation (10), we hav e R ∗ Σ = 1 M I(  L ; N T ) = 1 M M X i = 1 L X k = 1 H ( q k , i ) − 1 M H(  L | N T ) . (16) Note that the computational complexity of H(  L | N T ) grows exponentially with T due to exhausti ve enumeration of the state and observation sequences in B T and N T . Consequently , brute-force computation 9 on the exact value is intractable for large T . The computational complexity can be reduced via sampling B T and N T , and the solution for conditional entropies can be approximated by the empirical mean according to the following equation, H(  L | N T ) ≈ E z ∈ Ψ z n ∈ Ψ n h H(  L | N T = n ) i , (17) where Ψ z ⊂ B T and Ψ n ⊂ N T denote the set of su ffi ciently many samples on B T and N T such that the empirical mean becomes con v erged, respectiv ely . W e resort to Monte Carlo method, which keeps generating random states and observation sequences based on the initial state distribution, the transition probability matrices, and the observ ation emission matrices. For each state and observation sequence realization, we hav e that H(  L | N T = n ) = H(  T | N T = n ) , (18) where e ffi cient computation of the conditional entropy in Equation (18) can be conducted following [35]. D. P ower Allocation of Overlapped T r ansmission W e regard the achiev able rate as the objecti ve function of power allocation. Generally , the practical issue can be summarized as the follo wing two cases. Case 1: Giv en λ s , maximize the sum achiev able rate R ∗ Σ , subject to P L k = 1 λ k = λ s . Case 2: Giv en λ s , i and R jin f , for 1 ≤ j ≤ L , j , i , maximize the achie v able rate R ∗ i of layer i , subject to R ∗ j ≥ R jin f and P L k = 1 λ k = λ s . The numerical solution for L = 2 is provided in Section VI.C. IV . C hannel E stima tion and S ymbol D etection W e present the receiv er-side signal processing including channel estimation, symbol detection as well as joint detection and decoding. 10 Fig. 3. Illustration of partial pilot-based channel estimation for L = 2 , L p = 1. A. Channel Estimation Algorithm W e can employ pilot sequences to estimate the mean number of detected photoelectrons of each state. Howe v er , considering the pilots on all signal layers, the overhead is still non-negligible. In this work, the channel estimation can be performed based on pilot sequences on certain signal layers but not necessarily on all, which is called partial pilot-based channel estimation, as illustrated in Figure 3. W ithout loss of generality , we assume to transmit pilot sequences Z p = [ z p 1 , z p 2 , . . . , z p L p ] in L p layers, where z p i denotes the pilot sequence in layer i for 1 ≤ i ≤ L p and 0 ≤ L p < L . Let S p 1 , S p 2 , . . . , S p T p denote the state sequence for channel estimation, where T p denotes the number of chips. W e have that S p t = [ z p 1 , d t L e , . . . , z p L p , d t − 1 L e , z L p + 1 , d t − 1 L e , . . . , z L , d t − L + 1 L e ] T , and ˆ Λ is estimated based on EM algorithm. Let N p = [ N p 1 , N p 2 , . . . , N p T p ] denote the number of receiv ed photoelectrons in each chip for channel estimation, where N p is the observation sequence of S p t for 1 ≤ t ≤ T p . The estimation for ˆ Λ is processed by V iterations, and in each iteration the updating rule is provided as follows. E-step : In the v th iteration, based on the estimate result ˆ λ ( v − 1) s i in the ( v − 1)th iteration, the a posterior probability of S p t is giv en by Q ( v ) ( S p t = s i ) = P ( S p t = s i | N p , λ s i = ˆ λ ( v − 1) s i ) = P ( N p , S p t = s i | λ s i = ˆ λ ( v − 1) s i ) P s i ∈ B L \ L p P ( N p , S p t = s i | λ s i = ˆ λ ( v − 1) s i ) , (19) where B L \ L p =  P L p i = 1 z p i , d t − i + 1 L e e i + P L i = L p + 1 θ i e i | θ i ∈ { 0 , 1 } , L p + 1 ≤ i ≤ L  , and P ( N p , S p t = s i | λ s i = ˆ λ ( v − 1) s i ) =  τ t ˆ λ ( v − 1) s i  N p t N p t ! e − τ t ˆ λ ( v − 1) s i . (20) 11 Fig. 4. The trellis diagram for L = 3. M-step : Giv en a posterior probability Q ( v ) ( S p t = s i ) for the v th iteration, the ML-estimation for ˆ Λ ( v ) T = { ˆ λ ( v ) s i | s i ∈ B L \ L p } is giv en by ˆ λ ( v ) s j = P T p t = 1 Q ( v ) ( S p t = s i ) N p t P T p t = 1 Q ( v ) ( S p t = s i ) τ t , (21) where the preset initial ˆ Λ (0) must satisfy  ˆ λ (0) s i − ˆ λ (0) s j  λ s i − λ s j  > 0 for i , j and λ s i , λ s j . Pr oof: Please refer to Appendix E. B. HMM-Based Symbol Detection Based on HMM, the recei ver aims to detect state sequence  T according to the observation sequence N T and ( π 1 , A t , B t ). The trellis diagram for HMM is adopted to find the optimal state transition path maximizing the likelihood function or a posteriori probability . Figure 4 illustrates the trellis diagram for L = 3, where each state S t is expressed as { z k , d t − k + 1 L e | 1 ≤ k ≤ L } , and each branch between adjacent states corresponds to a non-zero element of A t . W e adopt V iterbi and Bahl-Cocke-Jelinek-Ravi v (BCJR) algorithms to maximize the likelihood function P ( N T |  T = s T ) and a posteriori probability P (  T = z T | N T ), respecti vely , and minimizes the error rate of sequence and symbol detection, respectiv ely . For V iterbi algorithm, we maximize the log-likelihood function of state sequence summarized as follows ˆ  T = arg max log P ( N T |  T = s T ) = arg max S t ∈ B L T X t = 1 N t log τ t λ S t − τ t λ S t , (22) where λ S t for S t ∈ B L can be obtained from channel estimation. 12 Letting L ( N t | S t ) = N t log τ t λ S t − τ t λ S t , we have that L ( N t |  t ) = L ( N t | S t ) + L ( N t − 1 |  t − 1 ) for 2 ≤ t ≤ T . Thus dynamic programming is adopted with the following updated equation max L ( N t + 1 |  t , S t + 1 , j ) = L ( N t + 1 | S t + 1 , j ) + max a i , j , t , 0  L ( N t |  t − 1 , S t , i )  , (23) which is initialized by L ( N 1 |  1 ) = L ( N 1 | S 1 ) ∼ N 1 log τ 1 ( λ 0 + λ S 1 ) − τ 1 ( λ 0 + λ S 1 ). The detected symbol sequence can be retrie ved via tracing back the optimal path. For BCJR Algorithm, we maximize the posterior probability for each symbol z k , i for 1 ≤ k ≤ L and 1 ≤ i ≤ M as follows ˆ z i , j = arg max log P ( z k , i | N T ) = arg max log P (  z k , i | N T ) ∼ arg max log P (  z k , i , N T ) , (24) where  z k , i = { S t | t = ( i − 1) L + k , ( i − 1) L + k + 1 , . . . , i L + k − 1 } . T o obtain P (  z k , i , N T ), we define the following probability functions α t ( s ) = P ( S t = s , N t ) , β t ( s ) = P ( N [ t + 1 , T ] | S t = s ) , γ t ( v , s ) = P ( N t , S t = s | S t − 1 = v ) , (25) where N [ a , b ] = { N t | a ≤ t ≤ b } . Note that we hav e P (  z k , i = $ k , i , N T ) = α ( i − 1) L + k ( s ( i − 1) L + k ) β iL + k − 1 ( s iL + k − 1 ) iL + k − 2 Y t = ( i − 1) L + k γ t ( s t , s t + 1 ) , (26) where $ k , i = { s t | ( i − 1) L + k ≤ t ≤ i L + k − 1 } . Furthermore, we hav e that γ t ( s t − 1 , i , s t , j , n ) = a t − 1 , i , j b t , j , n + 1 for s t − 1 , i , s t , j ∈ B L and n ∈ N . Then, the calculations of α ( s t ) and β ( s t ) are conducted according to the 13 follo wing recursi ve equations α t ( s t ) = X s t − 1 ∈ B L α t − 1 ( s t − 1 ) γ t ( s t − 1 , s t ) , β t ( s t ) = X s t + 1 ∈ B L β t + 1 ( s t + 1 ) γ t + 1 ( s t , s t + 1 ) . (27) The initial values are α 1 ( s 1 , i ) = π 1 ( s 1 , i ) b i , 1 , N 1 + 1 for s 1 , i ∈ B L and β T ( s T ) = π T ( s T ) for s T ∈ B L , where π 1 is gi ven by Equation (3); and π t ( s t + 1 , i ) = P ( S t + 1 = s t + 1 , i ) can be obtained by the following recursi ve equation, π t ( s t + 1 , i ) = X s t , j ∈ B L a t , j , i π t − 1 ( s t , j ) . (28) C. J oint Detection and Decoding W e adopt joint detection and decoding based on turbo processing. For ML and MAP decoding, the log- likelihood ratio ( L LR ) and log-aposterior ratio ( L AR ) are adopted as the input soft information to the soft channel decoder , respectiv ely . Let L LR ( v ) z k , i and L AR ( v ) z k , i denote the log-likelihood ratio and log-aposterior- ratio of z k , i after the v -th iteration, respecti vely . T ypically each iteration of the turbo processing consists of one ML / MAP symbol detection operation follo wed by V channel decoding iterations. For the ML-decoding, the initial L LR values are obtained by V iterbi algorithm as follows, L LR (0) z k , i = log P ( N T | z k , i = 1) P ( N T | z k , i = 0) = iL + k − 1 X t = ( i − 1) L + k log P ( N t | z k , i = 1) P ( N t | z k , i = 0) , (29) and the L LR of the i -th transmitted symbol in layer k in the v -th iteration is calculated by L LR ( v ) z k , i = log P ( N T | z k , i = 1) P ( N T | z k , i = 0) = iL + k − 1 X t = ( i − 1) L + k log E z k , i = 1 P ( N t | S t = s t ) E z k , i = 0 P ( N t | S t = s t ) , (30) where the expectation E z k , i = θ [ • ] for θ ∈ { 0 , 1 } is calculated based on a posterior probabilities by the ( v − 1)th iteration of channels in L \ k as follows E s t ∈S z k , i = θ [ • ] = X s t ∈S z k , i = θ Y j ∈L\ k P 1 − z j , d t − j + 1 L e ( v − 1) ( z j , d t − j + 1 L e = 0 | N t ) P z j , d t − j + 1 L e ( v − 1) ( z j , d t − j + 1 L e = 1 | N t )[ • ] , (31) where S z k , i = θ = { s t | e T k · s t = θ } ; z j , d t − j + 1 L e = e T j · s t ; and the a posterior probability of z j , d t − j + 1 L e after the ( v − 1)-th 14 iteration is giv en by P ( v − 1) ( z j , d t − j + 1 L e = 0 | N t ) = 1 − P ( v − 1) ( z j , d t − j + 1 L e = 1 | N t ) = 1 1 + exp  q j , d t − j + 1 L e 1 − q j , d t − j + 1 L e L LR ( v − 1) j , d t − j + 1 L e  . (32) For MAP-decoding, the initial L AR is determined by BLJR detection as follows L AR (0) z k , i = log P ( z k , i = 1 | N T ) P ( z k , i = 0 | N T ) ; (33) and the L AR of symbol z k , i from the v -th iteration is giv en by L AR ( v ) z k , i = log P ( v − 1) ( z j , d t − j + 1 L e = 1 | N T ) P ( v − 1) ( z j , d t − j + 1 L e = 0 | N T ) = log P ( N T | z k , i = 1) P ( N T | z k , i = 0) + log P ( z k , i = 1) P ( z k , i = 0) = L LR ( v − 1) z k , i + log q k , i 1 − q k , i , (34) where L LR ( v − 1) z k , i is computed according to Equations (30) and (31). Furthermore, the a posterior probability of MAP-decoding is gi ven by P ( v − 1) ( z j , d t − j + 1 L e = 0 | N t ) = 1 − P ( v − 1) ( z j , d t − j + 1 L e = 1 | N t ) = 1 1 + exp  L AR ( v − 1) j , d t − j + 1 L e  . (35) V . N umerical and S umula tion R esul ts In this section, we provide numerical and simulation results on the achiev able rates, power allocation, channel estimation as well as joint detection and decoding. A. Achie vable Rates Consider the superimposed transmission with L = 2 signal layers, where λ 1 = λ 2 = 10 and background radiation λ 0 = 0 . 01. W e ev aluate the sum achiev able transmission rate versus symbol number M and relati ve delay ρ 1 in Figure 5, where the scenario of ρ 1 = 0 , ρ 2 = 1 for perfect symbol boundary alignment is also sho wn for comparison. It is implied that introducing relati ve relays can enhance the achie vable 15 10 0 10 1 10 2 10 3 10 4 M 0 0.5 1 1.5 2 R (bit per symbol) 1 = 2 = 0.5 1 = 0.4, 2 = 0.6 1 = 0.3, 2 = 0.7 1 = 0.2, 2 = 0.8 1 = 0.1, 2 = 0.9 1 = 0, 2 = 1 Fig. 5. The achiev able sum rates with di ff erent relative delays. sum rate, and ρ 1 = 0 . 5 can maximize the sum rate, which can con v erge for M exceeding 10 2 , where an improv ement of 0 . 5 bit per symbol can be observed. Consider a more general scenario with possibly more than 2 signal layers, i.e., M = 1 × 10 4 , λ i = λ j = λ for 1 ≤ i < j ≤ L and background radiation λ 0 = 0 . 01. The achiev able sum rates for the case of L = 2 , 3 , 4 with the relati ve delays are sho wn in Figure 6, where the scenario of L = 1 without signal superposition is also shown for comparison. It is seen that the achiev able sum rate can be improved with su ffi cient receiv er- side signal intensity . Since the computational complexity of symbol detection gro ws exponentially with L , we can set a standard on the minimum L subject to at least σ bit per symbol gain ov er L − 1 signal layers. Accordingly , we can achiev e the optimal number of signal layers L ∗ corresponding to di ff erent λ . For example σ = 0 . 2; when λ < 3, L = 1 is optimal; when 3 ≤ λ ≤ 8, L = 2; when 8 ≤ λ ≤ 18, L = 3; and when λ > 18, L = 4. 16 2 4 6 8 10 12 14 16 18 20 0 0.5 1 1.5 2 2.5 3 R (bit per symbol) L = 4, 1 = 2 = 3 = 4 = 0.25 L = 3, 1 = 2 = 3 = 0.33 L = 2, 1 = 2 = 0.5 L = 1 Fig. 6. The achiev able sum rates of 2,3,4 signal layers. B. P ower Allocation W e consider the po wer allocation in Section III.D for L = 2. The first optimization problem is max  R ∗ 1 + R ∗ 2  , subject to λ 1 + λ 2 = λ s . Figure 7 plots the maximum achie vable sum rates and their optimal power allocation versus λ s . It is seen that as λ s increases, the optimal po wer allocation tends to become equal distribution, where the achiev able sum rate enhances as ρ grows from 0 . 1 to 0 . 5. The second optimization problem is to max R ∗ 1 , subjected to R ∗ 2 ≥ R 2 in f and λ 1 + λ 2 = λ s , as sho wn in Figure 8. Let P denote the intersection of lines R = R 2 in f and λ 1 + λ 2 = λ s ; and λ P denote the x -coordinates of P . The feasible solution for the problem is that λ 1 = λ s − λ P , λ 2 = λ P . C. J oint Detection and Decoding Assume that λ i = λ ave for 1 ≤ i ≤ L . The av erage symbol error rates for L = 2 and L = 3 of joint detection versus λ ave are illustrated in Figures 9(a) and 9(b), respectiv ely . Furthermore, we adopt a (12620 , 6310) LDPC code for each signal layer , where the parity check matrix construction and low- complexity message pass decoding follo w [36], [37] and [38]. The average bit error rates for L = 2 and 17 0 10 20 30 40 50 60 s -3 -2 -1 0 0.5 1 1.5 2 R * (bit per symbol) = 0.1 = 0.2 = 0.3 = 0.4 = 0.5 = 0.1 = 0.2 = 0.3 = 0.4 = 0.5 0.8 1 1.2 1.4 1.6 1.8 2 0.8 1 1.2 1.4 1.6 1.8 2 0.5 1 1.5 2 0.5 1 1.5 2 Fig. 7. The optimal power allocation and achiev able sum rate versus λ s . Fig. 8. The achiev able transmission rate of a single layer . 18 5 10 15 20 25 30 35 ave 10 -1 Symbol error rate Viterbi 1 = 0.1 BCJR 1 = 0.1 Viterbi 1 = 0.2 BCJR 1 = 0.2 Viterbi 1 = 0.3 BCJR 1 = 0.3 Viterbi 1 = 0.4 BCJR 1 = 0.4 Viterbi 1 = 0.5 BCJR 1 = 0.5 (a) The number of signal layer L = 2. 10 15 20 25 30 35 40 ave 10 -1 Symbol error rate Viterbi 1 = 0.33, 2 = 0.33 BCJR 1 = 0.33, 2 = 0.33 Viterbi 1 = 0.5, 2 = 0.25 BCJR 1 = 0.5, 2 = 0.25 Viterbi 1 = 0.5, 2 = 0.3 BCJR 2 = 0.5, 3 = 0.3 (b) The number of signal layer L = 3. Fig. 9. The symbol error rate of joint detection. 5 10 15 20 25 ave 10 -6 10 -4 10 -2 10 0 Bit error rate ML 1 = 0.1 MAP 1 = 0.1 ML 1 = 0.2 MAP 1 = 0.2 ML 1 = 0.3 MAP 1 = 0.3 ML 1 = 0.4 MAP 1 = 0.4 ML 1 = 0.5 MAP 1 = 0.5 (a) The number of signal layer L = 2. 5 10 15 20 25 30 35 ave 10 -6 10 -4 10 -2 10 0 Bit error rate Viterbi 1 = 0.33, 2 = 0.33 BCJR 1 = 0.33, 2 = 0.33 Viterbi 1 = 0.5, 2 = 0.25 BCJR 1 = 0.5, 2 = 0.25 Viterbi 1 = 0.5, 2 = 0.3 BCJR 2 = 0.5, 3 = 0.3 (b) The number of signal layer L = 3. Fig. 10. The bit error rate of joint detection and decoding with (12620 , 6310) LDPC code. L = 3 by joint detection and decoding versus λ ave are shown in Figures 10(a) and 10(b), respectiv ely . It is seen that for L = 2, ρ = 0 . 5 has the lo west error rate for both detection and decoding, which accords with the maximum achie vable sum rate. VI. E xperiment al R esul ts for 2- la yer - superimposed T ransmission W e conduct o ffl ine experiments on the 2-layer-superposition transmission for optical wireless scattering communication to experimentally ev aluate the proposed joint detection and decoding. At the transmitter 19 Fig. 11. Diagram of the experimental superimposed communication system. Fig. 12. Demonstration of the transmitter-side (left) and receiv er-side (right) test beds. side, a wa veform generator is adopted to produce OOK signals. A Bias-T ee is employed to combine the A C and DC signals to driv e the UV LED. At the receiv er side, a photomultiplier tube (PMT) is employed as the photon-detector , which is integrated with an optical filter in a sealed box. The UV signal of wav elength around 280nm can be detected, while the background radiation of other wa velengths is blocked. The PMT output signal is attenuated by an attenuator , amplified by an amplifier , and then filtered by a low-pass filter , which is then sampled by the oscilloscope. Finally , the photon counting processing, HMM-based MAP joint detection and decoding are realized in the receiv ed-side personal computer (PC) based on the sampled wav eforms from the oscilloscope. T able I shows the specification of experimental equipment, and Figures 11 and 12 illustrate the entire experimental block diagram and the test bed realizations, respecti vely . In the experiment, the background radiation intensity is around 150 photoelectrons per second in the indoor en vironment ( λ 0 ≈ 1 . 5 × 10 − 4 ). Furthermore, we adopt the following parameters for two signal layers: symbol duration T s = 1 µ s ; ρ = 0 . 5; uniform power allocation for 2 signal layers ( λ 1 = λ 2 = λ ave ); 20 T ABLE I S pecifica tion of device for experiment . UV LED Model TO-3zz PO#2036 W av elength 280nm Optical filter Peak transmission 28 . 2% Aperture size Φ 31 . 5mm × 28 . 3mm PMT Model R7154 Spectral response 160nm ∼ 320nm Dark counts < 10 per second Detection bandwidth > 200MHz the same parity check matrix construction and decoding algorithm of LDPC codes as those in simulation; and the uniform prior probabilities for 0 − 1 symbols. For each λ ave , we implement MAP joint detection and decoding and count the bit error rate based on the transmission of 1000 frames (1 . 262 × 10 7 random bits). W e experimentally e valuate channel estimation for L = 2 , ρ 1 = 0 . 5, where we exploit a 255-bit m sequence as a pilot sequence z p . For L p = 1, z p 1 = z p ; and for L p = 2, z p 1 = z p 2 = z p . The performance of channel estimation versus the number of iterations is illustrated in Figure 14(a), where the result of L p = 2 is from the ML estimation. It is implied that real time estimation for both L p = 0 and 1 can con ver ge to the ML solution; and assisted by the pilot sequence, the con ver gency of L p = 1 is faster than L p = 0. Furthermore, higher λ s i with large receiv er-side SNR can lead to faster con ver gence, which is close to the simulation result on the channel estimation with the same system parameters, as sho wn in Figure 14(b). Moreov er , the MAP detection with and without LDPC code (denoted as EXP) is ev aluated in Figure 13, where the simulation results with the same channel parameters (denoted as SL) is plotted for comparison. It is seen that the experimental results on the channel estimation, symbol detection and joint detection / decoding are close to the simulation results, which illustrates the feasibility of the proposed channel estimation and signal detection approaches in real communication scenarios. VII. C onclusion W e hav e proposed superposition transmission for optical wireless scattering communication based on HMM. W e have obtained the achie vable rates of proposed superposition transmission, and proposed 21 2 4 6 8 10 12 14 16 18 ave 10 -6 10 -4 10 -2 10 0 Error rate SER BCJR SL SER BCJR EXP BER MAP SL BER MAP EXP Shannon Limit Fig. 13. The average symbol and bit error rate of 2-layer superimposed communication from simulation and experimental measurements. 0 2 4 6 8 10 12 Iterative time 0 5 10 15 20 Blind estimation Pilot in layer 1 Pilot in layer 1 & 2 (a) Conv ergence of channel estimation from experiments. 0 2 4 6 8 10 12 Iterative time 0 5 10 15 20 Blind estimation Pilot in layer 1 Pilot in layer 1 & 2 (b) Conv ergence of channel estimation from simulations. Fig. 14. Con ver gence of channel estimation from both experiments and simulations. 22 the EM-based channel estimation and joint detection and decoding. The performance of the proposed approaches are v erified by numerical results. Moreo ver , for two- and three-layer transmission, both simulation and experimental results are employed to validate the feasibility of the proposed algorithms for channel estimation as well as joint detection and decoding. VIII. A ppendix A. Comparison of Achie vable Rates between OOK and 2-Pulse-P osition Modulations (2-PPM) The mutual information of single-use OOK modulation is giv en by I OOK ( X ; N ) = max 0 < q < 1 ( H  X X q ∈{ 0 , 1 } P ( X q ) P OOK ( N | X q )  − X X q ∈{ 0 , 1 } P ( X q ) H  P OOK ( N | X q )  ) , (36) and that of 2-PPM is giv en by I 2 − P P M ( X ; N 1 , N 2 ) = max 0 < q < 1 max 0 <τ< 1 ( H  X X q ∈{ 0 , 1 } P ( X q ) P 2 − P P M ( N 1 , N 2 | X q , τ )  − X X q ∈{ 0 , 1 } P ( X q ) H  P 2 − P P M ( N 1 , N 2 | X q , τ )  ) , (37) where X q ∼ { q , 1 − q } ; P OOK ( N | X q = 0) = λ N 0 N ! e − λ 0 , P OOK ( N | X q = 1) = ( λ 0 + λ 1 ) N N ! e − ( λ 0 + λ 1 ) , P 2 − P P M ( N 1 , N 2 | X q = 0 , τ ) = τ N 1 λ N 1 0 (1 − τ ) N 2 ( λ 0 + λ 1 ) N 2 N 1 ! N 2 ! e − τλ 0 − (1 − τ )( λ 0 + λ 1 ) , P 2 − P P M ( N 1 , N 2 | X q = 1 , τ ) = τ N 1 ( λ 0 + λ 1 ) N 1 (1 − τ ) N 2 λ N 2 0 N 1 ! N 2 ! e − τ ( λ 0 + λ 1 ) − (1 − τ ) λ 0 ; (38) λ 1 denotes the mean number of detected photoelectrons in each symbol duration; N , N 1 , N 2 denote the number of receiv ed photoelectrons; and τ denotes the duty ratio of the pulse in each symbol duration for 2-PPM. The achiev able rates of OOK and 2-PPM modulation are compared in Figure 15, where OOK modulation shows higher achiev able rate. 23 1 2 3 4 5 6 1 0.2 0.3 0.4 0.5 0.6 0.7 Achievable Rate 2-PPM OOK Fig. 15. The comparison between OOK and 2-PPM modulations with background intensity 1 × 10 4 per second. B. Pr oof of State T r ansition Matrix For k = ( t mod L ) + 1, we hav e d t + 1 − k + 1 L e = d t − k + 1 L e + 1, and d t + 1 − r + 1 L e = d t − r + 1 L e for r , k . Due to S t = [ z 1 , d t L e , z 2 , d t − 1 L e , . . . , z L , d t − L + 1 L e ] T , the r -th element of S t and S t + 1 must satisfy z r , d t − r + 1 L e = z r , d t + 1 − r + 1 L e for r , k . Consequently , The state transition probability P ( S t + 1 = s t + 1 , j | S t = s t , i ) = 0, if z r , d t − r + 1 L e , z r , d t + 1 − r + 1 L e ; and P ( S t + 1 = s t + 1 , j | S t = s t , i ) = P ( z k , d t + 1 − k + 1 L e | z k , d t − k + 1 L e ). Furthermore, z k , d t + 1 − k + 1 L e is independent with z k , d t − k + 1 L e , hence we hav e P ( z k , d t + 1 − k + 1 L e | z k , d t − k + 1 L e ) = q z k , d t + 1 − k + 1 L e k , d t + 1 − k + 1 L e (1 − q k , d t + 1 − k + 1 L e ) z k , d t + 1 − k + 1 L e . In addition, z i , d t − i + 1 L e = S t · e i for 1 ≤ i ≤ L . Therefore, z r , d t − r + 1 L e , z r , d t + 1 − r + 1 L e is equiv alent with  s t + 1 , j · e r    s t , i · e r  = 0, and P ( S t + 1 = s t + 1 , j | S t = s t , i ) can 24 be simplified into Equation (4). C. Pr oof of Chain Rules on Conditional Pr obabilities W e prov e the proposition based on the following chain rule on the probability of receiv ed signal giv en two users since the numbers of receiv ed photoelectrons in di ff erent chips are independent of each other , P ( N T | Z L ) = T Y t = 1 P ( N t | Z 1 , d t L e , Z 2 , d t − 1 L e , . . . , Z L , d t − L + 1 L e ) . (39) Consequently , Equation (12) can be proved by P ( Z k | Z L\ k , N T ) = P  Z k , 1 , Z k , [2 , L ] |{ Z i , d t 1 − i + 1 L e } , { N t 1 } , { Z i , d ˜ t − i + 1 L e } , { N ˜ t }  = P  Z k , 1 , Z k , [2 , L ] , { N t 1 } , { N ˜ t } | { Z i , d t 1 − i + 1 L e } , { Z i , d ˜ t − i + 1 L e }  P Z k , 1 P Z k , [2 , L ] P  Z k , 1 , Z k , [2 , L ] , { N t 1 } , { N ˜ t } | { Z i , d t 1 − i + 1 L e } , { Z i , d ˜ t − i + 1 L e }  = P  { N t 1 } | Z k , 1 , { Z i , d t 1 − i + 1 L e }  P  Z k , 1  P  { N ˜ t } | Z k , [2 , L ] , { Z i , d ˜ t − i + 1 L e }  P  Z k , [2 , L ]  P Z k , 1 P Z k , [2 , L ] P  { N t 1 } | Z k , 1 , { Z i , d t 1 − i + 1 L e }  P  Z k , 1  P  { N ˜ t } | Z k , [2 , L ] , { Z i , d ˜ t − i + 1 L e }  P  Z k , [2 , L ]  = P  { N t 1 } | Z k , 1 , { Z i , d t 1 − i + 1 L e }  P  Z k , 1  P Z k , 1 P  { N t 1 } | Z k , 1 , { Z i , d t 1 − i + 1 L e }  P  Z k , 1  P  { N ˜ t } | Z k , [2 , L ] , { Z i , d ˜ t − i + 1 L e }  P  Z k , [2 , L ]  P Z k , [2 , L ] P  { N ˜ t } | Z k , [2 , L ] , { Z i , d ˜ t − i + 1 L e }  P  Z k , [2 , L ]  = P  Z k , 1 | { N t 1 } , { Z i , d t 1 − i + 1 L e }  P  Z k , [2 , L ] | { N ˜ t } , { Z i , d ˜ t − i + 1 L e }  , (40) where Z k , [2 , M ] = [ Z k , 2 , Z k , 2 , . . . , Z k , M ], and the index es in volv ed in the brackets i ∈ L\ k , k ≤ t 1 ≤ k + L − 1 and k + L ≤ ˜ t ≤ k + M L − 1. Re-factorizing Equation (40), we have P ( Z k | Z M\ k , N T ) = P  Z k , 1 | { N t 1 } , { Z i , d t 1 − i + 1 L e }  P  Z k , 2 | { N t 2 } , { Z i , d t 2 − i + 1 L e }  P  Z k , [3 , M ] | { N ˜ t } , { Z i , d ˜ t − i + 1 L e }  , (41) where the indexes in volv ed in the brackets i ∈ L\ k , k + L ≤ t 2 ≤ k + 2 L − 1 and k + 2 L ≤ ˜ t ≤ k + M L − 1. Re-factorize Equation (41) for M − 2 times, we hav e P ( Z k | Z L\ k , N T ) = M Y j = 1 P  Z k , j |{ Z i , d t j − i + 1 L e } , { N t j }  , (42) where the index es in v olved in the brackets i ∈ L\ k and k + ( j − 1) L ≤ t j ≤ k + j L − 1. 25 D. Pr oof of conditional entr opies W e prov e the proposition by Equation (43) based on Proposition 1. H( Z k | Z L\ k , N T ) = X Z L ∈ B M L X N T ∈ N T P ( Z L , N T ) log 2 P ( Z k | Z L\ k , N T ) = X Z L ∈ B M L P ( Z L ) X N T ∈ N T P ( N T | Z L ) log 2 P ( Z k | Z L\ k , N T ) = X Z L ∈ B M L P ( Z L ) X N T ∈ N T P ( N T | Z L ) log 2 M Y j = 1 P  Z k , j |{ Z i , d t j − i + 1 L e } , { N t j }  = X Z L ∈ B M L P ( Z L ) X N T ∈ N T P ( N T | Z L ) log 2 M Y j = 1 P  Z k , j , { N t j }|{ Z i , d t j − i + 1 L e }  P  { N t j }|{ Z i , d t j − i + 1 L e }  = M X j = 1 X Z k , j ∈ B  X Z i , d ( t − i + 1) / L e ∈ B  k + ( j − 1) L ≤ t ≤ k + jL − 1 i ∈L\ k P ( Z k , j ) " Y i ∈L\ k P ( { Z i , d t j − i + 1 L e } ) # X { N t j }∈ N L P ( { N t j }| Z k , j , { Z i , d t j − i + 1 L e } ) log 2 P  Z k , j , { N t j }|{ Z i , d t j − i + 1 L e }  P  { N t j }|{ Z i , d t j − i + 1 L e }  = M X j = 1 X Z k , j ∈ B  X Z i , d ( t − i + 1) / L e ∈ B  k + ( j − 1) L ≤ t ≤ k + jL − 1 i ∈L\ k P ( Z k , j ) " Y i ∈L\ k k + jL − 1 Y t = k + ( j − 1) L P ( Z i , d t − i + 1 L e ) # X { N t j }∈ N L " k + jL − 1 Y t = k + ( j − 1) L P ( N t | Z k , j , { Z i , d t − i + 1 L e } ) # log 2 P  Z k , j , { N t j }|{ Z i , d t j − i + 1 L e }  P  { N t j }|{ Z i , d t j − i + 1 L e }  = M X j = 1 X Z k , j ∈ B  X Z i , d ( t − i + 1) / L e ∈ B  k + ( j − 1) L ≤ t ≤ k + jL − 1 i ∈L\ k P ( Z k , j ) " Y i ∈L\ k k + jL − 1 Y t = k + ( j − 1) L P ( Z i , d t − i + 1 L e ) # X { N t j }∈ N L " k + jL − 1 Y t = k + ( j − 1) L P ( N t | Z k , j , { Z i , d t − i + 1 L e } ) # log 2 P  Z k , j  Q k + jL − 1 t = k + ( j − 1) L P ( N t | Z k , j , { Z i , d t − i + 1 L e } ) P Z k , j ∈ B P  Z k , j  Q k + jL − 1 t = k + ( j − 1) L P ( N t | Z k , j , { Z i , d t − i + 1 L e } ) (43) T ypically , for single user transmission, the prior probability of the transmitted symbols remains constant, i. e, q i , j = q for 1 ≤ i ≤ L and 1 ≤ j ≤ M . Consequently , each term in P M j = 1 [ • ] remain constant for 2 ≤ j ≤ M − 1. When j = 1 or j = M , d t − i + 1 L e may equal 0 or M + 1, we define Z i , 0 = Z i , M + 1 = 0 for 1 ≤ i ≤ L due to the finite number of transmitted symbols. Neglecting the e ff ect of j = 1 and j = M , we 26 hav e that P M j = 1 P Z k , j ∈ B [ • ] = M P Z k , j 0 ∈ B [ • ], where j 0 can take any integer value in [2 , M − 1]; and l t − i + 1 L m =                                  j 0 , if 1 ≤ i < k , k + ( j 0 − 1) L ≤ t ≤ i + j 0 L − 1; j 0 + 1 , if 1 ≤ i < k , i + j 0 L ≤ t ≤ k + j 0 L − 1; j 0 − 1 , if k < i ≤ L , k + ( j 0 − 1) L ≤ t ≤ i + ( j 0 − 1) L − 1; j 0 , if k < i ≤ L , i + ( j 0 − 1) L ≤ t ≤ k + j 0 L − 1 . (44) Letting j 0 = 2, we hav e the following simplified form of Equation (43), H( Z k | Z L\ k , N T ) = M X Z k ∈ B  X Z i , 2 ∈ B X Z i , 3 ∈ B  1 ≤ i < k  X Z i , 1 ∈ B X Z i , 2 ∈ B  k < i ≤ L P ( Z k ) " Y 1 ≤ i < k P ( Z i , 2 ) P ( Z i , 3 ) Y k < i ≤ L P ( Z i , 1 ) P ( Z i , 2 ) # X { N k + L ,..., N k + 2 L − 1 }∈ N L " k + 2 L − 1 Y t = k + L P ( N t | Z k , { Z i , d t − i + 1 L e } ) # log 2 P  Z k  Q k + 2 L − 1 t = k + L P ( N t | Z k , { Z i , d t − i + 1 L e } ) P Z k ∈ B P  Z k  Q k + 2 L − 1 t = k + L P ( N t | Z k , { Z i , d t − i + 1 L e } ) . (45) E. Derivation of ˆ Λ ( v ) in the M-Step of Channel Estimation The likelihood function is giv en by L ( N p | λ s i = ˆ λ ( v ) s i ) = T p X t = 1 log X s i ∈ B L \ L p P ( N p , S p t = s i | λ s i = ˆ λ ( v ) s i ) ≥ T p X t = 1 X s i ∈ B L \ L p Q ( v ) ( S p t = s i ) log P ( N p , S p t = s i | λ s i = ˆ λ ( v ) s i ) Q ( v ) ( S p t = s i ) . (46) Letting ˜ L ( v ) ( N p | λ s i = ˆ λ ( v ) s i ) denote the last term of abov e inequality , we have that ˜ L ( v ) ( N p | λ s i = ˆ λ ( v ) s i ) ∼ T p X t = 1 X s i ∈ B L \ L p Q ( v ) ( S p t = s i )  N p t log τ t ˆ λ ( v ) s i − log N p t ! − τ t ˆ λ ( v ) s i  . Hence, the partial deri vati v e of likelihood function is giv en by ∂ ∂λ ( v ) s i ˜ L ( v ) ( N p = n | λ s i = ˆ λ ( v ) s i ) = T p X t = 1 Q ( v ) ( S p t = s i ) N p t ˆ λ ( v ) s i − τ t ! . (47) 27 Letting ∂ ∂λ s j L ( N p = n | λ s j = ˆ λ ( v ) s j ) = 0, we hav e that ˆ λ ( v ) s j = arg max ˜ L ( v ) ( N p = n | λ s i = ˆ λ ( v ) s i ) = P T p t = 1 Q ( v ) ( S p t = s i ) N p t P T p t = 1 Q ( v ) ( S p t = s i ) τ t . (48) R eferences [1] Z. Xu and B. M. Sadler , “Ultraviolet communications: potential and state-of-the-art, ” IEEE Commun. Mag. , vol. 46, no. 5, May 2008. [2] C. Xu, H. Zhang, and J. Cheng, “E ff ects of haze particles and fog droplets on NLOS ultraviolet communication channels, ” Opt. Expr ess , vol. 23, no. 18, pp. 23259–23269, 2015. [3] Y . Sun and Y . Zhan, “Closed-form impulse response model of non-line-of-sight single-scatter propagation, ” JOSA A , vol. 33, no. 4, pp. 752–757, 2016. [4] N. Raptis, E. Pikasis, and D. Syvridis, “Power losses in di ff use ultraviolet optical communications channels, ” Opt. Lett. , vol. 41, no. 18, pp. 4421–4424, 2016. [5] K. W ang, C. Gong, D. Zou, and Z. Xu, “T urbulence channel modeling and non-parametric estimation for optical wireless scattering communication, ” IEEE / OSA J. Lightwave T echnol. , vol. 35, pp. 2746–2756, Jul. 2017. [6] M. Davis, “Capacity and cuto ff rate for Poisson-type channels, ” IEEE T rans. Inform. Theory , vol. 26, no. 6, pp. 710–715, Jun. 1980. [7] A. D. W yner , “Capacity and error exponent for the direct detection photon channel. I-II, ” IEEE T rans. Inform. Theory , vol. 34, no. 6, pp. 1449–1461, Jun. 1988. [8] M. R. Frey , “Information capacity of the Poisson channel, ” IEEE T rans. Inform. Theory , vol. 37, no. 2, pp. 244–256, Feb . 1991. [9] A. Lapidoth and S. M. Moser , “On the capacity of the discrete-time Poisson channel, ” IEEE T rans. Inform. Theory , vol. 55, no. 1, pp. 303–322, Jan. 2009. [10] A. Lapidoth, J. H. Shapiro, V . V enkatesan, and L. W ang, “The discrete-time Poisson channel at lo w input po wers, ” IEEE T rans. Inform. Theory , vol. 57, no. 6, pp. 3260–3272, Jun. 2011. [11] K. Chakraborty and P . Narayan, “The Poisson fading channel, ” IEEE T rans. Inform. Theory , vol. 53, pp. 2349–2364, Jul. 2007. [12] K. Chakraborty , S. Dey , and M. Franceschetti, “Outage capacity of MIMO Poisson fading channels, ” IEEE T rans. Inform. Theory , vol. 54, pp. 4887–4907, Nov . 2008. [13] L. Lai, Y . Liang, and S. S. Shitz, “On the capacity bounds for Poisson interference channels, ” IEEE T rans. Inform. Theory , vol. 61, pp. 223–238, Jan. 2015. [14] H. Kim, B. Nachman, and A. El Gamal, “Superposition coding is almost always optimal for the Poisson broadcast channel, ” IEEE T rans. Inform. Theory , vol. 62, pp. 1782–1794, Apr . 2016. [15] N. Mehravari and T . Berger , “Poisson multiple-access contention with binary feedback, ” IEEE T rans. Inform. Theory , vol. 30, no. 5, pp. 745–751, May 1984. 28 [16] A. Lapidoth and S. Shamai, “The Poisson multiple-access channel, ” IEEE T rans. Inform. Theory , vol. 44, no. 2, pp. 488–501, Feb. 1998. [17] S. I. Bross, M. V . Burnashev , and S. Shamai, “Error exponents for the two-user Poisson multiple-access channel, ” IEEE T rans. Inform. Theory , vol. 47, no. 5, pp. 1999–2016, May 2001. [18] G. W ang, C. Gong, and Z. Xu, “Signal characterization for multiple access non-line of sight scattering communication, ” IEEE T rans. Commun. , vol. 66, pp. 4138–4154, Sept. 2018. [19] D. Raychaudhuri, “Performance analysis of random access packet-switched code division multiple access systems, ” IEEE T rans. on Commun. , vol. 29, no. 6, pp. 895–901, 1981. [20] H. Ding, G. Chen, A. K. Majumdar, B. M. Sadler , and Z. Xu, “Correction to Modeling of non-line-of-sight ultraviolet scattering channels for communication, ” IEEE J. Select. Ar eas Commun. , vol. 29, pp. 250–250, Jan. 2011. [21] Y . Zuo, H. Xiao, J. Wu, W . Li, and J. Lin, “Closed-form path loss model of non-line-of-sight ultraviolet single-scatter propagation, ” Opt. Lett. , vol. 38, pp. 2116–2118, Dec. 2013. [22] C. Gong and Z. Xu, “Channel estimation and signal detection for optical wireless scattering communication with inter-symbol interference, ” IEEE T rans. Wir eless Commun. , vol. 14, pp. 5326–5337, Oct. 2015. [23] C. Gong and Z. Xu, “LMMSE SIMO receiv er for short-range non-line-of-sight scattering communication, ” IEEE T rans. W ir eless Commun. , vol. 14, pp. 5338–5349, Oct. 2015. [24] C. Gong and Z. Xu, “Non-line of sight optical wireless relaying with the photon counting receiver: A count-and-forward protocol, ” IEEE T rans. W ir eless Commun. , vol. 14, pp. 376–388, Jan. 2015. [25] L. Rabiner and B. Juang, “An introduction to hidden Markov models, ” IEEE ASSP Mag. , vol. 3, no. 1, pp. 4–16, Jun. 1986. [26] Y . Ephraim and N. Merhav , “Hidden markov processes, ” IEEE T rans. Inform. Theory , vol. 48, no. 6, pp. 1518–1569, Jun. 2002. [27] X. Liu, C. Gong, B. Liu, S. Li, and Z. Xu, “Hidden markov model based signal characterization for weak light communication, ” IEEE / OSA J. Lightwave technol. , vol. 36, pp. 1730–1738, Sept. 2018. [28] T . K. Moon, “The expectation-maximization algorithm, ” IEEE Signal Pr ocessing Mag. , vol. 13, no. 6, pp. 47–60, Nov . 1996. [29] A. P . Dempster , N. M. Laird, and D. B. Rubin, “Maximum likelihood from incomplete data via the EM algorithm, ” Journal of the r oyal statistical society . Series B (methodological) , pp. 1–38, 1977. [30] G. D. Forne y , “The viterbi algorithm, ” Pr oc. IEEE , vol. 61, no. 3, pp. 268–278, Mar . 1973. [31] L. Bahl, J. Cocke, F . Jelinek, and J. Raviv , “Optimal decoding of linear codes for minimizing symbol error rate (corresp.), ” IEEE T rans. Inform. Theory , vol. 20, no. 2, pp. 284–287, Feb . 1974. [32] J. Hagenauer, E. O ff er, and L. Papke, “Iterati ve decoding of binary block and con volutional codes, ” IEEE T r ans. Inform. Theory , vol. 42, no. 2, pp. 429–445, Feb . 1996. [33] S. Benedetto, D. Divsalar , G. Montorsi, and F . Pollara, “A soft-input soft-output APP module for iterativ e decoding of concatenated codes, ” IEEE Commun. Lett. , vol. 1, no. 1, pp. 22–24, Jan. 1997. [34] T . M. Cover and J. A. Thomas, Elements of information theory . John W iley & Sons, 2012. 29 [35] D. Hernando, V . Crespi, and G. Cybenko, “E ffi cient computation of the hidden Markov model entropy for a giv en observation sequence, ” IEEE T rans. Inform. Theory , vol. 51, no. 7, pp. 2681–2685, Jul. 2005. [36] M. Y ang, W . E. Ryan, and Y . Li, “Design of e ffi ciently encodable moderate-length high-rate irregular LDPC codes, ” IEEE T rans. Commun. , vol. 52, no. 4, pp. 564–571, Apr . 2004. [37] M. P . Fossorier , “Quasicyclic low-density parity-check codes from circulant permutation matrices, ” IEEE T r ans. Inform. Theory , vol. 50, no. 8, pp. 1788–1793, Aug. 2004. [38] J. Chen, A. Dholakia, E. Eleftheriou, M. P . Fossorier , and X.-Y . Hu, “Reduced-complexity decoding of LDPC codes, ” IEEE T rans. Commun. , vol. 53, no. 8, pp. 1288–1299, Aug. 2005.