Joint source-channel coded multidimensional modulation for variable-length codes

Transcription

1 . RESEARCH PAPER. SCIENCE CHINA Information Sciences June 2014, Vol : :12 doi: /s Joint source-channel coded multidimensional modulation for variable-length codes CHENG Chen, TU GuoFang & ZHANG Can School of Electronic, Electrical and Communication Engineering, University of Chinese Academy of Sciences, Beijing , China Received October 10, 2013; accepted February 6, 2014; published online March 24, 2014 Abstract Resource-constrained networks such as deep space communications and mobile communications have the phenomena such us large-scale path loss, small-scale fading, and multipath effect, which are very hostile for information transmission. Combining Shannon s source-channel coding theory with multidimensional modulation, an en/decoding method is proposed in this article called variable-length symbol-level joint sourcechannel and multidimensional modulation. Utilizing symbol-level source a priori information, an optimized symbol-level L-dimensional M-ary phase shift keying (LD-MPSK) constellation mapping function has been derived, which can increase free Euclidean distance between modulated symbols and enhance the decoding performance. It has been shown in the simulation results that the proposed scheme can achieve more than 2dB signal-to-noise ratio gains compared with the conventional bit-level schemes under the condition of the same error rate. Keywords joint source-channel en/decoding, variable-length codes, multidimensional modulation, setpartitioning, constellation mapping Citation Cheng C, Tu G F, Zhang C. Joint source-channel coded multidimensional modulation for variablelength codes. Sci China Inf Sci, 2014, 57: (12), doi: /s Introduction In classical communication systems, source coding, channel coding, and modulating are usually implemented sequentially and independently, according to Shannon s source-channel separation theory [1]. Source coding is responsible for compressing data, which improves data transmission efficiency by removing data correlation and residual redundancy. Variable-length coding (VLC) is an effective technique to remove source redundancy, which is widely used in existing image/video coding standards such as the Joint Photographic Experts Group (JPEG), the Moving Picture Experts group (MPEG), and H.26x. However, it is very sensitive to channel errors and even a single-bit error may lead to large error propagation. The objective of channel coding is to correct channel errors, which improves the reliability of data transmission by adding some check bits. Most existing international channel coding standards, such as turbo codes and low density parity check (LDPC) codes, are bit-level en/decoding techniques. Therefore, when they are employed after symbol-level source coding such as variable-length codes (VLCs), they cannot efficiently use symbol-level source a priori information (SAI) to increase the free Hamming distance Corresponding author ( gft@ucas.ac.cn) c Science China Press and Springer-Verlag Berlin Heidelberg 2014 info.scichina.com link.springer.com

2 Cheng C, et al. Sci China Inf Sci June 2014 Vol :2 (FHD) between decoding paths. Modulation is responsible for improving bandwidth efficiency under limited bandwidth and power constraints. The existing modulation international standards such as M- ary phase shift keying (MPSK), M-ary quadrature amplitude modulation (MQAM), and L-dimensional M-ary phase shift keying (LD-MPSK) are widely used in mobile communications and deep space communications. However, they are all bit-level modulation techniques, which are not effective to exploit the characteristics of variable-length source symbols and as a result will not increase the free Euclidean distance (FED) between modulated symbols. Shannon s source-channel separation theory [1] states that there is an optimal code that can achieve arbitrarily small transmission error under the assumption that the transmission rate R is less than the channel capacity C. It also indicates that the optimal performance can be achieved by designing source coding and channel coding separately. However, the separation theory holds only with the following asymptotic conditions: 1) both source codes and channel codes are allowed infinite length and complexity; 2) the transmission conditions of a channel are well learned; and 3) it only holds for point-to-point communication systems. Practical systems are constrained by complexity and delay, which cannot meet these assumptions. In particular, Shannon s separation theory can hardly be proved for some practical systems [2]. These arguments have motivated the active research areas of joint source-channel coding/decoding (JSCC/JSCD) [3]. In recent years, VLCs are widely used in various image/video coding standards [3 7]. Using VLCs, the symbols that occur more frequently are assigned short code words and those that occur infrequently are assigned long code words. Following this general philosophy, we may devise a proper method to select and assign the code words for each source symbol to reach an optimal source code. Liu and Tu [4] noticed that the bit stream after VLC still has some residual redundancy, which can be used as source a priori information for error check and correction through iterative channel decoding. Following this idea, an efficient approach of JSCD for variable-length encoded turbo codes [4] was suggested, which used the symbol-by-symbol a posteriori probability (APP) decoding algorithm based on a symbol-level VLC trellis. Compared with the bit-level approach, this symbol-based approach achieved about 1.2 db coding gains in symble error rate (SER). Unfortunately, this approach has high decoding complexity, which will exponentially grow with the number of source symbols. For the purpose of decreasing the number of source symbols and increasing controlled source residual redundancy, a symbol-level JSCD approach was suggested in [4, 8]. This approach decreased the number of source symbols to 12 variable-length symbols. Using the a priori information of these variable-length symbols and adding controlled source residual redundancy simultaneously, the FHD of decoding paths was increased and the decoding performance was improved. The performance of communication systems depends on the source and channel en/decoding methods used. Besides, it can be easily affected by the FED of modulated symbols. To achieve further improvement, we consider modulation as an integral part of the encoding process and design in conjunction with symbol-level JSCD to increase the minimum Euclidean distance between pairs of coded signals. We have observed that most modulation techniques (such as MPSK and MQAM) are based on fixedlength bit-level modulation and do not provide error control and correction. Considering the use of coded signals for bandwidth-constrained channels, a performance gain would be expected without expanding the signal bandwidth, which can be achieved by increasing the number of signals over the corresponding uncoded system to compensate for the redundancy introduced by the code. This kind of coding is called coded modulation (CM), which mainly consists of trellis coded modulation (TCM) [9] and block coded modulation (BCM) [10]. TCM combines the functions of a convolutional coder of rate R = k/(k + 1) and an M-ary signal mapper that maps M = 2 k input points into a larger constellation of M = 2 k+1 constellation points. A coding gain is obtained by doubling the number of constellation points of the signals and set-partitioning of the constellation. Moreover, a method for increasing the dimensionality of TCM signals was suggested in [11]. This method was called trellis coded multidimensional phase modulation. The main concept in multidimensional TCM was to increase the number of symbols created in one processing period, with a number of coded bits being mapped to a vector of symbols (instead of a single symbol). Flexibility and higher bandwidth efficiency were obtained from this method. In addition,

3 Cheng C, et al. Sci China Inf Sci June 2014 Vol :3 better bit efficiency was possible. BCM is a CM method that combines the functions of a block coder and an M-ary signal mapper. It is similar to TCM yet has lower complexity using a block coder rather than a convolutional coder. In recent years, BCM is designed in conjunction with space-time codes, namely space-time BCM [12,13]. These approaches achieved low complexity and high bandwidth efficiency. Compared with BCM, TCM achieved a higher coding gain, however, with higher decoding complexity. On the contrary, BCM achieved a lower coding gain with relatively lower decoding complexity. In this article, we add a set-partitioning method of multidimensional TCM in the structure of BCM, and large coding gains have been obtained with low decoding complexity. In consideration of existing joint en/decoding approaches, symbol-level joint source-channel encoding/jscd approaches are more suitable for VLCs. However, they were not combined with modulation. Meanwhile, traditional CM approaches only fit to fixed-length coding methods, which cannot achieve optimal performance for VLC methods. The objective of this article is to present an approach that can jointly consider source-channel coding and multidimensional modulation for VLCs. Based on LD-MPSK, this approach uses the symbol-level SAI and adds controlled source residual redundancy to optimize symbol-level LD-MPSK constellation mapping. This joint coding approach plays roles of both channel encoding and modulating. By increasing both the FHD between decoding paths and the FED between modulated symbols, the transmission performance of a communication system can be improved and a joint en/decoding system with high coding gain and low decoding complexity is realized. 2 Symbol-level LD-MPSK modulation LD-MPSK modulation has been proposed in [11], where L denotes L dimensions of MPSK signals. The number of symbols transmitted in one processing period is equal to L. The procedures of conventional LD-MPSK modulation are as follows: 1) set-partitioning for LD- MPSK constellation points to increase the minimum Euclidean distance; 2) constellation mapping for the information bits; and 3) transmitting constellation points to the wireless channel. The basic idea of the symbol-level LD-MPSK modulation comes from the conventional LD-MPSK and symbol-level SAI of VLCs will be used. The symbols will be assigned in a proper way to improve the coding gain. That is, after constellation mapping, the interconstellation distance between points that occur frequently is larger than the interconstellation distance between points that occur infrequently. 2.1 Symbol-level adaptive coding Symbol-level adaptive coding is implemented for the purpose of reducing the number of variable-length symbols after variable-length source coding. It is concatenated after conventional variable-length source coding, which reduces the number of variable-length symbols to 12 based on the approach suggested in [8]. First, we set a number N, which is the number of variable-length symbols. The decoding complexity will be exponentially increased with the increase in N, while the statistical feature of the information bit stream after conventional source coding is related to the selection of N. Making a trade-off, in this study, we set N = 12 based on the suggestion given in [8], where the variable-length coded symbols are transformed into very few symbols using alternating run-length encoding. Second, symbol-level coding is implemented. A symbol sequence with symbol length n is generated as {s 1,s 2,...,s k,...,s n }, where s k {1,2,...,N} is the value of the kth symbol of the symbol sequence. Then, the a priori probabilities ofthese N kinds ofsymbolswillbe calculatedasp I = {p(1),p(2),...,p(m),...,p(n)},m {1,2,...,N}, which will be used as a symbol-level SAI for the following en/decoding. By symbol-level adaptive coding, the number of symbols in the information stream can be reduced to 12, that is, N = 12. In addition, symbol-level adaptive coding makes source residual redundancy in symbol sequence controllable. Symbol-level SAI and controlled source residual redundancy are both provided for decoding and demodulating, which will simultaneously increase FHD between decoding

4 Cheng C, et al. Sci China Inf Sci June 2014 Vol :4 paths and FED between modulated symbols. As a result, the decoding performance can be significantly enhanced. 2.2 Symbol-level constellation mapping of LD-MPSK We use a symbol-level constellation mapping method for the symbol sequence {s 1,s 2,...,s k,...,s n }, where s k is the kth symbol of a symbol sequence. This mapping method will be described as the following procedures. At the initial stage, we calculate the values of L and M for LD-MPSK through the number of symbols N. Every symbol corresponds to at least one constellation point (we use the one-to-one correspondence). Hence, we obtain the relation of N, L, and M as N M L. To increase the minimum Euclidean distance between constellation points, we set N < ML 2. (1) At the second stage, we partition the constellation into subsets in a way that the minimum Euclidean distance between signal points in a subset is increased 1). Denote p as the times of partitions, which should satisfy N M L /2 p < 2N. We then obtain M L log 2 2N < p log M L 2 N. (2) Take N = 12, for example. We have M L > 2N = 24 according to (1). Note that L denotes the dimensions of MPSK signals, and M are usually values expressed as the integer times of 2. We choose the values L and M on condition that the decoding complexity is as small as possible, that is, L = 2 and M = 8. Then, we have p = 2 according to (2). In a word, 2D-8PSK modulation will be implemented in the case of N = 12, and the partition times p = 2. After the pth level of partitioning, the LD-MPSK constellation with M L signal points is partitioned into M L /2 p subsets, where every subset has 2 p constellation points. The number of constellation points in one subset is denoted as N = ML 2 p. (3) One subset from the 2 p subsets will be arbitrarily selected as the constellation for symbol-level constellation mapping. By this means, every source symbol from the symbol sequence can be mapped to at least one multidimensional constellation point. There are redundant constellation points in the subset for the reason that N N < 2N. Hence we use the redundancy and the symbol-level SAI to design the constellation mapping function. According to the importance of symbols, symbols that have larger a priori probabilities are more important than those that have smaller a priori probabilities. Symbols are mapped to the constellation points so that the interconstellation distance between symbols that occur frequently is larger than the interconstellation distance between symbols that occur infrequently. The FED between modulated symbols will be increased by this approach, and thus the decoding performance can be enhanced. 3 Transmission scheme of joint source-channel coded multidimensional modulation The transmission scheme of joint source-channel coded multidimensional modulation is shown in Figure 1. At the encoder side, a symbol-level joint coding and modulating structure is proposed. A conventional source coder is first sequentially concatenated with the symbol-level joint coder and modulator, which is combined with a symbol-level adaptive coder and a symbol-level LD-MPSK modulator. At the decoder side, a corresponding symbol-level joint decoder and demodulator structure is proposed. 1)

5 Cheng C, et al. Sci China Inf Sci June 2014 Vol :5 Symbol-level joint coder and modulator Conventional source coder S Symbol-level adaptive coder s k Symbol-level LD-MPSK modulator x k p(s k ) Symbol-level SAI P I n k Channel Symbol-level joint decoder and demodulator H I y k Conventional source decoder Symbol-level Symbol adaptive decision S decoder L DM (y k ) s k Symbol-level LD-MPSK demodulator Figure 1 Transmission scheme of joint source-channel coded multidimensional modulation. At the encoder side, after conventional source coding the information stream S is used as the input of the symbol-level joint coder and modulator. The output of the symbol-level adaptive coder is the symbol sequence {s 1,s 2,...,s k,...,s n }, s k {1,2,...,N}. s k is the input ofthe symbol-levelld-mpskmodulator, which ismapped toacomplexchannelsymbol x k chosen from the LD-MSPK constellation χ by a constellation mapping function f where x k = f(s k ),x k χ, (4) χ = {(e jl12π/m,...,e jll2π/m ),l 1,...,l L = 0,...,M 1}. (5) Eq. (5) is the LD-MSPK constellation signal set. The constellation mapping function is designed using the symbol-level SAI P I, which will be discussed later. Transmitted over the channel, the received signal is y k = ρ k Es y k +n k, (6) where ρ k is the fading coefficient, E s is the symbol energy, and n k is the complex additive white Gaussian noise (AWGN) with one-sided spectral density N 0. In this article, we assume the channel as the AWGN channel, ρ k = 1, and the symbol energy E s = 1. Then, (6) is simplified as y k = x k +n k. (7) At the decoder side, the received signal y k will be further processed by the proposed symbol-level joint decoding and demodulating. The modulation soft-decision information L DM (y k ) can be obtained from the output of the symbol-level LD-MPSK modulator. Another symbol-level SAI H I will be used as the correcting information to modify L DM (y k ), which produces corrected soft-decision information L DM (y k). H I is generated from the source a priori information. Different from P I, which contains the absolute probability values of all N kinds of symbols, H I reflects the descending order of the probabilities containing the relative information of probability values. The corrected soft-decision information L DM (y k) is used for symbol decision, which generates the decoded symbol s k. After that, the decoding symbol sequence { s 1, s 2,..., s k,..., s n } is inputted to the symbol-level adaptive decoder, which finally gets the decoded information sequence S. 3.1 Derivation of symbol-level LD-MPSK constellation mapping functions As discussed above, the constellation mapping function f is optimized using the symbol-level SAI P I. In this section, we will discuss about the theory derivation of symbol-level LD-MPSK constellation mapping

6 Cheng C, et al. Sci China Inf Sci June 2014 Vol :6 functions. We have derived optimal/suboptimal mapping functions, which can meet different resource allocation requirements Optimal mapping p(s k y k ) is the probability of transmitting symbol s k on the condition that y k is received, that is, the APP of the transmission system, where y k = x k +n k = f(s k )+n k = g(s k ). According to Bayes formula, p(s k y k ) = p(y k s k ) p(s k ). (8) p(y k ) Given the received signal y k = g(s k = m) = gk m,m {1,2,...,N}, we obtain p(s t k y k = g m k ) = p(y k = g m k st k ) p(t) p(y k = g m k ), (9) where s t k is short for s k = t and p(t),t {1,2,...,N} is the source a priori probability of symbol t. The likelihood ratio is denoted as λ m (m,t) = p(sm k y k = g m k ) p(s t k y k = g m k ). (10) Plugging (9) into (10) yields the result λ m (m,t) = p(y k = fk m +n k x k = fk m) p(m) p(y k = fk m +n k x k = fk t, (11) ) p(t) where f m k is short for f(s k = m). Note that n k follows Gaussian distribution, then (11) is expressed as λ m (m,t) = exp{ 1 N 0 ( f m k f t k 2 2 n k f m k f t k ) } p(m) p(t). (12) During natural logarithm processing of (12), the log-likelihood ratio is which is a soft values of the receiving signal, and let N L m = L m (m,t) = lnλ m (m,t), (13) t=1,t m L m (m,t) (14) be the sum ofsoftvalue between receivingsignalmandthe othern 1receivingsignals, respectively. Assuming the symbol sequence is long enough, the frequency of occurrence of every symbol is approximately equal to the corresponding a priori probability. Hence L = N L m p(m), (15) m=1 which can be used as the measurement for the SER of demodulating. SER decreases with increasing L. The maximum of (15) is N N L max = max L m (m,t) p(m). (16) m=1t=2,t>m Plugging (14) into (15) yields the simplified form of (16) as L max max f N N m=1t=2,t>m f m k f t k 2 p(m), (17) which is a function of f. Then, we can obtain the mapping function x k = f(s k ) by maximizing L. Note that m,n {1,2,...,N}, and the computational complexity of (17) is (N!) 2 orders of magnitude. If the number of symbols N is large enough, the computational complexity of (17) is huge (e.g., when N 12,(N!) ). Hence, we have derived a suboptimal mapping function for the case of large N.

7 Cheng C, et al. Sci China Inf Sci June 2014 Vol :7 1 2 i N 1 N N 1 N 2 N P Figure 2 Permutation and grouping of the N symbols Suboptimal mapping For the case of large N, a suboptimal mapping method is considered as follows. At the first stage, we sort all of the N symbols by their source a priori probabilities in descending order as p(s k = 1) p(s k = 2) p(s k = i) p(s k = N), (18) where {1,2,...,N} is the permutation of {1,2,...,N}. Second, we divide {1,2,...,N} into q groups where N = N 1 +N 2 + +N q, as shown in Figure 2. The next stage is to map each symbol in a group to a constellation point, which will be described in detail as follows. First, the N 1 symbols in the first group will be mapped. That is, plug N 1 into (17) L max1 max f N 1 N 1 m=1t=2,t>m Second, the N 2 symbols in the second group will be mapped and L max2 max f N 2 N 2 m=n 1+1t=N 1+2,t>m f m k f t k 2 p(m). (19) f m k f t k 2 p(m). (20) After that, the rest groups will be mapped in the same way, and the mapping function x k = f(s k ) for the whole system is finally obtained. By this means, the number of symbols in one group is decreased, and as a result the computational complexity is decreased. However, this grouping method for optimizing mapping function is not an entirety design for N symbols; it is locally optimum for every group, yet suboptimal for the whole system. Along with the increase in the number of groups, the computational complexity will be decreased. However, the performance of the mapping function will be degraded. There is a trade-off between the computational complexity and the system performance. Choosing a proper number of groups according to the number of symbols and the experimental conditions, we can achieve the optimum allocation between performance and resource. 3.2 Symbol-level joint demodulating and decoding The soft-decision information of demodulation when receiving y k is L DM (y k ) = y k f m k 2, m {1,2,...,N}. (21) Optimizing (21) by the other symbol-level SAI H I, we obtain L DM(y k ) = max m (L DM(y k ) h(m)) = max m ( y k f m k 2 h(m)), (22) where correcting function h(m) H I = {h(1),h(2),...,h(n)},m {1,2,...,N} is a decreasing function of m {1,2,...,N}. The correcting information H I is the other symbol-level SAI that is different from P I, which reflects the descending order of the probabilities. Hence, it contains the relative information of probabilityvalues. Thesoft-decisioninformationiscorrectedbyH I, whichleadstoafurtherimprovement in system performance. The expression of H I can be obtained through computer simulation.

8 Cheng C, et al. Sci China Inf Sci June 2014 Vol : d min Figure 3 A subset of 2D-8PSK constellation (a) (b) Figure 4 2D matrix representation of 2D-8PSK constellation. (a) 2D-8PSK; (b) the subset of 2D-8PSK. 4 Simulation results To evaluate the performance of the proposed joint source-channel coded multidimensional modulation approach, we have performed three simulation experiments and compared the proposed symbol-level joint approach with other existing bit-level separate approaches. We set the symbol number N = 12. Then, according to (1), we set L = 2, M = 8, which means 2D-8PSK modulation is chosen for simulations. To increase simulation efficiency, we omit the procedure of symbol-level adaptive coding for information bit stream. Instead, an independent memoryless symbol sequence {s 1,s 2,...,s k,...,s n } with 12 symbols is randomly generated, where the probability of the mth symbol is 1/2 m, m = 1,2,...,N 1, N 1 p(m) = (23) 1 p(i), m = N. i=1 The 2D-8PSK constellation is shown in Figure 3, where the two 8PSK constellations are the two dimensions of the 2D-8PSK constellation. One 2D-8PSK modulated symbol is combined with two 8PSK modulated symbols selected from the two 8PSK constellations separately. The 2D-8PSK modulated symbols are expressed in a 2D matrix shown in Figure 4(a). According to (2), we get p = 2. After 2 times of set-partitioning for 2D-8PSK constellation, each 8PSK constellation is divided into two subsets (shown in Figure 3 as the filled dots and the empty dots, respectively). Therefore, four signal subsets are generated with 16 2D-8PSK constellation points for every subset. One of the constellation subsets will be arbitrarily selected as the corresponding constellation points for 12 symbols. In this study, the subset we have chosen for the simulation is shown in Figure 4(a) in bold and in Figure 4(b) as well, which is combined with the filled dots shown in Figure 3. Apparently, we have N = 16, according to the matrix of the subset shown in Figure 4(b). The expression of the correcting function h(m) H I is

9 Cheng C, et al. Sci China Inf Sci June 2014 Vol : (a) (b) Figure 5 Two different mapping functions for performance comparison. (a) Mapping function f 1 ; (b) mapping function f 2. h(m) = m α, m {1,2,...,N}, (24) where the value of the index α will greatly affect the decoding performance. We will discuss the effect of α in Exp. 3. The minimum Euclidean distance between constellation points of bit-level 2D-8PSK d 2 bit is shown in Figure 4(a) with line segment, that is, the square distance between the points 00 and 01. Similarly, the minimum Euclidean distance between constellation points of symbol-level 2D-8PSK d 2 sym is shown in Figure 4(b), that is, the distance between the points 00 and 02. Assume that the power of every constellation point in 8PSK constellation is 1, thus the minimum Euclidean distance of 8PSK constellation is d 2 min = d2 01 = 4sin 2 (π/8) = 0.586, which is shown in Figure 3. d 2 01 is the square distance between the constellation points 0 and 1 of 8PSK constellation. Now we have d 2 bit = d d 2 01 = = 0.586, d 2 sym = d2 00 +d2 02 = 0+2 = 2. (25) The minimum Euclidean distance of symbol-level 2D-8PSK is increased through set-partitioning and the selection of subsets. Moreover, the constellation mapping is crucial for the distance property, which is carried out for the symbol sequence. In our simulations, we select two different mapping functions for performance comparison. f 1 is derived from the direct mapping, which maps the 12 symbols to the first 12 constellation points out of 16 points without using the a priori information. This mapping method is shown in Figure 5(a), where stands for the redundant constellation points without mapping symbols. f 2 is derived from the suboptimal mapping method proposed in Section 3, which is obtained by maximizing L according to (17). Twelve symbols are divided into three groups with four symbols in every group, that is, N = N 1 +N 2 +N 3 = This mapping method is shown in Figure 5(b). Different simulation results will be obtained with different mapping functions, which will be discussed later in Exp Exp. 1: comparison with bit-level separate system This experiment is designed to compare the proposed symbol-level joint system with a bit-level separate system. We assumethe channelisan AGWNchannel, the length ofthe tested symbolsequence with 12different symbols is , and the probabilities of 12 symbols are shown in(23). For the bit-level separate system, the symbol sequence is coded as fixed-length symbols. The bit-level 2D-8PSK modulation is carried out, referred to Table 1 in [14]. For the symbol-level joint system, the proposed joint system shown in Figure 1 is used with the symbol-level 2D-8PSK. The mapping function is shown in Figure 5(a). Simulations are carried out for each E b /N 0 in db, where E b denotes the averageenergyper information bit and N 0 is the one-sided AWGN spectrum density. Figure 6 shows the SER results of both the bit-level

10 Cheng C, et al. Sci China Inf Sci June 2014 Vol : Bit 2D-8PSK [14] Symbol 2D-8PSK Alamouti 8PSK [12] Modified Alamouti 16QAM [13] Joint Alamouti 2D-8PSK Symbol error rate Symbol error rate E b /N 0 (db) Figure 6 Simulation results for bit-level separate system and symbol-level joint system E b /N 0 (db) Figure 7 The SER performance of Alamouti 8PSK, modified Alamouti 16QAM, and the proposed joint Alamouti 2D-8PSK. separate system and the symbol-level joint system. The results confirm to the distance property in (25). It has been found out that the proposed symbol-level joint system outperforms the bit-level separate system at all signal-to-noise ratio (SNR) levels. Hence, the proposed system is suitable for VLC codes. 4.2 Exp. 2: comparison with STBC fixed-length coding system In this experiment, we take the classical Alamouti space-time block coding (STBC) [12,13] into consideration. Alamouti STBC is a class of STBC, which is a simple two-branch transmit diversity scheme. Modified Alamouti 16QAM method [13] is based on the original Alamouti 8PSK method [12] and expands 8PSK constellation into 16QAM constellation by doubling the number of constellation points. As a result, the minimum Euclidean distance is increased and the coding gain is obtained compared with the original Alamouti method. We combine the proposed joint approach with Alamouti STBC system as joint Alamouti 2D-8PSK. Using the symbol-level SAI and the symbol-level 2D-8PSK modulation, the minimum Euclidean distance between constellation points and the FED between modulated symbols can be further increased. Simulations are carried out over a Rayleigh fading channel. The length of the tested symbol sequence with 12 different symbols is , and the probabilities of 12 symbols are shown in (23). We have set the same mode of two-branch transmit and one receiver for Alamouti 8PSK, modified Alamouti 16QAM, and the proposed joint Alamouti 2D-8PSK. Simulation results are shown in Figure 7. Because of the expansion of constellation, modified Alamouti 16QAM achieves about 1dB coding gains at the same SER compared with Alamouti 8PSK. Further on, the proposed joint Alamouti 2D-8PSK achieves about 2 db coding gains at the same SER compared with the modified Alamouti 16QAM, taking advantage from the combination of symbol-level 2D-8PSK. 4.3 Exp. 3: performance evaluation for parameters This experiment discusses about the impact of joint system parameters for system performance The impact of mapping function f and correcting function h Simulations are carried out in an AWGN channel, the length of the tested symbol sequence with 12 different symbols is , and the probabilities of 12 symbols are shown in (23). Figure 8(a) shows the simulation results of different combinations of mapping functions and correcting functions. There are four pairs of {f,α}, where f {f 1,f 2 } and α {0,0.5}. (α is the correcting factor, α = 0 means the correcting function h(m) = m 0 = 1 is constant and has no contribution to the system.)

11 Cheng C, et al. Sci China Inf Sci June 2014 Vol :11 Symbol error rate f 1, α = 0 f 1, α = 0.5 f 2, α = 0 f 2, α = 0.5 Rebuild symbol error rate f 1, α = 0 f 1, α = 0.5 f 2, α = 0 f 2, α = E b /N 0 (db) (a) E b /N 0 (db) (b) Figure 8 The performance with {f, α} pairs. (a) SNR performance; (b) RSNR performance = 0 = 0.25 α αα α = 0.5 = = 0 = 0.25 α αα α = 0.5 = 0.75 Symbol error rate Rebuilt symbol error rate E b /N 0 (db) (a) E b /N 0 (db) (b) Figure 9 The performance with different α. (a) SNR performance; (b) RSNR performance. Comparing the curves of the mapping functions f 1 and f 2 with the same correcting factor, the system with mapping function f 2 outperforms the system with mapping function f 1 about 1 2dB coding gains. These coding gains are achieved by the symbol-level suboptimal mapping method proposed in Section 3. With the same mapping function f, the system with a correcting function achieves further improvement. Comparing the curves of the correctingfactors α = 0 and α = 0.5 with the same mapping function, the system with correcting factor α = 0.5 outperforms the system with correcting factor α = 0 at low SNR and high SER levels. However, at high SNR and low SER levels, the system with correcting factor α = 0 yields better decoding performance. Thus, the correcting factor works well at high SNR level. However, the performance improvement reduces along with the increase in SNR. We have found that the best decoding performance is achieved at the {f 2,α = 0.5} pair, which is the combination of the suboptimal mapping function and the correcting function. Figure 8(b) shows the simulation results of rebuilt SNR (RSNR). The {f 2,α = 0.5} pair yields the best RSNR performance as well, especially at high SNR level The impact of correcting factor α Simulation conditions are the same as in (1). The parameters of the system are as follows: α {0,0.25,0.5,0.75} with fixed f. The simulation curves of the SER performance and the RSNR performance are shown in Figure 9(a) and Figure 9(b), respectively.

12 Cheng C, et al. Sci China Inf Sci June 2014 Vol :12 We have found that the best SER performance is achieved at α = 0.5, while the best RSNR performance is achieved at α = Considering both SER and RSNR performances, the integrated optimal performance of the system is achieved at α = 0.5. In this experiment, we only discussed about the correctingfactorwith the expressionh(m) = m α ; otherkinds ofexpressionsmay achievebetter performance. This will be considered in future research. 5 Conclusion In this article, an optimized symbol-level multidimensional constellation mapping function is derived using the symbol-level SAI. Through symbol-level LD-MPSK mapping, the FED between modulated symbols is increased. We have also used symbol-level adaptive coding to add controlled residual redundancy, which leads to the increase in FHD. Then, we propose a joint source-channel coded multidimensional modulation scheme for VLCs. It has been demonstrated from the simulations that the proposed scheme outperforms conventional bit-level schemes about 2 db for VLCs. Therefore, the proposed scheme can be applied to bandwidth-efficient and robust transmission of variable-length coded image and video data. Acknowledgements This work was supported by the National Natural Science Foundation of China (Grant Nos , ) and the Award Foundation of Chinese Academy of Sciences (Grant No ). References 1 Shannon C E. A mathematical theory of communication. Bell Syst Tech J, 1948, 27: , Vembu S, Verdu S, Steinberg Y. The source-channel separation theorem revisited. IEEE Trans Inf Theory, 1995, 41: Tu G F, Liu J J, Zhang C. Studies and advances on joint source-channel encoding/decoding techniques in flow media communications. Sci China Ser F-Inf Sci, 2010, 53: Liu J J, Tu G F, Zhang C, et al. Joint source and channel decoding for variable length encoded turbo codes. Eurasip J Adv Signal Process, 2008, 2008: Wu J, Chen S Z. Integrated joint source-channel symbol-by-symbol decoding of variable-length codes using 3-D MAP sequence estimation. Wuhan University J Nat Sci, 2007, 12: Kliewer J, Thobaben R. Iterative joint source-channel decoding of variable-length codes using residual source redundancy. IEEE Trans Commun, 2005, 4: Tervo V, Matsumoto T, Karjalainen J. Joint source-channel coding using multiple label mapping. In: Proceedings of 2010 IEEE 72nd Vehicular Technology Conference Fall, Ottawa, Tu G F, Liu X M, Zhang C, et al. Joint source-channel coding/decoding by combining RVLC and VLC for CCSDS IDC coefficients. In: Proceedings of 2010 International Conference on Networking, Sensing and Control, Chicago, Ungerboeck G. Channel coding with multilevel/phase signals. IEEE Trans Inf Theory, 1982, 28: Imai H, Hirakawa S. A new multilevel coding method using error-correcting codes. IEEE Trans Inf Theory, 1977, 23: Pietrobon S, Deng R, Laafanechere A, et al. Trellis-coded multidimentional phase modulation. IEEE Trans Inf Theory, 1990, 36: Alamouti S. A simple transmit diversity technique for wireless communications. IEEE J Sel Areas Commun, 1998, 16: Janani M, Nosratinia A. Single-block coded modulation for MIMO systems. IEEE Trans Commun, 2009, 57: Tran N H, Nguyen H H. A novel multi-dimensional mapping of 8-PSK for BICM-ID. IEEE Trans Wirel Commun, 2007, 6: