Irregular Mapping and its Application in Bit-Interleaved ... - IEEE Xplore

IEEE TRANSACTIONS ON BROADCASTING, VOL. 57, NO. 3, SEPTEMBER 2011

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Irregular Mapping and its Application in Bit-Interleaved LDPC Coded Modulation With Iterative Demapping and Decoding Zaishuang Liu, Kewu Peng, Tao Cheng, and Zhaocheng Wang, Senior Member, IEEE

Abstract—This paper proposes an irregular mapping technique, where a new labeling searched via modified adaptive binary switch algorithm (ABSA), mixed with a pre-fixed labeling, can provide near-optimal match to a given outer channel code. By using the proposed technique, the bit-interleaved low-density parity-check (LDPC) coded modulation systems with iterative demapping/decoding (BILCM-ID) could achieve near-capacity performance. With a slight modification on the LDPC coded modulation scheme of DVB-T2 standard, the proposed technique can improve the iterative demapping system by exploiting significant iterative gain. The superior performance is verified via extrinsic information transfer (EXIT) chart analysis and bit error rate (BER) simulation. Index Terms—Adaptive binary switch algorithm (ABSA), bitinterleaved coded modulation with iterative demapping/decoding (BICM-ID), irregular mapping, low-density parity-check (LDPC).

I. INTRODUCTION

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EW generation broadcasting systems require extension of coverage, improvement of receiving sensitivities and increase of transmission rates, which inevitably demands new coded modulation techniques providing high performance and efficiency. In recent years, bit-interleaved coded modulation (BICM) [1], [2] and its iterative demapping/decoding (ID) scheme [3]–[6] have presented a considerable momentum of further research, development and application. BICM is a bandwidth- and power-efficient technique consisting of serial concatenation of channel coding, bitwise interleaving, and constellation mapping [1]. Owing to its bitwise interleaver which introduces a relatively large diversity order, BICM can provide good error performance over fading channels [1], [2]. While in the channel coding area, low-density parity-check (LDPC) codes [7]–[9] have gained a lot of attention because of their design flexibility, decoding simplicity, high-throughput and especially the capability of achieving near-capacity performance, and therefore have been investigated and applied widely. As the key techniques, BICM and LDPC codes have been employed respectively by several communication/broadcasting standards represented by DVB-T2 [10]. BICM with iterative demapping/decoding (BICM-ID) was proposed by Li et al. [3], [4] and ten Brink et al. [5], [6] inde-

Manuscript received February 23, 2011; revised May 12, 2011; accepted May 19, 2011. Date of publication July 25, 2011; date of current version August 24, 2011. This work was supported by Standardization Administration of the People’s Republic of China (SAC) with AQSIQ Project 200910244. The authors are with Tsinghua National Laboratory for Information Science and Technology (TNList) and also with the Department of Electronic Engineering, Tsinghua University, Beijing 100084, China (e-mail: [email protected]; [email protected]; [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TBC.2011.2159530

pendently, wherein convolutional codes are usually employed as the channel codes. Unlike BICM with independent demapping, where Gray mapping is optimal in typical signal-to-noise ratio (SNR) region [2], the optimal constellation mapping, i.e., bit-to-symbol mapping (or called labeling), is relevant to the outer channel code for BICM-ID. Benefiting from the iterative gain between the demapper and decoder, BICM-ID may certainly outperform its BICM counterpart. It was soon recognized that the labeling is crucial for the BICM-ID design, especially to achieve near-capacity performance. A lot of previous work has been done to seek a good pair of channel code and constellation mapping [11]–[23], wherein extrinsic information transfer (EXIT) chart [12]–[15] was used as a powerful analysis tool. Early studies in the constellation mapping of conventional BICM-ID systems focused on designing the mapping rules according to certain criteria, or selecting the most appropriate one from several typical mapping rules [11], [12], [16]–[20]. However, there still remains a big distance away from the channel capacity especially in the schemes without the collaboration of unity-rate codes (URC), or called doping codes, which could remove the high error-floor in the conventional BICM-ID systems [21]. For labeling optimization, an important progress is the efficient systematic labeling search algorithm represented by binary switch algorithm (BSA) [22] and adaptive BSA (ABSA) [23]. Its contribution is to increase the degree of freedom (DOF) to obtain the labeling matching the channel code very well. Especially ABSA, which is closely associated with EXIT chart analysis, can find a labeling well matching the outer code and provide near-capacity performance to the conventional BICM-ID system with doping [23]. However, the labeling search algorithms so far have only been used to search the labeling of regular mapping, i.e., to employ only one labeling for a specific constellation. On the other hand, irregular mapping, i.e., to employ more than one labeling for a constellation, has been developed to optimize BICM-ID in a different way. Comparing with regular mapping, irregular mapping can provide an improved link adaptation capability and an increased design freedom to the BICM-ID system. So far, some irregular mapping schemes have been proposed [24]–[27], and it has been proved that the technique can also provide near-capacity performance to the conventional BICM-ID system. However, the problems such as how to simplify the irregular mapping scheme, how to find a good irregular mapping systematically, and how to reduce the computational complexity during labeling search, still need a further investigation. Although the conventional convolutional codes have been investigated in detail, there are other alternatives well suited for BICM-ID user scenarios. With the increasing demands of the transmission rate and quality, the application of high-order mod-

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ulation and high-performance channel code has become an inevitable trend of the development of coded modulation systems. The coded modulation scheme of DVB-T2 [10], as a typical example, employs not only high-order quadrature amplitude modulation (QAM) such as 64QAM and 256QAM, but also LDPC codes as its channel codes. As LDPC codes become more and more popular, the combination of BICM and LDPC codes, i.e., the bit-interleaved LDPC coded modulation (BILCM), has become a naturally accepted powerful coded modulation scheme, e.g., the DVB-T2 paradigm. Based on average mutual information (AMI), for BILCM-ID, Gray mapping is optimal for a perfect LDPC code, i.e., it achieves the channel capacity, providing its code length and its own decoding iterations all infinite [28]. In practical applications, the LDPC code is surely far from perfect, which means Gray mapping may be no longer the best choice for BILCM-ID. While Gray mapping is still optimal for the BILCM with independent demapping, some alternative labeling well matching the LDPC code can improve the performance of BILCM-ID by exploiting the iterative gain. Since the DVB-T2 scheme employs Gray mapping, it is mainly designed and optimized for the case with independent demapping. Unfortunately, its ID counterpart cannot obtain the attractive iterative gain. Besides the fact that the EXIT feature of LDPC code is difficult to match, the feasible BILCM-ID system has to confront the problem how to determine a good one from a huge amount of labeling candidates for high-order constellation. Thus, labeling optimization is still critical and challenging for BILCM-ID. To solve the problems above, this paper proposes a simplified irregular mapping scheme, combined with a labeling search algorithm modified from ABSA. It can provide near-capacity performance for BILCM-ID systems especially with high-order constellations, by providing an irregular mapping rule extremely matching the LDPC code. With a slight modification on the BILCM scheme of DVB-T2, the proposed technique can improve the performance of ID system significantly. The advantages are verified via EXIT chart analysis and performance evaluation. The rest of this paper is organized in the following structure. Section II describes the system model and introduces EXIT chart analysis. In Section III, we first review the previous irregular mapping research, after which a new irregular mapping scheme is proposed. Section IV reviews BSA and ABSA first, and then introduces modified ABSA for irregular mapping. EXIT chart analysis and simulation results of applying the proposed technique on the BILCM scheme of DVB-T2 are shown in Section V. Conclusions are drawn in Section VI. II. SYSTEM MODEL The transmitter and receiver modules of BILCM-ID are illustrated in Fig. 1. The transmitter is the same as that of BILCM. At the receiver, as a result of soft demapping and decoding algorithm, soft bit information are transferred in the form of logarithm likelihood ratio (LLR) between the demapper and decoder. The extrinsic information obtained as the result of the maximum a posteriori probability (MAP) algorithm for demapping/decoding is transferred to the decoder/demapper via de-interleaver/interleaver, and used as the a priori information for decoding/demapping, according to the turbo principle. For the

Fig. 1. BILCM-ID system model.

-ray constellation mapping, the extrinsic information of the th bit in the demapper’s output is calculated as

(1) where denotes the conditional probability density function of the received signal given the transmitted symbol , denotes the constellation subset with the th bit being , and denote the bits belong to the symbol . In 1999, ten Brink [12] first proposed using mutual information to describe the extrinsic information transfer between the demapper and decoder (or expressed as the inner and outer decoders more generally), and adopted the concept of EXIT curve. The demapping EXIT curve and the decoder’s inverted EXIT curve make up of a typical EXIT chart. Since then, EXIT chart has been widely used and become a powerful tool to analyze the convergence of iterative systems with serially concatenated inner and outer decoders [12]–[15], [19], [21], [23]–[27]. Since the performance of BICM-ID is evaluated by the convergence and asymptotic properties [14], which are represented by the threshold SNR and BER floor respectively, the analysis of its performance usually relies on the area property [15] of the EXIT chart, i.e., successful decoding is predicted by a tunnel between and , whereas the area between the two curves represents the capacity loss. It is recognized that EXIT curve-fitting is an essential way to design a good BICM-ID system. Thus, various efforts have been made to search for a better match between the two curves for minimizing the gap while still keeping the tunnel open. The typical BILCM system (with Gray or quasi-Gray mapping), benefiting from the powerful error control capability of the LDPC code, can achieve very good performance with independent demapping, as demonstrated in the DVB-T2 scheme. However, the decoding of a practical LDPC code cannot display perfect EXIT behavior because of multiple restrictions such as finite code length, finite decoding iterations, and specific code construction method and decoding algorithm. Therefore, it has a non-negligible gap from the channel capacity, which means an iterative gain could be exploited in a corresponding ID system. As shown in Fig. 2, the area between and indicates the unavoidable gap from the channel capacity. Nevertheless, in QAM cases, the of Gray mapping is almost flat, which means the iterative gain is very limited with Gray mapping. If the channel code and the constellation are both fixed, the performance of a BICM-ID system is mainly affected by the la-


Fig. 2. EXIT functions of DVB-T2 LDPC code and Gray mapping in 64QAM and 256QAM.

beling. For BILCM-ID systems, due to the powerful error control capability of LDPC codes, the majority of the LDPC performs a sharp increase in the beginning and a very slow uptrend afterwards, which requires the corresponding behaving in a similar form along the LDPC . Such a rigorous requirement makes the search of the labeling well matching the LDPC code a challenging work. III. IRREGULAR MAPPING A. Review of Irregular Mapping Irregular mapping is an efficient technique to change the toward an expected trend, and essentially inshape of creases the DOF via the adjustable mixing ratio of several specific labeling. Some irregular mapping schemes select a specific mapping rule to mix with Gray mapping, and determine the mixing ratio aided by EXIT chart analysis, e.g., MSEW mapping in [24], anti-Gray mapping in [25], and mapping in [26]. Another more complex scheme proposed in [27] combines 4 sorts of labeling and 3 URCs pair-wise into 12 schemes, and mixes them according to certain weights optimized via EXIT chart analysis. All the irregular mapping schemes above have an inherent drawback that the DOF is limited with the pre-fixed labeling, no matter in which ratios they are mixed. Although it may perform well in conventional BICM-ID, it is not effective if is difficult to match by adjusting the mixing ratio only, e.g., the LDPC paradigm. As the example of [27], increasing the number of the mixed labeling can improve the DOF, but both the computing cost of the weights search and the implementation cost of the system become high, which makes it restricted in practical applications. B. Proposed Irregular Mapping Scheme In order to solve the issues of existing schemes, this paper proposes a new irregular mapping scheme, which improves the DOF via systematic labeling search. For simplicity, the new

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scheme use two labeling mixed according to a certain ratio. The first labeling is pre-fixed, the second one (so called the new labeling), instead fixed, needs to be searched depending on the first labeling, the corresponding channel code and the mixing , which means ratio. The mixing ratio is defined as symbols employing the new labeling out of every there are symbols, while the others employ the first labeling. For a given outer code, the new scheme first selects a fixed labeling whose EXIT feature is close to that of the outer code, then sets a typical mixing ratio and searches for the new labeling. Assuming the demapping EXIT curves of the pre-fixed labeling and the new one are and respectively, the criterion of the search is that the EXIT curve of the irregular demapping, , matches the decoder’s ini.e., verted curve , and makes the gap between the two curves minimum while still keeps the tunnel open at a SNR as low as possible. According to the matching results, we decide whether to adjust the mixing ratio and do the search again. Obviously, an efficient computer search algorithm of the new labeling is the kernel of the scheme. This paper proposes a modified adaptive binary switch algorithm (ABSA) for the dynamic labeling search. IV. LABELING OPTIMIZATION A. Review of BSA and ABSA Binary switch algorithm (BSA) is previously used for index optimization in vector quantization [29], and then successfully applied to labeling search [22]. As a greedy algorithm, BSA tries to minimize the cost according to the cost function by switching every pair of labels. Based on the Chernoff upper bound for the pair-wise event error probability, i.e., the error probability of choosing symbol rather than the transmitted symbol , the influence of the labeling is described by (2)

for the AWGN channel, which provides a quantitative calculation of the cost. Here denotes average energy of transmitted symbol, and denotes the power spectrum of the white Gaussian noise. BSA in [22] considers only two marginal cases, i.e., the costs without any a priori information and with perfect a priori information. Moreover, the lack of the mechanism for weight adjustment makes it difficult to predict the weights properly to search out a labeling well matching the outer code. Thus, Yang and Xie et al. [23] proposed adaptive binary switch algorithm (ABSA) to overcome the inadequacy of conventional BSA and hunt for the labeling matching the outer code via instant EXIT chart analysis and adaptive adjustment of the weights after each BSA search. ABSA considers every case that the a priori information with different number of bits is known. The cost function is expressed as (3)

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where denote the cost with bits a priori information, and denote the corresponding weight. Still using the formula (2) to calculate the cost, ABSA treats the minimization of the cost function as an intermediate step to control the trend of EXIT curve, but not the ultimate goal. Instant adjustment of the demapping EXIT curve adaptively and re-searching the laby changing the weight beling is quite important in ABSA. For example, by increasing , a new labeling could be obtained, which leads the demapping EXIT curve around the point with bit a priori information to move up, while other parts of the curve far away to move down accordingly. B. Modified ABSA for Irregular Mapping Currently ABSA is only used for searching the labeling of regular mapping. Although it is powerful and possible to find the very single labeling best matching any outer code, its actual search result is restricted by the local optimum traps and the DOF inadequacy during the labeling search for regular mapping. Especially for the cases with high-order constellation such as 64QAM and 256QAM, and the powerful outer code such as LDPC code, it is still difficult to search out a labeling exactly matching the outer code by ABSA only. Combined with the irregular mapping scheme proposed in the previous section, a modified ABSA is proposed to solve the dynamic labeling search for irregular mapping. The basic procedures are summarized in Algorithm 1. Algorithm 1: Modified ABSA. Input: ; The pre-fixed labeling and its demapping ; The outer decoder’s inverted EXIT curve EXIT curve ; The mixing ratio . Output: New labeling , where the indexes are the labels and the elements are constellation points. 1: Initialize , e.g., randomly generate a labeling

3:

5:

;

do

for

do

for Try switching

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if Switch

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end

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end

and

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return

;

end

17: Find the middle one of the points nearest a priori information; 18:

, and its

;

19: end As the involvement of the irregular mapping technique, the modified ABSA can analyze the EXIT curve of the irregular demapping, which directly determines the optimization trend of the new labeling . In addition, some refinements can be employed to improve the efficiency, e.g., adjust the input adaptively to accelerate the search of the near-optimal curvematching, periodically re-initialize or change to avoid the local optimum traps, etc. Besides, to predefine and select reasonably according to specific situations can make it much easier to find the desired results, e.g., Gray or quasi-Gray mapfor BILCM-ID. ping can be selected as V. SIMULATION RESULTS In this section, the proposed irregular mapping scheme combined with modified ABSA is applied in the revised BILCM scheme of DVB-T2. Significant iterative gain can be exploited for the ID system with only a slight modification of the scheme, i.e., to replace Gray mapping with a suitable irregular mapping. The BILCM/BILCM-ID schemes and the simulation parameters are listed as follows:

DVB-T2 LDPC code with 64800-bit code length and 2/3 code rate; 64QAM and 256QAM constellations; AWGN channel and i.i.d. Rayleigh channel (with coordinate interleaving). Scheme A Independent demapping with Gray mapping; The 64QAM Gray mapping is shown in Fig. 3(a), while the 256QAM Gray mapping is bypassed; The maximum of the inner iteration of LDPC decoder is set to 50;

;

6:

8:

;

at some points do

2: while

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; initialize ;

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;

then and

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;

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end

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Calculate the demapping EXIT curve of

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Calculate the irregular demapping EXIT curve: ;

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if there is an EXIT tunnel then

;

Scheme B Iterative demapping with irregular mapping; The mixing ratio h = 1/2 for 64QAM, h = 1/4 for 256QAM; The pre-fixed labeling is Gray mapping; The new labeling L-64 for 64QAM is shown in Fig. 3(b), and the new labeling L-256 for 256QAM is shown in Fig. 4; The maximum of the inner iteration of LDPC decoder is set to 50; The maximum of the iteration between decoder and demapper is set to 10;


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Fig. 3. Labeling of 64QAM: (a) Gray mapping; (b) new labeling L -64.

Fig. 5. EXIT chart of the irregular demapping for 64QAM.

Fig. 6. EXIT chart of the irregular demapping for 256QAM. Fig. 4. New labeling L -256 of 256QAM.

Log-MAP algorithm [6] is used in the iterative demapping LLR-BP algorithm [30] is used in the LDPC decoding; Pseudo-random interleaver is used as the bit-interleaver instead of that in DVB-T2. The EXIT chart of Scheme A is depicted in Fig. 2. According to the prediction of EXIT chart analysis, the SNR threshold of successful decoding is about 13.59 dB for 64QAM and 18.25 dB for 256QAM, under ideal conditions. The EXIT chart of Scheme B for 64QAM is depicted in Fig. 5. The demapping EXIT curves of Gray mapping, -64 and the final irregular mapping are drawn. It is obvious that the irregular demapping curve matches the decoding counterpart well. A clear tunnel is shown in the enlarged figure. Its predictable SNR threshold can be less than 13.09 dB under ideal conditions, which means the gain predicted by EXIT chart analysis can be over 0.50 dB in 64QAM for the AWGN channel. The EXIT chart of Scheme B for 256QAM is depicted in Fig. 6, which displays similar

near-optimal curve-matching. And the predicted gain can be over 0.62dB in 256QAM. The BER performance comparison of the two schemes is shown in Figs. 7 and 8. Although the actual performances under non-ideal conditions both deteriorate inevitably, compared with Scheme A, in 64QAM, Scheme B can still obtain about 0.30 dB and 0.26 dB iterative gain for the AWGN channel and the Rayleigh channel respectively, at the BER of . And in 256QAM, Scheme B can obtain about 0.46 dB and 0.52 dB iterative gain for the AWGN channel and the Rayleigh channel respectively, at the BER of . The simulation results verify the advantages by applying the proposed irregular mapping technique to the LDPC coded modulation scheme of DVB-T2. VI. CONCLUSION In this paper, an irregular mapping technique combined with modified ABSA is proposed to provide near-capacity performance for BILCM-ID systems. A new labeling searched via modified ABSA, mixed with a pre-fixed labeling, can achieve a better match to a given LDPC code. Our modified ABSA, as a supplement and improvement of ABSA, can provide higher DOF and facilitate optimal labeling search for our proposed irregular mapping, with almost the same complexity as ABSA.

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Fig. 7. BER simulation results for 64QAM.

Fig. 8. BER simulation results for 256QAM.

With a slight modification on the BILCM scheme of DVB-T2, the proposed technique can improve the performance of ID system by exploiting significant iterative gain. The superior performance is demonstrated through EXIT chart analysis and BER simulation. With the requirement of high-order modulation and high-performance channel coding for future broadcasting systems, the proposed irregular mapping technique expects the prospect of broad application. REFERENCES [1] E. Zehavi, “8-PSK trellis codes for a Rayleigh channel,” IEEE Trans. Commun., vol. 40, no. 5, pp. 873–884, May 1992. [2] G. Caire, G. Taricco, and E. Biglieri, “Bit-interleaved coded modulation,” IEEE Trans. Inf. Theory, vol. 44, no. 3, pp. 927–946, May 1998. [3] X. Li and J. A. Ritcey, “Bit-interleaved coded modulation with iterative decoding,” IEEE Commun. Lett., vol. 1, no. 6, pp. 169–171, Nov. 1997. [4] X. Li and J. A. Ritcey, “Bit-interleaved coded modulation with iterative decoding using soft feedback,” Electron. Lett., vol. 34, no. 10, pp. 942–943, May 1998. [5] S. T. Brink, J. Speidel, and R.-H. Yan, “Iterative demapping for QPSK modulation,” Electron. Lett., vol. 34, no. 15, pp. 1459–1460, Jul. 1998. [6] S. T. Brink, J. Speidel, and R.-H. Yan, “Iterative demapping and decoding for multilevel modulation,” in Proc. IEEE Global Telecommun. Conf. (GLOBECOM), Nov. 1998, vol. 1, pp. 579–584.

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