DVB-S2 LDPC Decoding Using Robust Check Node ... - IEEE Xplore

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IEEE TRANSACTIONS ON BROADCASTING, VOL. 54, NO. 1, MARCH 2008

DVB-S2 LDPC Decoding Using Robust Check Node Update Approximations Stylianos Papaharalabos, Marco Papaleo, Student Member, IEEE, P. Takis Mathiopoulos, Senior Member, IEEE, Massimo Neri, Member, IEEE, Alessandro Vanelli-Coralli, Senior Member, IEEE, and Giovanni E. Corazza, Member, IEEE

Abstract—Broadband satellite services to fixed terminals are currently offered in the forward link by the 2nd Generation (2G) Digital Video Broadcasting Satellite (DVB-S2) standard. For this standard the use of powerful Low-Density Parity-Check (LDPC) error correcting codes has been adopted performing within approximately 1 dB from the Shannon capacity limit. This paper studies and compares for the first time in a systematic manner different approximation methods used in check node update computation of DVB-S2 LDPC decoding with the aim of reducing computational complexity. Various performance evaluation results are presented for a wide range of DVB-S2 parameters, such as LDPC codeword size, coding rate, modulation format and including several decoding algorithms. It is shown that the proposed check node update approximations have a robust behavior, i.e. the resulting performance is quite independent of the DVB-S2 modulation and coding parameters. It is further shown that these approximations perform very close to the optimal Sum-Product Algorithm (SPA) in degradation, which is less than 0.2 dB. Despite this small degradation, the reduction in computational complexity compared to the optimal SPA is significant and can be as high as 40% in computational time savings. Index Terms—Broadcasting, iterative decoding, Low-Density Parity-Check (LDPC) codes.

I. INTRODUCTION

L

OW-DENSITY parity-check (LDPC) codes were proposed initially by Gallager in the early 1960s [1] but they were not used for many years mainly because the technology was not mature for their practical implementation. In 1981 Tanner proposed a graphical representation of these codes, also known as Tanner graphs [2]. However, it was not until the late 1990s when they were rediscovered by MacKay [3] and others [4]–[6]. LDPC codes can achieve near Shannon capacity limit performance over the Binary Erasure Channel (BEC) as well as over the Additive White Gaussian Noise (AWGN) channel with moderate decoding complexity [3], [7]. Recently, they have been adopted by the new version of the Digital Video Broadcasting by Satellite (DVB-S2) standard [8]. In this standard, an inner LDPC code is incorporated with Adaptive Coding and Modulation (ACM) techniques as well as optimized high order modulation schemes for satellite

Manuscript received August 1, 2007; revised October 15, 2007. This work was supported in part by IST SatNEx-II FP6 Project (IST-027393). S. Papaharalabos and P. T. Mathiopoulos are with the Institute for Space Applications and Remote Sensing (ISARS), National Observatory of Athens, 15236 Athens, Greece (e-mail: [email protected]; [email protected]). M. Papaleo, M. Neri, A. Vanelli-Coralli and G. E. Corazza are with the Department of Electronics, Computer Science and Systems (DEIS), University of Bologna, 40136 Bologna, Italy (e-mail: [email protected]; mneri@deis. unibo.it; [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.2007.911365

transmission signals, such as the Amplitude Phase Shift Keying (APSK) signaling. It is interesting to note that the resulting capacity increase for the new DVB-S2 over DVB-S standard has been approximately 30% in broadcasting mode and more than 100% for unicasting (i.e. broadband) interconnections [9], [10]. Nowadays, the provision of digital broadcasting multimedia contents to mobile users has been a new emerging technology. For example, a special journal issue entitled ‘Mobile Multimedia Broadcasting’ focusing mainly on terrestrial systems has been recently published [11]. In this journal issue, it is noted that circulant LDPC codes have been adopted to provide Digital Terrestial/Television Multimedia Broadcasting (DTMB) services in China [12], [13]. The evolution of the new DVB-S2 standard to support fully mobility is another research topic currently under investigation [14]. It aims at offering broadband interactive services to mobile users situated on aircrafts, ships and high-speed trains targeting a niche market. LDPC codes can be decoded by the Sum-Product Algorithm (SPA), which is based on the message passing principle [3]. To reduce its complexity several decoding algorithms have been proposed over the past years. A detailed review of the currently available SPA reduced complexity decoding algorithms for LDPC codes can be found in [15]. Two practical implementation aspects of DVB-S2 LDPC decoding were proposed in [16] reducing the number of iterations as well as the computation operations. Some recent hardware implementations of the DVB-S2 LDPC codec can be found in [17]–[20]. Among the critical technical issues addressed in these references were the encoder/decoder architecture, the area size and the achieved overall throughput. A DVB-S2 digital receiver was reported in [21] dealing with several practical aspects, such as frame syncronization, timing acquisition, signal-to-noise ratio (SNR) estimation and digital predistortion. It should be noted that although in the previously mentioned references there have been various performance evaluation results published, this was done in a rather ad-hoc manner without using a systematic approach. Motivated by this observation, the purpose of this paper is to present for the first time a systematic and thorough performance evaluation study of DVB-S2 LDPC codes with different modulation and coding parameters in conjunction with reduced complexity decoding algorithms. Following [22], we have used either a Piecewise (PW) linear function with a few entries or a small Look-Up Table (LUT) to approximate the check node update computation as accurate as possible and applied them to DVB-S2 LDPC decoding. It is shown that the adopted approximations, which were optimized for pseudo-random based LDPC codes in [22], offer two potential advantages when applied to DVB-S2 systems: (i) robust behavior that is independent on the modulation and coding parameters, and (ii) good trade-off between performance and complexity. The structure of this paper is organized as follows. After this introduction, a brief overview of coding and modulation used in

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DVB-S2 standard is presented in Section II. Then, various algorithms (i.e. both optimal and sub-optimal) used to decode LDPC codes are described in Section III. The adopted approximations used in check node update computation of DVB-S2 LDPC decoding are introduced in Section IV. Various performance evaluation results assuming the DVB-S2 LDPC code with different modulation and coding parameters are presented in Section V and finally conclusions are made in Section VI. II. CODING AND MODULATION FOR THE DVB-S2 STANDARD Exploiting the increased feasibility of Application Specific Integrated Circuit (ASIC) technology, the DVB-S2 committee has chosen to adopt a solution where Forward Error Correction (FEC) encoding is performed into three steps using: (i) an outer systematic Bose-Chaudhuri-Hocquenghem (BCH) code; (ii) an inner LDPC code and (iii) a block bit-interleaver [8]. With reference to the outer encoding, BCH codes are well-known -error correcting block codes that prevent the propagation of undetectable errors generated by the inner LDPC decoder. LDPC codes are linear block codes with sparse parity-check rows, where matrix characterized by columns and is the information block length, is the codeword length and is the redundancy length added by the encoder. The LDPC codes standardized by DVB-S2 are irregular with variable bit node degrees and produce codewords of either 64800 or 16200 bits, i.e. long and short size codewords, respectively. The adoption of irregular codes is because bit nodes with high degrees collect more information from their adjacent check nodes so that they can be corrected after a small number of iterations. DVB-S2 LDPC codes have a well-defined structure, which is imposed of an appropriate parity-check matrix construction, by producing systematic codes in a very simple encoding operation, that is the use of Irregular Repeat Accumulate (IRA) codes [23]. of LDPC codes are Even though parity-check matrices needed for encoding sparse, in general generator matrices may be not. The latter matrices can be derived using the classical Gaussian elimination method. Furthermore, a non-sparse matrix can generate both storage and encoding complexity problems. For that reason, DVB-S2 has chosen to adopt methods that partially solve this problem by restricting a sub-matrix of the paritycheck matrix to be lower triangular and thus eliminating the need to derive a generator matrix [23]. In this way, linear encoding complexity is feasible. In more detail, the parity-check where is a staircase matrix has the form of matrix with column weight equal to 2, and is a random sparse matrix of size with column weight ranging between 3 and 13 depending on the coding rate. Another important feature is the periodicity of the sub-matrix , in order to reduce memory storage requirements by a factor bit nodes the check nodes of . In particular, for a group of connected to the first bit node need only to be specified, whereas can be the check nodes connected to -th bit node with determined by a pre-calculated formula [23]. In DVB-S2 the pais chosen. rameter III. OPTIMAL AND SUB-OPTIMAL REPRESENTATIONS OF THE SUM PRODUCT ALGORITHM (SPA) Let be a binary LDPC code described by a sparse , where is parity-check matrix of size

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the check nodes corresponding to the parity-checks of a bipartite graph and is the bit nodes corresponding to the encoded bits. Moreover, each bit node is connected to check nodes and each bit nodes, i.e. a regular LDPC check node is connected to denotes the set of code is being considered. In following, denotes the set check nodes connected to bit node and of bit nodes participating in the -th parity-check equation. In represents the set , excluding the -th addition, represents the set , excluding check node and the -th bit node. is Binary Phase Consider now that a codeword of size Shift Keying (BPSK) modulated and transmitted over the is the received codeword correAWGN channel. Also, sponding to the transmitted codeword with the same size . In general, the sum-product algorithm is composed of three steps: (i) initialization; (ii) iterative process and (iii) hard decision [15]. Step (ii) includes the check node and bit node updates, which are the core of the algorithm. These updates can be computed from

(1) and (2)

is the log-likelihood ratio (LLR) of the where message that the -th check node sends to the -th bit node, is based on all the bits checked by except , and the LLR of the message that bit node sends to check node , based on all the checks involving except , respectively. In is the a posteriori probability of each bit node addition, after transmission through the channel, i.e. where is the noise variance.In order to obtain a wide range of spectral efficiencies in DVB-S2 standard, 11 different coding rates (1/4, 1/3, 2/5, 1/2, 3/5, 2/3, 3/4, 4/5, 5/6, 8/9, 9/10) are available together with four different modulation schemes. Apart from the well-known -ary Phase Shift Keying (PSK) modulation formats, such as Quadrature PSK (QPSK) and 8-PSK, the hybrid amplitude and phase modulation format, referred to as -ary APSK, is mainly considered for two reasons: (i) increased bandwidth efficiency which is achieved for high and , and (ii) its ability values of , such as to better cope with typical non-linear distortion introduced by satellite transponders. Following the notation where denotes the number of constellation points on the -th ring, the considered in the DVB-S2 standard 16-APSK and 32-APSK modulation formats are composed of a double ring (4, 12) and a triple ring (4, 12, 16), respectively. Note that the constellation parameters are optimized as a function of the LDPC coding rate [8]. In the case of 8-PSK, 16-APSK and 32-APSK modulation formats, a simple block bit-interleaver is added after the LDPC encoding, in order to increase the code diversity. Coded bits are serially written column-wise into the interleaver, and then are serially read row-wise. rule [15]. The above algorithm is also known as the Apart from it, [15] provides a description of two more optimal algorithms with different check node update computation than

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TABLE I PIECEWISE LINEAR APPROXIMATION OF THE tanh(1) FUNCTION [22]

TABLE II LUT APPROXIMATION OF THE tanh(1) FUNCTION [22]

Fig. 1. Comparison of the two approximations of the tanh(1) function based on PW linear and LUT with the exact curve.

in (1). These are: (i) Gallager’s approach and (ii) Jacobian approach. In the first case, the check node update computation can be expressed as

(3)

denotes the signum function, the use of summawhere tions is preferred instead of multiplications and also another is used, i.e. function . In the second case, the check node update computation is applied into pairs of incoming messages in recursive form. It is based on the fact that for two and random variables and , with LLR values respectively, the following formula exists [15]

(4)

Fig. 2. Normalized computational time against the number of decoding iterations using different decoding algorithms for the DVB-S2 LDPC code. Short codeword of = 16200 bits, coding rate ( ) 2/3, 8-PSK modulation, AWGN channel and a maximum of 70 decoding iterations.

N

R

where denotes the logical Exclusive OR (XOR) operator. The logarithm in the right part of (4) can be stored in memory by using a Two-Dimensional (2D) LUT, usually with eight entities, in which both the sum and difference of the two LLR values and are required. For the min-sum algorithm, the correcting factor from (4) is omitted, so that the check node update can be computed from

(5) It can be observed that the LLR values from (5), i.e. for min-sum algorithm, are always greater than those from (3), i.e. for SPA. In the open technical literature there exist several approaches that improve the performance of the min-sum algorithm at the expense of some complexity increase. For instance, in [15] two algorithms were proposed, namely normalized min-sum (NMS) and offset min-sum (OMS) algorithms. Both are based


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N

Fig. 3. PER performance of DVB-S2 LDPC code, = 16200 bits, over the AWGN channel, a maximum of 70 decoding iterations, different coding rates and modulation formats. Four decoding algorithms are compared with each other: (i) PW linear approximation; (ii) LUT approximation; (iii) EX method (i.e. ideal performance) and (iv) min-sum algorithm.

on finding the minimum as shown in (5), but the former makes : use of a constant normalization factor

(6) whereas the latter makes use of a subtraction with a positive constant value :

(7)

IV. CHECK NODE UPDATE APPROXIMATIONS FOR DVB-S2 LDPC DECODING For the case of pseudo-random based LDPC codes, the effect on the BER performance when approximating the check node update computation from (1) with different methods was investigated in [22]. In particular, two different approaches were function and its inverse: (i) considered to compute the PW linear function with seven regions, and (ii) LUT with eight function (e.g. values. These two approximations of the see Tables I and II) together with the exact expression (EX) are illustrated in Fig. 1. The adopted approximations provide computational complexity savings with respect to the corresponding EX expressions. The PW linear approximation requires only one multiplication and one addition, whereas the LUT approximation requires no operations at all, apart from extra memory storage of eight values. The selection of the number of regions

(or equivalently the LUT size) was done in order to have a similar degree of discrimination to the LUT used in Log-MAP decoding of turbo codes, which makes use of eight stored values [24]. In order to compare the two approximations, i.e. PW linear and LUT, in terms of computational time savings, various computer simulation experiments were run assuming the DVB-S2 LDPC code with different codeword sizes, coding rates and modulation formats. In this respect, Fig. 2 illustrates the normalized computational time obtained for the following scenario: short DVB-S2 LDPC codeword of 16200 bits, coding rate equal to 2/3, 8-PSK modulation, AWGN channel, and a maximum of 70 decoding iterations. For comparison purposes, the computational time provided by the optimal Jacobian approach and the sub-optimal min-sum algorithm are also depicted. The curves shown in Fig. 2 have been normalized to the reference case, that is the EX method with a maximum of 70 decoding iterations. It can be seen that with respect to the EX method, the PW linear approximation is about 26% faster, whereas the LUT approximation is about 40% faster, respectively. In addition, the computational time of the Jacobian approach is about 12% lower as compared to LUT approximation, whereas the min-sum is the fastest algorithm, e.g. about 20% faster than the Jacobian approach. However, as it will be shown in the next Section, there is a small BER performance degradation associated with the min-sum algorithm. As expected, the simulation time increases linearly with the number of decoding iterations. Based on our experiments it is important to note that the computational time savings are not influenced significantly by the selection of LDPC codeword size, coding rate and modulation format. V. PERFORMANCE EVALUATION RESULTS In order to assess the performance of the two SPA approximations, the LDPC code considered in the DVB-S2 standard was adopted. Thus, following [8] an outer BCH code is assumed, which can detect and successfully correct up to 12 erroneous bits. The user packet length is 1504 bits (i.e. MPEG format),

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Fig. 4. BER performance of DVB-S2 LDPC code,

N = 16200 bits. Other system parameters are the same as in Fig. 3.

Fig. 5. PER performance of DVB-S2 LDPC code,


whereas both short (i.e. 16200 bits) and long (i.e. 64800 bits) LDPC codewords are selected. In the various performance evaluation results, which have been obtained by means of computer simulations, the coding rate varies from 1/2, 2/3 to 3/4 and the modulation format is based on either -ary PSK or -ary APSK, e.g. QPSK, 8-PSK, 16-APSK and 32-APSK, respectively. Consistent with the previous analysis, an AWGN channel is considered with a double-sided power spectral density and a maximum of 70 decoding iterations are performed at the decoder. Performance evaluation results obtained in terms of PER and BER are illustrated in Figs. 3 and 4 for the short LDPC codeword and in Figs. 5 and 6 for the long LDPC codeword, respectively. In all curves denoting the optimum SPA algorithm, which uses (1), is labeled as EX and is shown in solid lines.

The two considered approximations are labeled as PW and LUT curves and they are shown in dashed lines. Also, min-sum algorithm performance is shown in solid lines, but with some performance degradation compared to EX method (e.g. maximum , depending on the simulation parame0.8 dB at PER of ters scenario). It can be seen that for a target PER (or BER) of (or ) the loss of LUT with respect to approximately EX method is less than 0.2 dB, whereas the loss of PW with respect to EX method is less than 0.1 dB, respectively. Overall, the performance evaluation results obtained have shown that such behavior is independent of the code rate selection, modulation format and codeword size. Such observations clearly verify the robustness of the adopted approximations. In order to have a better view on the performance of different check node update algorithms, Table III depicts the required bit


Fig. 6. BER performance of DVB-S2 LDPC code,

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TABLE III PERFORMANCE COMPARISON AMONG DIFFERENT DECODING ALGORITHMS FOR THE DVB-S2 LDPC CODE, IN TERMS OF REQUIRED E =N AT PER (OR BER) (OR 10 ). SHORT CODEWORD OF N = 16200 bits, AWGN CHANNEL AND A MAXIMUM OF 70 DECODING ITERATIONS OF APPROXIMATELY 10

energy to noise power spectral density ratio at PER (or BER) of approximately (or ). In this table, the short LDPC codeword is considered together with different modulation and coding parameters. It can be observed that: (i) the Jacobian approach performs similarly with the EX method; (ii) the NMS algorithm performs similarly or slightly better than , the PW approximation, except for the last case (i.e. 8-PSK) in which a very small degradation is noticed and (iii) the OMS algorithm has very small degradation compared to the LUT approximation. Note that all performance evaluation results are comparable with those from [10], although 50 decoding iterations were used in this reference. By increasing the number of iterations to 70, only small performance improvement of less than 0.05 dB was observed. In all performance evaluation results a normalization approach to the function was considered, in order to approximate the approach to infinity and prevent decoding overflow [22]. This normalization was based upon the use of clipping when the input argument was greater than a predefined maximum value (e.g. 10), and explains the error floor removal to lower values in the corresponding PER/BER curves.

VI. CONCLUSION The aim of this paper was to study and compare, in a systematic manner, different check node update approximation methods used to decode LDPC codes specified by the DVB-S2 standard. Various results have shown small performance degradation compared to the optimal SPA but significant reduction in computational complexity was achieved. The different check node update approximations were shown to have robust performance, i.e. they were independent of the selection of DVB-S2 modulation and coding parameters. The observed performance/complexity trade-off can be useful in future applications of DVB-S2 systems, for instance when introducing mobility aspects, in which computational complexity savings could be of significant interest. REFERENCES [1] R. Gallager, Low-Density Parity-Check Codes. Cambridge: MIT Press, 1963. [2] R. Tanner, “A recursive approach to low complexity codes,” IEEE Trans. Inform. Theory, vol. 27, no. 5, pp. 533–547, 1981. [3] D. J. C. MacKay, “Good error-correcting codes based on very sparse matrices,” IEEE Trans. Inform. Theory, vol. 45, no. 3, pp. 399–431, 1999.

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