ABSTRACT. This work evaluates the bit error rate (BER) performance of various two-dimensional turbo product codes (TPCs). The turbo decoder is implemented ...
2011 IEEE GCC Conference and Exhibition (GCC), February 19-22, 2011, Dubai, United Arab Emirates
BER PERFORMANCE OF NONSEQUENTIAL TURBO PRODUCT CODES OVER WIRELESS CHANNELS S. A. Al Muaini, A. J. AlDweik, and M. A. AlQutayri College of Engineering, Khalifa University Sharjah Campus, P.O. Box 573, Sharjah, UAE. {sara.almuaini,dweik,mqutayri}@kustar.ac.ae tios (LLRs) of the received bits is requires accurate knowl edge of the channel state information. In the absence of such knowledge, the coding gain promised by SISO TPCs will not be attained. Consequently, hardinput hardoutput (HIHO) decoders have been proposed for applications where low complexity and short delay are required [4], [7]. Due to the unique structure of TPC codewords, it has been demonstrated in [7] that a noticeable coding gain im provement can be achieved using nonsequential (NSQ) decoding in HIHO TPCs. In this work, we investigate the BER performance of the NSQ decoding algorithm pro posed in [7] for various codes over AWGN and Rayleigh fading channels and compare the results with the standard sequential (SQ) TPC decoder. The BER performance eval uation aims at finding the effect of the code rate and code size on the coding gain of the NSQ and SQ decoders. This paper is organized as follows. In Section 2, the basic principles of iterative TPCs are briey introduced. Section 3 describes the decoding algorithm in details. Sec tion 4 demonstrates the numerical results of a nonsequential decoding algorithm and its performance compared to stan dard HIHO TPC over AWGN and Rayleigh fading chan nels. Section 5 provides the conclusion.
ABSTRACT This work evaluates the bit error rate (BER) performance of various twodimensional turbo product codes (TPCs). The turbo decoder is implemented using hardinput hard output data, which is impaired by additive white Gaussian noise (AWGN) and multipath fading. The effectiveness of the iterative TPC BER is evaluated using sequential and nonsequential decoding. Numerical simulation for differ ent TPCs have confirmed that the nonsequential decod ing in Rayleigh fading channels can offer double the cod ing gain offered in AWGN channels. Moreover, the extra coding gain achieved by the nonsequential decoder is in versely proportional to the code rate and the code length. Index Terms— Turbo codes, product codes, iterative decoding, hard decision decoding, Rayleigh fading.
1. INTRODUCTION Turbo error correction coding is a powerful channel cod ing scheme used for power limited systems such as deep space satellite communications. Turbo codes offer a per formance closer to the Shannon limit than any other class of error correcting codes [1]. Turbo product codes (TPCs), also known as a block turbo codes, have an excellent per formance at high code rates and can provide a wide range of block sizes [2], [3]. TPCs can be constructed using two or more simple linear codes either serially or in parallel; in order to achieve acceptable error performance with man ageable encoding and decoding complexity. To achieve the ultimate gain of TPCs, the decoder has to take soft input and produce soft output, and hence it is called softinput softoutput (SISO) decoder. The soft de coding is based on the ChaseII algorithm that requires a large number of hard decision decoding (HDD) operations for each row/column in the received matrix. Moreover, the ChaseII algorithm produces hard data which has to be converted to soft information before it can be utilized by the softinput decoder in the subsequent iterations [3]. The large number of HDD performed by the ChaseII al gorithm and the hardtosoft data conversion considerably increases the decoder computational complexity and de lay. Furthermore, the computation of the loglikelihood ra
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2. TURBO PRODUCT CODES TPCs are multidimensional arrays constructed from two or more linear block codes denoted as the component codes. Two dimensional TPCs are the most common among other TPCs where the product code is obtained using two sys 2 tematic linear block codes 1 (1 1 (1) min ) and (2 (2) () 2 min ) where and min are the codeword size, number of information bits and minimum Hamming dis tance, respectively. As depicted in Figure 1, the TPC is constructed as follows: 1. Place (1 £ 2 ) information bits in a matrix with 1 columns and 2 rows. 2. Encode the 1 bits in each row using code 2 to pro duce 1 channel bits in each row; 3. Encode the 1 bits in each column using code 1 to produce 2 channel bits in each column.
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The codeword size, number of information bits and minimum Hamming distance of the constructed product code are = 1 £ 2 , = 1 £ 2 and min = (1) (2) min £ min respectively. Hence the correcting capabil ities of TPC is given by [8]: $ % (1) (2) min min ¡ 1 = (1) 2
Transmitted matrix W
Rayleigh fading
where bc is the greatest integer smaller than or equal to
Received matrix R
Additive white Gaussian noise Z
Figure 2: Schematic model of a Rayleigh fading channel.
n1
with the entire code space and the decisions is made in favor of the codeword that is close in terms of Hamming distance to the received sequence.
k1 Checks on rows
3. THE DECODING ALGORITHM
k2
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A new decoding algorithm for HIHO decoding of TPCs is proposed in [7]. In this method, the received hard matrix is iteratively, but not sequentially decoded. This method proved to be efficient over AWGN channels where an ad ditional coding gain of more than 0.5 dB was reported. In this work, we extend the work presented in [7] to Rayleigh fading channels and investigate the performance for vari ous component codes. The classical sequential decoding of TPCs using HIHO decoders is summarized as follows:
Checks on Checks
1. Set the maximum number of iterations max ; Figure 1: The construction and codeword structure of TPCs.
2. Initiate the iteration counter = 1; 3. Decode every code component in horizontally;
Assuming that binary phase shift keying (BPSK) mod ulation f0 ! ¡1 1 ! +1g is considered, the received codeword R in a Rayleigh fading channel can be expressed as, R = ® ± W + Z (2)
4. Decode every code component in vertically; 5. = + 1; 6. If ( = max ) stop decoding, else, go to step 3.
where ± denotes the Hadamard elementbyelement prod uct, the Rayleigh fading matrix ® = (11 1 2 ) are independent and identically distributed (i.i.d) Rayleigh random variables, W = (11 1 2 ) is the trans mitted codeword matrix and Z = (11 1 2 ) is a zero mean additive white Gaussian noise (AWGN) with 0 2 variance. The channel model is presented in Fig ure 2. HIHO decoding of TPCs can be achieved by ap plying the conventional HDD to decode the component codes that forms the TPC. The decoding process starts by converting the received matrix R into binary matrix H = (11 1 2 ) 2 f0 1g then the decoding process starts rowbyrow (or columnbycolumn) till the last row. The resultant matrix is decoded in the same man ner but in the other direction. A complete HDD process over two dimensions represents one full iteration. Although this decoding process is suboptimal, it has much lower complexity compared to the maximum likelihood decod ing (MLD) where the received matrix has to be compared
Hence, performing unsuccessful decoding in row de coding will result in an inferior impact on performing col umn decoding. The result, in general, is an error ampli fication. The NSQ method avoids error amplification by selecting an appropriate reliability threshold that satisfy two requirements: high reliability in the first iteration and should be selected as low as possible. The value of is selected so that the error correction capability of the code vary at each iteration. Here, was set at ¡ 1 in the first full iteration. So the new algorithm decoding procedure is summarized as follows: 1. Set the value of reliability threshold ( = ¡ 1); 2. Set the maximum number of iterations max 3. Initiate the iteration counter = 1; 4. Decode every code component in horizontally;
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Obviously, these codes have similar code sizes and differ ent code rates that are equal to 065, 051 and 038, respec tively. It can be noted from this figure that the coding gain advantage of the NSQ HIHO decoder is proportional to the code rate. For example, the BCH (63 51 5)2 offered about 0.25 dB coding gain over the standard SQ HIHO decoder at BER of 10¡6 while the BCH (63 45 7)2 and BCH (63 39 9)2 offered about 0.1 and 0.05 dB of extra coding gain, respectively. Figure 4 investigates the impact of the code size on the additional coding gain achieved using the NSQ de coding as a function of the code size while the code rate is fixed at » 05. The TPCs codes considered are the BCH (31 21 5)2 (63 45 7)2 , and (127 85 13)2 and the channel is AWGN channel. As depicted in this figure, the BCH (31 21 5)2 achieved the maximum extra cod ing gain while the BCH(63 45 7)2 comes second and the BCH(127 85 13)2 comes third. Obviously, it can be con cluded that the additional coding gain of the NSQ decod ing is inversely proportional to the code size, which is con sistent with the results presented in [5], [7]. Similar to the AWGN channel case, the same TPCs were used in Rayleigh fading channels. Figure 5 presents the BER for a group of TPCs with different code rates and fixed code sizes. The results achieved confirm those ob tained for the AWGN case. It states that the extra coding gain offered by the NSQ decoders is inversely proportional to the code rate. Moreover, the extra coding gain in fading channels is larger than that in AWGN channels.
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Figure 3: The BER versus 0 for TPCs having the same code length and different code rates over AWGN channels. 5. For every code component in : if ( the number of estimated errors ^ & = 1) ) leave the code component unchanged, else, decode; 6. Decode every code component in vertically;
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7. For every code component in : if ( the number of estimated errors ^ & = 1) ) leave the code component unchanged, else, decode;
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8. = + 1;
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9. If ( = max ) stop decoding, else, go to step 3.
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Note that the proposed decoding algorithm requires negligible additional complexity since it uses a simple bi nary addition process for all the elements of ^
(127,85) NSQ [7]
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4. NUMERICAL RESULTS 10
Extensive Monte Carlo simulations are conducted to eval uate the BER performance of the SQ and NSQ decoding of TPCs. The component codes used are the binary BCH codes, each TPC is constructed using two identical com ponent code. All simulations are performed using five HIHO iterations. BPSK is considered as the modulation scheme and two channel models are investigated, namely the AWGN and Rayleigh fading channels. It is worth not ing that performing more than five iterations will have neg ligible effect on the BER. The reliability threshold of the NSQ decoders is set to ¡ 1. Figure 3 shows BER versus 0 over AWGN chan nels for the TPCs (63 51 5)2 (63 45 7)2 and (63 39 9)2 .
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Figure 4: The BER versus 0 for TPCs with differ ent code sizes but with the same code rates over AWGN channels. The BER performance of various TPCs with differ ent code sizes and code rates » 05 over Rayleigh fading
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Figure 5: The BER versus 0 for TPCs with the same code sizes but different code rates over Rayleigh fading channels.
Figure 6: The BER versus 0 for TPCs having differ ent code length and code rates » 05 over Rayleigh fading channels.
channels is presented in Figure 6. Similar to the AWGN case, it can be noted from this figure that extra coding gain offered by the NSQ is inversely proportional to the code size. Moreover, the extra coding gain in Fading channels is almost double that in AWGN channels.
codes: block turbo codes,” IEEE Trans. Commun., vol. 46, no. 8, pp. 10031010, Aug. 1998. [4] O. Sab, “FEC techniques in submarine transmission systems,” in Proc. OFC 2001, vol. 2, 2001, pp. TuF1 1 TuF13. [5] G. Bosco, G. Montorsi, and S. Benedetto, “A new al gorithm for “hard” iterative decoding of concatenated codes,” IEEE Trans. Commun., vol. 51, no. 8, pp. 12291231, Aug. 2003.
5. CONCLUSION This paper presented the BER performance of the TPCs using sequential and nonsequential decoding in AWGN and Rayleigh fading channels. The simulations were con ducted using two sets of TPCs, the first set consists of var ious TPCs having the same code sizes but different code rates. The second set consists of various TPCs having dif ferent code sizes but with almost the same code rates. The extensive simulation results have confirmed that the extra coding gain offered by the NSQ decoding is inversely pro portional to the code size and code rate. Moreover, the extra coding gain in Rayleigh fading channels can be as much as twice that in AWGN channel for the same code.
[6] J. Anderson, “Product codes for optical communica tions,” in Proc. ECOC 2002, pp. 12. [7] A. AlDweik, B. Sharif, “Nonsequential decoding al gorithm for hard iterative turbo product codes,” IEEE Trans. Commun. vol. 57, pp. 15451549, June, 2009. [8] J. G. Proakis, and M. Salehi, Digital Communications , 5th ed. New York: McGrawHill, 2007.
6. REFERENCES [1] C. Shannon, A mathematical theory of communica tion, Bell Syst. Tech. J., vol. 27, 1948. [2] R. Pyndiah, A. Glavieux, A. Picart and S. Jacq, “Near optimum decoding of product codes,” in Proc. IEEE GLOBECOM 1994, vol. 1, pp.339343. [3] R. Pyndiah, “Nearoptimum decoding of product
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