IEEE COMMUNICATIONS LETTERS, VOL. 13, NO. 5, MAY 2009
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A Geometric Approach for Turbo Decoding of Reed-Solomon Codes in QAM Modulation Schemes Cristhof J. Roosen Runge and Jos´e Roberto C. Piqueira
Abstract—This work studies the turbo decoding of ReedSolomon codes in QAM modulation schemes for additive white Gaussian noise channels (AWGN) by using a geometric approach. Considering the relations between the Galois field elements of the Reed-Solomon code and the symbols combined with their geometric dispositions in the QAM constellation, a turbo decoding algorithm, based on the work of Chase and Pyndiah, is developed. Simulation results show that the performance achieved is similar to the one obtained with the pragmatic approach with binary decomposition and analysis. Index Terms—Algorithm, constellation, Galois field, matrix coding.
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I. I NTRODUCTION
HE advent of turbo codes was in 1993 [1], providing tools for developing BTC (block turbo codes) algorithms [2], [3], with several practical applications [4]. Special attention has been dedicated to turbo decoding schemes based on Reed-Solomon (RS) codes, which are a non-binary subclass of Bose-Chaudhuri-Hocquenghem (BCH) codes, as reported in [5]. As RS codes are non- binary, in high spectral efficient modulation schemes such as QAM, the corresponding alphabet is composed of non-binary symbols [6]. There are works proposing turbo decoding schemes based on binary approach, i.e., with all the calculations and analyzes being conducted on a bit basis [3]. In this work, an algorithm based on the mapping relations between the Galois field elements, i.e., the modulation symbols and its associated geometry, is presented. This approach allows a direct treatment for the noise and symbols, providing easier manipulations for the RS turbo decoding scheme, with noise and symbols obtained directly from the reception channel. Consequently, a reduction in the amount of operations needed during the turbo decode process is obtained. In section II, a description of possible associations between the Galois field elements and constellation symbols is presented, in the form of a matrix product construction. In section III, the geometric turbo decoding algorithm is described. In section IV, the simulation results for a study case is presented. Section V contains the conclusions. II. R EED -S OLOMON T URBO M ATRIX C ODING
Reed-Solomon codes belong to a non-binary sub-class of BCH codes over a Galois field GF (2m ), with m integer [6]. The main parameters that characterize the codes are (n, k, d), where n is the length of the code words, k is the number Manuscript received January 15, 2009. The associate editor coordinating the review of this letter and approving it for publication was P. Cotae. The authors are with the Departamento de Engenharia de Telecomunicac¸ o˜ es e Controle, Escola Polit´ecnica, Universidade de S˜ao Paulo (e-mail:
[email protected]). Digital Object Identifier 10.1109/LCOMM.2009.090100
Fig. 1.
Product matrix.
of information symbols associated to the code word and d is the Hamming distance of the code. Each code symbol is an element of GF (2m ) [7]. Given two RS codes C1 and C2 characterized by (n1 , k1 , d1 ) and (n2 , k2 , d2 ), respectively, the product code C1 by C2 is a code Cp with (np , kp , dp ) parameters, where np = n1 xn2 , kp = k1 xk2 and dp = d1 xd2 . The Cp product matrix may be constructed as follows : • a k1 xk2 matrix is mounted with k1 xk2 information symbols; • the k1 rows are codified according to C2 ; • the n2 columns are codified according to C1 . A QAM modulation alphabet is formed with L symbols, where L is the cardinality of the alphabet and is normally a power of 2. Each symbol carries log2 L bits. In order to construct a geometric turbo decoding scheme, one can choose a RS code so that the modulation symbols have a multiplicity relation with the Galois field elements of the RS associated code. From now on, it is assumed that the RS codes and the QAM constellation are chosen so that C1 = C2 , and the GF (2m ) is considered to have 2m = L symbols. Consequently, there is a one-to-one relation between the constellation symbols and the Galois field, as the product matrix in Fig. 1 shows. Considering an AWGN channel and the described product matrix, each modulation symbol corresponds to a Galois field element. Consequently, the reliability associated to each coded received symbol depends on the sample noise measured in the output of the demodulator. Let S = (s1 , s2 , ..., sn ) be the vector associated to a row of the product matrix of a code word, and consider the Euclidian representation of the phase and quadrature components associated to si = (ai , bi ). Calling Z = (z1 , z2 , ..., zn ) the noise vector associated to S, and R = (r1 , r2 , ..., rn ) the AWGN
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IEEE COMMUNICATIONS LETTERS, VOL. 13, NO. 5, MAY 2009
Fig. 3.
Fig. 2. 16-QAM possible choices of candidate symbols to form subsets of candidate sequences.
channel output observation vector for the transmitted code word S, R = S + Z. Considering the notation and concepts presented in [8], ri represents the i-th soft output of the demodulator and sˆi , the corresponding symbol from the QAM modulation alphabet associated to ri , when hard decision output is used. The reliability associated to sˆi is proportional to the inverse of the noise vector absolute value, |ˆ ri − sˆi |. As the least reliable symbols are identified, it is possible, based on their position in the constellation geometry, to determine the set of nearest neighborhoods, according to the maximum likelihood criteria. This piece of information allows the construction of the set of candidates to be used in the soft turbo decoding algorithm, as explained in the next section. III. D ECODING A LGORITHM The turbo decoding procedure is based on the CHASE 2 algorithm [8], i.e., the group of candidate sequences is formed taking the position of the least reliable symbols received from the channel under consideration. A. Soft decision decoding Initially, the candidate sequence is formed as follows: • R is taken as a received vector corresponding to one row/column of the product matrix; • the least reliable received symbol in R is obtained, in order to compose the corresponding subset of candidates sequence, replacing this symbol by the nearest neighborhoods symbols in the QAM constellation, according to the minimum distance criterion, as shown in Fig. 2; • the same criterion is applied to the next least reliable symbol, in order to obtain the second subset of candidate sequences. The procedure continues until the number of candidates of the final set reaches a previously chosen value to be used in the turbo decoding process. Then, the decoding algorithm follows the classical approach proposed in [8], i.e, a hard decision decoding algorithm is applied to each sequence. As the candidate codeword set is obtained, the nearest codeword D from R is determined, applying a maximum likelihood decoding (MLD) algorithm.
Block diagram of the turbo-decoder at the k-th half-iteration.
B. Reliability of decoded symbols Once the winner codeword D is determined in the previous step by using MLD, the reliability of each symbol must be computed. For this purpose, a modified version of the turbo decoding algorithm proposed by Pyndiah [2] is used by manipulating the non-binary received symbols as described below. Each symbol di belonging to D has its associated extrinsic information wi calculated by: • if there is no codeword candidate C in the set of candidate sequences formed in the first step apart from D and such as cj = dj , in the j-position for j = 1, ..n, then: wj = (dj − ri )/ξ, with ξ being a weakening factor; • if there is a competing codeword C of D at minimum Euclidian distance from R, with cj = dj , then: wj = [((R − C)2 − (R − D)2 )/γ].[(dj − ri )/ξ]. As can be noticed, this calculation of the extrinsic information takes into consideration the existence of another word C in the set of candidates, measuring how far it is from the R received vector, in accordance with Pyndiah algorithm [2], [3]. The weakening factor ξ is associated to the reliability of the extrinsic information and is a parameter to be determined depending on the error probability [2]. The normalization factor, γ, depends on the constellation and on the chosen RS code. In the case of a 16-QAM, and a RS code over GF (24 ) with minimum Hamming distance chosen equal to 3, γ = (25.25)[(dE)2 ], with dE equal to the minimum Euclidian distance between the constellation adjacent symbols [7]. After this process, vector W associated with D is formed. The described process is applied to the whole matrix, firstly in the row/column directions, and sequentially in the other directions. The soft input for the decoding of the column (or row) in a k + 1 decoding step is given by: R[k + 1] = R + β(k).W (k).
(1)
In (1), β(k) is a scaling factor used to reduce the effects of the extrinsic information in the first decoding steps. Fig. 3 shows a simplified half-iteration diagram for the turbo decoding. IV. S IMULATION R ESULTS In order to evaluate the described algorithm, a simulation scheme using RS(15,13,2) block turbo code in a 16-QAM
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Results concerning the soft decision by using geometric approach are shown, too. In this case, the gain is about 1.5dB for BER = 10−6 , similar to the results obtained for the trellis soft RS decoded algorithm proposed in [11]. V. C ONCLUSIONS A turbo decoding algorithm using a geometric approach was proposed. Although this idea has not been tested for different codes and modulation schemes, the simulation result suggests that significant gains can be accomplished with this geometric method for turbo decoding. The non-binary treatment proposed implies a reduction in complexity of a factor up to m, depending on the mapping of the mounted scheme. Besides, the direct treatment over the noise and the received channel symbol from the channel with no re-mapping procedure during the process should simplify the hardware implementation. R EFERENCES
Fig. 4.
Simulation results.
modulation over an AWNG channel was implemented. The turbo decoding process used sixteen test patterns and six iterations. The noise vector is used as extrinsic information, weighted by an initial vector β = [0.06, 0.15, 0.24, 0.35, 0.48, 0.63, 0.9, 1] and, during the simulation, the reliability is supposed to increase. At the starting stage, the noise vector is 50% attenuated, corresponding to ξ = 2. Fig. 4 shows simulation results that are similar to the results obtained in [5], [9], [10]. The horizontal axis represents the signal to noise relation (Eb /N0 ), expressed in dB. Simulation result shows a 5dB gain for error bit rate (BER) equal to 10−5 and about 6dB for BER = 10−6 compared with a non-coded 16-QAM scheme.
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