Wireless Personal Communications (2005) 32: 1–7 DOI: 10.1007/s11277-005-7282-8
C
Springer 2005
Improved Performance of Maximum Likelihood Decoding Algorithm with Efficient Use of Algebraic Decoder P. G. BABALIS, P. T. TRAKADAS, T. B. ZAHARIADIS and C. N. CAPSALIS Division of Transmission Information Systems and Materials Technology, Department of Electrical and Computer Engineering, National Technical University of Athens, 9, Iroon Polytechniou, str. 157-80 Athens, Greece E-mail:
[email protected]
Abstract. A new soft decision maximum-likelihood decoding algorithm, which generates the minimum set of candidate codewords by efficiently applying the algebraic decoder is proposed. As a result, the decoding complexity is reduced without degradation of performance. The new algorithm is tested and verified by simulation results. Keywords: maximum likelihood decoding, algebraic decoder, communications systems
1. Introduction An efficient algorithm which performs soft decision maximum-likelihood decoding (MLD), using the exact probability for each codeword as a new likelihood metric and an efficient technique to generate the set of candidate codewords, has been proposed in [1]. This letter discusses a further improvement of the algorithm. The most important matter for a maximum-likelihood decoding algorithm is the derivation of a method, which generates the appropriate set of candidate codewords. Because of the use of algebraic decoder, this set should be as small as possible in order to minimize the decoding complexity. That is the subject of many studies reported [2–6], which focus on finding out optimal or near optimal MLD algorithms. In the proposed algorithm and in the one introduced in [1], the actual probability for each codeword is used and the most likely codeword is derived by the maximization of this probability. Also, Criterion 1 [1] is used to verify if a codeword from the set of candidates is the most likely or not. The difference between the two algorithms consists in the final set of candidates, which is generated by the algebraic decoder. The proposed algorithm exploits the fact that the decoding of some input sequences to the decoder produce codewords, which already exist in the set, and consequently, the algebraic decoder is not used. In the next section, we present the new decoding algorithm. In Section 3, simulation results are presented showing that the new algorithm considerably reduces the complexity. Concluding remarks are discussed in Section 4.
2. The Decoding Algorithm The set of candidate codewords is derived by the codewords which are produced by decoding of sequences v = y H + e, considering 2i erasure vectors whose elements e p1 , e p2 , . . . , e pi ,
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include all combination of 0 and 1. The value of parameter i is equal to the minimum value of l, which satisfies Criterion (2) [1]. We can notice that the 2i candidate codewords of set Ci may be either different or some of them can possibly be the same in the set. Thus, we can assume that, sometimes, the Ci will contain same codewords among the set. In this case, if we knew that a sequence y H + e gives, after its decoding, a codeword which has already been produced in the set Ci , we would not use the algebraic decoder and consequently the decoding complexity would be reduced. So, the derivation of a method which can possibly detect if an input sequence y H + e to the decoder gives a codeword existing in the set Ci , is our request. The following Lemma realizes that request. Lemma 1: Let c be a known codeword and Dc the set of positions where the decoder corrected the bits of the input sequence y H + ek and produced the codeword c. Putting in another way, Dc is the set of positions where sequences y H + ek and c differ. According to [1], the set p indicates the set of positions of the reorder elements of vector r. If an element of the first i of set p is the same with en element of set Dc there will be at least one input sequence, from the 2i − 1 remaining, which will be decoded to codeword c. Proof:
See the Appendix.
The above lemma secures that an input sequence to the decoder will be decoded or not to a known codeword. In the latter case, the use of algebraic decoder can be avoided and the decoding complexity is reduced. We now have to derive a procedure, which will be able to detect these input sequences to decoder. This procedure should be divided to two parts, first the examination of each sequence y H + ek , and then the connection with the decoding algorithm according to [1]. Let y H + ek and c be the input and output sequences of the decoder, respectively. We have already defined Dc as the set of positions where the previous sequences differ. We now define the vector Dc,y H +ek which has 1s at the positions where the two sequences differ and 0s elsewhere. This vector is stored at a two-dimensional array, D[dec][n], where parameter dec indicates the number of times that the algebraic decoder has been used. Then, we add (modulo-2 addition) vector ek to that row of the array in which Dc,y H +ek has been stored. The reason for making this addition will be cleared as soon as we describe the detection procedure in the following paragraph. Each input sequence to the decoder is checked whether it should be decoded according to the following procedure: We add vector ei to each row of the array and then, we calculate the Hamming weight. If the Hamming weight is smaller or equal to the error correction capability t of the code, the input sequence will not be decoded; else the input sequence will be decoded. This procedure is essentially equal with Lemma 1. Instead of checking sequence y H + ek with every codeword in the set, we check the difference of y H with every codeword. For this reason, we add ek to Dc,y H +ek , after decoding the input sequence y H + ek , in order to take the desirable difference between y H and the codeword which has been derived. According to the previous analysis the following Criterion is derived: Criterion 3: If there is even one vector, as a result of the addition between ei and every row of D[dec][n], whose Hamming weight is smaller or equal to the error correction capability t of the code, the input sequence y H + ek will not be decoded.
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Figure 1. Flowchart of the proposed algorithm.
The proposed decoding algorithm is based on the algorithm presented in [1], using Criterion 1 to check if the codeword is the most likely among the set, and Criterion 2 to find the set of candidate codewords. Criterion 3 is appropriately added to the original algorithm in order to improve its performance by taking advantage of the minimum use of algebraic decoder. The improved decoding algorithm can be summarized as follows (see Figure 1). 1. Decode y H + estep using algebraic decoder (the result is codeword c). 2. Compare Pr (c | y) with Pr (c∗ | y). 3. If Pr (c | y) > Pr (c∗ | y), check if codeword c satisfies Criterion 1 and either stop if it is satisfied, or else calculate the smallest value of l satisfying Criterion 2. 4. Check if y H + estep should be decoded according to Criterion 3. 5. Repeat this procedure until the search is completed for all the set of candidates.
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Figure 2. Comparison of the average decoding complexity between the two algorithms.
Figure 3. Comparison of the average decoding complexity between the two algorithms.
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The new algorithm is expected to give smaller decoding complexity since a more efficient use of algebraic decoder is exploited. Furthermore, this improvement should be greater for codes with larger error correction capability. 3. Simulation Results We simulated transmission of codewords through AWGN with antipodal signaling for five BCH codes with different length and rates. The simulation has been carried out for the decoding algorithm in [1] and the modified algorithm presented above. In Figures 2 and 3 we present the average decoding complexity for both algorithms for codes with length 15 and 31, respectively. The complexity required by the proposed algorithm is always less or equal, depending on the specific code, than that in [1]. We can also notice that for codes with the same block length the improvement is greater for codes with larger error correction capability. For example, the complexity of the proposed algorithm is slightly smaller than the one in [1] for BCH (15,11,3), but about 40–50% less, for low SNRs, for BCH (15,5,7). So, our assumption about the composition of the set of candidate codewords is verified and the decoding complexity is actually reduced. 4. Conclusions We have proposed a new soft-decision MLD algorithm that uses an additional Criterion in order to further reduce the decoding complexity of the algorithm introduced in [1]. That is achieved by the appropriate use of the algebraic decoder, minimiziming the number of times that is used by the algorithm. The results of computer simulation show that the average complexity is considerably reduced and hence the proposed decoding algorithm works effectively. APPENDIX: Proof of Lemma 1 According to Forney [5], a code with error correction capability t is able to correct all error patterns with maximum t errors. So, the difference between the input and output sequences to the decoder is d(y H + ek , c) ≤ t. We have defined Dc as the set of positions where these sequences differ. If an arbitrary element of the first i of set p is the same with an element of set Dc , the erasure vector having 1 at the common position of previous sets, let em , when added to y H , will give a sequence which will differ from c in t − 1 positions, d(y H + em ) ≤ t − 1 . That happens since the error, in y H + ek , which had been corrected by the decoder at the specific position, does not exist in y H + em since the 1 at this position of em changes the bit, after the modulo-2 addition, at sequence y H . Consequently, the sequence y H + em will be decoded to c.
References 1. P.G. Babalis, P.T. Trakadas and C.N. Capsalis, “A Maximum Likelihood Decoding Algorithm for Wireless Channels”, Wireless Personal Communications, Vol. 23, No. 2, pp. 283–295, 2002. 2. T. Kaneko, T. Nishijima, H. Inazumi and S. Hirasawa, “An Efficient Maximum-Likelihood-Decoding Algorithm for Linear Block Codes with Algebraic Decoder”, IEEE Transactions on Information Theory, Vol. 40, No. 2, pp. 320–327, 1994. 3. H. Tanaka and K. Kakigahara, “Simplified Correlation Decoding by Selecting Possible Codewords Using Erasure Information”, IEEE Transactions on Information Theory, Vol. IT-29, No. 5, pp. 743–748, 1983. 4. Gazelle and J. Snyders, “Reliability-Based Code-Search Algorithms for Maximum-Likelihood Decoding of Block Codes”, IEEE Transactions on Information Theory, Vol. 43, No. 1, pp. 239–249, 1997.
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5. G.D. Forney, “Generalized Minimum Distance Decoding”, IEEE Transactions on Information Theory, Vol. IT-12, pp, 125–131, 1966. 6. J.K. Wolf, “Efficient Maximum Likelihood Decoding of Block Codes Using a Trellis”, IEEE Transactions on Information Theory, Vol. 24, pp. 76–80, 1978.
Panagiotis G. Babalis was born in Athens, Greece, on January 3, 1974. He received his Diploma of electrical and computer engineering and the Ph.D. degree, both from National Technical University of Athens (NTUA), Athens, Greece, in 1996 and 2001, respectively. His main research interests include mobile satellite communications, modulation, and wireless communications systems coding. Dr. Babalis is a member of the technical Chamber of Greece.
Panagiotis T. Trakadas was born in Athens, Greece, on January 14, 1972. He received his Diploma of Electrical and Computer Engineering and the Ph.D. degree from National Technical University of Athens (NTUA), Athens, Greece, in 1996, and 2001, respectively. From 1998 to 2001, he participated in many European projects as a researcher. His main research interests include mobile communications systems and electromagnetic compatibility topics. Dr. Trakadas is a member of the Technical Chamber of Greece and IEEE Society.
Theodore B. Zahariadis received his Ph.D. degree in electrical and computer engineering from the National Technical University of Athens, Greece, and his Dipl.-Ing. Degree in computer engineering and information science from the University of Patras, Greece. Currently, he is the technical director of Ellemedia Technologies, where he leads R&D of end-to-end interactive multimedia services, embedded systems, and 3G/4G core network services. Since 1994 he has participated in many European co-funded projects. His research interests are in the fields of broadband wireline/wireless/mobile communications, interactive service deployment, management of IP/WDM networks, and embedded systems. He has published more than
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30 papers. He has been a reviewer and principal guest editor in many journals and magazines. He is a member of the ACM and the Technical Chamber of Greece.
Christos N. Capsalis was born in Greece, in 1956. He received the diploma in electrical and mechanical engineering from the National Technical University of Athens (NTUA), Athens, Greece, in 1979, the B.Sc. degree in economics from the University of Athens, Athens, Greece, in 1983, and the Ph.D. degree in electrical engineering from NTUA in 1985. He is currently a Professor at NTUA and Director of the wireless communications laboratory. His current research activities include wireless and satellite communications systems and EMC topics.
