A coding scheme for OFDM transmission is proposed, exploiting a previously unrecognised connection between pairs of Golay complementary sequences and second-order Reed-Muller codes. The scheme solves the notorious problem of power control in OFDM systems by maintaining a peak-to-mean envelope power ratio of at most 3dB while allowing simple encoding and decoding at high code rates for binary, quaternary or higher-phase signalling together with good error correction

Davis, James A

Jedwab, J

English

University of Richmond

University of RichmondUR Scholarship RepositoryMath and Computer Science Faculty Publications Math and Computer Science2-13-1997Peak-to-mean power control and error correctionfor OFDM transmission using Golay sequencesand Reed-Muller codesJames A. DavisUniversity of Richmond, jdavis@richmond.eduJ JedwabFollow this and additional works at: http://scholarship.richmond.edu/mathcs-faculty-publicationsPart of the Computer Sciences Commons, and the Mathematics CommonsThis Article is brought to you for free and open access by the Math and Computer Science at UR Scholarship Repository. It has been accepted forinclusion in Math and Computer Science Faculty Publications by an authorized administrator of UR Scholarship Repository. For more information,please contact scholarshiprepository@richmond.edu.Recommended CitationDavis, James A., and J. Jedwab. "Peak-to-mean Power Control and Error Correction for OFDM Transmission Using Golay Sequencesand Reed-Muller Codes." Electronics Letters 33, no. 4 (February 13, 1997): 267-68. doi:10.1049/el:19970205.Peak-to-mean power control and error correction for OFDM transmission using Golay sequences and Reed-Muller codes J.A. Davis and J. Jedwab Indexing terms: Frequency division multiplexing, Golay codes, Reed-Muller codes, Power control, Error correction A coding scheme for OFDM transmission is proposed, exploiting a previously unrecognised connection between pairs of Golay complementary sequences and second-order Reed-Muller codes. The scheme solves the notorious problem of power control in OFDM systems by maintaining a peak-to-mean envelope power ratio of at most 3dB while allowing simple encoding and decoding at high code rates for binary, quaternary or higher-phase signalling together with good error correction. Introduction: Orthogonal frequency division multiplexing (OFDM) modulation schemes offer many advantages for multicarrier trans-mission at high data rates over time dispersive channels [I], partic-ularly in mobile applications. The principal difficulty with such schemes is the need to control the peak-to-mean envelope power ratio (PMEPR) [2]. Any practical scheme must also allow good error correction, ease of encoding and decoding, and a high code rate. The scheme of [2, 3] uses block coding to transmit across the N carriers only those binary sequences with small PMEPR. How-ever, this entails an exhaustive search to identify the best sequences, requires large look-up tables for encoding and decod-ing, and leaves the problem of error correction unresolved. The scheme of [4] instead takes the transmitted codewords from a coset of a linear error-correcting code, choosing the coset representative or 'mask vector' by computationally intensive search in order to reduce the PMEPR. In this way the error correction properties are assured but the appropriate choice of linear code and coset repre-sentative for optimal PMEPR remains an open problem. The PMEPR of a binary or polyphase sequence of length N can be as large as N, but if the sequence is constrained to be a member of a Golay complementary pair then its PMEPR is at most 2, as recognised in [5]. (The aperiodic autocorrelation function of a se~uence (a:, a2, ... aN) for which a, E {O, I,_ ... , M-1} is C(u) = L, i~uexp(21y(a,-a;+u)IM) for u ~ 0, and a pair of sequences 1s a Golay complementary pair if the sum of their aperiodic autocorre-lations is zero for all u > 0.) We shall call any sequence which is a member of a Golay complementary pair a Golay sequence. There are at least 2mm! binary Golay sequences of length 2m [6, 7]. These sequences are potentially suitable for OFDM transmission, as mentioned in [3], but hitherto it has not been apparent that they possess sufficient intrinsic structure to form a practical coding scheme. Indeed most authors have contrasted the analysis of aperiodic sequence properties, for which constrained computer search is often the best known method [8], with that of periodic sequence properties, for which powerful algebraic methods such as group theory and character theory are available [9]. For background on coding theory, the reader should refer to [10, 11]. Second-order Reed-Muller codes: We consider 0-1 binary sequences of length 2m. Let x0 be the all-ones sequence. For i = I, 2, ... , m let x, be 2'-1 concatenated copies of the sequence comprising 2m-i O's followed by 2m-i l's. Then x 0 , x1, ••• , xm form the rows of a genera-tor matrix for the first-order Reed-Muller code RM(!, m), and these sequences together with the componentwise products x,x1 for I ~ i < j ~ m form the rows of a generator matrix for the second-order Reed-Muller code RM(2, m). Our central result is: Theorem 1: The codeword r,'):;1 x,uJXnu+i) + L.':'..--0c,x, is a binary Golay sequence of length 2m for any permutation re of { 1,2, .. ., m} and for any coefficients c, E {0,1 }. This shows how the 2mm! binary Golay sequences given by Golay's recursive and interleaving constructions [6] can be explic-itly represented as m!/2 distinct cosets ol RM(l, m), each contain-ing 2m+i codewords. The code consisting of all sequences identified in Theorem I is a subcode of RM(2, m) and therefore has a minimum distance of at least 2m-2, We can encode ltoglm!/2)J data bits as the choice of coset representative (for example, using a look-up table), and a further m+ I data bits directly as the c,. Received codewords can be efficiently decoded using standard hardware or software decod-ers for RM(2, m) (for example using majority-logic decoding [11]), recovering the coset representative from the coefficients of the terms x,x1. Corollary 1: ltoglm!/2)J+m+ 1 data bits can be encoded as 2m code bits such that all codewords have a PMEPR of at most 2, have a minimum distance of at least 2m-2, and belong to RM(2, m). For example, for m = 3 there are three choices of coset repre-sentative, namely x1x2+x2x1 = 00010010, x,x3+x2x3 = 00010100 and x 1x2+x1x3 = 00000110. We select one of two coset representa-tives (say the first two) according to the value of one data bit and add this coset representative to the encoded value L.,c,x, of four further data bits (ci,c2,c3,c4) to produce an 8-bit transmitted code-word. Although certain aspects of theorem 1 might, with hindsight and after careful consideration, be recognised in examples given in [7, 12], the connection with Reed-Muller codes and the consequent advantages for a practical coding scheme have not previously been noted. The coding scheme of corollary 1 uses only Golay sequences to ensure the PMEPR is at most 2. We can improve the code rate without unduly increasing the PMEPR by instead ordering the 2mtm-ll/2 coset representatives of RM(l, m) within RM(2, m), in increasing order of maximum PMEPR over the coset, and then selecting coset representatives from the ordered list. For example, by allowing any coset representative from the first half of the ordered list we can encode up to m(m+ 1)/2 data bits while retain-ing a minimum distance of 2m-2• In the case of length 16 code-words, this increases the number of data bits from 8 to 10 while retaining a mimmum distance of 4 by using 32 rather than 8 coset representatives, and yet the maximum PMEPR only increases from 2 to 4 (whereas it would be 16 if any transmitted sequence were allowed). Alternatively, we can increase the minimum distance in corol-lary 1 in exchange for a small reduction in code rate by choosing a subset of the m!/2 coset representatives for Golay sequences identi-fied in theorem 1. All such coding schemes retain a PMEPR of at most 2. The extreme version is to use just one coset representative so that the code is a single coset ofRM(l, m) with a minimum dis-tance of 2'"-1, encoding m+ 1 data bits (and taking advantage of special decoding algorithms for RM(l, m) [11]). Intermediate ver-sions can also be found. For example, in the case of length 16 codewords we can increase the minimum distance from 4 to 6 by reducing the number of data bits from 8 to 7. ELECTRONICS LETTERS 13th February 7997 Vol. 33 No. 4 267 Polyphase sequences: Theorem 1 generalises naturally to polyphase sequences: Theorem 2: The codeword 21-1 LT:i1 x,0l+x, 0+n+ L%o c, x, is a 21-phase Golay sequence of length 2m for any permutation n of {I, 2, ... , m} and for any coefficients c, E {O, 1, ... , 21-l}. This explicitly determines 211m+IJ.m!/2 21-phase Golay sequences and so provides a polyphase coding scheme analogous to Corol-lary 1. In the quaternary case 21 = 4 these sequences occur as m!/2 cosets of ZRM(l,m) in ZRM(2,m), each containing 4m+1 code-words (see [13] for the definition of the quaternary Reed-Muller code ZRM(r, m)). Received quaternary codewords can be effi-ciently decoded in the binary domain by applying the Gray map, under which ZRM(2, m) maps to RM(2,m+ 1) [13]. For higher phases 21 = 8, 16, ... , Theorem 2 indicates an appropriate defini-tion for the corresponding first-order and second-order Reed-Muller code. For all values of 21, theorem 2 provides a coding scheme in which Llog21(m!/2)J+m+1 data symbols can be encoded as 2m code symbols such that all codewords have a PMEPR of at most 2 and have a minimum Hamming distance of at least 2m-2• We can find similar variations on this scheme as described for the binary case, for example the code rate can be increased while maintaining the minimum Hamming distance. A very recent paper [14] reports the independent investigation of the use of Golay sequences in OFDM schemes. Translated into the language of the present paper, [14] essentially identifies a sub-set of the polyphase Golay sequences of theorem 2 involving m of the m!/2 coset representatives and arbitrary c,, noting that when only one coset representative is used the minimum Hamming dis-tance between codewords is 2m-1• However, [14] does not make the connection with first- or second-order Reed-Muller codes and in particular does not propose the use of efficient decoding tech-niques for Reed-Muller codes. [14] also does not determine the minimum Hamming distance when more than one coset represent-ative is used (except in the case m = 3). Conclusion: We have outlined a coding scheme in which the main criteria for a practical OFDM transmission system are simultane-ously satisfied. Details of the results announced here will be pro-vided in a forthcoming paper, including proofs of the claimed theorems 1 and 2, how to combine the identified Golay sequences into Golay complementary pairs, details of the decoding algo-rithms, and comparison of implementation options according to choice of code rate, PMEPR, minimum Hamming distance and number of phases. Acknowledgments: We are grateful to T. Wilkinson and A. Jones for helpful discussions that led to this research. J.A. Davis thanks Hewlett-Packard for generous hospitality and support during his sabbatical year 1995-6. ©IEE 1997 12 December 1996 Electronics Letters Online No: 19970205 J.A. Davis (Department of Mathematics and Computer Science, University of Richmond, Virginia 23173, USA) J. Jedwab (Hewlett-Packard Laboratories, Fi/ton Road, Stoke Gifford, Bristol BS12 6QZ, United Kingdom) References BINGHAM, J.A.C.: 'Multicarrier modulation for data transmission: an idea whose time has come', IEEE Commun. Mag., 1990, 28, pp. 5-14 2 JONES, A.E., WILKINSON, T.A., and BARTON, S.K.: 'Block coding scheme for reduction of peak to mean envelope power ratio of multicarrier transmission scheme', Electron. Lett., 1994, 30, pp. 2098-2099 3 WILKINSON, TA., and JONES, A.E.: 'Minimisation of the peak to mean envelope power ratio of multi carrier transmission schemes by block coding'. Proc. IEEE 45th Veh. Technol. Conf., Chicago, July 1995, pp. 825-829 4 JONES, A.E., and WILKINSON, T.A.: 'Combined coding for error control and increased robustness to system nonlinearities in OFDM'. Proc. IEEE 46th Veh. Technol. Conf., Atlanta, April-May 1996, pp. 904-908 5 POPOVIC, B.M.: 'Synthesis of power efficient multitone signals with flat amplitude spectrum', IEEE Trans. Commun., 1991, 39, pp. 1031-1033 6 GOLAY, MJ.E.: 'Complementary series', IRE Trans. Inf Theory, 1961, 7, pp. 82-87 7 GOLAY, M.J.E.: 'Sieves for low autocorrelation binary sequences', IEEE Trans. Inf Theory, 1977, 23, pp. 43-51 8 GOLAY, M.J.E., and HARRIS, D.B.: 'A new search for skewsymmetric binary sequences with optimal merit factors', IEEE Trans. Inf Theory, 1990, 36, pp. 1163-1166 9 POTT, A.: 'Finite geometry and character theory'. Lecture Notes in Mathematics 1601, (Springer-Verlag, Berlin 1995) 10 VAN LINT, J.H.: 'Introduction to coding theory'. (Springer-Verlag, Berlin 1992), 2nd Edn. 11 MacWILLIAMS, F.J., and SLOANE, N.J.A.: 'The theory of error-correcting codes' (North-Holland, Amsterdam 1986) 12 BUDISIN, s.z.: 'New complementary pairs of sequences', Electron. Lett., 1990, 26, pp. 881-883 13 HAMMONS, A.R., KUMAR, P.V., CALDERBANK, A.R., SLOANE, N.J.A., and SOLE, P.: 'The Z4-linearity of Kerdock, Preparata, Goethals, and related codes', IEEE Trans. Inf Theory, 1994, 40, pp. 301-319 14 VAN NEE, R.DJ.: 'OFDM codes for peak-to-average power reduction and error correction'. Proc. IEEE Globecom, London, November 1996, pp. 740-744 268 ELECTRONICS LETTERS 13th February 1997 Vol. 33 No. 4 

Peak-to-mean power control and error correction for OFDM transmission using Golay sequences and Reed-Muller codes

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Peak-to-mean power control and error correction for OFDM transmission using Golay sequences and Reed-Muller codes

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