The authors use binary linear UEP (LUEP) codes, in combination with a QPSK signal set and Gray mapping, to obtain new efficient block QPSK modulation codes with unequal minimum squared Euclidean distances. They give several examples of codes that have the same minimum squared Euclidean distance as the best QPSK modulation codes of the same rate and length. A new suboptimal two-stage soft-decision decoding is applied to LUEP QPSK modulation codes

Lin, Shu

Morelos-Zaragoza, Robert H

English

San Jose State University

Unequal Error Protection QPSK Modulation Codes

SJSU ScholarWorks

San Jose State UniversitySJSU ScholarWorksFaculty Publications Electrical Engineering1992Unequal Error Protection QPSK ModulationCodesRobert H. Morelos-ZaragozaUniversity of Hawaii at Manoa, robert.morelos-zaragoza@sjsu.eduShu LinUniversity of Hawaii at ManoaFollow this and additional works at: https://scholarworks.sjsu.edu/ee_pubPart of the Electrical and Computer Engineering CommonsThis Article is brought to you for free and open access by the Electrical Engineering at SJSU ScholarWorks. It has been accepted for inclusion in FacultyPublications by an authorized administrator of SJSU ScholarWorks. For more information, please contact scholarworks@sjsu.edu.Recommended CitationRobert H. Morelos-Zaragoza and Shu Lin. "Unequal Error Protection QPSK Modulation Codes" Faculty Publications (1992):1121-1125. doi:10.1109/ICCS.1992.255087I 1 1 '  Singapore ICCSllSlTA '92 Unequal Error Protection QPSK Modulation Codes Robert H. Morelos-Zaragoza and Shu Lin Department of Electrical Engineering University of Hawaii at Manoa 2540 Dole Street, Holmes 483 Honolulu, Hawaii 96822, USA Abstract Unequal error protection (UEP) codes find applica- tions in broadcast channels, as well as in other digital communication systems, where messages have differ- ent degrees of importance. In this paper, we use bi- nary linear UEP (LUEP) codes, in combination with a QPSK signal set and Gray mapping, to obtain new efficient block QPSK modulation codes with unequal minimum squared Euclidean distances. We give several examples of QPSK modulation codes that have the same minimum squared Euclidean distance (MSED) as the best QPSK modulation codes of the same rate and length. In the new constructions of QPSK modulation codes, even-length binary LUEP codes are used. Good LUEP codes are obtained when we combine shorter linear codes using either the well known lulu+?l- construction or the so-called construction X. Both con- structions have the advantage of yielding optimal or near optimal binary LUEP codes of short to moder- ate lengths, using very simple constituent codes, and may be used as constituent codes in the new construc- tions. LUEP codes lend themselves quite naturally to multi-stage decodings, using the decodings of compe nent subcodes. In this paper, we present a new subop timal two-stage soft-decision decoding of LUEP codes and apply it to LUEP QPSK modulation codes. 1 Introduction There are many practical applications in which it is re- quired to design a code that protects messages against different levels of noise, or messages with different levels of importance over a noisy channel with the same level of noise power. Examples of such situ- ations are: broadcast channels, multi-user channels, computer networks, pulse coded modulation (PCM) systems and source coding systems, among others. This research was supported by the NSF under Grant NCR-88813480 and by NASA under Grant NAG 5-931. Such a code is usually said to be an unequal error protection (UEP) code. In this paper, we propose to use binary linear UEP (LUEP) codes [l], in combina- tion with QPSK signal constellations, to obtain new efficient block QPSK modulation codes with unequal minimum squared Euclidean distances. That is, code sequences associated with the most important mes- sage bits are separated by a squared Euclidean dis- tance (SED) larger than the SED between code se- quences associated with less important message bits. In this paper two types of messages are considered, one more important than the other. Several examples of block LUEP QPSK modulation codes, having the same minimum squared Euclidean distance (MSED) as that of optimal QPSK modulation codes of the same rate and length [2-31, are presented. The paper is or- ganized as follows. In section 2, basic concepts and two constructions of LUEP codes based on specifying the generator matrix are presented. Section 3 deals with constructions of QPSK modulation codes and a suboptimal two-stage sofl-decision (TSSD) decoding of LUEP codes. An example is given which illustrates TSSD decoding of QPSK modulation codes. Finally, in section 4, conclusions on the results are presented. 2 When a code is used to provide multiple levels of er- ror protection, the conventional definition of minimum distance must be generalized. Since different levels of error protection are possible with a UEP code, a vec- tor of minimum distances, one for each level of error protection, needs to be defined. Let C be an ( n , k )  block code (not necessarily linear) over a finite alpha- bet A, n 2 k. That is, C is a one-to-one mapping from A' to A", i.e., Basic concepts of LUEP codes ~fi E A' 5 c(m) E A", 1121 Q 1992 IEEE In- __ Singapore ICCSllSlTA '92 where Based on Cl and Cz construct the following code, A k = A  X A  x ... x A .  As usual, an element m from A' is called a message, and an element E ( m )  from C is called a codeword. A' is known as the message set. Let A' be decomposed into the direct product of disjoint message subsets, A';, 1 5 i 5 e, such that and a message m E Ak can be expressed as p(C1, CZ) = { w) w = U 0 (U + V), U E c1, V E Cz} -k times Clearly, p(C1,Cz) is a (271, kl + kz) linear code with generator matrix, '= (  0 G z )  d = min{2di,max{di,dz}). GI GI The minimum distance of p(C1, Cz) is [8] Ak = A'' x At' x . . . x A'', m = (m,,mz,..*,mf), mi E A k ' ,  1 5 i 5 e, where each m i  is called the i-lh message part, 15 i _< e. The separation vector of C is defined as the &tuple S = (s~,sz,...,st), where si e min{d(E(m), ~'(m')) : mi # mi, mi,fi~i E A"), 1 5 i 5 e, where d ( X ,  X') denotes the Hamming distance between X and X' in A". Assume that C has all the compo- nents of its separation vector distinct and arranged in decreasing order, i.e., s1 > SZ > ... > S f ,  so that C is an (n, k) block code of minimum distance Q. Code C is said to be an (n, k) [-level LTEP code with separation vector for the message set A'] x A'z x . . . x A". This work concentrates on linear block multi-level error correct- ing codes. That is, A is taken to be the Galois field GF(q)  of q elements, where q is a power of a prime number. For a linear multi-level error correcting code, or LUEP code, denoted C, each element of the sepa- ration vector is given by si = min{wt(c(m)) : mi # 0, mi E GF(q)"} ,  C is said to be an (n, k) !-level LUEP code, with sepa- ration vector S = (SI ,  s2,. . . , s t ) ,  for the message space GF(q)'l x G F ( Q ) ' ~  x . . . x GF(q)".  S =  (Sl,SZ,.'.,St), A 11 i 5 e. (1) 2.1 LUEP codes specified by their generator ma t r ix  In this section, constructions of LUEP codes by ap- pending (or time sharing) cosets of subcodes in linear codes are presented. These constructions may be used to obtain constituent binary LUEP codes which, in conjunction with QPSK signals, yield efficient LUEP QPSK modulation codes (see section 3). 2.1.1 For i = 1,2, let C; be an (n, hi, d i )  linear code with generator matrix G;. Define the append operation be- tween two vectors U = ( U O ,  ul , .  . . , u ~ - ~ )  and V = ( V O ,  V I , .  . . , ~ " - 1 )  The lfilti + PI construction as A U0 V = (.U, til ,  " ' ,  U,,-l ,  VO,  W l ,  " ' (  t L 1 ) .  Theorem 1: p(C1,Cz) is a 2-level LUEP code with separation vector s = (sl,sZ), for the message space (0, 1}'1 x (0, 1}k2, where s1 2 min{2dl, max(d1, d z ) } ,  and sz 2 min{max{dl, d z } ,  d z }  = d z .  A A  Proof: See [7]. 2.1.2 Construct ion X [4] For i = 1 , 2 , 3 ,  let Ci denote a linear (ni, k;, d i )  binary code. Assume C3 C C, so that k3 5 kz and d3 >_ dz. Let Cx be the linear code whose generator matrix is where GI ,  [G:G;lT and G3 are the generator ma- trices of C1, CZ and C, respectively (Note that it is required that k1 = 122 - k3). Then C,y is an ( n l  + 713, kl + k3) linear code with minimum distance dx = min{d3, dl + dz) [SI. This method of combin- ing shorter linear codes to obtain a linear code with increased length and minimum distance is known as Construction X ,  and can be viewed as a generalization of the /filii + 1) construction. By an argument similar to that used to prove Theorem 1, we can prove the following result: Theorem 2: Cx is a 2-level LUEP code with sepa- ration vector S = (51, sZ), for the message space M = (0, l}kl x (0, 11'3, where si >_ di + dz, and 3 LUEP QPSK modulat ion codes In this section, we present a method of combining bi- nary two-level LUEP codes and QPSK modulation to achieve coded modulation schemes that offer two val- ues of minimum squared Euclidean distance, one for each message part to be protected. In this way, the most important message part will be mapped into code sequences with a larger squared Euclidean distance be- tween them than that corresponding to the less impor- tant message part. If data transmission is performed over an additive white Gaussian noise (AWGN) chan- nel, and the channel code is selected properly (i.e., efficient soft-decision decoding and small number of 1122 Q 1992 IEEE Singapore ICCSASITA '92 nearest neighbors), then at  high signal-tenoise ra- tios, we will have a smaller probability of bit error for the most important message part than for the rest of the message. To achieve the performance promised by a given minimum squared Euclidean distance, we present a new suboptimal two-stage soft-decision de- coding of two-level LUEP codes, that uses their trellis structure. 3.1 Constructions via Gray mapping In a QPSK signal constellation with Gray mapping be- tween labels and signal points, the squared Euclidean distance between signal points is twice the Hamming distance between the corresponding labels. We say that this QPSK signal constellation forms a second- order Hamming space [6]. By mapping 2-bit sym- bols into signal points in a QPSK signal set, via Gray mapping, we can combine (2n, IC1 + L z )  2-level LUEP codes and QPSK modulation to achieve a block coded modulation system that offers two values of minimum squared Euclidean distances, one for each message part. Some of the resulting LUEP QPSK block mod- ulation codes have the same minimum squared Eu- clidean distance as optimal QPSK block modulation codes with the same rate and length [2-31, while offer- ing in addition a larger minimum squared Euclidean distance between code sequences associated with most important message parts. The proposed construction is as follows: Let cb be a (271, kl + kz) binary LUEP code with separation vector 5 = (SI, SZ) for the message space {O, l}k l  x (0, l}ka. Consider a QPSK signal set S with the Gray mapping M between 2-bit symbols and S, 00 H O  01 H 1 11 H 2  10 - 3  Then C = M(Cb) is a 2-level LUEP QPSK block mod- ulation code of length n, dimension IC, rate R = k/2n (bits/dimension), and squared Euclidean separation vector S S E D  = (291,2SZ). In conventional coded modulation, given the mini- mum squared Euclidean distance (MSED) and rate of a modulation code, an asymptotic coding gain G is de- fined for an AWGN channel. For high signal-to-noise ratios, G equals the ratio of the MSED of the coded system to the MSED of an uncoded system transmit- ting at the same rate (number of bits per signal). Ac- cordingly, for each component of S S E D  we may define an asymptotic coding gain. In this work, we define an asymptotic coding gain vector as where, for i = 1,2, Gi = 10log1, [ 2si ] (dB) 4sin2 (r/2R) ~ 1123 Note that, as in the case of conventional coded mod- ulation systems, these asymptotic coding gains can only be reached if maximum-likelihood soft-decision decoding is available. To illustrate this construction method, in Table 1 some LUEP BPSK block modula- tion codes are listed. Some codes in Table 1 have the same minimum squared Euclidean distance as that of optimal block QPSK modulation codes with the same rate and length [2], and provide additional coding gain (or, equivalently, smaller probability of bit error) for the kl most important message bits. In Table 1, codes labeled with * are LUEP QPSK modulation codes ob- tained from the Iiilii + tl construction. Other codes are taken from [9]. It is interesting to note that a l l o p  timal block QPSK modulation codes found by Sayegh [2-31, of lengths 5 to 10, can be obtained from the [filii +VI construction combined with Gray mapped QPSK signal sets. All of these codes are actually LUEP QPSK modulation codes, and this appears to be the first time that this fact has been pointed out. 3.2 Two-stage soft-decision decoding Let C be an (n,IC) two-level LUEP code with separation vector (SI, s2) for the message space { O ,  l}kl x {0,1}k2. Then C can be represented as the direct sum of subcodes C1 and CZ, C = C1@ CZ, Le, C = {c = E l  + E2 : C l  E C1 and E Cz}, where CZ is an (n, kz, s2) subcode which contains all codewords of minimum weight of C ,  and C1 is an (n, k1, SI) subcode spanned by a system of coset r e p  resentatives of Cz in C. Let Ti be a trellis diagram for subcode Ci of C ,  i = 1,2. Then a trellis diagram of C can be expressed as the direct product of TI and 7'2, T = T1 @ 2'2. That is, states in T are pairs (SI, SZ), where si is a state in T,, for i = 1,2. The pair (SI, s2) is joined to all pairs ( s ; ,  si), in such a way that, for i = 1,2, si is joined to si in Ti [ll]. Viterbi maximum likelihood decoding algorithm can then be applied to T to estimate the most likely codeword of C using soft decisions. To reduce the number of computations in soft-decision decoding of a modulation code, a tech- nique called rnultr-stage decoding is usually employed. The proposed suboptimal two-stage soft-decision de- coding for two-level LUEP codes is as follows: 1. Using soft-decisions (squared Euclidean dis- tance) and the Viterbi algorithm, determine the closest path E l  in TI to the received sequence. At this decoding stage, the most significant message part is decoded. 2. Using soft-decisions and the Viterbi algorithm, determine the closest path E2 in E l  + T' to the re- ceived sequence, to estimate the less significant message part. Here E1 +Tz indicates that the branch metrics of E l ,  obtained in the first decod- ing stage, are used at  each decoding step of the Viterbi algorithm operating on trellis Tz. This two-stage soft-decision decoding is well known [2], [5], [lo], [I l l ,  [12]. However, thisappears to be the first Q 1992 IEEE - __  1 T  - Singapore ICCSllSlTA ‘92 time that multi-stage soft-decision decoding has been used for unequal error protection purposes. Although at  each stage the decoding is maximum-likelihood, the multi-stage soft-decision decoding method described above is suboptimal. At each decoding stage, the most likely path is estimated using only part (Ti) of the trel- lis T of C. This suboptimal multi-stage soft-decision decoding also increases the effective number of nearest neighbors, but this results in only a fraction of a dB in overall coding gain degradation [2], [lo], [ll], [12]. 4 Conclusions In this paper, we have introduced a new construction of QPSK block modulation codes for unequal error protection. These codes offer two values of minimum squared Euclidean distance (MSED), one for each mes- sage part. That is, code sequences associated with the most important message part are separated by a larger MSED. When these sequences are transmitted over an AWGN channel, a larger MSED translates into having a lower probability of error for the most important message symbols. We used Gray mapping on a QPSK signal set to o b  tain a second order Hamming space in which (an, IC) LUEP codes with separation vector I = (SI, s2) are mapped into (n,k) LUEP QPSK modulation codes with squared Euclidean separation S s f i ~  = (2sl,2s*). We have introduced a new suboptimal twestage soft- decision decoding for LUEP codes and shown its a g  plication in decoding LUEP QPSK modulation codes. Numerical results indicate that besides the well known penalty of a few tenths of a dB in overall coding gain, there is a larger cost to pay in coding gain for the most important message part, although this coding gain is still considerably larger than the overall coding gain. References [l] B. Masnick and J .  Wolf, “On Linear Unequal Er- tor Protection Codes,” IEEE Transactions on In- formation Theory, Vol. IT-13, No. 4, pp. 600-607, July 1967. (21 S.L. Sayegh, “A Class of Optimum Block Codes in Signal Space,” IEEE Transactions on Commu- nications, Vol. COM-34, no. 10, pp. 1043-1045, October 1986. [3] S.L. Sayegh, Private Communication (Tables of codes from reference 2), 1992. [4] N.J.A. Sloane, S.M. Reddy and C.L. Chen, “New Binary Codes,” IEEE Transactions on Informa- tion Theory, Vol. IT-18, No. 4, pp. 503-510, July 1972. [5] [5.] H. Imai and S. Hirakawa, “A New Multilevel Coding Method Using Error-Correcting Codes,” IEEE Transactions on Information Theory, Vol. IT-23, No. 3, pp. 371-377, May 1977. [6] F.R. Kschischang, P.G. de Buda and S. Pasupa- thy, “Block Coset Codes for M-ary Phase Shift Keying,” IEEE Journal on Selected Areas in Communications, Vol. 7, No. 6, pp. 900-912, Au- gust 1989. [7] W.J. Van Gils, “Linear Unequal Error Protection Codes from Shorter Codes,” IEEE Trans. Info. Theory, Vol. IT-30, No.3, pp.544-546, May 1984. [8] J.F. MacWilliamsand N.J.A. Sloane, The Theory of Error-Correcting Codes, Amsterdam: North- Holland 1978. [9] W.J. Van Gils, “Two Topics on Linear Unequal Error Protection Codes: Bounds on Their Length and Cyclic Code Classes,” IEEE Transactions on Information Theory, Vol. IT-29, No. 6, pp. 866- 876, November 1983. [lo] A.R. Calderbank, “Multilevel Codes and Mul- tistage Decoding,” IEEE Transactions on Com- munications, Vol. 37, No. 3, pp. 222-229, March 1989. [ll] F.Hemmati, “Closest Coset Decodingof lulu + V I  Codes,” IEEE Journal on Selected Areas in Com- munications, Vol. 7, No. 6, pp. 982-988, August 1989. [12] T. Kasami, T. Takata, T. Yamashita and S. Lin, “Suboptimum Decoding of Decomposable Block Codes,” submitted to IEEE Transactions on Information Theory, 1992. 1124 0 1992 IEEE I I l l  I Singapore ICCS/ISITA '92 Table 1: Some LUEP QPSK block modulation codes 2n k kl kz s1 s2 R (bits/dim) GI (dB) G2 (dB) 4 2 1 1 3 2  1 I 2  1.76 0.00 8 5 1 4 4 2  $8 3.28 0.27 2.03 0.27 2.71 0.95 3.98 3.01 * 0.64 10 7 1 6  4 2 7/10 3.65 1 0 7 4 3 3 2  7/10 2.40 0.64 3.06 1.30 4.77 3.01 * 3.98 3.01 3.96 0.95 0.95 2.71 1.56 3.32 5.44 3.98 3.01 4.07 3.10 1.20 14 11 1 10 4 2 11/14 4.21 14 12 1 11 3 2 617 3.51 1.75 8 5 4 1 3 2  5/8 8 6 1 5 3 2  314 10 5 1 4  5 4 112 10 8 1 7  3 2 415 12 6 1 5  6 4 112 12 6 2 4 5 4 112 1 2 9 1 8 4 2  314 12 9 4 5 3 2 314 14 7 1 6  7 4 112 14 7 4 3 5 4 112 1 4 8 1 7 5 4  417 14 11 4 7 3 2 11/14 2.96 1.20 3.01 * 1 2 1 0 1  9 3 2 516 * = LUEP QPSK code based on the lfilii + VI construction. 1125 

https://scholarworks.sjsu.edu/cgi/viewcontent.cgi?article=1003&amp;context=ee_pub

Unequal Error Protection QPSK Modulation Codes

Abstract

Similar works

Full text

Available Versions

San Jose State University

SJSU ScholarWorks