each pair of constellation points. The binary symbol error that corresponds to an edge is its edge label. For a constellation with 2 n points, each bi

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1 36th Annual Allerton Conference on Communication, Control, and Computing, September 23-2, 1998 Prole Optimal 8-QAM and 32-QAM Constellations Xueting Liu and Richard D. Wesel Electrical Engineering Department University of California, Los Angeles Los Angeles, CA and Abstract For low-rate trellis codes that do not utilize uncoded bits, a new approach to constellation labeling which maximizes the constellation edge prole can provide improved metrics. We investigate the edge prole optimality for 8-QAM and 32- QAM constellations. The labeling structures that achieve the optimal edge proles for 8-QAM and 32-QAM are presented. Explicit proofs of the optimal edge proles for each case are demonstrated. Also several other quasi-gray labelings for 32-QAM that are not prole optimal are given. 1 Introduction For low-rate trellis codes that do not utilize uncoded bits, the standard techniques of set-partitioning and Gray labeling are not well motivated. A new approach to constellation labeling maximizes the constellations edge prole [1], [2]. This approach produces better trellis codes (in the sense of the broadcast code design metrics) than the standard techniques. A low-rate 32-QAM trellis code design example in [2] shows how the new approach can yield improved metrics at low rates. This paper investigates 8-QAM and 32-QAM labeling strategies, identifying prole optimal labelings for each and proving their optimality. We also classify all quasi-gray labelings for 32-QAM into prole optimal and non-prole optimal labelings. Section 2 reviews the denition of the edge prole and some basic labeling lemmas [1] [2]. Section 3 presents the optimal edge prole 8-QAM constellation. Section 4 gives the structure and proof of optimal 32-QAM constellation. Section provides quasi-gray labelings for 32-QAM that are not prole superior. Conclusion is given in section 6, followed by an appendix containing proofs of several lemmas used in the paper. 2 Denition of the edge prole and basic labeling lemmas The denition of the edge prole and three lemmas about labeled constellations appear in [2], [1]. Consider the constellation to be a fully connected graph with an edge between This research is supported with a grant from Pacic Bell in conjunction with UC Micro grant

2 each pair of constellation points. The binary symbol error that corresponds to an edge is its edge label. For a constellation with 2 n points, each binary symbol error labels 2 n?1 edges. The edges with a given binary symbol error may not all have the same Euclidean distance. We are interested in the smallest distance associated with each edge label. We list denitions and basic lemmas below on which our following discussion are based. Denition 1 The edge prole of a labeled constellation is thonotonically increasing list of minimum squared distances, one for each nonzero edge label [2]. Denition 2 C 1 is prole superior to C 2 if the edge prole for C 1 is element by element greater than or equal to the edge prole for C 2, with at least one strict inequality [2]. Lemma 1 All the edges emanating from a given point must have dierent labels [1]. Lemma 2 Two dierent paths emanating from the same initial constellation point end at the same nal constellation point and form a cycle i they have the same edge label sum, i.e., the two dierent paths emanating from the same initial constellation point can't have the same edge label sum i they form a new path instead of a cycle [1], [3]. Lemma 3 If all points in a constellation are connected by paths consisting only of edges with thinimum edge length, then the set of edge labels of thinimum length edges must contain a basis for the set of all edge labels [1]. Another labeling lemma can be directly derived from Lemma 1 and Lemma 3. Lemma 4 If a basis has n distinct minimum-distance edge labels, then these basis vectors can label at most 2 n dierent constellation points. 3 Optimal edge prole of 8-QAM For the 8-QAM and 32-QAM constellations, since Gray codings are impossible [1], there are at least four distinct minimum-distance edge labels in 8-QAM and at least six distinct minimum-distance edge labels in 32-QAM. We refer to 8-QAM or 32-QAM constellation labelings with exactly four or six distinct minimum-distance edge labels as quasi-gray coded. For convenience, we denote thinimum-distance as one unit-distance in the rest of the paper. Figur shows how to construct an edge prole upper bound for 8-QAM. By starting from a central constellation point and drawing an edge to every other constellation point, an upper bound can be formed by these edges since all these edges are distinct and the edge labels appear exactly once among them. Figur gives the obtained upper bound for 8-QAM. The labeling structures that achieve this bound are given in Figure 2 where, and are linearly independent three-bit labels and e 4 must be either e 4 = (possible only in ) or e 4 = (possible in, and (c)). 4 Structure and proof of optimal 32-QAM For the 32-QAM constellation, an upper bound and the best known achievable prole are given in Tabl [2]. This upper bound is obtained by starting with the six distinct minimum distance edge labels and then using the same technique as 8-QAM. 2

3 E i d 2 min (E i) Figur: Edges for prole bound in 8-QAM. Edge prole bound for 8-QAM. e 4 e 4 e 4 e 4 e 4 (c) Figure 2: The optimal 8-QAM constellation labelings. In this paper, the tightness of this upper bound is investigated and a tighter bound which is the same as the best known edge prole is given. We begin with a lemma that partially characterizes the labeling structure imposed by six distinct minimum-distance edge labels. Lemma In any valid quasi-gray coded 32-QAM constellation labeling, one distinct minimum-distance edge label is equal to the XOR summation of three other distinct minimum-distance edge labels. Proof: In any valid quasi-gray coded 32-QAM constellation labeling, there are six distinct minimum-distance edge labels. Call these labels, e 2, e 3, e 4, e and e 6 where through e are the basis vectors. Then e 6 is some linear combination of these basis vectors, e 6 = k 1 k 2 e 2 k 3 e 3 k 4 e 4 k e ; fk i g 2 f0; 1g: (1) According to labeling lemmas 1 and 2, the valid edge labels for a unit-distance square are shown in Figure 3 where fi; j; l; mg 2 f1; 2; 3; 4; ; 6g. We will show that it is impossie i e i e i e l e i and are distinct. One is e 6. Figure 3: Valid edge labels in quasi-gray coded 32-QAM. ble to label a 32-QAM constellation only as in Figure 3. In other words, the structure shown in Figure 3 must appear in any valid 32-QAM constellation labeling. 3

4 Bound Best known Tabl: An upper bound for 32-QAM edge proles and the best known achievable prole. For a 32-QAM constellation, each row or column with six constellation points is labeled with at least three distinct unit-distance edge labels by Lemma 4. Now assuming we label 32-QAM only as in Figure 3 (denoted as Assumption 1), then all the horizontal edges in any column share the same label. Similarly, all the vertical edges in any row share the same label. Under Assumption 1, there arany types of labelings. In any such labeling, assume a minimum distance edge label to be a linear combination of any other vinimum distance edge labels. For any linear combination, a path (not a cycle) consisting of the edge label and its linear combination of edge labels can be found, which means this edge label is not equal to the corresponding linear combination by Lemma 2. This contradicts the fact that one of the six minimum-distance edge labels must be some linear combination of the basis vectors. Thus all such types of labeling structures under Assumption 1 are invalid, and therefore Figure 3 must appear in any valid quasi-gray coded 32-QAM labeling, i.e., e 6 = ; fx; y; zg 2 f1; 2; 3; 4; g: (2) Without loss of generality, we let, e 2, e 3, e 4, e and e 6 be the six distinct minimumdistance edge labels where through e are the basis vectors and e 6 = e 3 e 4 e : (3) Under these assumptions, according to lemmas 1 and 2, (e 2 ) can only label as Figure 3. If let one label rows, then will be used down each row. If also use one to label columns, then will be used across each column. Thus invalid subconstellation labelings as Figure 4 would then appear in 32-QAM labeling, in contradiction to Lemma 1. Thus we obtain another labeling lemma. e1 Figure 4: Invalid subconstellation labeling if labels both rows and columns. Lemma 6 In any valid quasi-gray coded 32-QAM, (e 2 ) can't be used to label both rows and columns. From Lemma, we can derive additional lemmas that characterize the structure of edge proles of quasi-gray coded 32-QAM constellation labelings. Lemma 7 In any valid quasi-gray coded 32-QAM constellation labeling, the structure of the edge prolust conform to Table 2, assuming thinimum-distance is one unit-distance (The proof is given in the Appendix). 4

5 The number of minimum-distance Possible The number of basis vectors in edge labels d 2 min edge labels? 1 1? 1 = 2, 4 2, 4, 8, 10? 2 +? 4 = 1 3, 1,, 9, 13? 3 + = 11 * 1 only possible here for the edge label e 6 = e 3 e 4 e Table 2: The edge prole structure of a valid quasi-gray coded 32-QAM constellation labeling. Lemma 8 In any valid quasi-gray coded 32-QAM, and e 2 can't both be used to label columns or both be used to label rows. (The proof is given in the Appendix). Lemma 9 When two or two e 2 appear in a horizontal or vertical path, an equivalent collapsed version of the subconstellation (i.e., treating the two points connected by or e 2 as one point) contains all labeling information as the original subconstellation. (The proof is given in the Appendix). Lemma 10 In a valid quasi-gray coded 32-QAM, if (e 2 ) appears once in a six-point row (column), then (e 2 ) must appear in the central position, and (e 2 ) can't appear twice in a six-point row (column). (The proof is given in the Appendix). Without loss of generality, let label rows and e 2 label columns. We now prove that the edge length prole in the second row of Tabl is the optimal edge prole. Theorem 1 The edge prole in the second row of Tabl is the optimal edge prole for quasi-gray coded 32-QAM. Proof: In any valid quasi-gray coded 32-QAM constellation, according to Lemmas 1, 2, 6, 8 and 10, is used to label rows and can appear once or three times in a six-point row; e 2 is used to label columns and can appear once or three times in a six-point column. Thus there are four cases we need to investigate: 1) one in a six-point row and one e 2 in a six-point column, 2) one in a six-point row and three e 2 in a six-point column, 3) three in a six-point row and one e 2 in a six-point column, 4) three in a six-point row and three e 2 in a six-point column. Using the symmetry of and e 2, we only need to consider the cases of 1), 2) and 4). Cas: appears once in a six-point row and e 2 appears once in a six-point column. The structure of 32-QAM for Cas is shown in Figure. First consider the labelings of the subconstellation circled by dotted lines in Figure where 6=, 6=, 6= and fx; y; j; kg 2 f3; 4; ; 6g by Lemma 1. According to Lemma 4, at least three distinct basis vectors must be from,, and. By Equation 3,,, and can't be four distinct basis vectors and among,, and, one of them must be equal to another one. By following the labeling lemmas, the three possible types of labeling for the subconstellation are shown in Figure -(d). Using the symmetry of and e 2, gures (c) and (d) are isomorphic. Thus we only need to consider the case of gures and (c). Figure 6 shows the two possible labeling structures for Cas, where,, and are distinct, fj, k, y, mg 2 f3, 4,, 6g and =. From gures 6 and, we can obtain Table 3 which represents the upper bound of edge prole of Cas.

6 ej e 2 e 2 ej (c) (d) Figure : 32QAM labeling structure of Cas., (c), (d) Subconstellation labelings of Cas. / e 2 e 2 e 2 e 2 / Figure 6: Two possiblinds of labeling structures for Cas. Case 2: appears once in a six-point row and e 2 appears three times in a six-point column. For this case, the labeling structure of 32-QAM is shown in Figure 7 where 6= 6=. First consider the labelings of the subconstellation inside the dotted rectangular box. ey (c) (d) e 2 e 2 em (e) (f) (g) Figure 7: 32QAM labeling structure of Case 2. -(g) six possible labelings for the subconstellation of Case 2. Six possible labelings for this subconstellation are shown in gures 7-(g) where,, and are distinct, fj, x, y, mg 2 f3, 4,, 6g and =. Correspondingly there are six possible constellation labelings which are shown in gures 8-(f). Using the labeling lemmas, from gures 8-(f), we obtain Table 4, which represents the upper bound of the constellation labelings respectively where gures 8(c) are isomorphic and so are gures 8(d). Using Lemma 7, Table 3 and Table 4, the upper bound of the edge proles of Case 1and Case 2-(f) can be obtained. We put the upper bound of Cas and Case 2 in Table where all these upper bounds of edge proles are dominated by Cas. 6

7 E in Cas d 2 min(e) E in Cas d 2 min(e) e 2 2 e e 2 2 e 2 2 e 2 2 e e 2 4 e 2 4 e 2 8 e e 2 8 e 2 8 e 2 8 e 2 8 e 2 e 2 e 2 e 2 e 2 e 2 e 2 e 2 e 2 e 2 Table 3: The upper bound of edge prole of Cas (Figure 6) and Cas (Figure 6). All these upper bounds of edge proles are achievable. The labelings which achieve these upper bounds will be discussed in the next section. Case 4: appears three times in a six-point row and e 2 appears three times in a six-point column. Figure 9 shows the 32-QAM labeling structure for Case 4. We apply Lemma 9 to get the equivalent subconstellation. All other blank edges must be labeled by using e 3 through e 6. In Figure 9, we need to label 9 dierent points by using e 3 through e 6, which have only three basis vectors. By Lemma 4, it is impossible to label Figure 9. Correspondingly it is not possible to label Figure 9. Therefore Case 4 can't appear in any valid quasi-gray coded 32-QAM. In conclusion, the edge prole bound for a quasi-gray coded 32-QAM constellation is the rst row of Table which is the same as the second row of Tabl. All valid quasi-gray coded 32-QAM constellation labelings which achieve this edge prole bound have an optimal edge prole. Further note that only Cas can achieve this edge prole bound. Six distinct classes of Quasi-Gray labeled 32-QAM The labeling [2] which achieves the optimal edge prole is shown below in octal ( on the left-hand-side). All other labelings which achieve the optimal edge prole are isomorphic and have the same labeling structure as Cas (illustrated in Figure 6). Several other quasi-gray labelings that are not prole optimal exist. Corresponding to the labeling structure of Cas (illustrated in Figure 6 ), one labeling which achieves this prole bound is shown below in octal (on the right hand side). All other labelings 7

8 e 2 e 2 e 2 e 2 e 2 e 2 (c) e 2 e 2 e 2 e 2 e 2 e 2 (d) (e) (f) Figure 8: Six possible labelings for Case 2. e 2 e 2 Figure 9: The labeling structure for Case 4. labeling structure for Case 4. The equivalent subconstellation achieving this prole are isomorphic and have the same labeling structure as Cas Corresponding to Case 2(c), Case 2(d), Case 2(e) and Case 2(f), the labelings which achieve their respective upper bounds of edge proles are given below from left hand side to right hand side. Furthermore they have isomorphic labelings and share the same labeling structures given in our proof

9 E in Case 2 (c) d 2 min(e) E in Case 2 (d) d 2 min(e) E in Case 2 (e) d 2 min(e) E in Case 2 (f) d 2 min(e) e 2 2 e 2 2 e 2 2 e e 2 2 e 2 2 e 2 2 e 2 2 e 2 2 e 2 2 e 2 2 e 2 2 e 2 2 e 2 2 e 2 2 e e 2 2 e e e 2 4 e 2 8 e 2 8 e 2 8 e 2 8 e 2 8 e 2 8 e 2 8 e 2 8 e 2 8 e 2 8 e 2 e 2 e 2 e 2 e 2 e 2 e 2 e 2 e 2 e 2 e 2 e 2 e 2 e 2 e 2 e 2 e 2 e 2 e 2 e 2 e 2 e 2 e 2 e 2 e 2 e Table 4: The upper bound of edge prole of Case 2. Cas Cas Case 2, (c) Case 2, (d) Case 2 (e) Case 2 (f) Table : The upper bounds for 32-QAM edge proles of Cas and Case 2. 6 Conclusion We have identied the optimal edge prole structures for both 8-QAM and Quasi-Gray coded 32-QAM constellations. Several other quasi-gray labeling structures for 32-QAM that are not prole optimal also have been given. For low rate codes, a better edge prole provides larger Euclidean distance which is demonstrated in a code design example by wesel [2]. We also conjecture that even high rate trellis codes benit from a superior edge prole when the number of states is large enough. But this is a moot point, since considering the level of complexity, it is more worthwhile to use turbo codes than trellis codes. Appendix Proof of Lemma 7 Proof: By Equation 3, some edge labels having two minimum-distance edge labels can be treated as having four minimum-distance edge labels and vice versa. Similarly some edge labels having threinimum-distance edge labels can be treated as having vinimumdistance edge labels and vice versa. Thus we group the cases of having two or four minimum-distance basis vectors in edge labels together and group the cases of having 9

10 REFERENCES Prole Optimal 8-QAM and 32-QAM Constellations three or vinimum-distance basis vectors in edge labels together. Since no edge label has the squared distance greater than 13 in 32-QAM edge proles shown in the bound, we only need to consider possible paths of edge labels with the squared distances less or equal to 13. For the case of two or four minimum-distance basis vectors in edge labels, there are six possible paths, which have d 2 min(e) equal to 2 or 4 or 8 or 10. The number of such edge labels is?? = 1. For the case of three or vinimum-distance basis vectors in edge labels, there are also six possible paths, which have d 2 min(e) equal to 1 or or 9 or 13. The number of such edge labels is?? 3 + = 11. Proof of Lemma 8 Proof: Using the symmetry of rows and columns, we only need to show that and e 2 can't both label columns. Assume both and e 2 label columns in 32-QAM, then they can't be used to label rows according to Lemma 6. Since each column with six constellation points must be labeled with at least three distinct minimum-distance edge labels by Lemma 4, then there is at least one column that is not labeled with either or e 2. Therefore we have to label the subconstellations with 12 points or 10 points by using e 3, e 4, e and e 6 (only three of them form a basis and can label at most 8 points), which is impossible. Proof of Lemma 9 Proof: This lemma is true because, e 2 can only be labeled as in Figure 3. So by applying the collapse rule and Lemma 2, the labeling information would not get lost in the collapsed version. Proof of Lemma 10 Proof: Using the symmetry of and e 2, rows and columns, we only need to consider the case of appearing in rows. If labels only one edge label and does not appear in the central position, then we have to label subconstellations with 28 or 22 points by using e 2, e 3, e 4, e and e 6 (only four of them form a basis and can label at most 16 points), which is impossible. Thus must appear in the central position if appears only once. If appears twice in a six-point row, there are ve possible labeling structures. Among these ve possible labeling structures, we have to label either subconstellations with 18 or 24 points or collapsed subconstellations with 18 or 20 points all by using e 2 through e 6 (only four of them form a basis and can label at most 16 points), which is impossible. Therefore can't appear twice in a six-point row in a valid quasi-gray coded 32-QAM. References [1] R. D. Wesel. Trellis Code Design for Correlated Fading and Achievable Rates for Tomlinson Harashima Precoding. PhD thesis, Stanford University, Aug [2] R. D. Wesel, C. Komninakis, and X. Liu. Towards Optimality in Constellation Labeling. In Proc. of Globecom '97, Phoenix, AZ, November [3] F. Harary. Graph Theory. Addison Wesley Longman, Incorporated,