Multilevel Codes and Multistage Decoding for Unequal Error Protection y
y
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Motohiko Isaka , Hideki Imai , Robert H. Morelos-Zaragoza , Marc P.C. Fossorier
]
and Shu Lin
]
The University of Tokyo, Institute of Industrial Science, 7-22-1 Roppongi, Minato-ku, Tokyo 1068558, Japan. Email:
[email protected] z LSI LOGIC Corp., 1551 McCarthy Blvd., M/S G-820, Milpitas, CA 95035-7451 USA. ] University of Hawaii at Manoa, Dept. of Electrical Engineering, 2540 Dole St. #483, Honolulu, HI 96822 USA. y
Abstract | In this paper, multilevel coded modulation with multistage decoding and unequal error protection is discussed. Three types of unconventional partitionings for 8-PSK and 16(64)-QAM constellations with UEP capabilities are analyzed. Based on the tight upper/approximated bounds on bit error probability, it is shown that more degrees of freedom in the construction of multilevel coding and multistage decoding can be accomplished by introducing asymmetries in PSK and QAM signal constellations. Generalizations to other PSK and QAM type constellations follow the same lines.
I. Introduction
Coding with unequal error protection (UEP) capability is a very attractive technique in communication and broadcasting systems because it gradually reduces the transmission rate. With multilevel block coded modulation and multistage decoding techniques [1], unconventional partitioning and/or nonuniform (asymmetric) signal constellations are natural strategies to provide UEP capability [2]. Previous work on coded modulation for UEP with asymmetric constellations has been presented in [2, 3, 4], all for the AWGN channel, and in [5] for the Rayleigh fading channel. However, no general theoretical analysis on the bit error probability has been given for multilevel block codes (although in [4] an upper bound for trellis codes is presented). In this paper, we discuss three types of unconventional partitionings (UCP) for multilevel codes (MLC) and multistage decoding (MSD) with UEP capabilities. Speci cally, we focus on 8-PSK and 16(64)-QAM constellations although its extension to other modulation formats follow the same line. In deriving the upper bounds on bit error rate (BER) based on union bound arguments for MLC/MSD with UCP of conventional constellations, we point out that two general approaches can be taken: one corresponds to the case when Pythagoras theorem holds, so that multi-dimensional decision regions can be decomposed into orthogonal 2-dimensional spaces [7]; in the other case, in which Pythagoras theorem does not
hold and multi-dimensional decision regions only can be cosidered [8]. The same UCP can be easily extended to asymmetric PSK and QAM constellations. Upper bounds on BER are derived by generalizing the results of the symmetric cases. However, as the distribution of nearest neighbors depends on the asymmetries, we observe that the union bound becomes in some cases very loose because of the important overlappings of decision regions. As a remedy, a tighter (and simple) approximated upper bound is derived based on Pythagoras theorem, by considering the asymmetries and the corresponding numbers of nearest neighbors. Further improvements are required for some UCP with QAM constellations because the overlappings of error regions de ned from the union bound may correspond to correct decisions at low SNR. Based on the derived tight upper/approximated bounds on BER, it is shown that more degrees of freedom in the construction of MLC/MSD can be accomplished by introducing asymmetries in PSK and QAM signal constellations. II. MLC/MSD for UEP with UCP
The MLC devised by Imai and Hirakawa [1] is a very powerful technique for constructing bandwidth ecient modulation codes systematically, with arbitrarily large minimum squared Euclidean distance (MSED). In particular, it provides the exibility to coordinate the distance parameters at each level. Furthermore, MLC constructed by this method allow the use of MSD procedures that provide good tradeo between error performance and decoding complexity. An 8-PSK (16-QAM) signal constellation is binary partitioned into L = 3 (4) levels. At each level, one of the subsets is selected by the corresponding bit of the component codeword. Let i2 and di represent the intraset distance (minimum squared Euclidean distance between signal points in disjoint subsets) and the minimum Hamming distance of the code Ci used at level-i, respectively, for i = 1; 2; 1 1 1; L. Then the MSED d(C ) of the overall modulation code is d(C ) = mini2f1;Lgfdi i2 g.
With traditional (Ungerboeck-type) MLC, partitioning is done in such a way that the intraset distance at the subsequent level is maximized. In this case, UEP capabilities can be obtained at high SNR by choosing d1 12 d2 22 d3 32 ( d4 42 ): (1) However, this approach does not work well at low SNR with MSD, especially when powerful component codes are used at the rst partitioning levels. This fact is due tod the large error dcoecients, which are proportional to 2 ; i = 1; 2 [9] (3 1 ; 2:25d2 ; 2d3 [10]) at rst and second (and third) index levels, where \2 (3; 2:25; 2)" is the average number of nearest signal points for a symbol. In summary, Ungerboeck-type set partitioning is not a good strategy to achieve UEP with MSD of MLC at low to medium SNR values due to the important increase in error coecients. As a result, UCP seem very promising to provide UEP capabilities with MLC and MSD. The three partitionings to be discussed in the following sections are depicted in Fig. 1 and Fig. 2 for 8-PSK and 16-QAM constellations, respectively. The set partitionings are represented in the gures in such a way that, at rst partitioning level, signal points are partitioned by color (black and white) and at the second level by symbols (square and circle). In block partitioning [8], signal points in a subset are contained in disjoint half planes at each level. It can be regarded as an opposite approach to Ungerboeck partitioning, in which the number of nearest neighbors is minimized at the expense of the intraset distance. As a result, UEP capabilities are easily achieved by using more powerful component codes at the rst levels of partitioning. This approach is suitable for L levels of error protection with the same levels of partitioning (2L-ary signal constellations). On the other hand, hybrid-type partitionings take the advantages of both partitioning methods to give l of error protection (1 < l < L). Two approaches can be considered. The direct hybrid or hybrid I partitioning [8] is obtained by applying block partitioning to the rst l 0 1 index levels, followed by L 0 l + 1 levels of Ungerboeck partitioning. This can be interpreted as enhancing the quality of the least important bits (LIB) by Ungerboeck partitioning once all other l 0 1 levels have been designed based on block partitioning. Mixed hybrid or hybrid II partitioning can be considered as another strategy to trade o the error performance of high index levels for an increase in the proportion of most important bits (MIB). The UCP in Figs. 1 and 2 can be extended to asymmetric 8-PSK and 16-QAM constellations. The 8-PSK constellation of interest in this paper is depicted in Fig. 3(a). We de ne as the angle between two signal points in a quadrant of an 8-PSK signal constellation, where
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Figure 2: 16-QAM constellation with: (a) block partitioning; (b) hybrid I partitioning; (c) hybrid II partitioning
ranges from 0 to 90 degrees. In the 16-QAM constellation depicted in Fig. 3-(b) point coordinates take values in the set f611 ; 612 g normalized to 121 + 122 = 1. Y
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Figure 3: Nonuniform constellations: (a) 8-PSK; (b) 16QAM In the following, let Cj be the binary (n; kj ; dj ) error correcting code applied as a component code to the j -th level (j = 1; 2; 1 1 1; L), and let A(wj) denote the number of codewords of weight w in Cj . III. Block partitioning
In block partitioning, [8] signal points in a subset are contained in disjoint half planes, as shown in Fig. 1(a) for 8-PSK. At the rst and second partitioning levels, the average number of nearest neighbor sequences is (1=2)w A(wi) with i = 1; 2, respectively, as opposed to 2w A(wi) for Ungerboeck partitioning [6]. This reduction in error coecient is realized at the expense of a nonincreasing intra-set distance at each level of the partitioning. Consequently, the rst decoding stage achieves an impressive coding gain, but the subsequent decoding stage perform quite poorly compared with Ungerboecktype partitioning. These characteristics can be re ned with an asymmetric constellation by choosing di erent values of in Fig. 1-(a) for 8-PSK. In this case, the intraset distance at each level becomes 12 = 22 = 4 sin2 ( 903600 ) and ). 32 = 4 sin2 ( 360 In the following, we consider 8-PSK signaling. For the rst (second) level code C1 (C2 ), decoding can be achieved by using the projection of the received signal components on the Y 0 (X 0)coordinate axis. As the decodings of both levels are equivalent, we only focus on the rst level decoding. Without loss of generality, the all-zero codeword is assumed to be transmitted at the rst level. Also, for MSD, any n-tuple can be chosen in the remaining levels.
Let w denote the Hamming weight of the decoded codeword cw for the rst level code C1 . With MSD, the decision regions occupy a w-dimensional space separated by the decision hyperplane Y1 + Y2 + 1 1 1 + Yw = 0. For the labeling of Fig. 1-(a), an error event occurs whenever the transmitted signal sequence is corrupted by AWGN noise so that the received sequence falls inside the region de ned by Y1 + Y2 + 1 1 1Yw < 0. Given that for the non-zero positions of cw , i components of the corresponding signal sequence have projection value 11 and the other w 0 i components have projection value 12 , the squared Euclidean distance between the transmitted signal sequence and the decision plane is given by [8] 1 (2) d2 (i) = (i11 + (w 0 i)12)2 ; P
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where 11 and 12 vary with the choice of . Note that the number of such hyperplanes is 0wi1 among the total 2w decision planes. Union bound arguments show that the BER, with a nonuniform 8-PSK constellation and block partitioning, at level j , j = 1; 2, is upper bounded as follows, Pbj Pb?j
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w w w (j) 0w X Q Aw 2 n i=0 i w=dj n X
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2REb d2 (i) N0 P
(3) where R = (k1 + k2 + k3 )=n is the overall MLC rate in bits/symbol , Eb =N0 denotes the energy per bit to noise ratio and Z 1 2 1 e0n =2 dn: (4) Q(x) = p 2 x Based on the above results, we should point out that two general approaches have to be taken when deriving union bounds for MLC with MSD. In the rst approach, the decoder is able to associate with each signal point a distinct decision region so that each code sequence representing a codeword of weight w is associated with a distinct 2w-dimensional decision region. In that case, these 2w-dimensional decision regions simply correspond to the underlying 2-dimensional constellation considered replicated in w orthogonal dimensions, so that Pythagoras theorem holds. The union bound follows from Pythagoras theorem. This method applies to conventional Ungerboeck-type partitioning [6], as shown in [7]. However, for many unconventional partitionings such as those of Fig. 1, di erent code sequences share the same decision regions, so that Pythagoras theorem no longer holds. A di erent approach is therefore necessary to evaluate the distance contributions to the union bound, as that taken in evaluating (2) [8, 12].
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The upper bound on BER for the third level can be derived by a simple union bound arguments. The derived upper bounds have been veri ed to be tight by comparing them with simulation results. Hence we can discuss the performance of block partitioning with variable parameters of the constellations based on the upper bounds. The required Eb=N0 to achieve Pb = 1005 , where the bound is suciently tight, is represented in Fig. 4 as a function of for a nonuniform 8-PSK constellation. The component codes are the (64,18,22), (64,45,8) and (64,63,2) extended BCH (e-BCH) codes at the rst, second and third levels, respectively. The overall rate is 1.96875(bit/symbol). The required Eb =No to achieve the same bit error probability with uncoded QPSK is also shown as reference. Compared with uniform 8-PSK constellations ( = 45 degrees), a lower bit error probability can be achieved at the rst and second levels in nonuniform 8-PSK constellations if is smaller than 45 degrees, due to the enhanced intraset distances. On the other hand, the performance of the third level degrades because the signal points within a quadrant get closer for small values of . For this choice of component codes, good trade-o s in error performance between the 3 stages are possible for < 60. For > 60, the error performance of stage-2 becomes worse than that of stage-3, provided correct decisions at stage-2 were made. Consequently, as shown in Fig. 4, errors propagate so that both stages have about the same error performance. 30 "uncodedQPSK" "level1-(64,18)eBCH" "level2-(64,45)eBCH" "level3-(64,63)eBCH"
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ond level, as shown in Fig. 1-(b). Therefore, the error performance of the rst level is the same as for block partitioning. On the other hand, the intraset2 distance of the third partitioning level is enhanced to 3 = 2. Moreover, with symmetric modulation, the number of average nearest signal sequences at second level increases to (3=2)w A(2) w . By proper choice of the component codes, the error performances of the last two levels can be balanced so that two levels of error protection are achieved. For < 45 degrees and hybrid I partitioning, the average number of nearest signal points in Fig. 3-(a) is one at the second partitioning level. However, for half the points (\inner points") in each half-plane corresponding to the decoding of stage-1, a second neighbor exists, which results in increased multiplicity, as shown in the following. On the other hand, for > 45 degrees, only one of the two points has a nearest signal point while both have one non-nearest neighbor. For a nonuniform 8-PSK constellation, the upper bounds on the BER for the rst and third levels are the same as for block partitioning after modifying the values. We can derive a union bound on BER of the second stage by generalizing the approach of block partitioning in the 2w-dimensional decision space [12]. The union bound of the second stage can be well approximated by considering e ective error coecient. De ne D12 and D22 (D12 < D22) as the squared Euclidean distance to two neighboring signal points for \inner constellation", respectively. For < 45, the e ective number of nearest neighbors for an \inner constellation"can be estimated as 1+ L , where L denotes the log-likelihood ratio representing the contribution of non-nearest neighbor with respect to the nearest neighbor [12], D22 exp 0 RE N0 4 REb 2 2 L = D12 = expf0 4N0 (D2 0 D1 )g: (5) RE exp 0 N0 4 Considering the other point has only one nearest neighbor, the e ective error coecient can be calculated as (2) w (1 + L =2) Aw for < 45, and based on the same arguments (1=2 + 1=L )w A(2) w for > 45, both including the symmetric case ( = 45; L = 1) as a special case. We veri ed that the approximated bound overlaps with the union bound at all SNR values and is tight, because Pythagoras theorem applies for the second decoding stage of the partitioning considered. In Fig. 5, the required Eb=N0 to achieve Pb = 1005 is calculated based on the derived bounds for the hybrid I partitioning of 8-PSK and uncoded QPSK. The component codes are the (64,18,22), (64,45,8) and (64,63,2) e-BCH codes at the rst, second and third levels, respectively and the overall code rate is R = 1:96875 (bits/symbol). For < 40, the multilevel code has two b
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Figure 4: Required Eb=N0 at Pb = 1005 with nonuniform 8-PSK and parameter : block partitioning The error performance of asymmetric QAM constellation with block partitioning can be analyzed based on the same arguments [8, 14]. IV. Direct hybrid (Hybrid I) partitioning
Hybrid I partitioning [8] for 8-PSK constellation is realized by introducing Ungerboeck partitioning at the sec-
space associated with the error event of weight w considered. Although not exact since Pythagoras theorem is not valid in deriving the union bound, this approach provides a simple and tight approximation, whose validity can be justi ed by simple geometrical considerations. Note that the truncated bound assuming only one nearest quadrant (L = 0) does not provide an upper bound and comes below the simulation results.
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Figure 5: Required Eb=N0 at Pb = 1005 with nonuniform 8-PSK constellations and parameter : hybrid I partitioning
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levels of protection, while for 40 < < 60, error propagation from level-2 to level-3 is not as severe. The required Eb=No for the third level is constant within this range, because the intraset distance of the third partitioning level is constant regardless of . If becomes too large, then the error performance of the rst level dominates the other levels and errors propagate. The same partitioning and bounding method can be applied to QAM constellations [8, 13]. V. Mixed hybrid (Hybrid II) partitioning
Another partitioning scheme, referred to as Hybrid II [11] is depicted in Fig. 1-(c) for 8-PSK. Compared with block partitioning, the intraset distance at the second level is enhanced at the cost of increasing the number of decision regions for the rst level. In the limiting case = 0, this partitioning becomes an Ungerboeck mapped QPSK constellation. With hybrid II partitioning, upper bounds for the second and third stage decoders directly follow from the union bound. However, at the rst stage decoding, the e ect of non-nearest quadrants cannot be ignored when decreases. The union bound can be derived again based on a generalization of the analysis for block partitioning in the 2w-dimensional decision space. Similar approximations as in Section IV are possible after de ning [12]
2 exp 0 RE N0 12 REb 2 = expf0 (12 0 121 )g: (6) L = RE 2 N 0 exp 0 N0 11 b
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This approximation follows from applying Pythagoras theorem based on Fig. 3-(a) to the (2w)-dimensional
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Figure 6: Required Eb=N0 at Pb = 1005 with nonuniform 8-PSK and parameter : hybrid II partitioning In Fig. 6, the required Eb=N0 to achieve Pb = 1005 is depicted for the hybrid II partitioning of 8-PSK and uncoded QPSK. The component codes are the (64,18,22), (64,57,4) and (64,51,6) e-BCH codes at the rst, second and third levels, respectively. The overall code rate is 1.96875(bit/symbol). Three protection levels are possible for < 35, and only two for 35 < < 60. For 16-QAM, the union bound and its approximation for the rst decoding stage based on the previous approach becomes loose, especially at low SNR. This is due to the fact that the overlappings of the decision regions not only are non-negligible, but also correspond to correct decisions due to the \inner"constellations (\0011"and so on), as suggested from Fig. 2. To overcome this problem, a new tight approximated upper bound is proposed for 16-QAM constellations based on the consideration of e ective nearest neighbors in each two dimensional error regions as in [13]. In Fig. 7, the required Eb =N0 at Pb = 1005, is calculated as a function of 11 . Recall that 11 = 1=p10 0:32 for the symmetric 16-QAM constellation. Component codes used are the (64,10,28), (64,45,8), (64,36,12) and the (64,36,12) e-BCH code for level-1,2,3 and 4, respectively, with the overall rate R = 1:984375 bits/symbol. For small 11 , all the four levels show virtually the same performance because of the error propa-
gation from the rst (and second) level(s). On the other hand, it can be observed that the rst and second levels require smaller Eb=N0 at a constant BER as 11 gets larger while the performance of the third and fourth levels deteriorate. 20
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, pp.1-5, November 18-22, 1996.
[5] N. Seshadri and C.-E. W. Sundberg,\Multi-level block coded modulation with unequal error protection for the Rayleigh fading channel," European Transactions on Telecommunications, Vol. 4, No.3, pp. 325-334, May-June 1993.
[7] U.Wachsmann, R.F.H. Fischer and J.B.Huber,\Multilevel Codes: Theoretical Concepts and Practical Design Rules, " submitted to IEEE Trans. Inform. Theory (revised July 1998).
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[8] R.H. Morelos-Zaragoza, M.P.C. Fossorier, S. Lin and H. Imai \Multilevel block coded modulation with unequal error protection," in Proc. of IEEE International Symposium on Information Theory (ISIT97), p.441, Ulm, Germany, June 29-July 4, 1997, and submitted to IEEE Trans. on Inform. Theory, March 1997.
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the Fifth Communication Theory Mini-Conference, GLOBECOM'96, London, U.K.
[6] G.Ungerboeck, \Channel coding with multilevel/phase signals, "IEEE Trans. Inform. Theory, vol.IT-28, pp.55-67, Jan. 1982.
"uncodedQPSK" "level1-(64,10)eBCH" "level2-(64,45)eBCH" "level3-(64,36)eBCH" "level4-(64,36)eBCH"
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[4] G.Taricco and E.Biglieri,\Pragmatic unequal error protection coded schemes for satellite communications", in Proc. of
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Figure 7: Required Eb=N0 at Pb = 1005 with nonuniform 16-QAM and parameter 11 : hybrid II partitioning VI. Conclusion
Multilevel coded modulation with UEP capabilities and multistage decoding has been presented for asymmetric 8-PSK and 16-QAM constellations and three types of unconventional partitionings. Theoretical bounds on BER have been discussed for each partitioning. The union bounds are very tight for block partitioning, but become loose in some cases for other partitionings, mainly due to important overlappings of decision regions. To overcome this problem, a tighter simple approximation technique has been proposed. The resulting approximated bounds are tight, and in conjunction with the union bound, can closely predict the performance of MSD of MLC designed from asymmetrical constellations. Although this paper focuses on asymmetric 8-PSK and 16-QAM constellations with certain degrees of regularity, these results can be extended to many more general and arbitrary signaling constellations. References [1] H. Imai and S. Hirakawa, \A new multilevel coding method using error-correcting codes,"IEEE Trans. Inform. Theory, vol. IT-23, no.3, pp.371-377, May 1977. [2] A.R. Calderbank and N. Seshadri, \Multilevel codes for unequal error protection," IEEE Trans. Inform. Theory, vol.39, no.4, pp.1234-1248, July 1993. [3] L.F. Wei, \Coded modulation with unequal error protection," IEEE Trans. on Commun., vol.41, no.10, pp.1439-1449, Oct. 1993.
[9] Y. Kofman, E. Zehavi and S. Shamai (Shitz), \Performance Analysis of a Multilevel Coded Modulation System," IEEE Trans. on Commun., vol. COM-42, pp. 299-312, February-MarchApril 1994. [10] T.J. Lunn and A.G. Burr,\Number of neighbours for staged decoding of block coded modulation,"Electron. Lett., Vol.29, pp.1830-1831, October 1993. [11] M. Isaka, R.H. Morelos-Zaragoza, M.P.C. Fossorier, S.Lin and H.Imai \Coded modulation for satellite broadcasting based on unconventional partitioning,"IEICE Trans. on Fundamentals, vol. E81-A, no.10, pp.2055-2063, Oct 1998. [12] M. Isaka, R.H. Morelos-Zaragoza, M.P.C. Fossorier, S. Lin and H. Imai \Error performance analysis of multilevel coded asymmetric 8-PSK modulation with multistage decoding and unequal error protection," in Proc. of IEEE International Symposium on Information Theory (ISIT98), p.230, Cambridge, MA, August 16-21, 1998. [13] M. Isaka, R.H. Morelos-Zaragoza, M.P.C. Fossorier, S. Lin and H. Imai \Multilevel coded 16-QAM modulation with multistage decoding and unequal error protection," in Proc. of IEEE Global Telecommunication Conference (GLOBECOM98), pp.3548-3553, Sydney, Australia, Nov. 8-12, 1998. [14] M. Isaka, R.H. Morelos-Zaragoza, M.P.C. Fossorier, S. Lin and H. Imai \Multilevel coded asymmetric modulation with multistage decoding and unequal error protection," submitted to IEEE Trans. on Commun., Sep. 1998.
