# The art of error correcting coding - Moreloz R.H.

ISBN 0471-49581-6

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Once a decision is made in the first stage, it is passed on to the second stage. The decoder in the second stage uses the trellis of the second component code with information from the first stage. For 8-PSK modulation, if the decoded bit in the first stage is bi — 0, then the received signal sequence is unchanged. If the decoded bit is b\ = 1, then the received signal is rotated by 45°. Again, branch metrics are distances (correlations) from the subsets selected at the

COMBINING CODES AND DIGITAL MODULATION

183

Es/No (dB)

Figure 112 Simulation results of MLD versus two-stage decoding for pragmatic 8-PSK

modulation.

second partitioning stage — given the decision at the first decoding stage — to the receive signal sequence.

Finally, based on the decisions in the first two decoding stages, the decoder of the third component is used. The branch metrics are the same as for BPSK modulation. There are four rotated versions of the BPSK constellation, in accordance with the decisions in the first two decoding stages. Therefore one approach is to rotate the received signal according to the decisions on bib-2 and use the same reference BPSK constellation. This is illustrated in Figure 119.

For medium to large code lengths, hybrid approaches may be the way to go for ultimate MCM performance, with powerful turbo codes used in the top partition levels and binary codes with hard-decision decoding assigned to lower partition levels. These combinations can achieve excellent performance [WFH].

9.3.2 Unequal-error-protection with MCM

Because of its flexibility in designing the minimum Euclidean distances between coded sequences at each partition level, MCM is an attractive scheme to achieve unequal error protection (UEP). However, great care has to be exercised in choosing the bits-to-signal mapping, so that the desired UEP capabilities are not destroyed. This issue was investigated in [MFLI, IFMLI], where several partitioning approaches were introduced that constitute generalizations of the block [WFH] and Ungerboeck [Ungl] partitioning rules.

In these hybrid partitioning approaches, some partition levels are nonstandard while at other levels partitioning is performed using Ungerboeck’s rules [Ungl], In this manner, a

184

THE ART OF ERROR CORRECTING CODING

Ó

Figure 113 16-QAM constellation for pragmatic TCM with two-stage decoding.

Figure 114 Example of MCM with 8-PSK modulation.

good tradeoff is obtained between error coefficients and intra-level Euclidean distances. To achieve UEP capabilities, the Euclidean distances at each partition level are chosen such that

diSf > d2d\ > ? ? ? > dv5l- (9.7)

For 1 < ³ < v, let vt (ui) be the codeword of Ct in correspondence to a A;,;-bit message vector ù, and let s — s(u) and s' = s(u') denote coded 2l'-ary modulation signal sequences corresponding to message vectors é = (é³,è2, - ? ? ,uv) and á' = (é[,é2, ? ? ?, u'v), respectively. The Euclidean separations [YI] between coded sequences at the ã-th partition level, for ³ = 1, • ? •, v, are defined as

Sj = min{d(s,s') : ù ô = u'j,j < ³} , (9.8)

with si = d\ äô s2 = d2S2, ? ? •, s„ = d„Si. For transmission over an AWGN channel, the set of inequalities (9.7) results in message vectors with decreasing error protection levels.

It is known from [WFH] that Ungerboeck’s partitioning rules [Ungl] are inappropriate for multistage decoding of multilevel coded modulations, at low to medium signal-to-noise ratios, because of the large number of nearest neighbor sequences (NN) in the first decoding stages.

Example 101 Figure 120 shows simulation results of the performance of a three-level coded 8-PSK modulation with the (64,18,22), (64,57,4) and (64,63,2) extended BCH codes (ex-BCH codes) as component codes Ci,i = 1, 2,3, respectively. The Euclidean separations are

185

O' Q

* *

0 0

6 Q t>2

• o t>3

0

1

Figure 115 Trellises of component codes of an example MCM with 8-PSK modulation.

si = 12.9, S2 = s3 = 8, for 18 and 120 information bits, respectively (asymptotic coding gains of 8.1 dB and 6 dB, respectively). The adverse effects of the number of NN (or error coefficient) in the first decoding stage are such that the coding gains are greatly reduced.

In the following, a UEP scheme based on nonstandard partitioning is presented. The reader is referred to [WFH, MFLI, IFMLI] for details on multilevel coding design for both conventional (equal error protection) and UEP schemes.

Nonstandard partitioning

The block partitioning [WFH] shown in Figure 121 (a) is used to construct three-level coded 8-PSK modulation schemes with UEP. In the figure, the color black is used to represent signal points whose label is of the form O&2&3, with b-2,fe3 e {0,1}. Similarly, the color white is used for points with labels 1&2^3- A circle indicates that the label is of the form 61O63, bi, bz ˆ {0,1}, while a square is used to represent signal points with labels 61163.

It can be seen from Figure 121 (b) that in order to determine the value of the first label bit, 61, only the X-coordinate is sufficient. If a signal point is on the left-hand half plane (X < 0) then it corresponds to b\ = 0, otherwise it corresponds to 61 = 1. In the same way, the Y-coordinate suffices to determine the value of the second label bit &2 • If a signal point lies in the upper half plane (Y > 0), then 62 = 0, otherwise b-2 = 1. This property of block partitioning allows the first and second levels to be decoded independently or in parallel. A similar observation led to the development of parallel decoding (PD) for multilevel codes with Gray mapping in [Schr],

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