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

ISBN 0471-49581-6

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Figure 112 shows simulation results of MLD (denoted with the legend “TC8PSK.23.SSD”) and two-stage decoding of pragmatic TC-8PSK (legend “TC8PSK_23_TSD”). With two-stage decoding, a loss in performance of only 0.2 dB is observed compared with MLD.

A similar transformation can be applied in the case of M-QAM, the difference is that the transformation is based solely on the I-channel and Q-channel symbols. That is, there is no need to compute the phase. An example is shown in Figure 113 for TC 16-QAM with ? = 2 coded bits per symbol. The coded bits are now the indexes of cosets of QPSK subsets. The transformation of 16-QAM modulation (? = 2) is given by a kind of “modulo 4” operation:

1*1 < 2; x > 2; x < 2,

\y\ < 2; y > 2; y< 2.

Finally, it is interesting to note that a pragmatic TCM system with a turbo code as component code was recently proposed in [WM|.

x =

180 THE ART OF ERROR CORRECTING CODING

Figure 108 Two paths at minimum squared Euclidean distance in the trellis of Example 96.

Figure 109 Block diagram of an encoder of pragmatic TCM.

9.3 Multilevel coded modulation (MCM)

In the Imai-Hirakawa multilevel coding scheme [IH], the 2"-ary modulation signal set is binary partitioned in u levels. The components of codewords of v binary component codes Ci, 1 < i < v, are used to index the cosets at each partition level. One of the advantages of MCM is the flexibility of designing coded modulation schemes by coordinating the intra-set Euclidean distances, Sf, i = 1,2, • • •, v, at each level of set partitioning, and the minimum Hamming distances of the component codes. Wachsmann et al. [WFH] have proposed several design rules that are based on capacity (by applying the chain rule of mutual information) arguments. Moreover, multilevel codes with long binary component codes, such as turbo codes or LDPC codes, were shown to achieve capacity [WFH, For8].

It also worthwhile noting that, while generally binary codes are chosen as component codes, i.e., the partition is binary, in general the component codes can be chosen from any finite field GF{q) matching the partition of the signal set. Another important advantage of multilevel coding is that (binary) decoding can be performed separately at each level. This multistage decoding results in greatly reduced complexity, compared with MLD for the overall code.

COMBINING CODES AND DIGITAL MODULATION

181

Figure 110 Block diagram of a two-stage decoder of pragmatic TCM.

9.3.1 Constructions and multi-stage decoding

For 1 <i <v, let Ci denote a binary linear block (n, ki, di) code. Let V{ = (vn, n,2, ? • •, vln) be a codeword in Ci, 1 < i < v. Consider a permuted time-sharing code 7r(|Ci IC21 • • • \CV\), with codewords

v = (V11V21 • ? ? V„1 V12V22 ? ? ? Vv2 ??? VinV2n ? ? ? vvn) .

Each i/-bit component in v is the label of a signal in a 2"-ary modulation signal set S. Then

s(v) = (s(vnv2i ? ? -t;„i),s(t;i2U22 ' ‘ ’^2), • • ?, s(vlnv2n ? ? ? vun)

is a sequence of signal points in S.

The following collection of signal sequences over S,

K^{s{v) : wG7r(|C1|C2|"-|a|)},

forms a v-level modulation code over the signal set S, or a !/-level coded 2^-ary modulation. The same definition can be applied to convolutional component codes.

The rate, or spectral efficiency, of this coded modulation system, in bits/symbol, is R = (ki + k2 + ? ? ? + kv)/n. The MSED of this system, denoted by Dq( A), is given by [IH]

D2C(A) > min {diSf}. (9.6)

Example 99 In this example, a three-level block coded 8-PSK modulation system is considered. The encoder structure is depicted in Figure 114. Assuming a unit-energy 8-PSK signal set, and with reference to Figure 103, note that the MSED at each partition level are 5} = 0.586, 51 = 2 and <5f = 4.

The MSED of this coded 8-PSK modulation system is:

Dq(A) = min{di5l, d25\, ^3^} = min{8 x 0.586,2 x 2,1 x 4} = 4,

and the coding gain is 3 dB with respect to uncoded QPSK The trellises of the component codes are shown in Figure 115. The overall trellis is shown in Figure 116.

182

THE ART OF ERROR CORRECTING CODING

Figure 111 Partitioning of an 8-PSK constellation (? = 2) and coset points.

As mentioned before, one of the advantages of multilevel coding is that multistage decoding can be applied. Figures 117 (a) and (b) show the basic structures used in encoding and decoding of multilevel codes. Multistage decoding results in reduced complexity (e.g., measured as number of branches in trellis decoding), compared to MLD decoding (e.g., using the Viterbi algorithm and the overall trellis.) However, in multistage decoding, the decoders at early levels regard the later levels as uncoded. This results in more codewords at minimum distance, i.e., an increase in error multiplicity or number of nearest neighbors. The value of this loss depends on the choice of the component codes and the bits-to-signal mapping, and for BER 10 2 ~ 10 '5 can be in the order of several dB.

Example 100 In this example, multistage decoding of three-level coded 8-PSK modulation is considered. The decoder in the first stage uses the trellis of the first component code Ci- Branch metrics are the distances (correlations) from the subsets selected at the first partitioning level to the received signal sequence, as illustrated in Figure 118.

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