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

ISBN 0471-49581-6

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Example 97 Figure 107 shows the modified state diagram for the 4-state TC 8-PSM modulation of Example 96. The branches in the trellis have been labeled with integers corresponding to the eight phases of the modulation signals. To compute the weight enumerating sequence T(x), the same procedure as in Section 5.3 is applied. Alternatively, by directly analysing the trellis structure in Figure 108, it can be deduced that the MSED between coded sequences is

Dc = min{?>par-

DL} = 3-172,

which, when compared to an uncoded QPSK modulation system with the same spectral efficiency of 2 bits/symbol, gives an asymptotic coding gain of 2 dB.

9.2.4 Pragmatic TCM and two-stage decoding

For practical considerations, it was suggested in [Vit4, ZW] that the 2"-ary modulation signal constellations be partitioned in such a way that the cosets at the top two partition levels are

COMBINING CODES AND DIGITAL MODULATION

2v-ary modulation

177

Figure 104 General encoder of rate-(V — l)/v trellis-coded modulation.

Figure 105 Encoder of a 4-state rate-2/3 trellis coded 8-PSK modulation.

associated with the outputs of the standard memory-6 rate-1/2 convolutional encoder. This mapping leads to a pragmatic TCM system. With respect to the general encoder structure in Figure 104, the value of k = 1 is fixed, as shown in Figure 109. As a result, the trellis structure of pragmatic TCM remains the same, as opposed to Ungerboeck-type TCM, for all values of v > 2. The difference is that the number of parallel branches v — 2 increases with the number of bits per symbol. This suggests a two-stage decoding method in which, at the first stage, the parallel branches in the trellis “collapse” into a single branch, and a conventional off-the-shelf Viterbi decoder used to estimate the coded bits associated with the two top partition levels. In a second decoding stage, based on the estimated coded bits and the positions of the received symbols, the uncoded bits are estimated. Figure 110 is a block diagram of a two-stage decoder of pragmatic TCM.

In [MM], a symbol transformation is applied to the incoming symbols that enables use of a Viterbi decoder without changes in the branch metric computation stage. The decoding procedure is similar to that presented in [CRKO, PS], with the exception that, with symbol transformation, the Viterbi algorithm can be applied as if the signals were BPSK (or QPSK) modulated. This method is described below for M-PSK modulation.

Specifically, let (x, y) denote the I and Q coordinates of a received M-PSK symbol with amplitude r = y/x2 +y2 and phase <j> = tan _1 (y/x). Based on 0, a transformation is applied such that the M-PSK points are mapped into “coset” points labeled by the outputs of a rate-1/2 64-state convolutional encoder.

178

THE ART OF ERROR CORRECTING CODING

Figure 106 Trellis structure of a rate-2/3 trellis coded 8-PSK modulation based on the

encoder of Figure 105.

For TCM with M-ary PSK modulation, M = 2", v > 2, let ? denote the number of coded bits per symbol3, where ? = 1,2. Then the following rotational transformation is applied to each received symbol, {x. y), to obtain an input symbol (x1, y') to the VD,

X³ = r cos — Ô)] ,

ó1 = Ã8³ï[2‘'_ˆ(0-Ô)]³ (9.5)

where Ô is a constant phase rotation of the constellation that affects all points equally. Under the transformation (9.5), a 2m-CPSK coset in the original 2"'-PSK constellation “collapses” into a coset point in a 2^-PSK coset constellation in the x' — y' plane.

Example 98 A rate-2/3 trellis-coded 8-PSK modulation with 2 coded bits per symbol is considered. Two information bits (u1,U2) are encoded to produce three coded bits (u2,V2,vi), which are mapped onto an 8-PSK signal point, where (v-2. vy ) are the outputs of the standard rate-1/2 64-state convolutional encoder4. The signal points are labeled by bits (u2, V2-, Vi), and the pair (v2, Vi) is the index of a coset of a BPSK subset in the 8-PSK constellation, as shown at the top of Figure 111.

In this case ô' = 2ô and, under the rotational transformation, a BPSK subset in the original 8-PSK constellation collapses to a coset point of the QPSK coset constellation in the x' — y' plane, as shown in Figure 111. Note that both points of a given BPSK coset have the same value of ô'. This is because their phases are given by ô and ô + òò.

3 The case ? = 2 corresponds to conventional TCM with 8-PSK modulation. The case ? = 1 is used in TCM with coded bits distributed over two 8-PSK signals, such as the rate-5/6 8-PSK modulation scheme proposed in the DVB-DSNG specification [DVB],

4 v\ and V2 are the outputs from generators 171 and 133, in octal, respectively.

COMBINING CODES AND DIGITAL MODULATION

4^

179

a = d(0,4) = 2 b = d(0,l) = 0.765 c = d(0,2)= 1.414 d = d(0,3) = 1.848

Figure 107 The modified state diagram of a 4-state TC 8-PSK modulation scheme.

The output of the VD is an estimate of the coded information bit, ui. In order to estimate the uncoded information bit, u2, it is necessary to re-encode u\ to determine the most likely coset index. This index and a sector in which the received 8-PSK symbol lies can be used to decode U2- For a given coset, each sector S gives the closest point (indexed by u2) in the BPSK pair to the received 8-PSK symbol. For example, if the decoded coset is (1,1) and the received symbol lies within sector 3, then u2 = 0, as can be verified from Figure 111.

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