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

ISBN 0471-49581-6

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(9.4)

10

00

4 bits

Extended Hamming (8,4, 4) code

4 symbols

11*

*01

Figure 101 Block coded QPSK modulation using an extended Hamming (8,4,4) code.

174

THE ART OF ERROR CORRECTING CODING

Eb/No (dB)

Figure 102 Simulation results of a block coded QPSK modulation using an extended Hamming (8,4,4) code. AWGN channel.

Therefore, while the use of an expanded 2"-ary modulation causes the distance between signal point to decrease, a properly selected error correcting code can make sequences of signal points at a minimum distance larger than that of an uncoded system, with the same spectral efficiency.

9.2 Trellis-coded modulation (TCM)

Proposed by Ungerboeck in 1976, the main idea in TCM is to perform mapping by set partitioning. A basic trellis structure, associated with the state transitions of a finite-state machine, is selected and signal subsets mapped to trellis branches. For systems that require high spectral efficiency, uncoded bits may be assigned to parallel branches in the trellis.

9.2.1 Set partitioning and trellis mapping

Bit labels assigned to the signal points are determined from a partition of the constellation. A 2"-ary modulation signal set S is partitioned in v levels. For 1 < ³ < v, at the ³-th partition level, the signal set is divided into two subsets <Sj(0) and 5j(l), such that the intra-set distance, Sf, is maximized. A label bit bi ª {0,1} is associated with the subset choice, Sfibi), at the ã-th partition level. This partitioning process results in a labeling of the signal points. Each signal point in the set has a unique I'-bit label b\b> and is denoted by s(b\, b2, ? • ? -, b„). With this standard (Ungerboeck) partitioning of a 2"-ary modulation signal constellation, the intra-set distances are in nondecreasing order Sf < <5f < • • • < 6%. This strategy corresponds to a natural labeling for M-PSK modulations, i.e., binary representations of integers, whose

COMBINING CODES AND DIGITAL MODULATION

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value increases clockwise (or counter-wise). Figure 103 shows a natural mapping of bits to signals for the case of 8-PSK modulation, with <52 = 0.586, = 2, and <5| = 4.

Ungerboeck regarded the encoder “simply as a finite-state machine with a given number of states and specified state transitions”. He gave a set of pragmatic rules to map signal subsets and points to branches in a trellis. These rules can be summarized as follows:

Rule 1:

All subsets should occur in the trellis with equal frequency.

Rule 2:

State transitions that begin and end in the same state should be assigned subsets separated by the largest Euclidean distance.

Rule 3:

Parallel transitions are assigned signal points separated by the largest Euclidean distance (the highest partition levels).

The general structure of a TCM encoder is shown in Figure 104. In the general case of a rate (v — \)/v TCM system, the trellis structure is inherited from a k/(k + 1) convolutional encoder. The uncoded bits introduce parallel branches in the trellis.

Example 96 In this example, a 4-state rate-2/3 TCM system is considered. A constellation for 8-PSK modulation is shown in Figure 103. The spectral efficiency is p = 2 bits/symbol. A block diagram of the encoder is shown in Figure 105. The binary convolutional code is the same memory-2 rate-1/2 code that was used in Chapter 5. Note from Figure 106 that the trellis structure is the same as that of the binary convolutional code, with the exception that every branch in the original diagram is replaced by two parallel branches, associated with the uncoded bit tt].

9.2.2 Maximum-likelihood decoding

The Viterbi algorithm2 can be applied to decode the most likely TCM sequences, provided that the branch metric generator is modified to include parallel branches. Also, the selection of the winning branch and surviving uncoded bits should be changed. The survivor path (or trace-back) memory should include the (u — 1 — k) uncoded bits, as opposed to just one bit for rate-1 jn binary convolutional codes. It is also important to note that in 2"-ary PSK or QAM modulation, the correlation metrics for two-dimensional symbols are of the form xpxr + ypyr, where (xp, yp) is a reference signal point in the constellation and (xr, yr) is the received signal point. All other implementation issues discussed in Sections 5.4 and 7.2 apply to TCM decoders.

9.2.3 Distance considerations and error performance

The error performance of TCM can be analysed in the same way as for convolutional codes. That is, a weight enumerating sequence can be obtained from the state diagram of the TCM

2 The Viterbi algorithm is discussed in Sections 5.4 and 7.2.

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THE ART OF ERROR CORRECTING CODING

010

Figure 103 Natural mapping of an 8PSK constellation.

encoder, as in Section 5.3. The only difference is that the powers are not integers (Hamming distances) but real numbers (Euclidean distances). Care needs to be taken of the fact that the state transitions contain parallel branches. This means that the labels of the modified state diagram contain two terms. See [BDMS].

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