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

ISBN 0471-49581-6

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Let X denote a 2i/-ary signal constellation with minimum distance DTuni. For i =

1.2. • ? ?, u, let f.' (x) be the i-th bit of the label of a signal point x, and let Xt‘ C X the subset of signal points with labels such that the i-th bit has a value be {0,1}.

9.4.1 Gray mapping

A one-to-one and onto binary map m from {0,1}" to X is a Gray mapping if, for all i = 1, 2, • • •. u, and b 6 {0.1}, each x ˆ XI has at most one nearest neighbor y ? T}', at distance Dm\n, where b' = b © 1. Gray mapping is the key component of a BICM system. Its main function is - ideally - to produce an equivalent channel that has v parallel, independent and memory less, binary channels. Each channel corresponds to a position in the label of a signal x ? X. For each codeword at the output of the binary encoder, the interleaver assigns at random a position in the label of the signals to transmit the coded bits.

190

THE ART OF ERROR CORRECTING CODING

000

001 '

011 '

010

100

O101

110

(a)

-A2-Ai A, A2 (b)

(c)

Figure 121 An 8-PSK constellation with block partitioning: (a) labeling; (b) X coordinate projections; (c) decoder structure, (ci denote estimated codewords in Ct. i = 1, 2, 3.)

9.4.2 Metric generation: De-mapping

Before the description of how metrics for an MLD decoder are generated, some notation is needed. Let r denote the channel output after transmission of x. Assuming uniform input distribution, the conditional probability of r given ?l(x) = b is

p{r\ll{x) = b) — ^ p{r\x)p(x\E{x) = b) = ^2 P(r\x)- (9-12)

xex x&xi,

Let i denote the position of the coded bit Vj in the label of x7r(J). At each time j, let Vj be a code symbol and xn^ the interleaved signal point, received as after transmission over a noisy channel.

The receiver then produces bit metrics

A*(rvr(j);b)=log j p(rw

(J) I

(9.13)

for b — 0,1 and i = 1, 2, • ? •, v.

An MLD algorithm, such as the Viterbi algorithm, uses the above metrics and makes decisions based on the rule

COMBINING CODES AND DIGITAL MODULATION

191

Eb/No (dB)

Figure 122 Simulation results of a three-level coded 8-PSK modulation with UEP capability. BCH component codes and block partitioning.

Figure 123 A bit-interleaved coded modulation system.

Vi = argmax ^ A*(r„U], Vj). (9.14)

i

As before, a max-log approximation of (9.13) is possible, resulting in the approximated bit metric,

Al(Ur (j)-b) = maxlogp(rff(i)|a;). (9.15)

xex%b

9.4.3 Interleaving

With transmission over an AWGN channel, a short interleaver will suffice. The main purpose is to break the correlation introduced by the 2"-ary modulation signal set, which carries v bits

per signal. Therefore, an interleaver of length equal to a few times v is enough to approach

best performance [CTB2], Note that this interleaver has nothing to do with the interleaver that a turbo code or a block product code would use.

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THE ART OF ERROR CORRECTING CODING

9.5 I\irbo trellis-coded modulation (TTCM)

Conceptually, there are various approaches to the combination of turbo codes, or product codes with interleaving, and digital modulation: Pragmatic coded modulation [LGB], turbo TCM with symbol interleaving [RW1, RW2] and turbo TCM with bit interleaving [BDMP2].

9.5.1 Pragmatic turbo TCM

Motivated by the extraordinary performance of turbo coding schemes, in 1994 [LGB], another pragmatic coded modulation scheme was introduced. Its block diagram is shown in Figure 124. The main feature is, as in pragmatic TCM, the use of the turbo encoder and decoder operating as in binary transmission mode. This requires careful computation of the bit metrics, as in the case of BICM.

v- k k

Figure 124 Combination of a turbo encoder and digital modulation [LGB]

9.5.2 Turbo TCM with symbol interleaving

In 1995, Robertson and Worz proposed the use of recursive systematic convolutional encoders, such as those proposed by Ungerboeck [Ungl], as components in an overall coding system similar to that of turbo codes. A block diagram of this scheme is shown in Figure 125. As can be seen from the diagram, interleaving operates on symbols of v bits, instead of on bits for binary turbo codes. There is a need to puncture redundant symbols, due to the two paths of modulated signal points. A careful component code (no parallel transitions) and interleaver design (even positions to even positions, odd-to-odd; or even-odd and odd-even) are required. In terms of iterative decoding, note that the systematic component cannot be separated from the extrinsic one since they are transmitted together in one symbol. However, the LLR can be separated into an a-priori and a systematic-and-extrinsic part. Care must be taken so that the information is not used more than once in the component decoders. This is the reason why redundant symbol puncturing, in the form of a selector, is needed at the output of the encoder [RW2]. Figure 126 shows a block diagram of an iterative decoder for turbo TCM.

9.5.3 Turbo TCM with bit interleaving

In 1996, Benedetto et al. [BDMP2] proposed symbol puncturing rules such that the outputs of the encoder contain the information bits only once. Moreover, as opposed to symbol interleaving and puncturing of redundant symbols, multiple bit interleavers were proposed. A

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