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Figure 5.7. General structure of encoder/modulalor for trellis-coded modulation.
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lation size to 2n+1. If m < n, parallel transitions are allowed in the trellis; if m = n, no parallel transition exists. The assignment of signal subsets to state transition in the trellis is based on the three heuristic rules devised by Ungerboeck [133], which are summarized as follows: 1. Parallel state transitions are assigned signal points separated by the greatest Euclidean distance. Note that parallel transitions in the trellis are the characteristics of TCM containing one or more uncoded information bits. 2. The transitions originating from the same state or merging into the same state in the trellis are assigned subsets of signals that are separated by the greatest Euclidean distance. 3. All signals are used in the trellis diagram with equal frequency, and with a certain degree of regularity and symmetry. It should be pointed out that, for a trellis-coded modulation scheme, the performance will be dominated by the minimum free Euclidean distance between codewords that can be expressed as [134] Dfree = min[d , dfree ] (5.19)
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where d is the minimum distance between parallel transitions and dfree denotes the minimum distance between nonparallel paths in the TCM trellis diagram. In the special case of m = n, the subsets contain only one signal, and hence there are no parallel transitions.The nal TCM design criterion, thus, is to maximize the minimum free Euclidean distance. Note that the set partitioning is performed with the goal of maximizing the rst quantity d, whereas the trellis of the constituent code is designed to maximize the second quantity dfree. With an appropriate set partitioning and trellis
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design, the overall minimum distance of the code will be large enough to overcome the loss from the constellation expansion (due to the redundancy in the code), and provide a signi cant coding gain. In the following example, we present the trellis-coded 8PSK modulation and determine the coding gain. Example 5.1 (TCM Design Example; Example of 8PSK TCM Design in AWGN Channels) Let us consider the use of the 8PSK signal constellation 2 in conjunction with rate- 3 trellis codes. The coded 8PSK signal set employs the signal points in Figure 5.6. The uncoded 4PSK, which is used as a reference in measuring coding gain, employs the signal points in either subset B0 or B1. Figure 5.8 is a state transition (trellis) diagram for uncoded 4PSK modulation with (a) one and (b) four trellis states and (c) coded 8PSK modulation with four trellis states. The trivial one-state trellis diagram is shown only to illustrate uncoded 4PSK from the viewpoint of TCM. The subsets D0, D2, D4, and D6 are used as the signal points for illustration purposes. Every connected path through a trellis represents an allowable signal sequence. In both systems, starting from any state, four transitions can occur, as required to encode two information bits per modulation interval (2 bps/Hz). The four parallel transitions in the one-state trellis diagram of Figure 5.8a for uncoded 4PSK do not restrict the sequences of 4PSK signals that can be transmitted; that is, there is no sequence coding. Hence, the optimum decoder can make independent nearest-signal decisions for each noisy received 4PSK signal.The smallest distance between the 4PSK signals is 2e , which we denote it as d0 and call the free distance of uncoded 4PSK modulation for the sake of using common terminology with sequence-coded systems. Each 4PSK signal has two nearest-neighbor signals at this distance. In the four state trellis of Figure 5.8b, the 8PSK signals are assigned to the transitions in the four-state trellis in accordance with the principle of mapping by set partition as discussed above. Note that each branch in the trellis corresponds to one of the four subsets C0, C1, C2, or C3. For the 8-point constellation, each of the subsets C0, C1, C2, and C3 contains two signal points. As illustrated in Figure 5.6, the distances d0, d1, and d2 are de ned as the minimum distance of constellation points in the subsets A, B0, and C0, respectively. It can be easily seen that the squared Euclidean distance between parallel transitions is d2 = 2 e . On the other hand, any two signal paths in the trellis of Figure 5.8c that diverge in one state and remerge in another after more than one transition have at least squared Euclidean distance d2 + 2d2 = d2 + d2 between them. 0 1 0 2 For example, the paths with signals 0 0 0 and 2 1 2 have this distance. The distance between such paths is greater than the distance between the signals assigned to parallel transitions. Hence, the minimum Euclidean distance separation between paths that diverge from any state and remerge at the same state in the four-state trellis is d2 = 2 e . This minimum distance in the trellis code is called the free Euclidean distance and is denoted by Dfree. When compared with the Euclidean distance d0 = 2e for the uncoded 4PSK modulation, we observe that the four-state trellis code gives a coding gain of 3 dB.
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