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By Peter W. Hawkes (Ed.)

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Extra info for Advances in Electronics and Electron Physics, Vol. 79

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Since trellis codes operate on effectively independent uniformly distributed semi-infinite input sequences, transient states increase the complexity of a trellis code without affecting performance. Figure 21 demonstrates several types of transient behavior. In Fig. 21a, state 1 is transient. 21. Several types of FGH transient behavior: (a) State 1 is transient; (b) states 1 and 2 are transient for at most two time steps; (c) state 3 is transient for only one time step; (d) states 4 and 5 are transient.

C . Initial State Notice that the initial state of the SO-realization is always the identity, even if a nonidentity initial state was used in Eqs. (49) and (50) to generate the weighting pattern. The following theorem generalizes this idea. Theorem 2. An FGH with initial state so = x can be transformed into an input-output equivalent FGH with initial states, = e. The transformation is given by the following equations: a*(s) = a(s) b*(u) = x-’ * b(u) * a(x) C*(S) = c(s) d*(U) = d ( u ) c(x) (59) (60) (61) Proof) Application of Eqs.

Output-equivalent codes are interchangeable for almost every purpose, with one major exception. As discussed in Section VI, rotational invariance is a function of a code's input-output behavior, not its set of reachable output sequences. Hence, when designing rotationally invariant codes, input-output equivalence is operative, infinite input memory systems are important, and the input labeling on the trellis diagram should always be indicated. This has been neglected by several authors. ALGEBRAIC SYSTEMS, TRELLIS CODES, AND ROTATIONAL INVARIANCE 27 u=o; y=u P u = l ; y=b (a) FIG.

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