By N. Cooke

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E). Catastrophic trellis codes are undesirable because they allow a single channel error to propagate into a potentially infinite number of decoding errors. Forney showed that every LSC is equivalent (in a certain sense) to a finite input memory LSC with a finite input memory inverse. This may not be true for trellis codes in general, and it is certainly not true when rotational invariance is considered, as can be seen in Section VI. Although it is fairly simple to check an arbitrary trellis for an FIM inverse using a recursive algorithm, a structural check such as the one found by Massey and Sain (1967, 1968)for LSCs is not known for FGHs.

21b, as stated in the following theorem. Theorem 4. Regardless of the initial state and input sequence, a given FGH cannot be in a transient state after a certain finite amount of time t’ (defined in Eq. Moreover, the set of communicating states reached at time t ’ depends only on the initial state, not on the input sequence. Pro08 We will first prove the result for observable FGHs. Choose any initial state so E S and generate a weighting pattern from so using Eqs. (49) and (50). Let l(u) = min{k I T,(o) = T+*(u) t for all i 2 k) = max I(u) (63) VOU t’ (62) = min I(u) veL1 u‘ = argmin l(u).

At each time step, the elements of the state set are represented as a vertical column of dots, called nodes. This is shown in Fig. 6 with S = { A , B , C , D } . Because trellis codes are normally time-invariant, only one time step of the trellis is included in a diagram. The state transitions allowed by g(0) are depicted as branches connecting the nodes at time k to the nodes at time k 1. There are I UI branches leaving each node, one for each input element. The example in Fig. 6 has U = (0,l).

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