By Peter W. Hawkes (Ed.)

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

**Example text**

In contrast, an LSC in the zero state is “unexcited”: A zero input will always produce a zero output and the system will remain in the zero state. For LSCs, however, the concept of initial state can be entirely avoided by using minimal realizations and by considering two-sided rather than one-sided infinite input sequences (Forney, 1970). D. Minimality and Transient States Given an FGH weighting pattern, it’s useful to know how many states the corresponding trellis code will have. D). ALGEBRAIC SYSTEMS, TRELLIS CODES, AND ROTATIONAL INVARIANCE 43 Theorem 3.

Calculating Free Euclidean Distance An error occurs in the decoder when an incorrect path becomes more likely than the correct one. Hence, to minimize the probability of error, all paths should be as “far apart” as possible. As discussed in Section ILH, the asymptotic error rate for a code is determined (primarily) by dfree,the Euclidean distance between the closest two paths. Calculating the free Euclidean distance of a code is similar to maximum likelihood decoding. To calculate drree,one must find the closest two encoder paths.

Some reflection shows that the following statements about an observable trellis code are equivalent: 1. The code is catastrophic. 2. The code has an infinite input memory inverse. 3. The encoder state cannot be determined solely by examining a finite number of encoder outputs. 4. There exist two distinct trellis paths with identical output sequences. 5. There exists an infinitely long error event with a finite Euclidean distance. 6. There exists an infinite number of error events with the same (finite) Euclidean distance.