By Gray-Meyer

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Now any decoder can be thought of as also making an estimate t 1 0,oo) of th~ encoded sequence t 1o,oo) transmitted ~wer the channel simply by considering t 1o,oo) to be the result of encoding i[O,oo) . 7) t[O,oo) -t[O,oo) in the sequence of decoded transmission errors. In fact, we can show the general situation as in Fig. 2 where it is clear that which encoder inverse is used is of no l [O,ool= i Fig. [O,col = i [O,ool + Q i[O,ool+ [O,ool ~ho,col ! omposition" of the Decoder for a FCE. significance.

Catastrophic Fixed Convolutional Encoders In this section, we probe more deeply into the meaning of a catastrophic FCE as defined previously in Section 8 in terms of the signal flowchart of the FCE. We first note that our definition there is equivalent to saying that an FCE is catastrophic if and only if there is an information sequence i 10 ,oo) with WH (i(O,oo)) = oo that produces an encoded sequence t[O,oo) for which WH (t10 ,oo)) < oo • To sec this equivalence, we note that if the signal flowgraph has a closed loop of weight 0 besides the self-loop at the zero state, then the information sequence i 10 ,oo) which drives the encoder to a state on the former loop and around the loop forever after is an infinite weight sequence whcih produces a finite weight encoded sequence.

2. 5) is unity. 5) is 2y'e (1- €) where € is the crossover probability of the BSC. 6} where dH denotes the Hamming distance between the indicated N-tuples. 6) is often surprisingly close to the true~ for MLD of the code~. x 2 ) as the reader can verify by some detailed examples. e. unless P(y 0) = P(y ll) for ally. 7) with the transmission gain A(z) for a non-catastrophic FCE to obtain an upper bound on MLD of the trellis code defined by this particular FCE. L. e. that the MLD chooses a path through the trellis which departs from the correct path at the root node.

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