e E7.2. Solution: Figure E7.4 shows the trellis diagram when the Viterbi Algorithm is performed. It shows (1) the number-of-errors computation for each branch, (2) the best parent selection at each node, (3) the very best final node selection; and (4) backtracking to determine the best path through the trellis. From this best path, the decoder decides that the best output is (0 0). Channel Coding and Decoding: Part 2-Convolutional Coding and Decoding receive: error = 2 (total 2) error = 2 (total 2) best parent = node 1 best parent (total 0) = node 1 (total 0) best parent = 3 (total 0) best parent = 3 (total 1) best parent = O best parent = O (total 1) (total best parent = 2 01 error = 1 (total 1) error = 1 (total 2) (total 2) best parent = 2 (total 1) Figure E7.4 Using the VA to determine best path 7.3 Performance of the Convolutional Coder Now that we know how the coder and the decoder in convolutional coding/decoding work, we want to evaluate their performance. Channel coders and decoders are intro- duced with the intention of correcting bit errors, SO we'll determine how well a convolutional coder is able to correct bit errors. In general, a convolutional coder can correct e bit errors in about every 4k bits, where the value k is the solution to 2k = (number of states (nodes)) and e is the value computed according to: (7.3) The key to determining how well the trellis decoder is able to correct bit errors is dmin' It tells you how easy it could be to mistake one path for another in the trellis. Specifically, d is the smallest distance between any two paths with the same start and end node in a trellis. Alternatively, this dmin min value is equal to the smallest distance between the all-zeroes path and another path through the trellis that starts at node 0 (time 0) and ends at node 0 (any later time). Chapter Seven For example, consider Figure 7.12. There you see a trellis and you see the all- zeroes path (the path that when followed gives you an output that is all 0 bits). You see another path highlighted that 