 state a branch begins at, and the state it ends in. 4. Finally, if we know the "state" and the "input," then we have all the information we need to figure out the output Vk (assuming no noise). That output is drawn above each branch. "state" "input" Figure 9.16 Illustrating the creation of rk in a simpler case Chapter Nine time O time 1 time 2 -1.5A -1.5A -0.5A -0.5A Figure 9.17 A trellis describing the creation of rk (without the noise) Here, I'm going to use a "common sense" argument (rather than a lengthy statis- tical one) to explain how the optimal decoder uses the trellis of Figure 9.17 to get the output I' : 1. Since the noise n'k is easily shown to be a Gaussian random variable with zero mean, then that, by definition, means that the average noise is 0. 2. It makes sense, then, that given the sequence of received V (k = 0, 1, 2, ..., L-1), which on average contain noise nk that equals 0, you could try to match the r's to the best path of noise-free outputs in the trellis. That is, you would want to find the path in the trellis whose outputs (drawn with no noise) are closest to the V (k = 0, 1, 2, ..., L-1) that you received. Once you've found this path, then you can figure out, by following the branches, what inputs Ik (k = 0, 1, 2, L-1) to put out from the receiver. Here's an example. Let's say that we receive the two values V1 = 1.7A and V2 = 1.7A. We want the receiver to output the best guess on the sent bits. To do this, we turn to our trellis, and we find the path through the trellis with outputs that are closest to V1 = 1.7A and V2 = 1.7A. Looking at each path through the trellis, we find out how close each path is-see Figure 9.18(a) through (d). We then decide that the "bottom lines" path of Figure 9.18(d) is the closest. The branches on this path indicate that the outputs are I = A and I2 = A, which tells us that the input bits are m = (1,1). Our receiver outputs these bits. This optimal rest-of-the-receiver is commonly called an MLSE (Maximum Likeli- hood Sequence Estimator). In general, searchi