 on other noise samples at other times (for example, n/+1). So we put in a filter called a whitening filter which took in V' and put out tr,, where (9.56) and n'k is a random variable that is independent of other noise samples at other times (for example, n'k+1 ). From here, we are ready to build the final part of the optimal receiver. To start, we take a better look at Vk First, if we write out the sum in long- hand, we find that we can rewrite Vk in the following form: (9.57) Now, that means we can visualize V being built as shown in Figure 9.15. There, we have the I's being stored in a shift register; each Ii is multiplied by Un-i' and then these are added together. Finally, the noise is added on. IO I 1 IL-2 IL-1 k-(L-1) Vk-(L-2) Figure 9.15 Illustrating the creation of rk (coming out of the whitening filter) Channel Filtering and Equalizers To keep the construction of the rest of the receiver easy to understand, let's consider a simple example. Let's say (9.58) That is, we're assuming that all the other Vk-i terms are 0, SO we have a simpler equation to work with. We'll also assume (1) V = 1 and V1 = 0.5, and (2) that we have BPSK modulation, in which case IK is either A or -A. The V in this simpler case corre- sponds to Figure 9.16. Now, as marked in that figure, we can see Vk created as follows: An "input" Ik comes in, it moves the "state" (what's at position 1) over to position 2 (creating I-) Multiplication by V'O and V1, and the addition of noise, creates the final output Vk. We can draw a trellis diagram describing the creation of the output Vk if the noise were not present. This diagram is shown in Figure 9.17: 1. The "state" (in Figure 9.16) is represented by nodes, the dots drawn at each time-at any one time, this state is either -A or +A. 2. The "input" Ik (in Figure 9.16) is represented as a branch-if the input is Ik = - A, the branch is solid; if the input is Ik=A, the branch is dotted. 3. When an input comes in (that is, a branch is drawn), it changes the state. You can see this indicated by the