a very difficult and complicated task. It's a problem engineers are still working to resolve. Channel Filtering and Equalizers But there is a way around this problem. We can introduce another filter, a digital filter called a whitening filter, Wk, SO that, after passing r'k through Wk: we end up with (9.51) (9.52) (9.53) where nk=nk*Wk is now a Gaussian random variable that is independent of all other Gaussian random variables and SO on. That is, we can filter out the depen- dence of the random noise on other random noise samples using a filter Wk. It really doesn't serve us to go through the details of how we construct the whitening filter Wk, as it all comes out of statistical literature, and other texts delight in sharing that with you. Let's just get right to the key result (how to build the whitening filter!), SO we have what we need to keep surging ahead with our construction of the optimal rest-of-the-receiver For this one paragraph, I'm going to assume some knowledge of discrete time processing. If you don't have it, this paragraph may be hard to understand, and in that case (if this paragraph is important to you) you'll want to take a look at a book on discrete time processing. The whitening filter W is obtained from the equation w(e)") (9.54) Any W(ejw) that satisfies Equation (9.54) will do, although it is conventional wisdom to choose the W(ejw) that is also causal and stable (that is, all poles fall within the unit circle). So, the first step in the optimal rest-of-the-receiver is adding a whitening filter Wk. This filter changes the noise samples for us-it makes each noise sample independent of other noise samples, which is key in building an optimal receiver. (Engineers still are unsure of ways to build optimal receivers otherwise.) 9.5.3 The Final Part of the Optimal Receiver So far, we've built the optimal receiver front end, which receives r(t) and puts out (9.55) Chapter Nine We realized we had a problem with rk, with the noise samples at time k, NK. This noise is a random variable that depends