 far. It is used if the channel introduces an unknown value a-this time a is not described as a random value but rather a is just some unknown value. For example, the channel may add a constant a to the sent signal, as shown in Figure 10.2, and the value a added on is not well-described as a random value, but just some unknown, fixed number. Engineers wanted to create an estimate that would be as close to a as statistically possible. They found a way to create this estimate mathematically, and called the estimation method they invented maximum likelihood (or ML) estimation. The esti- mate for a that this method provides is called âML . âML estimate is created according to the following mathematical equation: = argmax p(ra), (10.10) or, equivalently, (10.11) modulator receiver t)=s(t)+a+n(t) some unknown value Figure 10.2 A channel introduces an unknown value Example 10.1 Let's say that you pick up some unknown value a corrupted by a zero-mean unit- variance Gaussian noise. That is, let's say you find: r=a+n (E10.1) where (E10.2) Chapter Ten Determine the best estimate of a. Solution: We're dealing with an unknown value of a here, SO in this case we turn to Equation (10.11). Specifically, this equation tells us that we require p(ra) if we're going to come up with an estimate for a, SO let's start by getting p(ra). The key is realizing that p(ra) is the likelihood of having v (for example, V = 1) show up, given a (for example, a = 0.2). Now, since = a + n, I'll get V when a is sent only if N=r- - a (so for our example values, n = 1 0.2 = 0.8). Mathematically, p(ra)=p(n=r-a) (E10.3) Now, plugging in the known distribution for noise (Equation E10.2), this means (E10.4) Okay, now we have p(ra). We're ready to come up with our estimate using Equation (10.11). (E10.5) (E10.6) (E10.7) (E10.8) (E10.9) Therefore, if I see r=a+n=0.7, with n defined according to equation (E10.2), I'll guess (estimate) that 0.7 is in fact the value of a. Estimation and Synchronization 10.3 Evaluating Channel Phase: A Practical Example 10.3.1 