ed about how N1 and N2 are independent of one another. This means that we can write p(n) = p(n)p(n2). Throwing this into our equation leads to (5.38) Next, we know that the noise terms are Gaussian, with 0 mean, which means we write Plugging in this information leads us to a new equation, S, = arg max - (5.39) Getting It from Here to There: Modulators and Demodulators Now, here comes a neat statement that you may not have heard before. If you're looking for the value of X that maximizes f(x), then you can alternatively look for the value of x that maximizes In f(x). Applying this idea to our equation gets us to S, = argmax l (5.40) = argmax (5.41) To get this equation, I just used two simple rules of math: (1) Inxy = 1nx + Iny and (2) In exp x = X. Next, I can remove the terms that are not a function of Si, since they'll be the same for each and every Si, and won't help me in deciding which Si makes the term biggest. Doing that gets me to (5.42) Then, I'll multiply though by the constant 20, which leaves me with (5.43) Now, here's a little mathematical trick: I can choose the value that maximizes x, or equivalently, I can choose the value that minimizes -X. Using this, I can rewrite the equation according to (5.44) That is, in shorthand notation S = argmin (5.45) And that, at last (phew), is the final equation for how the decision device goes about picking Si . Now, in the very common case where all the S,'s are equally likely- that is, p(s) is a constant-then this equation can be rewritten as (5.46) Chapter Five And here lies this really nice and easy interpretation. This equation simply says to tell the decision device to choose the Si closest to r. Take a look at Figure 5.27. All Equation (5.46) says in this case is choose S1, because it's closest to r. Now isn't that an intuitively pleasing result? All that math, and it all comes down to: "Given r, choose the S that's closest on the plot." Isn't it nice when math and common sense meet? Wait a minute isn't that just engineering? 5.4.5 How to Build It Let's loo