elihood that the noise N1 x - Acos(a). Similarly, with V2 = Asin(a) + N2, the likeli- hood that V2 = y given a is the likelihood that N2=y = - Asin(a). With these two things in mind, we can write (10.21) Now, with the noises N1 and N2 independent of one another, then we can express the equation above according to (10.22) Next, we can continue to apply what we know to this equation. We know that the noise N1 and N2 are Gaussian random variables, with mean 0 and variance on2-that is, mathematically, p = Substituting this Equation (10.22), we have (10.23) Using the simple mathematical property exp(a)exp(b) = exp(a + b) we end up with (10.24) This equation represents p(r)a). We now have p(a) and p(ra), which is all we need to compute the MAP estimate p(a). The estimate corresponds to the value computed from the equation (10.25) From a (10.19), p (a) is constant 1/7 to for any phase in [0, 2). Being constant, does not affect the choice of maximum value âMAP (that is, the value maximizing p(ra)-1is the value maximizing p(r)a)). Hence, write equation same we can our as = argmax p(ra) (10.26) 288 Chapter Ten Substituting the values we have from Equation (10.24) for p(r)a) we end up with: (10.27) (10.28) (10.29) We can remove all multiplier terms that are not a function of a because these terms won't affect the optimization. This leads us to amap = argmax exp (10.30) Now, we use a little mathematical trick. We can find the value that maximizes x or we can find the value that maximizes ln (x). Either way, we will end up with the same value. This tells us that our equation can be rewritten according to amap = argmax In (10.31) = argmax Acosa+rAsin a] (10.32) Estimation and Synchronization Borrowing from a bit of math not shown here-take my word for it-you can rewrite the equation according to (10.33) (This comes about because Lab is the same as [a(t)b(t) dt where a's and b's represent a (t) and b(t) on an orthonormal basis.) Now, since a is the value that maxi- mizes the righthand term, it is equivalent to the value a that 