Our Example and Its Theoretically Computed Estimate Let's now look at a practical example of how to come up with an estimate of a given r. Take a look at Figure 10.3, where we have a transmitter that is sending a cosine wave S (t)=Acos((o.1) over a short time interval. The channel introduces a phase offset a and adds a noise n(t). As a result, the received signal is r(t)=Acos(Wt+a)+n(t),0<t<T (10.12) where the channel phase a is known to be a random value uniform in the range [0, 2), and the noise n(t) represents an additive white Gaussian noise. The receiver greets the incoming signal with a receiver front end, which maps r(t) to the vector = by mapping it on to the orthonormal basis {0,(t), 02(t)} where and Specifically, as seen in Chapter 5, the receiver front end computes (10.13) where (10.14) CHANNEL phase offset s(t) =Acos(Wct),O<t<TE = r(t) =Acos(Wct+a)+n(t),O<t<TE a is uniform random value between [0, 2). Figure 10.3 Channel introducing phase offset and noise Chapter Ten (10.15) Using a little bit of math, this simplifies to r =(11, 12) (10.16) where (10.17) (10.18) We'll assume T = 2 to simplify our presentation. It can be shown (but you'll have to take my word for it) that n1 and n2 are both Gaussian random variables with mean 0 and variance On2, and N1 and N2 are independent. We'll now figure out the MAP estimate given the = of equations (10.16) to (10.18). To get this estimate we will need two things, p(a) and p(ra). Let's start out by evaluating these. 1. p (a): We were told that a is a random variable, uniform in the range of [0, 2). Writing this statistically, we know that the distribution of a, p (a), corresponds to (10.19) otherwise 2. p(ra): We will need to figure this one out statistically, as follows: We start with p(ra)=p(r,12)a) (10.20) That is, p(ra) is the probability that V1 takes on a particular value given a, and the probability that V2 takes on a particular value given a. Estimation and Synchronization Now, we know that V1 = Acos(a) + N1. So, given a, the likelihood that r=x is the lik