ress this problem mathemati- cally to make it easier. Any r(t) can be represented on an orthonormal basis, and can be written instead as a vector r. We saw this idea in Chapter 5. For example, the signal r(t) in Equation (10.3) can be fully represented as the vector r = (11,12) using the orthonormal basis {0,(+), Q2(t)} where and Specifically, the signal r(t) in Equation (10.3) can be expressed as the vector r described by r = (1, 12 r2 (10.4) where (10.5) Estimation and Synchronization (10.6) The problem we want to solve can now be stated as: Given r, we want to find an estimate of a. We do not just want to estimate a in the special case of Equations (10.4) to (10.6), but we want to do this for any r containing a random value a. In general, we have an r equation, and we have a random value a contained in that r equation. We want to estimate that a. 10.2.2 What We Need to Get an Estimate of a Given r There are two or three things required to create an estimate of a given r. They are all statistical quantities. 1. p(r)a): The exact value of r depends on: the random value a; the value of the signal sent; and the random value of the channel noise. The value of r, then, is a random value that depends on three values. For a given a, r becomes a random variable that depends only on the signal sent and the noise. That is, for a given a, r can be described as a random variable with a distribution that we will label p(ra). We require knowing or being able to evaluate this distribution before we can create an estimate of a, given r. 2. p (a): a, being a random value, is characterized by a probability density function p(a). We also need to know this in order to compute an estimate of a. 3. (r): As I explained in point 1, r is a random value, and that means it can be described by a probability density function p(r). In some cases, generating the value of a will require we know the function p(r). Next, we'll see how to use these values to create an estimate of a, given r. There are three main ways to come up with this estimat