satisfies the equation (borrowing from first-year calculus) (10.34) Taking this derivative leads us to (10.35) That means that our estimate is the one that, given r(t), forces the integral of Equation (10.35) to 0. A physical device that finds this âMAP given r(t) is shown in Figure 10.4. Here, you see a term, sin where à is the current estimate of the phase a, multiplies r(t). This product is passed through an integrator, which outputs the value (10.36) output decision close to sin(Wct+a) = Acos(Wct+a)+n(t),O<t<TE device not close to update a and send sin(wct+a) it back Figure 10.4 A device to evaluate MAP MAP 290 Chapter Ten This value enters into a decision device (typically implemented using a voltage- controlled oscillator, or VCO for short). If the integral is 0 (or very close to it), then the decision device decides that the current à estimate is the desired estimate . the value is not close to 0, then the decision device updates the value of à and sends it back to see if it now makes the integral of Equation (10.35) equal to 0 (or very close to it). In this way, given r (t) = Acos(Wt+a)+n(t),te[0,Tg],wecan generate an estimate of a. 10.3.2 The Practical Estimator: the PLL In practice, engineers estimate the phase a using a device very close to the one shown in Figure 10.4, namely the device shown in Figure 10.5, called a phase-locked loop, or PLL for short. Looking at Figure 10.5, we see that the PLL is very close to the estimator of Figure 10.4. Here are the differences: (1) the integrator has been replaced by a filter F(f); and (2) the decision device has been implemented using a VCO. decision device built output = Acos(Wct+a)+n(t),Ot<TE using VCO sin(@ct+a) in(@ct+a) Figure 10.5 Phase-locked loop Figure 10.6 helps to explain the workings of the PLL. At the input we have (1)=Acos(e),++a)+n(1) (10.37) To keep the written explanation of the PLL workings simple, I'll simply ignore the noise for the sake of the verbal description. We'll be assuming that the input is r(t)=Acos(w) (10.38) Now, as this r(t) co