TE (10.48) where a represents a random phase offset and n(t) represents an additive white Gaussian noise. We then built a device that creates the estimate à of a. We also saw an alternative version of this estimator, called the PLL, which is commonly deployed by most engineers. But in some communication systems, we receive (10.49) Estimation and Synchronization where O is a value in the set and represents information bits stored in the transmitted phase. This is what we receive if the transmitter corre- sponds to an MPSK modulator. You can see how this received signal is created by taking a look at Figure 10.7. In this section, we are interested in finding a way to create an estimate for a, given the received signal shown in Equation (10.49), rather than the r(t) shown in Equation (10.48). CHANNEL phase modulator offset s(t) =Acos(wct+0), Acos(@c+0+a)+n(t) Figure 10.7 A practical received signal Typically, the way engineers generate the estimate of a, given r(t), is shown in Figure 10.8. The idea, as explained in that figure, is this: use a processor that gets rid of 0, and then use the PLL shown in Figure 10.5. Only one question remains: how do we get rid of the phase O? One device commonly employed to do this is shown in Figure 10.9. In comes (t)=Acos(w.t+0+a)+n(t) (10.50) processor: get rid of sin(@ct++) Acos(Wct+0+a)+n(t) Figure 10.8 Creating an estimate of a, a, given r(t) Chapter Ten centered sin(Mwet+a) Acos(@ct+0+a) r"(t) 12 AMcos(Mw.t+M.a) r'(t) = [r(t)]M AMcos(M@ct+2k.M+M.a) + low freq. terms Figure 10.9 Processor to remove O where O is a value in the set To simplify the presentation of the workings of Figure 10.9, we'll ignore the noise. That means we can write the input as (10.51) where k=0 or 1 or or M - 1. After passing this input through the power-of-M device, we end up with (10.52) (10.53) + (lower frequency terms) This signal is passed through a bandpass filter, which cuts out all the terms except the term centered around frequency Mo--that is, except the first term of Equation (10.53). As a resu