mes in, it gets multiplied by the sinusoid sin (w,t+â) (10.39) Estimation and Synchronization Acos(Wct+a).sin(ct+a = Acos(Wct+a),O<t<TE A sin(a - sin(Wet + a) sin(@ct+a) Figure 10.6 The PLL explained where à is the current estimate of our channel phase. The output signal from that product is r'(t)=Acs(t+a)sin(t) (10.40) (10.41) This signal then goes through the filter F(f), which acts as a low-pass filter, cutting out the high-frequency term in r((t) (the first term in v (t)). As a result, the output corresponds to (10.42) We will assume that F(f) is close to 1 (f(t) is close to S(t)) in the low-frequency range, and hence the r'(t) is well-approximated by r'(1)==sin(a-a) (10.43) This signal then enters into the VCO, which acts as a decision device, deciding based on its input whether to update the current estimate à, or keep it and be done. Specifi- cally, the VCO is a device that determines the rate of change of the phase estimate à, according to (input to VCO) (10.44) 292 Chapter Ten (10.45) Now, if the phase estimate à = a, then plugging this into the equation above, day==0 (10.46) That is, the estimate à is not changed. The VCO decides to hold onto its current estimate. Alternatively, let's say the value of à is close to a. In that case, sin (a-a)=a-a, - and we have (10.47) If à is smaller than a, then di e be positive, which means that the VCO is going to increase the estimate â. If, on the other hand, à is larger than a, then da is negative, which means that the VCO will decrease the value of â. In this way, the VCO acts as a decision device, making changes in à until à = a. This is, practically speaking, the way engineers generate an estimate of the unknown phase a. There are a number of tweakings to the PLL that are sometimes performed, and under different conditions they make different choices for the F(f). But these details are left for another time-and another book. 10.3.3 Updates to the Practical Estimator in MPSK Let's consider what we've done SO far. We received the signal r(r)=Acos(w.-a)+n(t). OSI<