lt, the output of the bandpass filter is "(t) = = cos (10.54) The PLL tracks the phase of this term and, hence, it ends up with the phase esti- mate à = Ma. We use a divide-by-M to generate the desired phase estimate of a. Estimation and Synchronization 10.4 Conclusion This chapter describes the world of estimation, also known as synchronization. It is an important part of telecommunications because, when sending a signal from transmitter to receiver, the channel often adds unwanted parameters along the way, which we label a. We need to be able to estimate these parameters SO we can remove them from the received signal. This chapter provided a general framework for creating the estimate à of any given parameter a. Specifically, there are three ways to generate this esti- mate. The first method is known as MMSE, the second as MAP, and the third as ML. We then considered an important practical example, namely the estimation of channel phase. We saw how to estimate phase using the MAP method and we saw how engi- neers implement a device similar to the estimator created from the MAP method. Chapter Ten Problems 1. A transmitter sends a. A receiver picks up the value r, which is characterized (Q10.1) Draw a block diagram of a receiver that performs ML detection. 2. A transmitter sends a value a, which is a random variable characterized by Figure Q10.1. A receiver picks up the value V. (a) If r is characterized by p(r|a=+1)=-exp(-lr) (Q10.2) (Q10.3) draw a block diagram of a receiver that performs MAP detection. (b) If V is characterized by (Q10.4) Figure Q10.1 The pdf of a 1/2 (S(t+1) Estimation and Synchronization (Q10.5) draw a block diagram of a receiver that performs MAP detection. 3. A receiver observes r=a+n (Q10.6) where a is a Gaussian random variable with mean 0 and variance s2, and n is a Gaussian random variable with mean 0 and variance s12. (a) Find the MMSE estimate. (b) Find the MAP estimate. 4. A transmitter sends (t)=Acos(wt+0i),iTst<(i+1)T = (Q10.7) where 0:=0° = or 180° and receives (Q10.8) Draw a bl