st a noise; it also acted as a filter with impulse response c(t). In this chapter, we again add to what channels do-we make them more realistic. 10.1 Introduction In addition to noise and/or filtering, channels also add unwanted parameters to the sent signal. For example, take a look at Figure 10.1. There you see that the modulator sent the signal s(t)= Acos(Wt+0j),iTt<(i+1)T (10.1) which might, for example, be a QPSK signal. Many channels are well-modeled as follows: CHANNEL phase frequency timing Modulator offset offset offset Figure 10.1 Channel effects 280 Chapter Ten (1) the channel adds a timing offset T, a phase offset 0, a frequency offset Ao;; and (2) the channel adds a noise n(t). Combining these effects leads to the received signal (10.2) Many receivers, given this received signal, try first to estimate and remove the timing offset T, the phase offset 0, and the frequency offset Aw. Then, all that is left is the sent signal in the presence of noise n(t). They use the demodulators we saw in Chapter 5 to detect the signal in the presence of noise n(t). This chapter is all about how a receiver estimates T, 0, Aw and any other unknown parameter, SO that it can remove these parameters from the received signal (leaving just the signal in the presence of noise). This process of estimating unknown param- eters at receivers is called estimation or synchronization. We will begin with a general explanation of how to estimate any parameter that a channel might introduce to a signal, and then we will explain different techniques that the receiver might use to estimate and remove a channel phase O. 10.2 Estimation: Part 1 10.2.1 Our Goal In this section, we consider the following problem. We receive a signal r(t) with a random value a (with probability density function p(a)) contained in it. For example, we may receive the signal r(t)=Acos(Wt+0;+a)+n(t),iTt<(i+1)T (10.3) where a represents a random phase offset introduced by the channel. We want to find a way to estimate the value of a in r(t). There is a way to exp