) Chapter Nine provide an equation for the zero forcing linear equalizer and an equation for the MMSE linear equalizer. Assume No=0.5. Solution: Writing the received signal in the form of a convolution, we have + 0.25 8k-2) + Nk (E9.24) Comparing this with Equation (9.59), we recognize that (E9.25) which, using z-transforms, corresponds to (E9.26) Now, for the zero forcing linear equalizer, we turn to Equation (9.61) which in this case leads us to (E9.27) and, for the MMSE linear equalizer, we turn to Equation (9.66), which this time leads us to (E9.28) 9.7 Other Equalizers: the FSE and the DFE In addition to the linear equalizer, sometimes called LE for short, engineers designed other cheap alternatives to the optimal whitening filter followed by MLSE. I'll provide just a brief overview here, and I'll let you, in graduate courses or out of general inter- est, read other books that describe the workings of the other alternatives to the whitening filter and the MLSE. First, there is the fractionally spaced equalizer, or FSE. If someone introduces you to this equalizer, don't be at all intimidated. It is, in truth, just a linear equalizer in disguise, for which they have found a way to do all of the filtering in the digital domain, where it is cheaper and easier anyway. Secondly, there is the differential feedback equalizer, or DFE. In this case, what you do once you have rk from the receiver front end is shown in Figure 9.21. Basically, you have what looks like a linear equalizer followed by a decision device. The main difference here is that a feedback filter has been added below the decision device. Channel Filtering and Equalizers Very briefly, you receive (9.67) (9.68) (9.69) You use the Ck to get rid of the second term and the d to get rid of the third term. Decision device FEEDFORWARD EQUALIZER FEEDBACK EQUALIZER Figure 9.21 The DFE 9.8 Conclusion The channel was different. Instead of giving you what you sent with a noise-i.e., r(t) = s(t) + n(t)-it gave you something else. It gave you what you sent, filtere