d by a channel filter, plus a noise-that is, r(t) = c(t) * s(t) + n (t). Because of that, we had to introduce a whole chapter. We first found a new way to express the modulator, and we were able to general- ize it SO that it did what we called "pulse shaping." Then we looked at receivers. First, we tried one out, and we found out that it had problems with noise amplification, except in the case when we had a channel filter that was flat over the frequencies of transmis- sion. Then we found the optimal receiver front end. From there, we built an optimal receiver, which was made up of a whitening filter and what is called an MLSE. Finding that costly to build, engineers came up with a new design. They built the linear equal- izer in two forms (zero forcing and MMSE). That's the meat of this chapter, in a nutshell. Chapter Nine Problems 1. (a) Provide an equation s(t) describing the output of a 16-PSK modulator with pulse shaping g(t). Make sure that your s(t) is in a form similar to Equation (9.6). (b) Repeat (a) for a 16-ASK modulator. 2. Consider a system where we have a transmit filter g(t), a channel filtering CB(t) = S(t), and the receiver of Figure 9.6 with filter f(t). The total filtering effect is x(t) (the convolution of all the filter effects). (a) Draw a block diagram of the communication system, including the modula- tor and the receiver front end. Indicate the value of the signal before the decision device. (b) It is decided that x(t) will be a raised cosine function. Use a table of values to plot the raised cosine function x(t) of equation (9.21). On one column, use t values spaced by T/4, and on the other column provide the value of x(t). Do this over the range [-2T, 2T]. Provide two plots, for roll-off factors of 0.5 and 1.0. (c) Use your plot and your answer to (a) to explain how the raised cosine function x(t) allows you to avoid ISI. (d) Provide a possible selection for the filter g(t) and the filter f(t) SO that the total filter x(t) is the raised cosine filter. 3. Find three possible choices fo