which corresponds to n==I,x(kT-iT)+n, (9.17) where Nk = n"(kT) or, taking the kth term out of the sum, we can write this as (9.18) Desired Intersymbol Noise Information Interference (ISI) (The decision device is exactly the same one we built in Chapter 5, taking in an rk and putting out a guess on the symbol sent.) Chapter Nine 9.3.2 Making the Receiver a Good One We'd like to have the receiver work SO that it creates rk = I k ; in other words, we'd like for the kth received sample from the sampler in Figure 9.6 to correspond exactly to the kth information symbol sent at the transmitter. If we could do that, then the decision device would, given rk=1k, = always make the correct decision on Ik Looking at Equa- tion (9.18), let's see what it would take to make rk = Ik. It would take three things: 1. x(0) = 1 2. x(kT-il)=0,k=i 3. nk=0. We can't make the noise nk=0, = but recall that and g(t) is a filter put in at the transmitter, while f(t) is a filter we put in at the receiver. Since we controlg(t) and f(t), we can control x(t). So, we could choose g(t) and f(t) such that (9.19) which means we would satisfy points 1 and 2 above. If f(f) and g(t) are selected such that x(t) satisfies Equation (9.19), x(t) and/or the communication system is said to satisfy the Nyquist criteria for zero ISI (intersymbol interference). Here are some choices for x(t) that satisfy Equation (9.19). The sinc scenario: The first well-known choice for x(t) that satisfies Eq. (9.19) is (9.20) A plot of this x(t) is shown in Figure 9.7(a). In the frequency domain, this signal is shown in Figure 9.7 (b). The raised cosine scenario: Another well-known choice for x(t) is called the raised cosine function, in which case x(t) corresponds to (9.21) where a is called the roll-off factor and corresponds to a value between 0 and 1. Three raised cosine functions are shown in Figure 9.8(a), and their frequency response is shown in Figure 9.8(b). Channel Filtering and Equalizers sin (t/f) x(t) = sinc(t/T) X(f) = F{sinc(t/-)} 11/2T Figure 9.7 x(t) = sinc (