 the rest-of-the-receiver is the value: (9.41) Let's take a more careful look at r'k and see what we've got to work with. Here comes the math that allows us to better understand rk Taking the r'k of Equation Channel Filtering and Equalizers (9.41), and substituting we up with (9.42) (9.43) (9.44) (9.45) where Nk = = n'(t)h(t-kT)dt Look at the integral in Equation (9.45). Let's brief look at the integral of Equation (9.45) and x(t), you'll be able to understand what I mean when I write: (9.46) which, letting -,=x((k-i)T) can be rewritten as (9.47) There are two other ways we can express this rk The first is a shorthand nota- tion from digital signals and systems literature, which tells us we can write rk according to (9.48) where Ik**k denotes the sum of Equation (9.47) and represents a discrete-time convolution. The second way we can write the V' of Equation (9.47) is to write out the sum longhand, leading to (9.49) Chapter Nine That is to say, the output we are seeing, r's corresponds to the output generated by the shift register, multiplier, and adder shown in Figure 9.14. There, the I's are stored in a shift register of length L, and each I is multiplied by Xk-ii then they are all added together and a noise is added onto them. I I I IL-2 IL-1 XK (L-1) ((L-2) Figure 9.14 Illustrating pictorially the value of r' coming out of the receiver front end and into the "rest of the receiver" 9.5.2 A Problem with the Input, and a Solution There is a problem with rk, something that it makes it very hard to build an optimal rest-of-the-receiver. We have that (9.50) where [n(t)h(t-kT): dt. Herein lies our problem. The n(t) in the Nk integral is additive white Gaussian noise, and it is filtered by h(-t) (that is, integrated with h(t - kT)). The output, Nk, can be shown to be a Gaussian random variable that is correlated with noise samples at other times-that is, Nk is correlated with Nk-1 and Nk-2' and SO on, as well as being correlated with Nk+1 and Nk+2' and SO on. This noise correlation makes building an optimal receiver 