nput r(t), so, they've obviously got the same input. Now, if I can show you that the value A in Figure 5.29 is equal to r1, and the value B in Figure 5.29 is equal to VN, then these receiver front ends will also have the same output, and SO they do exactly the same thing. It's all math, showing that A is equal to V1, and here we go: A=h(t)*r(t) li=T where h(t)=q,(T-t) (5.47) (5.48) (5.49) (5.50) (5.51) (5.52) Following the exact same set of mathematical arguments, we can show that B = VN. So there you have it, the receiver front end of Figure 5.29 has the same input and output as the receiver front end in Figure 5.28, and SO they do the exact same thing. We can replace one by the other, and still build the equivalent receiver. And that's really all Figure 5.29 says. The Matched Filter Receiver - Version 2 Yep, they made another one, and it's somewhat popular, SO we need to look at it. It's going to take some math to figure out how to build this receiver, and a bit of math to explain the actual implementation, but hang in there. This next receiver implementation starts out by rewriting the way a decision device makes its decisions. When last we looked, we saw decision devices making decisions according to (5.53) Now, we're going to write out the first term, which leads us to (5.54) assuming r = (r,12) Next, we're going to evaluate (r which this time leads us to (after a couple of lines of math) Chapter Five S = argminr2-2(2-20np(s)) (5.55) Now, the V term is the same for all Si, and as a result it won't affect our decision on which S to select. So we can drop it, which leads us to S = argmin = - (5.56) Now, for a math trick we've used once before: Choosing the value minimizing -X is the same as choosing the value maximizing X. Using this little trick, let's write: S = argmax (5.57) S = argmax (5.58) S = argmax (5.59) (5.60) where Now we've got a new equation describing how the decision device, given r, makes a choice on which Si to output. I'll show you how engineers build a demodula- tor that uses a decision devi