 use, but it will be a useful one in helping us understand the problems a receiver faces when it picks up a received signal r(t) of Equation (9.9), a signal that includes channel filtering. 9.3.1 The Proposed Receiver Take a look at the receiver in Figure 9.6. It takes the incoming signal r(t) and first multiplies it on top by a cosine and on the bottom by a sine. This leads to two signals which are fed into a filter f(t), which passes frequencies around 0 Hz but cuts out higher frequency terms. Chapter Nine cos(Wct) rc(t) Decision device r'(t) r"(t) rs(t) sin(Wct) Figure 9.6 Proposed receiver for new r(t) First, the multiplication by a cosine leads to (after about three lines of math): (9.10) + (high frequency terms that will be cut out by f (t)) and the multiplication by a sine leads to (after a different three lines of math) (9.11) + (high frequency terms that will be cut out by f (t)) where NC (t)=n(t) = cos wct and n,(t)=n(t)sinw, To simplify the presentation and not have to bring two different signals with us in the rest of our discussion, we can represent this as a single complex signal, namely r'(t)=r.(1)+ ir,(t) (9.12) which leads to (9.13) + (high frequency terms) Channel Filtering and Equalizers (9.14) + (high frequency terms that will be cut out by f (t)) where n' (t) )=nd(+)+jn,(t). This is the signal we show entering the channel filter f(t). The multiplication by sine and cosine in essence: (1) removes the Wct term from the received signal (that is, returns the signal to baseband); and (2) allows us to write the signal in a handy complex notation. Next, we apply the filter f(t). This leads to (9.15) (9.16) where x(t)=h(t)*f(t)=g(t)*cp(t)*f(t) and "(t)=n'(t)*f(t). This tells us that our sent signal, after passing through a channel filter and having noise added on to it, and then passing through a receiver filter f(t), basically is the sent signal with the combined effects of the filtering and the noise. We now sample the signal at times kT (k = 0, 1, 2, ...). At time kT, we end up with the output rk 