ow Fading Multipath Channels The usual DS/BPSK receiver would multiply y(t) with the PN waveform c(t) and then compute the cosine component of the carrier, (1.160) where r(t) = =c(t)y(t) = s(t) c(t)J(t) (1.161) OSIT. (1.162) Evaluating (1.160) using (1.161) and (1.162) gives (1.163) where (1.164) is a Gaussian random variable with variance N/12 and = - + (1.165) where do is the value of the data signal d(t) during the interval (0, Tb). Suppose in (1.159) the multipath delay T satisfies (1.166) where T is the PN chip time. Then for each t, c(t) and c(t - T) are inde- pendent and n' is a sum of independent random variables which can be approximated as a Gaussian random variable with variance no greater than a2S/(2Wss). Since typically S<<J is assumed, for all realistic values of a, this multipath noise term n'o is negligible compared to the noise term no which is due to jamming. Thus we have the approximation ro=doVE,+ no (1.167) for multiplath delay T satisfying (1.166). When the multipth delay T is greater than the chip time T, there is neg- ligible degradation due to multipath. This also applies when the direct path experiences slowly varying frequency non-selective fading as discussed in the previous section. In general, however, it is possible to do better. Assuming the channel is slowly varying SO that the multipath parameters a, T, and 0 Coherent Direct-Sequence Systems are known to the receiver, the receiver can multiply y(t) with c(t - T) and find the cosine component corresponding to the coordinate (1.168) This results in the cosine component relative to the multipath signal of the form (ignoring th direct path noise term) (1.169) where (1.170) is a Gaussian random variable with variance N/2. Thus there are two out- puts of the channel given by ro and r1- Except for some unrealistic waveforms for the jammer, the correlation between no and n1 is zero and thus these are independent Gaussian random variables. The optimum decision rule5 based on ro and r1 is to compare (1.171) with zero as in (1.19). The bit 