error probability is thus (1.172) This bit error probability is better than using a conventional DS/BPSK receiver which only uses ro in its decision. Condition (1.166) for multipath delay results in a diversity system where two independent channel outputs are available. Thus DS/BPSK spread-spec- trum signals not only provide protection against jamming but also can resolve multipath and take advantage of the natural diversity available. For a multipath channel with L paths and a resulting channel output (1.173) it is possible to compute at the receiver the L outputs (1.174) where -) - is given in (1.168). This assumes the receiver has complete knowledge of the multipath statistics which include amplitudes {a1}, delays 5This can be obtained by using the maximum-likelihood rule of comparing p(r,, = -1) with Slow Fading Multipath Channels {T1}, and carrier phases {01}. If now all # j (1.175) then we have the approximation l = 1,2,...,L (1.176) where {n1} are independent Gaussian random variables with variance Nj/2. Here the optimum decision rule is given by (1.19) with (1.177) and the bit error probability is s-o(v(2) (1.178) where Eb is the energy of any single multipath signal when the amplitude term is unity (a = 1). The energy due to all multipath terms is (1.179) and (1.178) can be rewritten as (1.180) Next suppose that each multipath signal has a slowly varying independent fade. The L receiver outputs are then given by 1=1,2,...,L (1.181) where {A1} are independent fade random variables. Conditioned on A = = (A1,A2,...,A1) ..., (1.182) the bit error probability and its Chernoff bound are given by Pb(A) (1.183) Coherent Direct-Sequence Systems Assuming A has probability density function pi() for each l = 1,2,...,L the average bit error bound is (1.184) which for Rayleigh amplitudes with =1,2,...,L (1.185) becomes (1.186) where E is the average signal energy in the l-th multipath signal given by E1bl1,2,,L (1.187) Note that when all the multipath energy terms are identical the exact expression for Pb is given by 