t such as a groundwave at HF, which appears in shorter ranges between transmitter and receivers, the signal out of the chan- nel is assumed to have Rician fading statistics. In the following assume Rayleigh fading only. Since the transmitted signal x(t) has bandwidth Wss centered at carrier fre- quency wo radians, it can be represented in terms of samples of the inphase and quadrature components of the signal at sample times {n/Wss: n = - 1,0,1,2, }. Thus the channel output can be modelled as (1.194) where at any time t the envelope terms are independent Rayleigh random variables and all phase terms are independent and uniformly distributed over [0,2]. Also, since the total multipath spread is Tm, for all practical purposes we can truncate the number of terms to L = WssIm + 1. (1.195) The assumption regarding the slowly varying nature of the channel where (1.192) holds means that An(t) and On(t) are constant during the bit time (1.196) and the same situation as before with the finite number of distinct multipath components shown in (1.173) occurs here. Coherent Direct-Sequence Systems For this case the receiver can compute outputs for individual multipath signals as follows, n=1,2,...L (1.197) provided the phase terms {0n} are known at the receiver. Also assuming envelopes {An} are known, the optimum receiver compares - EAr (1.198) to zero to make the binary decision a. Next examine the optimum receiver structure implied by (1.197) and (1.198). Note that (1.197) can be rewritten as n = 1,2,...,L (1.199) and SO (1.198) becomes (1.200) Figure 1.17 shows a block diagram for the optimum demodulator. The ideal tapped delay line receiver of Figure 1.17 attempts to collect coherently the signal energy from all the received signal paths that fall within the span of the delay line and carry the same information. Because its actions act like a garden rake this has been coined the "Rake receiver" [23]. The bit error bound using the ideal Rake receiver is given by (1.186). An exact expression is given by (see Proakis [3], Ch