apter 7) (1.201) where 1=1,2,. ,L. (1.202) The ideal Rake receiver assumes complete knowledge of the phase and envelope terms which appear in the correlation functions When the fading is slow this estimate is quite good. A simpler form of the optimum receiver can be obtained using complex baseband representations for the radio signals on a carrier frequency of Slow Fading Multipath Channels wo radians. In general, the radio signal f(t) with a carrier frequency wo has representation =g(t)cos[wyt + n(1)] = Re{F(t)eiart (1.203) where g(t) and n(t) are real-valued functions and F(t) = g (t)ein(c) (1.204) is the complex baseband signal representing the radio signal f(t). Using script letters to represent complex baseband signals (t;0) = J(t) = (1.205) where as(1206 Assuming the carrier frequency is much greater than the signal bandwidth, the form for r in (1.200) is given by (1.207) The receiver structure of Figure 1.17 can implement this complex form of the optimum receiver by replacing y(t) by F (t) and Anc(t)P(t; On) by Ane-jonc(t) for each n. We now examine a way of estimating Ane-jo, which is required in a practical Rake receiver. Note that c(t)L(t) (1.208) A2c(1)(p(1;2) Figure 1.17. Ideal Rake receiver. Slow Fading Multipath Channels Recall that c(t) is independent of c(t - (n - l)/Wss) for each value of t when n # l and thus, (1.209) where do is the data bit in (0, Tb) and ni is a Gaussian random variable. This suggests that the estimate for Ane-jon be given by the conjugate of (1.209). This estimate, however, includes the data bit do. Assuming Ane-jon remains unchanged over 2Tb seconds, an estimate can be based on the pre- vious Tb second channel output signal. This estimate is shown in the com- plex form of the Rake receiver illustrated in Figure 1.18. +1 DATA BIT OF PREVIOUS DECISION Figure 1.18. Rake with estimates. Coherent Direct-Sequence Systems Thus far the results in this section have applied only to uncoded DS/BPSK signals in slow fading multipath channels. With coding and ideal interleavers and deint