or in terms of Fourier transforms H(w) = = S*(w)e-just. (1.168) Suppose now that s(t) corresponds to an M-chip segment of a PN waveform, i.e., To = MT Then, (1.169) where dn is the polarity (+1) of the n-th chip and p(t) is the basic chip pulse shape. For a baseband matched filter, we would have p(1)= (1.170) otherwise whereas for a band-pass matched filter, (1.171) otherwise. Taking the Fourier transform of (1.169) and substituting its complex conju- gate into (1.168) results in H(w) = (1.172) which has the implementation illustrated in Figure 1.26 as previously stated. As a prelude to our discussion of the discrete time version of the matched filter acquisition system, consider the output x(t) of the baseband matched filter for an input x(t), i.e., (1.173) Of course, for x(t) = s(t),x(t) achieves its maximum value at t = , namely, (1.174) For any other x(t), e.g., a time shift of s(t), and s(t) as in (1.169), we can write (1.173) as = MT)ds (1.175) Pseudonoise Code Acquisition in Direct-Sequence Receivers Suppose now that we sample x(t) as time instants t = (M = 0, 1, 2, , i.e., at a multiple N of the chip rate. Then M N ((n-1)N+i)T = DDX (1.176) where (1.177) and Dj is the j-th subchip polarity which is related to the n-th chip polarity dn by __i=1,2,...N (1.178) Equation (1.176) is in the form of a discrete time correlation that can be implemented either in analog or digital form using a tapped delay line or shift register respectively (see Figures 1.27a and 1.27b). In the latter case, the contents of the shift register, which holds the signal samples digitized to one bit, and the holding register, which permanently contains the fixed seg- ment of the code used for comparison, are correlated by comparing them stage by stage, generating a "+1" if the two stages match and a"-1" if they don't, and summing the resulting set of "1's" and "-1's." Using either the analog or digital correlator of Figure 1.27a or 1.27b as matched filters in Figure 1.25b results in the discrete time version of the matched filter rapid 