e then looks for the maximum y for which the phase-plane trajectories will eventually reach the (0, y/G) point, i.e., the DLL will phase and frequency lock. The acquisition trajectories for G = 10, = 1/V2, N = 2 and N = 4 are shown in Figures 2.16 and 2.17, respectively [4], [14].7 It is found that the maximum normalized search rate for N = 2 is y = 1.0, while for N = 4, the maximum normalized search rate is y = 0.5. Furthermore, the open loop gain has little effect on the trajectories for G > 10 [14]. The second question raised above can be dealt with by using the defini- tion of Y as a starting point and computing the acquisition transients as a function of time and then finding the acquisition time. Figure 2.18 shows the transient response of the DLL for G = 100, S = 1/V2, N = 2 and N = 4. The search rate is chosen to be y = 0.9 for the first case and y = 0.45 for the second case, respectively. The graphs show that the acquisition time (time required for the transient to subside within x = 0.1) for N = 2 is shorter than that for N = 4. The actual acquisition time can be computed from the definition of the normalized time given in (2.67) and the relation between the single-sided loop bandwidth BL and the loop radian natural frequency Historically acquisition trajectories were first found in [1] and [3] but were later shown to be in error [4]. y =1.5 y=1.0 y=0.9 j=0.5 5 = 0.707 Figure 2.16. Acquisition trajectories for N = 2 (reprinted from [14]). Pseudonoise Code Tracking in Direct-Sequence Receivers j=0.5 G = 10 5 = 0.707 Figure 2.17. Acquisition trajectories for N = 4 (reprinted from [14]). w n Since from [16] (2.75) then for critically damped loops Wn = BL1.5303 tack= (2.76) Table 2.3 summarizes the results for several different values of loop band- width. Finally, the actual maximum search velocity in chips/sec (which can be interpreted as the maximum allowable drift in the code) can be found from the definition of the normalized search velocity y. As a function of the sin- gle-sided loop noise bandwidth BL