of the DLL and TDL code tracking loop, there are primarily two questions of interest: 1. What is the maximum relative code rate offset (due to code Doppler, clock instabilities, etc.) between the received and locally generated PN codes SO that the received signal can still be acquired? Equivalently, what is the maximum search rate (velocity) to achieve acquisition? 2. How long do the transients last, i.e., how long does it take to acquire? The answer to the first question may be obtained (in a noise-free envi- ronment) by examination of the phase-plane trajectories. These trajectories are plots of normalized code delay error rate & versus normalized code delay error E1 with normalized time as a parameter along the curve. To obtain these trajectories for the DLL, we begin by rewriting the system equation of noise-free operation, as given by (2.15), in the normalized form Po(y-x)=GFo(po)Dn(x) (2.66) where Thwit y s e (2.67) with Vn the radian natural frequency of the loop. For a second-order loop with linear closed loop transfer function [16] 1 + 25s/w, wnGF(s) (2.68) s + w, GF(s) the product of the loop gain G and the filter transfer function F(s) becomes (2.69) (2.70) where Z is the loop's damping factor. For a critically damped loop Pseudonoise Code Tracking in Direct-Sequence Receivers (5 = 1/V2), (2.70) simplifies to (2.71) which, upon substitution into (2.66), yields G+j=x/G+x+Dnx) = + V2D'n(x)x. (2.72) the dot now denotes the derivative with respect to normalized time (t) and the prime denotes the derivative with respect to the normalized transmis- sion delay (x). The solution of the second-order partial differential equation in (2.66) for the phase-plane trajectories is facilitated by defining x/x = dx/dx, which results in Dx)1/x-- (2.73) For a constant search velocity, y = 0 and the above equation simplifies to (2.74) Statistical methods (e.g., Newton's method [19]) of solving differential equations can now be applied to (2.74) to compute the trajectories in the (x,x) plane for given initial conditions. On