Third-Order Delay-Locked Loop: Mean Time to Lose Lock and Optimal Parameters

For the first time a systematic approximation for the mean time to lose lock (MTLL) of a coherent third-order PN-code tracking loop has been derived. Such loops are essential in various spread spectrum systems (Global Positioning System, GPS, for example). The computation of the MTLL is based on the singular perturbation method. The application of this method to the coherent delay-locked loop (DLL) yields an approximate expression for the MTLL. Therefore, with the proposed loop model we are able to analyze this third-order system at a level that gives a well understanding of the nonlinear loop behavior and the exit phenomenon. The influence of a loop offset due to an acceleration rate (jerk) between transmitter and receiver on the optimal filter parameters is described by comparing MTLL and tracking error performance. As intuitively might be expected it turns out that acceleration rate and code rate are exchangeable in the sense that a lower code rate allows a higher acceleration rate (and vice versa) for the same signal-to-noise ratio in order to maintain the same performance. In a case study, GPS code tracking for objects with high jerk is briefly discussed.