Deriving the first and the second moments of the normalized timing error of first order early-late tracking loops using linear analysis and real time simulation for wireless receivers moving away linearly from the transmitter

In this paper we provide a review for the time domain analysis of the normalized timing error dynamics of a wireless receiver Early-Late tracking loop (A loop that controls the timing of a pseudo-noise code that is locally generated in the receiver). This review shows that the timing error dynamics modelling is cumbersome and can be simplified by linearizing the tracking loop. The linearization assumption makes the use of Laplace transform to model the timing error dynamics possible. The Laplace transform is used to derive formulas for the normalized timing error first and second moments for the case of linear varying time delay inputs of first order Early-Late tracking loops. The derivation of these moments facilitates modelling the timing error dynamics by using probability density functions that implement these moments to calculate the probability that the tracking loop will fail to handle the timing error dynamics. A simulation is carried out and verified that the numerical evaluation of the proposed formulas matches the simulation output to some tolerable error. The formulas are used to assess the performance of Early-Late tracking loops when subjected to linear variable time delay. The results show that the velocity of the wireless receiver, the received signal power and the chip rate are the main factors that affect the timing error first moment (the mean). The second moment (the variance) of the timing error is affected by the received signal power and the tracking loop gain.