The influence of Doppler effects on the tracking performance of a noncoherent second-order delaylocked loop (DLL) operating on a data modulated signal is investigated. For the performance analysis we consider the tracking accuracy (steady state error and jitter) of the linear DLL and the reliability of the nonlinear loop. The nonlinear analysis concerning the loop reliability makes use of an asymptotic expansion for the mean exit time (mean time to lose lock, MTLL) which has been derived by applying the singular perturbation method. In particular, we give optimal loop parameters and the optimal bandwidth of the bandpass filter in the loop arms to achieve a maximum MTLL. Due to Doppler effects the optimal loop parameters are rather different using either linear or nonlinear analyses.
I IntroductionCodetracking loops are essential for various directsequence spread-spectrum (DSSS) systems in communication and navigation. Noncoherent loops are rather important for such systems because they are insensitive to a data modulation and they do not presume a reliable carrier tracking prior to code synchronization as do coherent loop. Especially the later fact is essential since it is usually possible to ensure a reliable code tracking even for low signal to noise ratios (SNR), where conventional carrier synchronizers fail. Furthermore, second-order loops are preferred to first-order loop since an acceleration of the receiver relative to the transmitter (Doppler rate) causes a steady state tracking error whereas for first-order loops the error grows to infinity [l].For example, noncoherent second-order DLL are used in satellite communication and navigation systems such as the 'lkacking and Data Relay Satellite System (TDRSS)[2], or the Global Positioning System (GPS) [3]. Alternatively, this loop could be used also in a code division multiple ac-(CDMA) mobile communication system. For the performance analysis we consider two criteria: the t r d i n g accuracy (at high SNR) and the MTLL (at low SNR). While the tracking error of the linearized loop would be a good performance measure for many applications, the probability of a loss of lock is increasing rapidly at low SNR and large Doppler rates due to the loop nonlinearity. In this case the tracking accuracy is leas important since it does not take into account an eventual loaa of lock which should be as rare as possible for a good overall system performance.The DLL has been analyzed extensively in the literature and the MTLL is well known to be an important performance measure (see e.g. [4] -[SI). While the calculation of the MTLL yields explicit expressions for firstorder loops, the solution for second-or higher-order loops is not known today. The singular perturbation method is a powerful tool t o approximate the MTLL by an a s y m p totic expansion of the exact solution. This method was applied to calculate the MTLL of the coherent secondand third-order DLL [9], [IO], the second-order modified codetracking loop [ll] and the noncoherent second-order DLL [12]. It turns out ...
