V.I. Borisov, V.I. Shestopalov, A.E. Limarev, T.F. Kapaeva

In this paper timing delay is evaluated for multiple access wideband communication systems. Two well-known early-late gate tracking loops are considered, i.e. coherent and noncoherent loops. Average time before tracking loss is a performance measure. The mathematical model is developed in the form of stochastic dynamic system whose potential is an antiderivative of the tracking loop discriminator function. On basis of the boundary problem for Fokker-Planck-Kolmogorov equation an integral expression of the average time before tracking loss is obtained using the dynamic system potential. Furthermore, it is shown that the potential has multiple local minima (stability zones) depending on signal delay time in a multiple access system. The case with two signals is considered in detail. In the paper analytical as well as numerical methods are used for average time evaluation before tracking loss. Asymptotic estimates are obtained for large signal-to-noise ratios. It is also demonstrated that the derived solution coincides with well-known Kramers formula for bistable systems when general hypergeometric functions have asymptotic representation in a coherent case. Kramers formula is defined by the information about the third potential derivative in a noncoherent case. Using the numerical results average time before timing loss is plotted versus signal-to-noise ratio and potential profile is plotted as a function of multiple access parameters. The conclusion concerning the methods of tracking loop performance enhancement is drawn from the analysis of the numerical results.