Title :
Analytical Computation of Mean Time to Lose Lock for Langevin Delay-Locked Loops
Author :
Groh, Ingmar ; Gentner, Christian ; Selva, Jesus
Author_Institution :
German Aerosp. Center (DLR), Inst. for Commun. & Navig. (KN), Wessling, Germany
fDate :
11/1/2012 12:00:00 AM
Abstract :
This paper presents a novel method for the analytical mean time to lose lock (MTLL) computation of coherent second-order Langevin delay-locked loops (DLLs). Analytical MTLL computation is a key task for DLLs, since the computational complexity of numerical MTLL simulations is far too high in many operating ranges of the second-order Langevin DLLs. To obtain the crucial MTLL values analytically without simulations, we rewrite the Langevin stochastic differential equation (SDE) as a vector-valued Ornstein-Uhlenbeck (OU) SDE. It includes a Gaussian noise term, which yields as a solution of the vector-valued OU SDE a time-variant Gaussian distribution. Thus, the complementary error function yields the loss of lock probability and thereby the MTLL. If we replace the complementary error functions by suitable exponential approximations, we obtain a simple MTLL expression with an exponential function as dominant term. The simple exponential MTLL expression yields the optimum loop parameters corresponding to the maximum MTLL. Simulation results confirm that the optimum loop parameters corresponding to our analytical MTLL computation method and to the simplified exponential approximation coincide. Besides the crucial analytical MTLL results, the OU random processes yield additionally the likewise crucial analytical jitter results.
Keywords :
Gaussian distribution; Gaussian noise; approximation theory; computational complexity; delay lock loops; differential equations; probability; random processes; stochastic processes; Gaussian noise term; Langevin stochastic differential equation; OU random processes; analytical MTLL computation method; analytical mean time-to-lose lock method; coherent second-order Langevin delay-locked loops; complementary error function; computational complexity; exponential approximation function; lock probability loss; numerical MTLL simulations; optimum loop parameters; second-order Langevin DLL; time-variant Gaussian distribution; vector-valued OU-SDE; vector-valued Ornstein-Uhlenbeck SDE; Computational modeling; Delay; Eigenvalues and eigenfunctions; Mathematical model; Noise; Random processes; Ornstein-Uhlenbeck random process; Spread-spectrum system; code synchronization; loss of lock phenomenon;
Journal_Title :
Communications, IEEE Transactions on
DOI :
10.1109/TCOMM.2012.082812.100629