Steady-state analysis of delay-locked loops tracking binary Markovian sequences

Abstract

We analyze stationary phase tracking error of delay-locked loops (DLL) in direct spread code division multiple access (DS-CDMA) using Markovian spreading sequences. The phase tracking error is caused by noise generated inside of DLLs by multiple access interferences. When binary Markovian sequences are used, the noise is not considered as white Gaussian noise. This makes analysis of the tracking error difficult. In this paper, we describe DLLs by stochastic difference equations and derive forward evolutional equations of the probability distribution of the states of DLLs. Applying path integral analysis to the evolutional equations, we obtained stationary distribution. We found from the distribution that Markovian spreading sequences with negative eigenvalue were effective in decreasing stationary phase tracking error of not only a type of DLL in asynchronous CDMA but also DLLs in chip-synchronous CDMA.

An analysis of delay-locked loop tracking binary Markovian sequence in the presence of multiple access interference and channel noise

“…As a result, LPF is described by the last equations of Eqs. (24) is equivalent to a linear model of the archetype DLL dealt with in [14]. For the linearized archetype DLL, the superposition principle is applicable to respective responses of the DLL to MAI and channel noise.…”

“…For the linearized archetype DLL, the superposition principle is applicable to respective responses of the DLL to MAI and channel noise. The influences of the MAI and the channel noise are investigated respectively by the path integral analysis [14] and by deriving Fokker-Planck Equation [19]. In this paper, products Assume that the DLL state (ε(n), v d (n), v f (n)) is given at time n and that the switching time of c (1) ,…”

“…These two parameters k o and τ f are decided from the view points of the frequency precision of practical VCO, the sensitivity of the DLL to MAI and channel noise, and the dynamical tracking performance [14]. As we derived Eq.…”

“…The time spent for computing the variance by the method in Section 4 was about 1/1000 compared to that by the path integral method [14] when no channel noise was added to the DLL. Figure 5 indicates that replacing i.i.d.…”

“…Recently, it is reported that the preceding analyses can not evaluate the dependency of the synchronization error on the statistical characteristics of spreading sequences and that a kind of path integral estimates the error with high precision [14]. However, because of high computational complexity of the path integral method, the DLLs to be analyzed are restricted to archetype DLLs which suffer from MAI but are free from communication channel noise.…”