Optimal design of a noncoherent second-order delay-locked loop using the exit-time criterion

Abstract

For the first time an approximation based on the singular perturbation method for the mean time to lose lock (MTLL) of a noncoherent second-order delay-locked loop (DLL) has been derived. Such a loop is essential in direct-sequence spread-spectrum systems. The influence of loop offset due to a constant acceleration (Doppler rate) between the transmitter and receiver as well as the optimal choice of the loop parameters maximizing the MTLL are given. Furthermore, the impact on the bit error probability due to the synchronization error is studied.

Mean time to lose lock of the first‐and second‐order modified code tracking loop used in spread spectrum systems

“…It is therefore sufficient to maximize The dependence of E,oPt on yd occurs because G ( E ) depends on y& It is interesting to note that the optimal loop parameters are pure constants, independent of the others. This agrees with earlier results of other tracking loops, [3,28,29,31,32,33], and shows also the importance of the time scaling which led to such a specific loop description (10). Together with (9), (1 1) and (31) the optimum rescaled MTLL normalized by the bit time Tb turns out to be 60pt = 0.373, Knowing IuoPtI we can immediately deduce the relationship for the optimal loop bandwidth Bk, (1 1): (33) Hence, the optimal bandwidth must be chosen according to the processing gain and the velocity vo.…”

“…it is easy to show that the state equation of the first-or-Vol.5, No. 3 May-Jun. 1994 der MCTL can be related to the Langevin equation of motion which describes a specific second-order system.…”

“…Furthermore, it was also used to calculate the MTLL of the coherent second-and third-order DLL, [27,29,32]. In this paper, we extend the concepts and ideas of [3] and [32] to analyze the second-order MCTL. We calculate its MTLL under the constraint of a channel noise and a Doppler rate caused by an acceleration of the transmitter relative to the receiver.…”

Optimal design of an all-digital chip timing recovery loop for direct-sequence spread-spectrum systems

“…Threshold extension by optimal loop design was shown. This method can be extended to solve the second order loop case, similarly to [13,17,18] and can also be used for the modified DDLL [7] in DSSS systems over frequency selective fading channels.…”

“…The normalised equation is preferred since only in this equation one can properly compare the noise intensity with the restoring force. We will show below that in many (as a matter of fact almost all) practical cases, the noise intensity is small compared to the restoring force, even in the threshold region [13][14][15][16][17][18], thus allowing us to use the singular perturbation method in our analysis. For the Brownian motion wðtÞ, we have [12] ðdwðtÞÞ 2 $ dt, due to the quadratic variation of the Brownian motion.…”