Long-term frequency stability requirements and optimal design of a code tracking delay-lock loop

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

“…However, from Equation (3), see also Figure 3(a), g 0 ðz f Þ ¼ 0 ðE f Þ ¼ 0 for E f AE 1:5, thus for the case without drift, u ¼ 0, the pre-exponent of t à L cannot be calculated using the classical Equation (44). This case was investigated in detail in Reference [25] and presented in Reference [14], see also Reference [15]. While the exponential term remains as in Reference (44), the pre-exponential term is different (there is no term of g 0 ðz f Þ).…”

“…Note that in References [14,15] the system equations were understood in Stratonovich sense [12] since the loops were analogue ones while in our case the loop equation should be understood in the Ito sense since the loop originated for the difference Equation (7), thus a small deviation due to the Wong-Zakai correction [22] is expected. However, this correction is of order and as was seen above (Table 1) can be ignored compared to g, thus only a negligible influence on the mean exit time is expected.…”

“…In order to use the 'conventional' singular perturbation approach [13,23,24] to calculate the MTLL, the noise intensity factor should be independent of the state variable E. Since this is not the case here, we will transform Equation (33) following Reference [15]. …”

“…The use of the singular perturbation approach for the computation of the MTLL in a second order optimal PLL was introduced in [13]. The MTLL for code-tracking loops were presented in [14][15][16][17][18][19]. Preliminary results for the band-limited noncoherent digital code-tracking loop were presented in [20].…”

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

“…Thus, the singular perturbation method can be applied [SI which gives rise to an approximation for the MTLL. The applications of this method to the noncoherent first-order and to the coherent second-order DLL were described in [9] and [ll], respectively. This paper deals with the noncoherent second-order DLL which is widely used in direct-sequence spread-spectrum systems to synchronize the local spreading code to the received pseudonoise (PN) code under the constraint of a constant acceleration (Doppler rate) between the transmitter and receiver and an additive white Gaussian noise (AWGN) channel.…”