###### Abstract
Therefore, narrower bandwidths are known to be a good solution in weak signal processing because they suppress noise effects. However, both longer \(T\) and narrower \(B\) are less robust to signal dynamic stress. Investigation of the trade-off between the sensitivity and the loop update interval to choose an optimal \(T\) was performed for a signal tracking Kalman filter with a special emphasis on the discrete nature of the loop, where the carrier phase measurements were asynchronous with the loop update [9]. Furthermore, designing a DPLL relies heavily on transforming a continuous-time system into a discrete-time system, performed by Laplace to a \(z\)-domain mapping. However, if the loop update interval is large relative to the most significant time constant of the system, the discrete-time system may not be an accurate representation of the corresponding continuous-time system. That is, the larger loop update interval makes the system less stable, and the updated output no longer gives an accurate measure of the output [10]. This means that there is another type of factor, similar to transforming continuous-time systems into discrete data systems. The DPLL will be equivalent to its analog counterpart only if the product \(BT\) (the so-called normalized loop bandwidth) is close to zero [5; 11]. For sufficiently small values of \(BT\) (e.g., \(BT\) < 0.1), this can provide an adequate starting point for the analysis and design of DPLLs. However, as \(BT\) increases, the deviation of the equivalent loop noise bandwidth and the closed-loop pole locations of the designed DPLL from the desired ones in the continuous-time domain eventually leads to the instability of the loop. Therefore, both the limit on \(T\) and \(B\) should be considered together [11; 12]. Another method (controlled-root method) was proposed to derive the loop filter constants from the loop roots based on the direct physical meaning of the loop noise bandwidth and the root-specific decay rate or damping. In this method, the deviation of discrete-time systems from continuous-time systems for large values of \(BT\) is avoided. However, the maximum achievable \(BT\) for a stable loop is limited to 0.4 for a rate-only feedback-type numerically controlled oscillator (NCO) [11; 12]. Note that this condition is satisfied for most communication systems because their \(BT\) value remains within this range. However, for some new applications in GNSSs, such as weak signal tracking, \(T\) values larger than one navigation data bit duration (e.g., 20 ms) are preferred for high sensitivity after the navigation data bit transition is solved correctly. Moreover, the use of pilot signal components in new GNSSs provides a chance to use the longer \(T\), which will inevitably require larger \(BT\) values. In this study, although two of the DPLL design methods (minimization, controlled-root) are introduced above, another DPLL design approach, designing the filter in the analog domain and then transforming it to the digital domain by using three types of digital transform methods (step-invariant, impulse-invariant, and bilinear), is used. As it is simpler than designing directly in the digital domain or applying controlled-root formulation, it is still widely employed as an engineering standard in many GNSS receiver designs, especially for high sensitivity. In addition, the computational delay (also called as transport lag), a time delay due to the calculation process for the loop update, is assumed not to exist for the first time of the analysis and is considered later. The paper concentrates exclusively on the theoretical upper and lower limits of \(BT\). The approach taken here for the upper limit is to choose a value of \(BT\) that satisfies the stability condition based on root loci in the \(z\)-domain. Each digital transform method is analyzed for the utilization of the loop filters and the NCOs. In addition, the lower limits of \(BT\) are obtained considering the effect of the \(BT\) on the oscillator quality and dynamic stress of the DPLL.