###### Abstract  
Determining the loop noise bandwidth and the coherent integration time is essential and important for the design of a reliable digital phase-locked loop (DPLL) in global navigation satellite system (GNSS) receivers. In general, designers set such parameters approximately by utilizing the well-known fact that the DPLL is stable if the normalized bandwidth, which is the product of the integration time and the noise bandwidth, is much less than one. However, actual limit points are not fixed at exactly one, and they vary with the loop filter order and implementation method. Furthermore, a lower limit on the normalized bandwidth may exist. This paper presents theoretical upper and lower limits for the normalized bandwidth of DPLL in GNSS receivers. The upper limit was obtained by examining the stability of DPLL with a special emphasis on the digital integration methods. The stability was investigated in terms of \(z\)-plane root loci with and without the consideration of the computational delay, which is a delay induced by the calculation of the discriminator and the loop filter. The lower limit was analyzed using the DPLL measurement error composed of the thermal noise, oscillator phase noise, and dynamic stress error. By utilizing the carrier-to-noise density ratio threshold which indicates the crossing point between the measurement error and the corresponding threshold, the lower limit of the normalized bandwidth is obtained. coherent integration time; global navigation satellite system (GNSS) receiver; loop noise bandwidth; normalized bandwidth; phase-locked loop (PLL); signal tracking loop; stability analysis 2023235887.10.  
within an aircraft and a missile are benefit from a large \(B\) and small \(T\) because of the large gain (to alleviate the dynamic stress) and faster update rate. On the contrary, in the case of the static but weak signal conditions such as indoor users can employ the small \(B\) and large \(T\) to raise the post-correlation signal-to-noise ratio while suppressing the noise effect. A software implementation of this based on a simple numerical integrator for the PLL of a pseudo-noise spread spectrum receiver to accommodate a given set of dynamics specifications was previously presented [4]. A systematic survey of the theoretical and experimental work on this topic for the period between 1960 and 1980 can be found in [5]. Typically, longer \(T\) in correlators, which are equivalent to longer loop update intervals in loop filters, provide the additional processing gain required for the detection and processing of signals in harsh environments, specifically weak indoor signals and integrated navigation systems [6]. The benefits of the longer \(T\) using the aiding technique are multi-path mitigation in the Doppler domain, cross-correlation protection by data wipe-off, and reduced squaring loss. The use of ancillary sensors with relevant processing facilities was proposed to extend the \(T\) to several seconds by providing better-matched replica signals [7]. However, there is no further processing gain beyond a certain \(T\) because of the spatial and temporal decorrelation of the received signal, the residual instability of the receiver oscillator, and data bit transitions, all of which result in a narrowing of the bandwidth [8]. The limits on the \(T\) due to the spatial decorrelation of multipath signals were derived and experimentally verified using a generalized multipath scattering model [6]. The results showed that the processing gain obtained from longer \(T\) saturates after the antenna has moved a certain distance, which is typical for indoor propagation. In addition, one of the dominant error sources in PLL is dynamic stress error, which is a barrier to further reducing the loop bandwidth \(B\). In particular, removing the dynamic stress for a moving platform receiver using external aiding information would be a logical first step towards decreasing the \(B\) and enhancing the tracking sensitivity in weak signal environments.