Discrete vs. Continuous Carrier Tracking Loop Theory, Implementation, and Testing with Large BnT

The majority of textbooks (e.g., Kaplan 1996) appear to provide an analog or continuous representation on the theory of carrier tracking loops based on continuous tracking loop equations. Such a representation may be adequate to explain the tracking loop phenomenon when the noise bandwidth (Bn) integration time (T) product is much less than 1 where we are able to predict and explain the observed input/output relationship based on continuous carrier tracking theory (or equations). But, as soon as this product gets large than 1 then there appears to be an inconsistency between observed results and those predicted from the continuous tracking loops (Stephens & Thomas 1995). In order to investigate this inconsistency and be able to explain the lack of consistency between the published theory and experimental data we implemented the carrier-tracking loop for Open Source GPS using the same integration time for the carrier and code loops equal to the 20 ms data period. First, we tracked in accumulated carrier phase by setting the loop bandwidth approximately equal to 25 Hz, which produces a BnT of 0.5. Initially the carrier tracking loop appeared to work well but, later on the position and to a larger extent the velocity fixes accumulated more and more error. Second, since it was known that this loop works well with an integration time of 1ms it spurred us on to look into this phenomenon in greater detail. It is known that, when BnT is close to or above one, field test results are not consistent with the theoretical expectation (e.g., Kaplan 1996) any more. It is obvious that the theory should be corrected in this case to reflect our learned knowledge from the observations. In (Gao 2007) it is given an understanding on this issue and an explanation for signal tracking errors of both FLL and PLL in a digital GNSS receiver. From an understanding of these new formulas, one can see that, even when the receiver clock noise and the signal-Doppler tracking error are constant, they generate larger and larger signal tracking errors when BnT increases. However, this phenomenon is not observed in the classic GPS tracking theory. It was suggested that this could be symptomatic of a carrier-tracking loop with a large BnT. This effect is a part of G. Gao’s Ph.D. research in which he has come up with a suggested solution to this phenomenon (Gao 2007). Furthermore, in this paper, Dr. Progri provided a complete derivation of the stability condition of the discrete-time tracking loop based on the condition of the stability of the system function and also of the discretetime unit pulse function. We first performed the analysis of this phenomenon and then verified it with data from tests with the Open Source GPS software receiver emulation (theoretical or ideal signal) and then with the GP2015/2021 hardware (with a real GPS signal coming from a GPS satellite). The results presented include our theoretical approach and the experimental data. According to our analysis it appears that for 0 < BnT < 0.34 the carrier tracking is stable; however, for 0.34 < BnT ≤ 0.5 the carrier tracking is marginally stable; i.e., it stable in short term or for a few seconds and unstable in the long term or for many seconds, minutes, hours etc. From the experimental data, recorded by Mr. Kelly, we were able to confirm that for BnT ≤ 0.3 the carrier tracking loop was stable, for BnT = 0.5 the carrier tracking loop was marginally stable and BnT = 1 the carrier tracking loop was unstable.