Keywords

1 Introduction

For navigation applications or ambiguity resolution, codephase center corrections (CPC) – also known as group delay variations (GDV) – are becoming more important, like e.g. in ambiguity resolution with Precise Point Positioning. They are delays in the received codephase of the GNSS signal and can have an impact on the ambiguity resolution in code-carrier linear combination (LC), e.g. Melbourne-Wübbena, when the magnitude of the respective antenna CPC is in the range of the LC wavelength (Kersten and Schön 2017). A concept to estimate CPC and also absolute phase center corrections (PCC) has been developed at the Institut für Erdmessung (IfE) – an antenna calibration facility accepted by the International GNSS Service (IGS) – in close cooperation with Geo++ (Menge et al. 1998; Wübbena et al. 2000; Böder et al. 2001) and is constantly improved and optimized (Kröger et al. 2021; Kersten et al. 2022). PCC are required to ensure a highly accurate position in GNSS applications, like precise point positioning. The estimation of PCC of an antenna under test (AUT) is done with a robot in the field. The estimation process is independent of the reference antenna’s PCC; thus referred to as an absolute calibration. This approach is adapted for the estimation of CPC. IfE estimated CPC of the GPS L1 signal for low cost and geodetic antennas were shown in Kersten and Schön (2017), whereas Breva et al. (2019) presented first multi-GNSS CPC.

Related work on CPC/GDV comprises first CPC estimation with a robot done in 2008 (Wübbena et al. 2008), where a Kalman filter based on undifferentiated observations was used in a real-time process. At Deutsches Zentrum für Luft- und Raumfahrt (DLR) the electromagnetic behaviour of aeronautic antennas is estimated in an anechoic chamber. The group uses this information to calculate antenna dependent pseudorange errors as well as their GDV (Caizzone et al. 2019). Work at TU Dresden is based on code-minus-carrier linear combinations (CMC) to estimate satellite and receiver GDV together in a network approach (Wanninger et al. 2017; Beer et al. 2019). Wübbena et al. (2019) published absolute GDV for 36 antennas, Beer et al. (2021) were able to estimate absolute GDV for GNSS-satellite antennas with their CMC approach.

In the goal of estimating absolute CMC accurately and repeatedly, the noise of the codephase observations within the calibration process needs to be reduced significantly without deforming the CPC pattern. Therefore, Breva et al. (2022) presented an alternative data preprocessing strategy by using the empirical mode decomposition. Another opportunity for the noise reduction is to modify different receiver settings to ensure a stable tracking of the GNSS signals within the fast robot motion.

The present paper is structured as follows: after a brief overview about the antenna calibration and tracking loops of the GNSS receiver in Sect. 2, the impact of different settings of the GNSS tracking loops on codephase observations is studied in Sect. 3, which are used for the CPC estimation. In Sect. 4 the resulting CPC pattern with different receiver settings are shown as well as their repeatability. Section 5 concludes this paper.

2 Theoretical Background

2.1 Antenna Calibration at IfE

The Institut für Erdmessung uses an antenna calibration robot for estimating absolute phase and codephase center corrections. CPC are antenna dependent delays of the received codephase. They are varying with azimuth (\(\alpha )\) and elevation (el) of the incoming satellite signal and are divided into a codephase center offset (CCO) and codephase center variation (CPV). The CPC can be calculated by

$$\displaystyle \begin{aligned} \begin{aligned} CPC(\alpha^k,el^k) = &-CCO \cdot \mathbf{e}(\alpha^k,el^k) \\&+CPV(\alpha^k,el^k)+r. \end{aligned} \end{aligned} $$
(1)

Here, the CCO is projected onto the line-of-sight unit vector e towards the satellite k. The constant parameter r cannot be estimated without additional information. The rank deficit is removed by defining a certain datum (\(CPC(z=0) = 0\)).

To estimate absolute CPC, an antenna under test is mounted on top of the robot nearby a reference antenna (see, Fig. 1). Each antenna is connected to a GNSS receiver (e.g. Septentrio PolaRx5TR), which are synchronized by an external frequency standard (Stanford Rubidium FS725). This setup forming a baseline of around 8 m, which allows calculating receiver-to-receiver single differences (SD). By time differencing the SD (\(\Delta SD\)), almost all error sources are either cancelled out or reduced to a negligible magnitude. The antenna pattern information of the AUT are obtained by tilting and rotating the antenna around a fixed, certain point in space with the robot:

$$\displaystyle \begin{aligned} \Delta SD^k(t_i) = CPC^k_{AUT}(t_{i+1})- CPC^k_{AUT}(t_i) + \epsilon \end{aligned} $$
(2)

It is noted that in addition to noise, also multipath (MP) are gathered in the parameter \(\epsilon \). Its amount highly depends on the antenna gain: MP at antennas with symmetric gain behaviour is cancelled out by rotating the antenna horizontally. In tilting robot sequences, the differences in MP between two epochs are still present and consequently contained in the \(\Delta SD\). Detailed investigations on MP within robot-based antenna calibration are currently done in the DFG MAESTRO project. The time differenced single differences are used as the input for the estimation process based spherical harmonics. A detailed description of this approach can be found Kröger et al. (2021) and Kersten and Schön (2017).

Fig. 1
figure 1

Antenna calibration setup at IfE with the robot (foreground) and the nearby reference station (background)

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2.2 Tracking Loops of GNSS Receiver

The task of GNSS receiver’s tracking loops is to continuously track the GNSS signals received from the antenna and determine the aggregated observations like code, Doppler and carrier-phase for the navigation processing. In Fig. 2 (left) the main components of GNSS receivers are depicted. After amplifying the satellite signal, a down conversion to an intermediate frequency (IF) is achieved. Afterwards, the analog IF is converted via the analog-to-digital converter (ADC) to a digital IF signal. In the signal processing, the inphase and quadrature parts of the digital IF are correlated with the replica signals, analysed by the loop discriminators and filtered to steer the numerically controlled oscillator (NCO). The focus in this contribution is on the tracking loops, a detailed description of all main components can be found e.g. in Häberling (2016) or Kaplan and Hegarty (2017) and will not be discussed here.

Fig. 2
figure 2

(left) Overview of the GNSS receiver’s main components and (right) a detailed look into the tracking loops, which are parts of the signal processing based on Häberling (2016)

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The basic principle of tracking loops are presented in the right of Fig. 2. They are located in the signal processing part of the GNSS receiver and starts right after the signal acquisition algorithm, where the GNSS signal is roughly located in the \(\{ \tau , f_D \}\) search space, where \(\tau \) is the propagation time and \(f_D\) indicates the Doppler frequency of the satellite signal. The loops improve these rough estimates and continuously track changes in these parameters from that point forward (Misra and Enge 2006). Three kinds of tracking loops exist: The delay lock loop (DLL), the phase lock loop (PLL) and the frequency lock loop (FLL). Each loop consist of a discriminator, a loop filter and a numerical or voltage controlled oscillator (NCO, VCO). The main goal of the loops is to align the replica signals, generated by the local oscillator, to the incoming signals. Their outcomes are the code delay, frequency and phase of the satellite signal, which correspond to the codephase, carrier phase and Doppler observables in GNSS processing.

The replica signal \(u_2(t)\) could be shifted in its code time, its phase or in its frequency to the incoming signal \(u_1(t)\). The discriminator compares \(u_2(t)\) and \(u_1(t)\) to estimate possible delays or shifts, at which the code time delay \(\Delta \tau \) is estimated by the DLL, the phase shift \(\Delta \theta \) by the PLL and the frequency shift \(\Delta f \) by the FLL. The discriminator output signal \(u_d(t)\), which depends on the used discriminator function (e.g. early-minus-late for DLL), passes the loop filter next. The filter reduce the noise in order to produce an accurate estimate of the original signal as output \(u_f(t)\). A detailed description and analysis of loop filters and their parameters are depicted in Sect. 3. The last component of the loop is the VCO/NCO. This oscillator of a DLL creates a new replica signal \(u_2(t)\) by slowing down or speeding up the clock that controls the speed of the replica code generator by the amount of \(\Delta \tau \). In case of FLL or PLL, it is synchronizing the frequency and the phase of \(u_f(t)\) with the frequency and phase of \(u_1(t)\) by \(\Delta f\) and \(\Delta \theta \). Afterwards, the delays and shifts are estimate again by the discriminator. When these parameters are equal to zero or to a constant value, the loop is in a locked state.

The first operating tracking loop is the DLL. This loop provides the prompt correlation measurement required by the PLL and FLL. It must accurately estimate \(\Delta \tau \) before the PLL begins to track. The FLL typically starts to operate, when the \(C/N_0\) of the GNSS signals is too weak for a PLL operation.

3 Analysis of Different Receiver Settings

3.1 Practical Experiment

In order to analyse the impact of different receiver settings on the observations and the resulting CPC pattern, an antenna calibration experiment was set up. The goal of this experiment is to find the optimal receiver settings to optimize the tracking of codephase signals within the fast, challenging robot motion in a way that all CPC information are maintained.

Therefore, a Novatel antenna NOV703GGG.R2 NONE (S/N: 12420040) was mounted on the robot on 7th and 8th June 2022. Moreover, the Leica antenna LEIAR25.R3 LEIT (S/N: 9330001) was used as a reference. Each antenna was connected to one of two identical Septentrio PolaRx5TR GNSS receivers that are linked to the same external frequency standard FS725. In addition, the Novatel AUT was also connected to the IFEN software receiver. With this, a standard antenna calibration procedure was running with four individual sets (hereinafter called \(P_1\) to \(P_4\)) with a duration of about 4 to 6 h (Table 1).

Table 1 Duration of individual calibration sets \({P_1}\) to \({P_4}\)
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In general, hardware receivers have a limited amount of changeable settings for common users, like the bandwidth or order of the tracking loops. The directly changeable settings for the here used Septentrio receivers are the bandwidth of the DLL (default: 0.25 Hz) and the PLL (default: 15 Hz), as well as their coherent integration time (default: DLL 100 ms; PLL 10 ms). For further analysis, the default settings are used. The software receiver allows changing over 170 receiver settings in post-processing by using the same digitalized data stream. A list of all settings can be found in IFEN GmbH (2019). To enable a suitable solution, several receiver settings are defined beforehand, like the correlator mode or correlator type. Here, the focus is on the impact of the DLL bandwidth and the DLL order on the time differenced single differences. Furthermore, the impact of an aided or unaided DLL, by using the default bandwidths for FLL (narrow: 1 Hz; wide: 10 Hz) and PLL (narrow: 9 Hz; wide: 50 Hz), is investigated.

3.2 Impact on Time Differenced Single Differences

To improve the estimation of CPC, less noisy codephase observations (\(\Delta SD\)) are required. As mentioned in Sect. 2.2, the goal of loop filters is the noise reduction of the observations. Therefore, the noise bandwidth (\(B_N\)) and the filter order (FO) of the filter can be modified.

First, the impact of different DLL \(B_N\) on the \(\Delta SD\) of the GPS C1C signal is presented in Fig. 3. In grey, the \(\Delta SD\) observed with the Sx3 software receiver are presented for the first calibration set \(P_1\). For a better comparison, the \(\Delta SD\) observed with the classical Septentrio receivers are depicted in red. The parameters for both plots are the same, except the noise bandwidth. A \(B_N\) of 1 Hz is used in the left figure and a \(B_N\) of 0.25 Hz in the right figure. Obviously, decreasing the \(B_N\) leads to less noisy observations.

Fig. 3
figure 3

Time differenced single differences of the GPS C1C signal of the first calibration set \(P_1\). The \(\Delta SD\) from the Septentrio receivers are presented in red and the \(\Delta SD\) of the software receiver with different settings* are depicted in grey. *(left) \(B_N\) = 1 Hz, FO = 2, unaided DLL. (right) \(B_N\) = 0.25 Hz, FO = 2, unaided DLL

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The second analysis belongs to the impact of different DLL filter orders. After Kaplan and Hegarty (2017) first order loop filter are sensitive to velocity stress, second order to acceleration stress and third order to jerk stress. In Fig. 4 the \(\Delta SD\) with FO = 1 (left) and FO = 2 (right) are shown. By comparing both figures, the receiver sensitivity to robot motion (velocity) is clearly visible, because of the very noisy \(\Delta SD\) from first order DLL. A DLL filter order of 2 and a \(B_N\) of 0.1 Hz leads to comparable \(\Delta SD\), observed with hardware receivers.

Fig. 4
figure 4

Time differenced single differences of the GPS C1C signal of the first calibration set \(P_1\). The \(\Delta SD\) from the Septentrio receivers are presented in red and the \(\Delta SD\) of the software receiver with different settings* are depicted in grey. *(left) \(B_N\) = 0.1 Hz, FO = 1, unaided DLL. (right) \(B_N\) = 0.1 Hz, FO = 2, unaided DLL

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It should be noted, that previous studies consider an unaided DLL, so that the codephase tracking is done only by the DLL. An aided DLL uses information from the carrier tracking loops (FLL/PLL) to effectively remove the dynamic stress from the code loop. In this case, the aided DLL \(B_N\) can be as small as 0.005 Hz (Misra and Enge 2006). Figure 4 shows the differences between a second order unaided DLL (left) and a first order aided DLL (right) with a \(B_N\) of 0.05 Hz. The \(\Delta SD\) of an unaided second order DLL with a small \(B_N\) are less noisy than the \(\Delta SD\) acquired with the Septentrio receiver. The aided DLL leads to similar behaviour of software and hardware receiver, however with higher overall noise (Fig. 5).

Fig. 5
figure 5

Time differenced single differences of the GPS C1C signal of the first calibration set \(P_1\). The \(\Delta SD\) from the Septentrio receivers are presented in red and the \(\Delta SD\) of the software receiver with different settings* are depicted in grey. *(left) \(B_N\) = 0.05 Hz, FO = 2, unaided DLL. (right) \(B_N\) = 0.05 Hz, FO = 1, aided DLL

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4 Repeatability of Estimated CPC Pattern

Figures 6, 7, 8 show the CPC computation results for the hardware and the software receiver with different settings. The left side of Figs. 6, 7, 8 shows the estimated mean CPC pattern from the Novatel antenna for the GPS C1C signal, and the right side shows the absolute differences between two individual calibration sets \(P_1\) to \(P_4\) as a cumulative histogram. Here, only CPC values above 5\(^\circ \) elevation in the antenna frame are considered. Each figure shows the results when using a different set of receiver settings:

  • (1) Using Septentrio hardware receiver and default settings (Fig. 6).

    Fig. 6
    figure 6

    (left) Estimated mean pattern of the Novatel antenna observed with (1)* hardware receiver and (right) absolute differences between estimated pattern (CPC \(\leq \) 5\(^\circ \)) from two different calibration sets (\(P_1\)–\(P_4\)) as cumulative histogram. *default settings

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    Fig. 7
    figure 7

    (left) Estimated mean pattern of the Novatel antenna observed with (2)* software receiver and (right) absolute differences between estimated pattern (CPC \(\leq \) 5\(^\circ \)) from two different calibration sets (\(P_1\)–\(P_4\)) as cumulative histogram. *\(B_N\) of 0.05 Hz, FO of 1 and aided DLL

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    Fig. 8
    figure 8

    (left) Estimated mean pattern of the Novatel antenna observed with (3)* software receiver and (right) absolute differences between estimated pattern (CPC \(\,{\leq }\,5^\circ \)) from two different calibration sets (\(P_1\,{-}\,P_4\)) as cumulative histogram. *\(B_N\) of 0.05 Hz, FO of 2 and unaided DLL

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  • (2) Using Sx3 software receiver with a \(B_N\) of 0.05 Hz, FO of 1 and aided DLL (Fig. 7).

  • (3) Using Sx3 software receiver with a \(B_N\) of 0.05 Hz, FO of 2 and unaided (Fig. 8).

The mean CPC pattern estimated with (1) and (2) have a very similar behaviour, however, the repeatability with (2) is worse with 95.4% of the differences below 111.72 mm, whereas 95.4% are below 99.2 mm for (1). This can be explained by the higher overall noise in the \(\Delta SD\) calculated with (2) (Fig. 5, right). It can be assumed, that Septentrio receivers using an aided DLL. By using an unaided second order DLL with a small bandwidth (3), the repeatability of the antenna calibration sets are significantly improved (95.4% of the observations below 65.6 mm). However, the estimated pattern differs from the pattern estimated with (1). This result shows very well the trade-off between noise performance and signal dynamics. Smaller bandwidth results in a smaller noise and accordingly better repeatability of the antenna calibration, when using an unaided DLL. But, a too small bandwidth leads to a different pattern, because the signal dynamics can no longer be tracked so well. By using an aided DLL, the signal dynamics are captured by the carrier tracking loops, which leads to stable tracking performance, but with higher noise and consequently a decrease in the repeatability. It should be noted, that the set duration of 4 to 6 h are very short for CPC estimation. Previous experiments with longer calibration duration, e.g. 12–14 h, show better repeatability.

5 Conclusion

In this contribution, the impact of different receiver settings on the antenna calibration was presented. In order to improve the CPC estimation, the overall noise of the observations needs to be reduced. One opportunity is to understand and modify the receiver tracking loop settings, especially the loop filters. Therefore, an experiment was carried out during a standard calibration of a Novatel antenna. Beside of the Septentrio hardware receiver, also the Sx3 software receiver from the IFEN company was used in a zero-baseline configuration. This experiment shows, that the interaction between noise bandwidth, filter order and the choice of an aided or unaided DLL plays an important role in the data acquisition of the antenna calibration. The trade-off between a small bandwidth and consequently less overall noise and the signal dynamics, caused by the fast robot motion, must be chosen carefully. For example, a small bandwidth in an unaided DLL can increase the repeatability significantly, but the correctness of the CPC pattern can be lost. The default settings of the Septentrio hardware receivers have a good performance for the CPC estimation, however, a smaller bandwidth than the default 0.25 Hz can be chosen if an aided DLL is used.