Carrier phase bias correction for GNSS space-time array processing using time-delay data

GNSS signals are so vulnerable that they can be easily jammed by accidental or malicious interference because of the low transmission power of satellites. Space-time array processing (STAP) technology is the most effective method for anti-jamming, but undesired bias is introduced by STAP, leading to signal distortion. Traditional methods usually calibrate antennas and apply distortionless algorithms to solve this problem. However, calibration is burdensome and cannot be widely used, and most distortionless algorithms require information on signal direction. Thus, these methods are not applicable in general array receivers. We propose the time-delay carrier phase bias estimation (TD-CPBE) algorithm for STAP bias correction, which does not need antenna calibration or signal direction information. TD-CPBE locates moments of jamming alterations, estimates the STAP-introduced carrier phase bias by open-loop tracking, and corrects bias with a phase shift in the tracking procedure. The simulation results show that TD-CPBE can effectively correct the STAP carrier phase bias with a residual error of less than 3°, which is close to that of the traditional distortionless algorithm, and the open sky experiment further verifies the proposed algorithm. Since TD-CPBE requires much less prior information, it is a very effective and applicable STAP carrier phase bias correction method.


Introduction
With the development of Global Navigation Satellite Systems (GNSS) and their relative technologies, positioning precision has been enhanced to the centimeter scale. Accurate carrier phase measurement is the key factor of precise navigation, which is the foundation of carrier phase differential technology and ensures the submeter precision of positioning. However, GNSS signals are so vulnerable that they can be easily jammed by accidental or malicious interference because of the low transmission power of the satellites. Jamming is one of the most serious threats to GNSS high-precision applications, deteriorating or even disabling carrier phase measurements. There are many ways to eliminate jamming; among them, space-time array processing (STAP) technology, which digitally weights and sums the antenna array data, is the most effective approach since it can deal with both narrow-band and wideband interference (Gao et al. 2016;Li et al. 2023). Unfortunately, undesired bias is introduced in the carrier phase because of array processing, and typically, the bias is more than 40° (Marathe et al. 2015), which limits the application of array technology in highly precise receivers.
To solve the phase bias problem of STAP, researchers have proposed many methods in recent years. These methods can be classified into analog bias calibration and digital bias mitigation methods. In the analog domain, one of the main challenges in STAP bias correction is the relative directional characteristics, which are significantly influenced by the patterns of antennas. Therefore, some precise antennas have been specially designed, and the bias can be limited to 1 cm by careful calibration (De Lorenzo et al. 2012). Calibration usually needs to be precarried out in a microwave anechoic chamber or an open sky, interference-free environment (Anantharamu et al. 2012). The calibration results are stored as a lookup table for different direction signal compensation when the STAP is working (Daneshmand et al. 2014). In these calibration methods, an inertial navigation system (INS) is also often used to provide a reference position 113 Page 2 of 12 (Backén et al. 2008). However, even if analog instruments were ideal and bias-free, digital STAP processing would also distort GNSS signals (Dai et al. 2017). Thus, many algorithms have been designed to optimize the traditional STAP procedure. For example, Daneshmand (2013) utilized the periodicity of C/A code and proposed a distortionless constraint for STAP weights, which ensures that the STAP is digitally bias-free. O'Brien et al. (2011) suggested that the S curve after code correlation contains all of the STAP bias information, so a correction algorithm was applied to the S curve to mitigate bias. Chen et al. (2015) modeled the STAP as an equivalent filter and constrained its linear phase response, realizing distortionless digital processing. These digital bias mitigation algorithms are only effective when the antenna pattern and signal direction are provided, which limits their application in low-cost general array receivers.
Recently, different algorithms aimed at STAP carrier phase bias correction have been proposed and studied. These algorithms are suitable for blind null steer beamforming and do not need prior information. In addition, they are based on the assumption that the carrier phase is approximately continuous, and the algorithms reasonably correct STAP extra abnormal bias after array processing (Zhang et al. 2012). For instance, Jia et al. (2018) applied eigen decomposition to an autocorrelation matrix of input data, estimated the GNSS signal direction, and constrained the carrier phase continuity to mitigate the STAP bias. However, this algorithm only works in linear antenna arrays. In addition, these bias correction algorithms still lack verification using real data experiments.
Here, we propose a time-delay carrier phase bias estimation (TD-CPBE) algorithm for STAP bias correction. Based on the phase continuity assumption, TD-CPBE estimates and mitigates STAP bias in tracking the procedure using time-delay data. Simulations and open sky experiments are carried out to verify the proposed algorithm, and the results show that the STAP carrier phase bias can be effectively mitigated with a residual bias less than 3°. Since the antenna pattern and direction information are unnecessary, TD-CPBE is quite easy to apply in low-cost general array receivers.
First, the mathematical model of space-time array processing is introduced, and the characteristics of STAP carrier phase bias are analyzed. Then, the TD-CPBE algorithm is illustrated in detail, and TD-CPBE is simulated and compared with a traditional distortionless algorithm to verify its effectiveness. Then, an open sky experiment is carried out to further test its performance. Finally, the conclusion is presented in the last section. Figure 1 shows a model of an adaptive antenna array with N elements (Gupta et al. 2016). A n (f , , ) represents the frequency response of the n th element in the ( , ) direction, where and denote the elevation and azimuth angles, respectively. Each antenna element is backed by front-end electronics, represented by F n (f ) , which downconverts the signal to the baseband and performs the analog-to-digital conversion. After passing through the front end, the digitized signal is then filtered by an M-tap adaptive FIR filter, in which the weight can be denoted as w nm and (·)* stands for conjugate transposition. The frequency response of each FIR filter can also be expressed as W n (f ) in the frequency domain.

Mathematical mode of STAP carrier phase bias
The output data can be denoted as: where T s is the time interval between two taps, which is usually equal to 1/f s , and f s is the sampling frequency. The input data X(f) contain the desired GNSS signal S(f), the jamming signal J(f), and the noise N(f). Taking one satellite, for example, X(f) can be written as: where P is the number of jamming events. There are many algorithms to generate STAP weight w nm , and one of the most widely used methods is to apply the power inversion (PI) criterion to achieve a minimal output power of array processing (Capon et al. 1969). The N × M weight matrix calculated by the PI algorithm can be illustrated as: where R xx is the NM × NM autocorrelation matrix of the input data, c n stands for the reference tap, which can be typically set as an NM × 1 vector as (4), and α is a complex constant, which is given as: Each element of W PI corresponds to the weight w nm in (3). By the weighted sum of (1), the jamming signals can be effectively eliminated if P, the number of jamming signals, does not exceed N-1 (Jie et al. 2022).
In fact, space-time processing can be treated as a filter, which is related to the signal injecting direction ( , ) as: After STAP, the GNSS receiver demodulates and despreads the output signal, including operations such as multiplication with the local carrier and coherent integration with local code. Therefore, the correlation output (considering the signal only) can be denoted by: where A 0 is the amplitude of the signal, c is a constant, Φ(f ) is the ideal power spectrum of the C/A code, is the code phase, is the carrier phase, and ̂ and ̂ are their corresponding local estimations.
In GNSS signal processing, the receiver adjusts the local phase estimation ̂ and ̂ so that (8) has a maximum amplitude. It can be proven that if the STAP is ideal, which means that H(f , , ) = 1 , |R( , )| reaches its maximum value when ̂= and ̂= . However, when H(f , , ) ≠ 1 , the equivalent filter H introduces bias to the GNSS signals.
To further analyze the STAP bias, (7) can be expressed in the form of the amplitude response multiplied by the phase response as (9), and the phase response is further expanded using a Taylor series: where i stands for the ith Taylor coefficient. In this case, (8) can be transformed into (10) as: Equation (10) shows that the phase response of H causes STAP bias, and the zero-order coefficient 0 introduces carrier phase bias, while 1 and higher-order coefficients affect the code phase. In addition, 0 does not relate to frequency, so the carrier phase bias can be treated as a phase shift of the correlation function. Therefore, some researchers have proposed

Time-delay carrier phase bias estimation algorithm
The challenge in STAP bias correction is that it strongly relates to analog instruments and varies with signal direction. Traditional methods focus on instrument calibration and compensation and STAP digitally distortionless processing, which sacrifices the blind null-steer characteristic of the PI algorithm and cannot be applied in most low-cost array receivers without prior information support.
On the other hand, supposing the receiver works in a jamming-free environment at the beginning, the output carrier phase should be approximately stable and continuous. If several jamming signals appear, the continuity of the carrier phase would be interrupted, and STAP completely causes extra undesired bias. Based on this assumption, the TD-CPBE algorithm is designed to estimate abnormal carrier variation and correct it using the phase reversing shift method (Kalyanaraman et al. 2010). Rewrite (10) as (11): Define R 0 and R nm as a simplified correlation function that does not contain the variable as: Then, the ideal correlation function R Ref and STAP correlation function R STAP can be denoted as (14) and (15) Considering that the signal is stably tracked in a jammingfree environment, −̂≈ 0 ; thus, (14) and (15), it can be seen that ∠R N×M stands for the total carrier phase bias, including analog and digital bias.
In GNSS signal processing, a carrier phase discriminator is used to determine the carrier phase difference between the received signal and local replication: where imag[*] and real[*] stand for the imaginary and real parts of R( , ) , respectively. Suppose that before the K th correlation epoch, the receiver has not been jammed, and the output of (17) mainly contains three parts, the true phase difference −̂ , the noise error , and the dynamic error e , which can be denoted as: When the jamming is turned on at the (K + 1) th epoch, the STAP bias ∠R N×M is introduced, and the output of the discriminator turns into (19) as: The TD-CPBE algorithm is designed to estimate ∠R N×M based on three assumptions: Assumption 1: The dynamic state of the receiver does not change acutely in K + L correlation epochs.
Assumption 2: The STAP bias does not change acutely in L correlation epochs.
Assumption 3: The noise error is Gaussian, or its average is zero.
These three assumptions hold in most static or lowdynamic receivers if K and L are at the millisecond level. Therefore, the dynamic error e in Δ can be estimated using the output of the discriminator in a jamming-free interval (Xie et al. 2021): in which −̂≈ 0 is assumed when the received signal is stably tracked. After the jamming turns out at the (K + 1) th epoch, the input signal carrier phase can be expressed as the phase of K th plus the integration of the Doppler frequency as: where t c is the interval of correlation time, L is the index of the epoch, and f d is the signal Doppler frequency.
On the other hand, the number-controlled oscillator (NCO) of the tracking loop is stopped from the (K + 1) th epoch, and open-loop tracking is carried out to generate the local carrier phase so that it is not adjusted to track the biased input signal. The open-loop tracking phase can be denoted as: where f d is the estimation of the Doppler frequency at each epoch. Based on Assumption 1, each ̂E st [K + l] is approximately precise in a short interval: The STAP bias ∠R N×M can be estimated as (27), in which Assumption 2 is applied to suppose that ∠R N×M does not change in this L epoch interval, while the noise error [K + l] is averaged based on Assumption 3.
After tracking and estimation operations, the receiver returns to the (K + 1) th epoch, corrects the input signal, and retracks the data. The output correlation function after correction can be denoted as: where the phase shift item e −j∠R N×M is used to correct the STAP carrier phase bias.
Thus far, the theory of the TD-CPBE algorithm has been illustrated clearly. It is known from this deduction that the proposed algorithm is carried out after STAP and correlation, without the need for analog or signal direction information, so it is quite applicable in low-cost array receivers. However, more details should be explained to apply Fig. 2 Flow chart of TD-CPBE algorithm, which contains 9 steps and 2 judgments TD-CPBE. The procedure of the TD-CPBE algorithm is illustrated in Fig. 2 and explained in detail as follows.
Step 1: Input STAP data y(t). In this step, y(t) is the output data of STAP as (1) but expressed in the time domain.
Judgment 1: Judge whether the STAP interference cancellation ratio (ICR) exceeds Th 1 . The ICR is calculated by dividing the input signal power P in by the STAP output signal power P out as: The ICR stands for the working state of STAP, while Th 1 is set empirically. Generally, the jamming power is much higher than that of noise, and the ICR does not exceed 5 dB if the receiver works in a jamming-free environment. Therefore, if Judgment 1 is false, which means the receiver is not jammed, the procedure skips to Step 8. Otherwise, the procedure continues to Step 2.
Step 2: Correlation & obtain Δ . The receiver demodulates y(t) with the local carrier, correlates with the local code, and obtains the carrier phase difference using (17).
Judgment 2: Judge whether the correction phase ∠R N×M needs to be updated. There are many reasons for carrier phase variation, such as motion or clock fluctuation. Serious error will be introduced if the receiver incorrectly modifies the carrier phase. Therefore, TD-CPBE defines ∠w N×M as the index for STAP bias detection, which is calculated by: where arg[*] means calculating the phase of the complex number. It is known that ∠w N×M only relates to the input data and STAP, which excludes other reasons for phase variation. Then, the variation in ∠w N×M is monitored to locate the jamming changing moment by TD-CPBE using (31): in which the average of ∠w N×M from the last correction epoch P to current epoch k is calculated to smooth noise variation. Th 2 is also set empirically and is usually less than 5° if the jamming circumstance has not changed. Therefore, if Judgment 2 is false, which means ∠R N×M needs no update, the procedure skips to Step 9. Otherwise, it continues to Step 3.
Step 3: Estimate dynamic error ̂e . (23) is used to estimate ̂e based on Assumption 1, supposing that the carrier phase has been effectively corrected in the previous K epochs.
Step 4: Open-loop tracking for L epochs. TD-CPBE stops updating NCO from the (K + 1) th epoch if (31) is true, and (25) is used to generate the local carrier phase in open-loop tracking. In this case, f d of these L epochs should be estimated before applying (25). First, the Doppler frequency of the previous K epochs is used to linearly predict the Doppler frequency dop est [K + l] of the subsequent L epochs. Then, f d is calculated using (32), in which G 1 is the loop gain of the carrier phase tracking loop defined by (33): where is the damping coefficient and n is the characteristic frequency. K and L are important parameters affecting the performance and should be chosen according to the dynamics of receivers. If K and L are set too long, the estimation of the Doppler frequency and open-loop tracking may be inaccurate, but if they are too short, the noise cannot be properly smoothed. In our test, K = 100 ms and L = 20 ms are suitable in most static or low dynamic situations.
Step 5: Update the STAP bias ∠R N×M . Since the jamming circumstance has changed, the STAP carrier phase bias is consequently altered as well. Therefore, ∠R N×M is updated according to (27).
Step 6: Seek back to the (K + 1) th epoch data.
The data of open-loop tracking are stored in TD-CPBE, which starts from the (K + 1) th epoch and ends at the (K + L) th epoch. After estimating ∠R N×M , the algorithm seeks back to the jamming changing moment to correct the carrier phase bias and repeats the tracking procedure from that epoch.
Step 7: Phase correction. TD-CPBE corrects the carrier phase of y(t) by phase shifting as (34): and this ∠R N×M correction is applied to each epoch of y(t) until the next time jamming changes so that bias estimation updates.
Step 8: Reset the correction phase. If Judgment 1 is false, which means the receiver is not jammed, ∠R N×M can be reset to zero to eliminate any accumulated error.
Step 9: Correlation & update NCO. If there is no jamming signal or TD-CPBE has effectively corrected the carrier phase, closed-loop tracking can be continued, and the NCO of the tracking loop continues to update.
Thus far, the procedure of TD-CPBE has been introduced and explained in detail, and the performances of the algorithm is presented in the next section. (32)

Simulation analysis of TD-CPBE
A four-antenna circular array is simulated to verify the performance of TD-CPBE, and the structure of the array is shown in Fig. 3, in which A 1 is the center reference antenna. The distance between the antenna and A 1 is half of the carrier length λ, which is approximately 11.8 cm for the BeiDou B3 frequency.
In addition, the nonideal characteristics of analog channels are simulated. The amplitude response and phase response are shown in Fig. 4, in which the amplitude values of the four antennas fluctuate between -0.5 dB and 0.5 dB, while the phases are approximately linear.
To simulate the dynamics of the receiver, the frequency of the signal changes, starting at 50 Hz, linearly increasing at 50 Hz/s for 2 s, and then remaining stable for another 2 s. The main parameters are listed in Table 1. On the other hand, two wideband jamming signals are set to interfere with signal receiving. The main parameters of the jamming signals are listed in Table 2.
First, the variables for judgment are illustrated in Fig. 5. It can be seen from the figures that when the receiver is not jammed, the ICR and ∠w N×M are close to zero, but when jamming is applied, these two variables consequently change. In addition, the noise variation of ∠w N×M can be significantly smoothed after filtering, as shown in the bottom of Fig. 5. Therefore, the ICR and ∠w N×M are effective indices for interference change detection.
The performance of TD-CPBE is shown in Fig. 6. To make the comparison clearer, we simulate four kinds of signal processing. The first is a signal received by A 1 without jamming or STAP, which is set as a reference.   The second is a signal processed with the traditional PI-STAP algorithm without correction. The third is a signal processed with an improved minimum variance distortionless response (I-MVDR) algorithm proposed by Chen et al. (2015), which is theoretically distortionless if the bias of analog instruments is neglected. The fourth is a signal processed with the proposed TD-CPBE algorithm. In Fig. 6, the latter three simulation results subtract the reference, eliminating some public errors to show the bias performance. It can be seen from the solid blue line in Fig. 6 that PI-STAP introduces bias in GNSS signal tracking. On the one hand, the Doppler frequency of PI-STAP is erratic when the jamming circumstance changes and converges to an unbiased state in a short time (Xie et al. 2019). On the other hand, the carrier phase suffers step bias, which reaches approximately 45° in this simulation. In contrast, the I-MVDR and TD-CPBE effectively correct the STAP bias, as shown by the red triangle and yellow star dashed lines. The maximum frequency and phase bias are 0.09 Hz and 2.24° for I-MVDR and 0.06 Hz and 2.28° for TD-CPBE, respectively. More statistical results are shown in Table 3, and it can be seen that the performances of these two algorithms are similar. However, considering that TD-CPBE needs neither the information of analog instruments nor the direction information, it is much more applicable in low-cost general array receivers.

Open sky experiment of TD-CPBE
To further verify the TD-CPBE algorithm, an open sky experiment is carried out. In this test, an antenna array is installed on a car for GNSS signal collection, while jamming signals are simulated and added in a software-defined receiver (SDR). This method is often used in GNSS jamming experiments (Mosavi et al. 2017) to avoid jamming studies impacting civil GNSS receivers. The structure of the array is shown in Fig. 7, and it includes 7 antennas (the far side of the picture), a frontend converter, an A/D converter, and a lighting port output. It should be mentioned that only data from 4 antennas are used to enhance the processing efficiency; specifically, antennas 1, 2, 4, and 6 compose a circular array.
The output baseband data of the array are stored on a personal computer (PC) and processed by the SDR. The complete experiment platform is shown in Fig. 8. The testing car and its route are shown on the top of Fig. 9. The car remains static at first and then accelerates. The bottom of Fig. 9 shows the Doppler frequency of the received signal during 4 s of car motion.
For the jamming simulation, suppose that the jamming sources are far enough away from the receiver, and the direction of jamming can be treated as fixed (Daneshmand et al. 2018). Therefore, three jamming signals are generated and added to the array baseband data, whose parameters are listed in Table 4.
The key parameters of the SDR are the same as those in Table 1. Since the information of the analog instrument and signal direction are unknown, the I-MVDR algorithm is not applicable in this open sky experiment. Therefore, only the other three kinds of signal processing are analyzed, in which the PI-STAP and TD-CPBE processing results after subtracting the jamming-free tracking results are presented in Fig. 10. Figure 10 shows that STAP introduces frequency jerks and carrier phase bias, and the maximum frequency jerk reaches 3 Hz, while the variation in phase bias is more than 80°. In contrast, TD-CPBE effectively corrects the STAP bias, with a maximum frequency bias of 0.42 Hz and phase bias of 8.21°, respectively. In addition, the variation in the TD-CPBE bias is much more stable than that in the STAP bias, which fluctuates at approximately 0 Hz and 0°. Additional statistical results are listed in Table 5. It should be emphasized that the information on analog instruments and  the signal direction are not used in this open sky experiment. Therefore, TD-CPBE is a totally blind adaptive algorithm that is very useful in general array receivers.
However, the open sky experiment results in Fig. 10 and Table 5 are worse than those of the simulation in Fig. 6 and Table 3, which is mainly due to the difference in noise level caused by STAP in the real world. In the simulation test, the carrier noise ratio (CNR) is set to 45 dBHz, and it can be seen from Fig. 6 that the extra CNR loss caused by STAP is neglectable in the simulation. In contrast, in the open sky experiment, the bias fluctuates more severely in Fig. 10, even in the beginning 1 s when there is no jamming. Meanwhile, the processing of the antenna array further increases the noise level, and the bias variation after STAP becomes more intense. In this case, TD-CPBE can only maintain the bias performance when jamming changes and cannot improve the CNR loss caused by STAP. Thus, the discrepancy between the simulation and open sky test can be concluded for array processing.
There are several drawbacks of TD-CPBE that should also be mentioned. Since the algorithm corrects bias by reverse phase shifting the input data, it only corrects the phase bias at a single frequency. For the code phase bias, it is known from (10) that it contains the bias of several frequencies; therefore, TD-CPBE is invalid for code phase bias. Second, TD-CPBE can only correct bias within 180° since (17) uses the arc tangent function for phase discrimination. For biases beyond the half circle, cycle slips should be detected and corrected by other specific algorithms. Third, it is also known from the deduction that TD-CPBE is effective only if the STAP bias is stable in L epoch open-loop tracking, which means that the direction of jamming does not change sharply at this time. A 20 ms stable interval is sufficient for TD-CPBE and is not a challenge in most lowdynamic scenes.

Conclusion
STAP technology is the most effective method for GNSS anti-jamming, but undesired bias is introduced and distorts the signals. This study proposes a time-delay carrier phase bias estimation (TD-CPBE) algorithm for STAP bias correction that does not need antenna calibration or signal direction information compared to that in traditional methods.
TD-CPBE locates the moment of jamming change, maintains open-loop tracking for several integration epochs, estimates the STAP introduced carrier phase bias using time-delay data, and finally corrects it by reverse shifting the data phase after STAP. The simulations and open sky experiments show that TD-CPBE has similar performance to that of the distortionless improved MVDR algorithm. The residual phase bias of TD-CPBE is less than 3° in the simulation and less than 10° in the open sky test, in which the discrepancy is mainly because of the rising noise level caused by array processing. Since TD-CPBE does not need prior calibration and information, it is especially suitable for low-cost general array receivers. In future work, the combination of a TD-CPBE and GNSS/INS ultracoupling system can be studied, considering that INS can provide highly precise short-time measurements for open-loop tracking with the TD-CPBE algorithm.
Publisher's Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. Yuchen Xie received his Ph.D. degree in Satellite Navigation Technology at the National University of Defense Technology in 2022. He is currently a lecturer in the College of Electronic Science and Engineering, National University of Defense Technology, Changsha, Hunan, China. His current research interests include GNSS signal processing and GNSS anti-jamming. Zhe Liu received his Ph. D degree in Satellite Navigation Technology at the National University of Defense Technology in 2015. He is currently a lecturer in the College of Electronic Science and Engineering, National University of Defense Technology, Changsha, Hunan, China. His current research interests include GNSS signal processing and GNSS multipath signal mitigation technology.
Feixue Wang received his PH.D. degree in Electronic Science and Technology at the National University of Defense Technology in 1998. He is currently a professor and a Doctor Tutor at the College of Electronic Science, National University of Defense Technology. His research interests include Beidou positioning a n d G N S S e v a l u a t i o n technology.