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**A Novel DE-KFL for BOC Signal Tracking Assisted by FRFT in a Highly Dynamic Environment**

*Yiran Luo, Beijing Institute of Technology, China, and University of Calgary, Canada; Lei Zhang, Beijing Institute of Technology, China; Naser El-Sheimy, University of Calgary, Canada*

**Location:** Spyglass

This paper presents a novel Double Estimator Kalman Filter Loop (DE-KFL), which is assisted by Fractional Fourier Transform (FRFT), for tracking high-dynamic BOC signals. New BOC signals that can enhance positioning performance and spectrum compatibility are currently widely used for the modern Global Navigation Satellite System (GNSS). Bump Jump (BJ), BPSK-LIKE and Double Estimator Technique (DET) algorithms are often applied to GNSS receivers to cope with the multi-peaks of the Auto-Correlation Function (ACF) of BOC signals. The BJ technique will cost long time in detecting signal power, while the means of BPSK-LIKE will theoretically induce in a 3dB-power loss when processing baseband signals in comparison with it. DET is introduced with a more excellent performance than the former algorithms to cope with the process of tracking for various BOC signals. Since the sub-carrier tracking loop of DET has the performance of a comparatively narrow pull-in range, it is more likely to be out of lock under dynamic stress. The Partially Matched Filter (PMF) is introduced to implement the acquisition process of BOC signals in this paper. Fractional Fourier Transform (FRFT) is also used to estimate the acceleration of the signal and it will enhance the performance of the new DE-KFL when tracking BOC signals in a highly dynamic environment. Simulations show that presented algorithms dramatically remove the effect of dynamic stress caused by the carrier acceleration. The DE-KFL can effectively enhance the robustness of the tracking loop when tracking BOC signals in a highly dynamic environment.

Objectives:

The advent of modernized and new Global Navigation Satellite Systems (GNSS), including Chinese BeiDou II system, GPS modernization of the US and GALILEO satellite navigation system of the European Union (EU), has enhanced the application of BOC (Binary Offset Carrier) modulation based on positioning, navigation and timing (PNT). Bump Jump (BJ), BPSK-LIKE and Double Estimation Technique (DET) algorithms are three most effective and popular means to cope with BOC signals. There are two more channels, very early (VE) and very late (VL) channels, introduced to the tracking loop apart from early (E), prompt (P), and late (L) channels when using the BJ technique. In this case, the receiver will cost longer time in detecting the signal power of P, VE, VL. There will also be more than two vice-peaks of Auto-Correlation Function (ACF) of the high-order BOC signal, which dramatically reduce the accuracy of measurements of tracking and positioning. Based on the introduction of BPSK-LIKE technique, the means to cope with baseband BOC signals can be the same as the average means to acquire and track the baseband BPSK-modulated signals and the incoming signal with only Single Side-Band (SSB) of the BOC splitting spectrums in frequency domain will be correlated with the signal replica. However, such technique will cause more power loss and the signal with lower CNR will be more difficult to be acquired or locked. Besides, the introduction of BOC modulation will no more be viewed as the fact that is dominated in multipath mitigation and anti-jamming. Compared with BJ and BPSK-LIKE, DET is more suitable to process BOC signals. The Sub-carrier Locked Loop (SLL) in DET has a narrower pull-in range in comparison with the conventional Delay Locked Loop (DLL) since the code chip of the square-wave sub-carrier is narrower than the chip of the Pseudo Random Noise (PRN) code. Hence, the code loop of a GNSS receiver is more likely to be out of lock when using DET method to cope with BOC signals in a highly dynamic environment.

To enhance the performance of acquisition and tracking of satellite navigation signals in a highly dynamic environment, proposes to estimate the code phase error, carrier phase error and Doppler shift error based on maximum likelihood (MLE) method, by which numerically controlled oscillator (NCO) of the local signal can be replicated more accurately. However, the MLE approach is lack of computational efficiency because of the high dimensionality and nonlinearity of the resulting cost function, which means that it is difficult to be applied to a GNSS receiver. The Discrete Chirp-Fourier Transform (DCFT) is also presented to estimate the Doppler frequency shift and Doppler shift rate of the incoming signals. The simulations in the paper indicate that when coping with the incoming intermediate frequency signal (IF) in a highly dynamic environment in GNSS receivers, comparatively low power attenuation will occur even simultaneously at the environment with a low carrier-to-noise ratio (CNR) of 25dB-Hz and at the status with long coherent integration time. In this case, the peak of the signal power in frequency domain also would not attenuate significantly and high-dynamic signals can be acquired as well. However, the higher the level of Doppler frequency rate is, the larger the acceleration difference between the incoming signal and the signal replica will be. Therefore, the large acceleration of the carrier could cause a high level of acceleration estimating error. A previous paper proposes an algorithm that the Fractional Fourier Transform (FRFT) is used to estimate carrier acceleration to track high-dynamic navigation signals. Almost all sorts of the dynamic stress of navigation signals can be compensated since the acceleration could be estimated by the means of FRFT, and it leads to an excellent performance that GNSS receivers could carry out robust signal tracking under the severe environments of high dynamics.

This paper proposes a novel Double Estimator Kalman Filter Loop (DE-KFL), which is assisted by Fractional Fourier Transform (FRFT), for tracking high-dynamic BOC signals. The power of linear frequency modulated (LFM) signal could be concentrated in the fractional Fourier domain. In this case, the acceleration of the carrier can be exactly estimated by the peak value of the signal power. Hence, the fact that almost all sorts of the dynamic stress error will be compensated when tracking high-dynamic BOC signals can significantly enhance the tracking robustness of the new DE-KFL. The ranging code periods of both the data and pilot components in GPS L1C signal, containing the modulation components of BOC (1,1) and BOC (6,1), are 10230 chips, and code rate of L1C is 1.023 Mb/s. For this case, the traditional algorithms of serial-search acquisition and code-phase parallel-search acquisition cannot be introduced to cope with BOC signals any more. Other proposed and more advanced algorithms, such as XFAST, Matched Filter (MF) and Partially Matched Filter (PMF), can be applied to BOC-signal GNSS receivers. The mentioned PMF and BPSK-LIKE technique will be used at the premier stage of the presented algorithms in this paper.

Methodology:

The Doppler shift and PRN code chip number of the incoming signal can be successfully estimated in the process of PMF. Supposing that there is no jitter component in the incoming signal, the sign of the data symbol is ignored and square-wave sub-carrier is approximated as a sine function onto the carrier which makes theoretical analysis of the BOC signal more intuitive.

There is a quadratic term in the signal model. FFT process therefore fails when performing the estimation of this LFM signal. The high-dynamic stress error introduced by the acceleration will dramatically drop the availability of GNSS receiver based on baseband signal tracking. The basic process of the FRFT is to perform a rotation with angle for the two-dimensional plane in the time-frequency domain, and the original time-frequency plane will be converted to the fractional Fourier plane. Hence, the power of the LFM signal diverged in time-frequency domain would be concentrated in the new fractional Fourier domain. By this means, Doppler shift rate of the navigation signal could have access to be exactly estimated. The digital computation of FRFT is implemented to estimate the acceleration of the carrier. The acceleration which can compensate the initial code phase error and the Doppler error caused by high-dynamic stress will be applied to assist DE-KFL tracking to enhance the tracking robustness in a highly dynamic environment.

There are two stages before the process of signal tracking in this paper, PMF for acquiring the code phase and the Doppler frequency, as well as FRFT for estimating the acceleration. Then, DE-KFL will be used for BOC signal tracking in a highly dynamic environment. The DE-KFL which is optimum to process the high-dynamic BOC signal can estimate the carrier phase error, code phase error, sub-carrier phase error, Doppler angular velocity and Doppler angular velocity error on condition that the system model can be accurately constructed as well as the noise variance can be given. The state vector of signal model contains carrier phase error, code phase error, sub-carrier phase error, carrier angular frequency and carrier angular frequency rate, while the observation vector is made up of carrier phase error, code phase error and sub-carrier phase error. In the process of observation extraction, the two-quadrant-arctangent discriminator is used for the estimation of carrier phase error, and the early-late amplitude discriminator is introduced to deal with the estimations of both code phase error and sub-carrier phase error. The process noise matrix can usually be given by the empirical value. This paper only takes the effect of Doppler frequency rate random walk into consideration on state estimation for an ideal simulating situation. The noise matrix can be given by the proposed ESto model which will be modelled by noise channel.

Since the PRN code period for the BOC signal is much long, one-time FFT process cannot cope with the entire PRN code chips. Hence, it is necessary to deposit digital sampled signals of the length of the BOC PRN code period into a series of registers when performing the acquisition process. In this case, the entire chips of the BOC PRN code period can finally be completely coped with. The statement of the implementation process of the DE-KFL assisted by FRFT can be depicted as:

Step 1: Deposit the IF digital sampled signals of the length of the BOC PRN code period into registers;

Step 2: Set P of PMFs in each channel and an N-point FFT is implemented after zero-padding for the P cumulative sums, and P is not larger than N;

Step 3: Detect the power of N-point FFT outputs. If the power of FFT outputs exceeds the threshold, go to Step 4. Otherwise, go back to Step 2;

Step 4: Deliver the acquired coarse code phase and Doppler to the FRFT channel to perform the estimation process of the acceleration;

Step 5: Use estimated acceleration to compensate the coarse code phase and Doppler. Therefore, the compensated Doppler and code phase can be obtained;

Step 6: The compensated code phase, compensated Doppler and estimated acceleration are applied to update the initial state variable and the local NCO of carrier, code, and sub-carrier. Then, start the tracking process with the DE-KFL. If the tracking loop is out of lock due to some certain interference, go back to step 2 again.

Actual results:

The main aim of this paper is to verify the performance of the new DE-KFL when tracking BOC signals in a highly dynamic environment, so the CNR of incoming BOC signals is in an average level, 43dB-Hz, to maintain a high level of probability when detecting the power of BOC signals in simulations. Supposing that the Doppler shift rate is exactly estimated by FRFT and the high-dynamic stress error is completely compensated. The tracking performances of DE-KFL in various sorts of accelerations are compared in simulations.

The DE-KFL can perform robust tracking process in a row for the BOC signals with accelerations of 10g to 100g. Furthermore, the accuracy levels of state variables for DE-KFL do not dramatically vary with the increasing trend of the acceleration. Overall, the proposed algorithm can significantly remove the effect of high-dynamic stress on tracking loop. The code chip of BOC (6,1) sub-carrier is shorter than that of BOC(1,1), and the fact leads to the consequence that the high-dynamic BOC(6,1) signal is more likely to be out of lock. The BOC (6,1) signals are identical to the BOC (15,2.5) signals in terms of the modulation means, so DE-KFL assisted by FRFT can also remove the high-dynamic stress error of high-order BOC signals in a highly dynamic environment.

Conclusions:

This paper proposes a novel Double Estimator Kalman Filter Loop (DE-KFL) assisted by FRFT for tracking BOC signals in a highly dynamic environment. The acceleration estimated by the process of FRFT will be obtained at the stage of signal acquisition. Then, the estimated acceleration will assist the tracking process of DE-KFL and it can markedly enhance the availability of the GNSS receiver based on tracking the high-dynamic BOC signals. This paper verifies that the presented algorithm can dramatically remove the effect of high-dynamic stress to the tracking loop of the BOC signal and simultaneously it makes the GNSS receiver maintains a robust tracking process with the DE-KFL in a highly dynamic environment. It can also be confirmed that the DE-KFL assisted by FRFT when tracking the high-order BOC signal has as the excellent performance as the algorithm when tracking the BOC(1,1) signal does.

Significance of my work:

The presented algorithm can dramatically remove the effect of high-dynamic stress to the tracking loop of the BOC signal and it makes the GNSS receiver maintains a robust and high-accuracy tracking process with the DE-KFL in a highly dynamic environment. The presented algorithm can be used for both low-order and high-order BOC signal high-dynamic tracking loops.

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