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**Performance Evaluation of Two Kinds of Evil Correlation Function Generation Methods for Signal Deformation Monitoring**

*Xiang Wang, Xiaowei Cui, Gang Liu, Zhenyu Tian, Department of Electronic Engineering, Tsinghua University, China; Mingquan Lu, Department of Electronic Engineering, and Beijing National Research Center for Information Science and Technology, Tsinghua University, China*

**Location:** Beacon B

Peer Reviewed |

Peer Reviewed

1 INTRODUCTION

Satellite-Based Augmentation Systems (SBAS’s) have been developed to provide wide-area differential corrections and integrity messages to promote accuracy and enhance integrity for Global Navigation Satellite Systems (GNSS’s) toward safety-critical application scenarios. Signal deformations, which were considered to be caused by imperfections or malfunctions of signal generation hardware onboard satellites, are important presupposed objects of integrity monitoring. Conventionally, acquisition and tracking of GNSS signals are achieved by utilizing the good auto-correlation characteristics of Pseudo-Random Noise (PRN) codes, bringing about correlation function to be the handle of signal processing. Further as had been observed, deformed signals should always make correlation functions distorted, misleading different Early minus Late (E-L) discriminators with different configurations, thus invalidating or even worsen differential processes. In order to monitor correlation functions with potential excessive distortions, SDM algorithms based on multi-correlator techniques were developed, making judgement on basis of slopes and symmetries measured by several symmetrically deployed E-L correlator pairs. Multi-correlator-based SDM methods have been operational steadily on SBAS’s such as Wide Area Augmentation System (WAAS) in North America and European Geostationary Navigation Overlay Service (EGNOS) system in Europe.

Robust operational signal deformation monitoring (SDM) relies on well-designed SDM algorithms. However, an SDM algorithm under development could not expect to be tested with real data because of the rarity of Evil WaveForm (EWF) incidents. Therefore, simulations have been working as a main technical routine on which SDM algorithm developments rely. There are three external conditions for SDM algorithm designers to take into account. The first is receiver configuration space, standing for the requirement on ability of protection to maximum differential pseudo-range error (maxPRE) between reference and various user receivers within specified configuration space not exceeding a given Maximum Error Range Residual (MERR). The second is Threat Space of potential signal deformations, reflecting ability to address and cover various deformed signals, which is a maximum set of distortions probably generated by the objective signal. Being the key condition of an SDM algorithm design to validate the former two, the third is Threat Model of potential signal deformations, simulating style of manifestations of all the possible EWFs on code sequences or correlation functions. To sum up, an SDM algorithm under development should be tested with simulations by carrying out SDM detections and maxPRE calculations toward various anomalous correlation functions generated under given Threat Model with particular EWF parameters within properly discretized Threat Space under specification to make sure the resulted Maximum Undetected Differential Error (MUDE) from all the maxPREs of undetected EWF points not exceeding the given MERR. Therefore, generation of Evil Correlation Functions (ECFs) is the foundation of SDM algorithm design, hence the robustness of SBAS and safety of life further.

Traditionally for the mainstream multi-correlator-based SDM algorithm assessment, ECFs are directly generated according to a group of formulae derived in Doctorate Thesis of Phelts R.E., instead of deformed code signals followed by correlating processes. There might exist some equivalent or simplified processes to reduce computational burden. Additionally, this formula-based generation method was also applied by Pagot J.B. in SDM algorithm assessment for EGNOS. In order to remove effects of the aforementioned potential equivalent and simplified processes, a filter-based generation method of ECFs is proposed in this article, basing on the fact that fast development of computing and memory hardware has made the computational burden of EWF-generating and cross-correlating processes affordable. Besides, both formula- and filter-based methods will be compared qualitatively by comparing shapes of correlation functions to find potential laws of differences between them, and quantitatively by carrying out massive simulations to obtain differences in their impacts on SDM performances.

2 CONTEXTS OF THIS STUDY

2.1 EWF Modeling

Since SVN-19 event in 1993, the first observed occurrence of signal deformation, researches on modeling and mitigation of EWFs came into sight. Several early Threat Models, including ones simple as a sinusoidal wave or a delayed replica added on nominal signal, and ones complicated as Most Evil WaveForm (MEWF) model, were proposed. In 2000, International Civil Aviation Association (ICAO) adopted 2nd-Order Step (2OS) model as the standard threat model in Standards and Recommended Practices (ICAO SARPs), which categorizes the modeling of EWFs into three modes, described by:

(1) Digital Deformation Mode (TM-A): indicates the occurrence of failures on digital components of signal generating hardware. One parameter, ?, in length of chip, is used corresponding to the ratio of a lead or lag amount to a nominal chip on each falling edge of ideal code sequence.

(2) Analog Deformation Mode (TM-B): represents the existence of failures on analog components. Two parameters used are fd in Megahertz (MHz), corresponding to the damped frequency of oscillation on each transition of ideal code sequence, and ? in Meganeper per second (MNp/s), corresponding to the damping factor, respectively.

(3) Combination Mode (TM-C): means a combination of digital and analog failures, and uses all the three parameters above.

2.2 Traditional Generation Method of ECF

As was summarized in INTRODUCTION, SDM acts as conjunction between domains of signal and ranging by carrying out SDM detections and maxPRE calculations on potential ECFs that potential EWFs might manifest themselves as. This equivalence between generations of deformed signal and ECF brought much convenience to SDM algorithm design during the end of last century when capability of computing and memory hardware was primitive to perform precise EWF modeling and massive multiply accumulation operations.

A group of formulae were derived to generate anomalous correlation functions. For TM-A, equivalence of effects between lead and lag of chip falling edges was proposed and a piecewise form of lag-induced ECF was derived. Additionally, slope corrections for different PRN codes with different Peak-Flush Side-Lobes (PFSLs) are introduced by type-normalization process. While for TM-B, convolution of ideal correlation function and impulse response of 2nd-order damped oscillation was decomposed into three steps, namely a differential process of ideal correlation function, a convolution process of resulted piecewise accumulative step responses and 2nd-order impulse response, and an integral process of resulted piecewise accumulative 2nd-order impulse response. As a result, explicit expression of ECFs from TM-B was derived. Similarly, process of TM-C is a cascade of those of TM-A and TM-B. This traditional pattern of ECF generation has been proved to be effective to robust SDM algorithm design for single-frequency SBAS’s and performing well in GNSS integrity monitoring implementation.

3 A NOVEL FILTER-BASED GENERATION METHOD OF ECF

As a matter of fact, a correlation function under steady tracking is obtained from the cross-correlation between incoming signal and local nominal replica. So is a ECF from a deformed signal. Authentically, EWF simulations should be performed by firstly generating an infinite-bandwidth distorted waveform from an ideal one with a deforming filter that is given a set of EWF parameters, and then running correlating process with the generated deformed and nominal waveforms, resulting a distorted cross-correlation function. Thanks to great progress on hardware and software hence strong power on massive and parallel computing, the relevant computational burden is much more affordable than before, indicating that filtering and correlating processes should be no longer budget consuming. This procedure is believed to be what a more precise simulation should run as.

Signal generating hardware on board a GNSS satellite could be divided into digital and analog components. The logic of TM-A modeling relies on the perspective of taking the digital components as a whole, which is represented by a generalized digital-deforming filter or a function. For a ranging code suffering digital distortion, the corresponding infinite-bandwidth digital chip waveform is generated by just adding an amount of lead or lag, which is a fraction of chip length and represented by ?, on each falling edge of ideally rectangular waveform, resulting distorted digital chip waveform and cross-correlation function. For a nominal digital chip waveform, ?=0 is applied. In this part, the proposed filtered-generation method is identical to the traditional formularized-generation method. In the same way as digital components, the analog components of signal generation hardware on board a GNSS satellite are deemed as a 2nd-order filter, of which the step response is expressed as definition of TM-B. The input of this analog-deforming filter is the output of the former digital-deforming filter/function. Especially for TM-B modeling, ideal rectangular waveforms are used as this input by applying ?=0. While for TM-C modeling, TM-A-deformed waveforms are used as input of the 2nd-order analog-deforming filter.

4 SIMULATIONS AND ANALYSES

4.1 Qualitative Comparisons

Some PRNs of GPS L1 C/A signal are tested for comparisons between formularized and filtered generation methods. All the thirty-two PRNs of GPS L1 C/A signal could be divided into three classes in accordance with magnitudes of PFSLs of correlation functions. Broad class contains PRNs of 7, 15, 17, 21, and 24, with PFSLs of 63/1023. Skinny class contains PRNs of 8 and 22, with PFSLs of -65/1023. The rest twenty-five PRNs belong to Nominal class with PFSLs of -1/1023. PRNs of 15, 22, and 29 are selected for three classes respectively.

For digital distortions, effects of both generation methods are identical. While for analog distortions, it could be observed from comparisons that the asymmetries and fluctuations on correlation functions from both generation methods appear quite similar, only differentiated by a constant coefficient under either TM-B or TM-C. However, detailed differences and ratios from proposed filtering-generated correlation functions to traditional formulary-generated ones under both TM-B and TM-C within a scaled correlation-peak region show that neither differences nor ratios are constant for any one of the Broad (PRN-15) or Skinny (PRN-22) correlation functions, and only the Nominal (PRN-29) ones have near constant ratios. Preliminary judgements could be made as follows:

(1) All of the difference-curves are asymmetric, introducing discrepancies between tracking errors of two differently generated correlation peak, and probably also differential pseudo-range biases. It is indicated that correlation function generation method might affect ranging performance of a signal in simulation.

(2) All of the ratio-curves are also asymmetric and fluctuate more violently in L-halves than in E-halves generally, except for Nominal correlation functions under TM-B. This may mitigate or worsen the asymmetries of all kinds of correlation functions. Correlation coefficients between ideal and distorted correlation functions aligned to prompt are compared. For range -0.25 to 0.25 chips where detection metrics are defined, filtering-generated Broad correlation peaks have slightly smaller correlation coefficients than formulary-generated ones do, indicating worsened asymmetries, comparing to sustained and mitigated asymmetries for Nominal and Skinny correlation peaks respectively.

(3) It could also be observed that ratios for Nominal correlation peaks are generally consistent, particularly for TM-B, which indicates detection-metric values being correspondingly consistent across both generating methods. While for Broad and Skinny correlation peaks, the ratios vary significantly, hence with biases between pairs of detection-metric values.

4.2 Anticipated Quantitative Comparisons

Anticipated simulations on PRNs of 15, 22, and 29 with both formularized- and filtered-generation methods would be performed for assessment. Procedure of assessment shall be designed before, along with appropriately discretized Threat Space and receiver configuration space with accordance to ICAO specifications. Simple ratios, symmetric difference ratios, and symmetric sum ratios of in-phase correlator values of multi-correlators from range of -0.25 to 0.25 chips with intervals of 0.01 chips except Prompt to that of Prompt correlator are applied as detection metrics.

Anticipated results of above assessment might be regarded from two perspectives.

(1) For SDM of traditional single-frequency SBAS, threshold to MUDE values, i.e. MERR, is set to 6.08 meters for User Differential Ranging Error Index (UDREI) of 4. At this level, performances of both methods might be similar because maxPREs are as insensitive to detections as MERR raises.

(2) While for SDM of next generation of Dual-Frequency Multi-Constellation (DFMC) SBAS, Dual-Frequency Ranging Error (DFRE) for DFREI of 4 is 3.64 meters, and MERR should be further deflated by a factor of 2.26 to about 1.61 meters because of DF inflation effect of nominal code noise and multipath. At this level, maxPREs are much more sensitive than at the level of single-frequency SDM. Thus, performances of both methods are expected to be different, no matter whether worsen or improved.

5 CONCLUSIONS

For DFMC SBAS, which is under standardization and would be operational in recent years, SDM is progressively important in integrity monitoring because signal deformation might act as the largest source of uncertainties in ranging error after elimination of 1st-order ionospheric delay. Therefore, SDM algorithms for DFMC SBAS should be better designed and more precisely evaluated. Since computing and memory hardware for parallel or heterogeneous parallel processing has obtained quite great progress, the novel proposed filtered-generation method of ECFs is suggested to be applied in later SDM algorithm assessments for its compliance to Threat Model definitions and accuracy in deformation modeling.

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