Leslie Montloin

Laurent Azoulai

Airbus

Anaïs Martineau

Carl Milner

Christophe Macabiau

GNSS multipath failures modes analysis for airport surface operations

Processing and Navigation Research Group (SIGNAV) of ENAC in Toulouse, France. The PhD project is in collaboration with the radio navigation group of Airbus. She is currently working on multipath modeling for taxi and parking operations.

Dr. Anaïs MARTINEAU graduated in 2005 as an electronics engineer from the Ecole Nationale de l'Aviation Civile (ENAC) in Toulouse, France. Since 2005, she has been working at the signal processing lab of the ENAC where she carries out research on integrity monitoring techniques. She received her Ph.D. in 2008 from the University of Toulouse.

Dr. Carl MILNER is an Assistant Professor within the Telecom Lab at the Ecole Nationale Aviation Civile. He has a Master's degree in Mathematics from the University of Warwick, a PhD in Geomatics from Imperial College London and has completed the graduate trainee programme at the European Space Agency. His research interests include GNSS augmentation systems, integrity monitoring, air navigation and applied mathematics.

Dr. Christophe MACABIAU graduated as an electronics engineer in 1992 from the ENAC in Toulouse, France. Since 1994, he has been working on the application of satellite navigation techniques to civil aviation. He received his PhD in 1997 and has been in charge of the signal processing lab of ENAC since 2000.

ABSTRACT

Global Navigation Satellite System (GNSS) is currently used in civil aviation to provide aircraft with position and velocity estimates from en-route to Precision Approach (PA) operations. Extending the use of GNSS to automatic taxi and parking remains a challenge. Indeed, during surface operations, GNSS pseudo range measurements suffer from higher multipath errors than whilst in flight because of signal reflections from the aircraft structure, and from additional sources of multipath that are the airport surface and obstacles surrounding the airborne antenna [START_REF]Development of a GPS deterministic multipath simulator for a, efficient computation of the positioning errors[END_REF]. The standardized multipath error model [START_REF]Minimum operational performance standards for Global Positioning System / Aircrafts Based Augmentation System airborne equipment[END_REF] used by integrity monitoring algorithms during in-flight operations is not valid for surface operations. Current integrity monitoring algorithms are thus not designed to protect users from the effects of multipath during surface operations. Hence it is necessary to develop an integrity monitoring algorithm designed to properly detect multipath ranging errors and protect users from the effects of multipath during surface operations. Several steps are needed to set up such an algorithm. The first step [START_REF] Montloin | GNSS Multipath Error Model for Surface Operations[END_REF] was to model the raw code ranging errors due to multipath from:

the aircraft structure and the airport surface, the aircraft structure, the airport surface, and obstacles surrounding the airborne antenna, in both static and dynamic conditions. The second step, which is the main objective of this paper, is to define multipath faults modes that may potentially affect GNSS pseudo range measurements during surface operations. Firstly, the navigation algorithm considered for the faults modes identification is presented. The algorithm is a tight-coupling GNSS+Inertial Navigation System (INS) Kalman filter. The accuracy performances of the navigation algorithm in both static and dynamic configurations are obtained by simulations. Next, performances are compared to the accuracy requirements that are demanded for the control function. This function will be used to perform automatic taxi and parking operations. Secondly, analytical models of the horizontal position error at the Kalman filter output are provided in both static and dynamic conditions. These models are adapted to taxi and parking operations. The impact of multipath ranging errors on the horizontal position error is underlined. In the static case, the multipath ranging errors are deterministic multipath ranging biases. Ranging biases induce a deterministic horizontal position error. Parameters of the navigation algorithm influencing the multipath-induced deterministic positioning error are presented and their influences are discussed. In the dynamic case, the multipath ranging errors are modeled as the sum of a deterministic bias and a stochastic error that follows a centered Gaussian distribution. The multipath stochastic errors induce an inflation of the variance of the horizontal position error. Parameters influencing the variance of the horizontal position error are presented and their influences are discussed. Thirdly, the concept of multipath faults is defined. A practical methodology aiming to identify single and double multipath faults is proposed. The methodology requires implementing and simulating the horizontal position error models presented herein. Based on this methodology, the static single and double multipath faults for the control function during taxi operations are identified. Next, the dynamic single multipath faults for the control function during taxi and parking on the apron operations are identified.

INTRODUCTION

GNSS is currently used in civil aviation to provide aircraft with position and velocity estimates from en-route to PA operations. GNSS is also currently used during surface operations for the position awareness (also known as routing [START_REF]ICAO-Manual on advanced surface movement guidance and control systems (A-SMGCS)[END_REF]) function. This function aims to display the airport map and the routes to follow on navigation displays. The challenge for the incoming years is to extend the use of GNSSs to automatic surface operations. This will require the future airport navigation systems to combine guidance and control functions [START_REF]ICAO-Manual on advanced surface movement guidance and control systems (A-SMGCS)[END_REF]. The guidance function will provide indications to the pilot to follow the assigned route [START_REF] Guilloton | A methodology to Elaborate Aircraft Localization Requirements for Airport Navigation[END_REF]. The control function will allow the pilot to drive in all weather conditions using steering indications [START_REF] Guilloton | A methodology to Elaborate Aircraft Localization Requirements for Airport Navigation[END_REF]. Compared to the position awareness function, the guidance and control functions require high performance (meter level) on the aircraft positioning service in terms of accuracy and integrity [START_REF] Guilloton | A methodology to Elaborate Aircraft Localization Requirements for Airport Navigation[END_REF]. Integrity is a measure of the trust which can be placed in the correctness of the information provided by the total system. It includes the ability of a system to provide timely and valid warnings when the system must not be used for the intended operation [START_REF]International Civil Aviation, Aeronautical Telecommunications, Volume I Radio Navigation Aids[END_REF]. One of the main issues in extending the use of GNSSs to these stringent functions is multipath. Multipath is the reception of echo replicas of the desired signal by the GNSS airborne antenna. For in-flight operations, the structure of the aircraft itself is the dominant source of multipath error. A multipath ranging errors model has been standardized and is currently used GNSS integrity monitoring algorithms for in-flight operations [START_REF]Minimum operational performance standards for Global Positioning System / Aircrafts Based Augmentation System airborne equipment[END_REF]. However, during taxi and parking operations, additional sources of multipath errors may affect the pseudo-range measurements [START_REF]Development of a GPS deterministic multipath simulator for a, efficient computation of the positioning errors[END_REF]: the airport surface, which can be modeled in airport environments by tar or grass to represent taxiways, obstacles on the airport surface surrounding the GNSS airborne antenna such as aircrafts and buildings.

These additional sources of multipath result in two main consequences. Firstly, multipath replicas are one of the dominant contributors of error for surface operations.

Multipath errors on raw code pseudo range measurements may reach few m [START_REF] Montloin | GNSS Multipath Error Model for Surface Operations[END_REF]. This may lead to multipath errors in the horizontal positioning domain of up to few dm, which is significant with respect to the required meter level accuracy. Hence multipath may significantly degrade the accuracy performance of the GNSS-based navigation systems during airport surface operations. Secondly, the standardized multipath error model is not valid for surface operations. Current integrity monitoring algorithms are thus not designed to protect users from the effects of multipath during surface operations. Considering these issues, the main objective of our project is to propose an integrity monitoring algorithm capable to support guidance and control functions during surface operations and designed to protect users from the effects of multipath. The design of such an algorithm requires an understanding of the multipath faults modes as well as their probability of occurrence [START_REF] Van Dyke | GPS integrity failure modes and effects analysis[END_REF]. In this project, "multipath faults" are qualitatively defined using the GPS integrity fault definition provided in [START_REF] Van Dyke | GPS integrity failure modes and effects analysis[END_REF]: a multipath fault is a multipath ranging error inconsistent with the nominal error distribution due to a fault condition and which can lead to a position error larger than the maximum tolerable error for a given pair function/operation. The proposed integrity monitoring algorithm will be capable to detect multipath faults with a probability of missed detection that depends on the probabilities of occurrence of the multipath faults and on the integrity requirements.

In order to achieve the main objective presented herein, several steps are needed. The first step aimed to provide multipath ranging errors models adapted to surface operations [START_REF] Montloin | GNSS Multipath Error Model for Surface Operations[END_REF]. The second step objectives constitute the main goals of this paper that are:

to propose a GNSS-based navigation system capable to support the accuracy requirements for surface operations, to identify the multipath faults for control function during taxi and parking on the apron operations.

This paper is organized as follows:

-The first section defines the operations and functions targeted in this publication as well as the operational performance required by these operations/functions. It also provides the general architecture of the GNSS-based navigation system considered for the rest of the paper. Finally, this part introduces the main assumptions made on the GNSS measurements used in the navigation system and the main notations used in the paper. -The second section provides a brief summary of the static and dynamic multipath ranging errors models that are adapted to surface operations and developed in [START_REF] Montloin | GNSS Multipath Error Model for Surface Operations[END_REF].

Next, for each static and dynamic configuration, position errors models adapted to taxi and parking on the apron operations are proposed. These errors models are specific to the navigation algorithm presented in the first section, that is to say the Kalman filter. -The third section provides the simulation results.

Firstly, the simulations conditions are exposed. Secondly, the simulation-based methodology to assess the accuracy performance of the GNSS-based navigation system described in the first section is presented.

Simulations results are compared to the accuracy requirements proposed in [START_REF] Guilloton | A methodology to Elaborate Aircraft Localization Requirements for Airport Navigation[END_REF] for the control function during taxi and parking on the apron operations. Thirdly, the paper explains how the position error models established in the second section are used in simulations to identify the multipath faults. Multipath faults are then identified for the control function during taxi and parking on the apron operations. -Finally, the main results derived in this paper and future works are presented in the conclusion.

I. SYSTEM MODEL

This section aims to provide the general architecture of the positioning algorithm used in the paper and to introduce the notations that will be used.

This publication focuses on two types of surfaces operations: taxi and parking on the apron. The taxi on the taxiway operation begins when the aircraft exits the runway or the high speed taxiway [START_REF]The role of the Global Navigation Satellite System (GNSS) in supporting airport surface operations[END_REF]. It ends when the aircraft enters the apron area and begins the parking on the apron operation. The apron area is defined as the area intended to accommodate aircraft for purposes of loading or unloading passengers away from airport terminals [START_REF]ICAO-Manual on advanced surface movement guidance and control systems (A-SMGCS)[END_REF]. The apron operation ends when the aircraft enters the gate region and moves on taxilanes [START_REF]The role of the Global Navigation Satellite System (GNSS) in supporting airport surface operations[END_REF]. In addition, only control function is considered in this paper. The control function will allow the pilot to drive in all weather conditions using steering indications [START_REF] Guilloton | A methodology to Elaborate Aircraft Localization Requirements for Airport Navigation[END_REF].

The positioning system and the integrity monitoring system must be chosen in order to support the operational requirements in terms of accuracy and integrity for the control function during taxi and parking on the apron. These requirements are provided in Table 1. In this Table 1: Operational requirements on the aircraft positioning system for control function [START_REF] Guilloton | A methodology to Elaborate Aircraft Localization Requirements for Airport Navigation[END_REF] In order to meet the meter level accuracy and integrity operational requirements indicated in Table 1, few solutions could be envisaged. Firstly, the use of corrections broadcasted by Satellite Based Augmentation System (SBAS) satellites could help to reduce the errors affecting GNSS pseudo range measurements. However, the visibility of SBAS satellites may be degraded in airport environments. Secondly, the use of multi constellation and multi frequency Ground Based Augmentation Systems (GBASs) could also help to reduce the positioning error. Multi constellation multi frequency Aircraft Based Augmentation Systems (ABASs) based on the hybridization of GNSS measurements with external sensors is a third solution to meet the meter level accuracy and integrity operational requirements. Since this solution will most likely be available before the operation of multi constellation and multi frequency GBAS systems in airport environments, a multi sensors GNSS+INS tight coupling positioning algorithm is considered in this paper and is represented in Figure 1. In this paper, a double constellation GPS+Galileo is considered. Note that the 24 GPS and 27 Galileo satellites are considered to be static during a surface operation that lasts few tens of s up to few min. Indeed, a preliminary analysis shows that the angular variation of the satellite elevation and azimuth angles does not exceed few tenths of degrees during a time period of few min. These angular variations are neglected in this paper. Each GPS and Galileo satellite is assumed to broadcast GPS L1C and GPS L5 signals. Each Galileo satellite is assumed to broadcast Galileo E1C and Galileo E5 signals.

GPSL1C/L5 and GalileoE1/E5a raw (or unsmoothed) code iono free pseudo range measurements are used to estimate the aircraft position. 4 types of errors affect GNSS raw code pseudo range measurements that are:

-The multipath error. The raw code multipath ranging error is denoted as in Figure 1

II. POSITION ERROR MODELS

Several publications underline the impact of aircraft dynamics on the multipath ranging errors affecting the raw code pseudo range estimates [START_REF]Development of a GPS deterministic multipath simulator for a, efficient computation of the positioning errors[END_REF] [START_REF] Macabiau | Characterization of the Earth-Surface Multipath Error for Aircraft GPS Receivers[END_REF]. Hence, a previous publication proposes separate raw code multipath ranging error models for both static and dynamic configurations [START_REF] Montloin | GNSS Multipath Error Model for Surface Operations[END_REF]. Since the position error models at the Kalman filter output depends on the multipath ranging error models, separate position error models for both static and dynamic configurations are provided in this publication.

II.1. Static Case

II.1.1. Multipath ranging error models

The multipath ranging error models adapted to the static case and developed in [START_REF] Montloin | GNSS Multipath Error Model for Surface Operations[END_REF] are briefly presented below. Let's assume that the aircraft is static in the airport as represented in Figure 2.

Case (a)

Case (b) Figure 2: static aircraft in an airport Considering that the multipath sources and the satellite "j" are static, and considering that the Delay Locked Loop (DLL) is in steady state, the raw code multipath ranging error that affects the pseudo range measurement between the satellite "j" and the airborne antenna remains constant in the time domain:

Eq.1 where:  is the DLL output code multipath ranging error in steady state.

The amplitude of highly depends on the multipath sources that induce the error on the pseudo range measurement between a satellite "j" and the airborne antenna. More precisely, 2 cases can be distinguished and are illustrated in Figure 2: -Case (a): the pseudo range measurement is affected by multipath from the airport surface and the aircraft structure, -Case (b) : the pseudo range measurement is affected by multipath from the airport surface the aircraft structure and obstacles surrounding the airborne antenna.

The value of related to both cases are indicated in Table 2.

Case Value of Case (a)

( ) . It also depends on the relative orientation of the aircraft fuselage with respect to the satellite. This angle is denoted as in Figure 2 (a).

Case (b)

The values of for a wide range of elevation angles have been simulated on GPS L1C, GPS L5, Galileo E1C and Galileo E5a measurements with the deterministic multipath ranging error simulator described in [START_REF]Development of a GPS deterministic multipath simulator for a, efficient computation of the positioning errors[END_REF]. Values of are plotted in Figure 3 for dual frequency measurements GPSL1C/L5 and GalileoE1/E5a and for . Note that , and are illustrated in Figure 2.

II.1.2. Position error model

The goal of this subsection is to model the horizontal position estimate error at the Kalman filter output when the aircraft is static in the airport. The GNSS raw code pseudo range measurement error vector at time is given by:

[ ] ⏟ [ ] ⏟
Eq.2 where:  is the number of GNSS raw code pseudo range measurements used in the Kalman filter to estimate the aircraft position at time ,  is the GNSS raw code pseudo range measurement error vector due to the receiver noise ranging error, the residual satellite clock error and ephemeris errors and the tropospheric delay estimation error,  is a biases vector that represents the GNSS raw code multipath ranging errors vector.

is developed in Section II.1.1.

Note that, in Eq.2, the total ranging error is assumed to be the sum of the ranging errors due to all error contributors. Eq.2 is true if we consider that the tracking loops can be approximated by linear models. This assumption is valid in our application since the ranging errors standard deviations of the different contributors of errors are sufficiently low regarding the parameters of the tracking loops.

From APPENDIX, at time follows a Gaussian distribution characterized by a deterministic vector denoted as [ ] (

) and by a covariance matrix denoted as [ ] ( ):

( [ ] [ ])
Eq.3

where:

 [ ]
is the horizontal position estimate error covariance matrix that would have been obtained if ,

 [ ] (( )( [ ̂ ]) )
 , and are the Kalman matrices presented in APPENDIX, is the Kalman state vector and ̂ is the a posteriori state vector estimated by the Kalman filter,  [ ] ,  ( ) represents the first 2 lines of vector ,  ( ) represents the position error in the North direction at time ,  ( ) represents the position error in the east direction at time .

From Eq. 3, the multipath ranging biases vector induces a horizontal deterministic position error [ ] which is illustrated as follows. Let's assume that, during the simulation, the satellite constellation is considered as fixed. During the period of the simulation, GNSS raw code pseudo range measurements are used to estimate the static aircraft position. Indeed, preliminary simulations have shown that the number of visible satellites in Toulouse Blagnac airport, France, depends on the instant of the day and varies between 10 and 15 satellites considering a double constellation GPS+Galileo. The Kalman filter is assumed to be in steady state during the time period of the simulation. For the 13 GNSS measurements are affected by multipath from the airport surface and from the aircraft structure.

From , the GNSS signal received from satellite "2" is suddenly affected by an additional multipath from one obstacle of the airport until the end of the simulation. This results in a raw code multipath ranging bias on the pseudo range from satellite "2" denoted as . Hence, from Section II.1.1., is given by:

[ ( ) ( ) ] Eq.4 [ ( ) ( ) ]
Eq.5 From Eq.3, [ ] can be considered as the response of an equivalent filter to the raw code pseudo range biases vector . The coefficients of the equivalent filter depend on the Kalman matrices . Since GNSS satellites and aircraft are considered as static, since the expected covariance matrix of the raw code pseudo range measurement errors is constant during the simulation, and since the Kalman filter is in steady state, the coefficients of the equivalent filter remain roughly constant. Hence, the deterministic position estimate errors in the North and East directions converge to time-constant values. Moreover, the transient period of the Kalman filter, defined in this subsection as the time period between and the time epoch corresponding to the stabilization of the deterministic position errors in the time domain, lasts roughly 30s. This transient time mainly depends on the expected values of the raw code pseudo range measurement errors standard deviations that are given to the Kalman filter. In our simulation, these values are low (few dm). The Kalman filter mainly "relies on" the measurement model, the Kalman gain is thus relatively high, and the transient period is thus relatively long.

II.2. Dynamic Case

II.2.1. Multipath ranging error models

The multipath ranging error models adapted to the dynamic case and developed in [START_REF] Montloin | GNSS Multipath Error Model for Surface Operations[END_REF] are briefly presented below. Let's assume that the aircraft performs a straight line trajectory with a constant speed in the airport as represented in Figure 2 (red line).

The values of highly depends on the multipath sources that induce the error on the pseudo range measurement between a satellite "j" and the airborne antenna along the trajectory. As for the static case, 2 cases can be distinguished and are illustrated in Figure 2:

-Case (a): the pseudo range measurement is affected by multipath from the airport surface and the aircraft structure, -Case (b) : the pseudo range measurement is affected by multipath from the airport surface the aircraft structure and obstacles surrounding the airborne antenna.

The model of along the trajectory related to both cases are indicated in Table 3. presents high variations along the trajectory and is modeled by a stationary non centered Gaussian law ( ) Table 3: raw code multipath ranging errors models in dynamic configuration

Case

Case (a)

From Table 3, the raw code multipath ranging error due to multipath from the airport surface and from the aircraft structure (Case (a)) is a constant bias (

) along the trajectory. Details about the value of ( ) are provided in Section II.1.1.

Case (b)

The raw code multipath ranging error due to multipath from the airport surface, the aircraft structure and obstacle(s) (Case (b)) presents small scales (dm level) strong amplitude variations (up to m level amplitude) along the trajectory. This is due to the high spatial variations of the multipath ranging errors in the airport area where the field scattered by the illuminated façade of the obstacle(s) is concentrated [START_REF] Montloin | GNSS Multipath Error Model for Surface Operations[END_REF]. This area is called "impact zone". Hence can be considered as a stochastic error. More specifically, [START_REF] Montloin | GNSS Multipath Error Model for Surface Operations[END_REF] shows that along a straight line trajectory in an impact zone follows a stationary Gaussian distribution characterized by a mean and a standard deviation . mainly depend on: -The obstacle(s) characteristics (sizes, materials and shapes), -The satellite elevation angle ( ), -The orientation of the façade of the obstacle(s) with respect to the direction of propagation of the incoming GNSS signal ( ).

Note that , and are illustrated in Figure 2. Due to the effect of the DLL on the raw code ranging errors, the errors are time-correlated along the trajectory. In this paper, is assumed to follow a first-order Gauss-Markov process characterized by an auto-correlation time of the inverse of the DLL bandwidth, that is to say .

II.2.2. Position error model

The goal of this subsection is to model the horizontal position estimate error at the Kalman filter output when the aircraft performs a constant-speed straight line trajectory in the airport in the airport. The GNSS raw code pseudo range measurement error vector at time is given by:  [ ] can be computed by recurrence from APPENDIX.

[ ] ⏟ [ ]
Note that Eq.6 and Eq.7 have been derived assuming that the tracking loops can be approximated by linear models.

As explained in Section II.1.2, this assumption is valid in our application.

An important remark is that, in Eq.3 and Eq.7, a model of the true behavior of the position error is proposed assuming a sub-optimal Kalman filter. In other words, the filter does not contain the exact model of the measurement process. Indeed, the Kalman gain is calculated using a measurement error covariance matrix that does not contain the model of multipath errors.

From Eq. 7, the stochastic multipath ranging vector induces an inflation of the horizontal position error covariance matrix which is illustrated as follows. Let's consider an aircraft performing a straight line trajectory in an airport. During the simulation, a constant number of ranging measurements [START_REF]Integrity monitoring applied to the reception of GNSS signals in urban environments[END_REF] are used to estimate the aircraft position. The Kalman filter is assumed to be in steady state.

-For 13 GNSS measurements are affected by multipath from the airport surface and from the aircraft structure.

-From , the GNSS signal received from satellite "2" is suddenly affected by additional multipath from one obstacle of the airport until the end of the simulation. This results in a raw code multipath ranging error on the pseudo range from satellite "2" , where

III. SIMULATIONS RESULTS

In Section II, the impact of raw code multipath ranging errors on the position error has been assessed. Multipath ranging errors have been classified as follows:

-Errors due to multipath from the aircraft structure and the airport surface. These errors are called "nominal multipath errors" in the following. -Errors due to multipath from the aircraft structure, from the airport surface and from obstacles. They may affect temporally one or several pseudo range measurements, regarding the relative location of the satellites, the aircraft and the obstacles of the airport. These errors are called "abnormal multipath errors" or errors "inconsistent with the nominal error distribution" in the following.

The main goals of this Section are to assess the accuracy performance of the positioning algorithm provided in Figure 1 under nominal conditions and to identify the multipath faults modes that may occur during surface operations.

Both accuracy performance analysis and faults modes identification have been performed assuming particular models for the nominal errors affecting the inertial sensors (gyroscopes and accelerometers) measurements and the GNSS raw code pseudo range measurements. These errors models are presented in the next subsection.

III.1. Simulations conditions

III.1.1. Inertial sensors measurements errors

The main errors that affect gyroscopes and accelerometers measurements are listed as follows [START_REF]Study of the contribution of GNSS/UNS hybridization to GNSS integrity monitoring for civil aviation applications[END_REF]:

-The Gaussian gyroscope/accelerometer measurement noise , -The gyroscope/accelerometer bias , -The gyroscope/accelerometer scale factor , -The gyroscope/accelerometer misalignment . The measured absolute non-gravitational acceleration / angular velocity of the aircraft expressed in the mobile frame at time can be written as [START_REF]Hybridisation of a GPS Receiver with Low-Cost Sensors for Personal Positioning in Urban Environment[END_REF]:

( )
Eq.9

where:

 [ ]
The values of the parameters describing the errors terms are provided in Table 4. These values are typical values used in civil aviation. In Table 4, "Std" stands for standard deviation.

Error Value of the parameter describing the error

Gyroscope Accelerometer

Std: Std: 1 st order Gauss-Markov process Std: Autocorrelation time: 1500s 1 st order Gauss-Markov process Std: Autocorrelation time: 3000s

Chosen uniformly in [ ]

Chosen uniformly in

[ ] chosen uniformly in [ ] chosen uniformly in [ ]
Table 4 : Characteristics of the inertial sensors measurements errors models

III.1.2. GNSS measurements errors

Raw code multipath ranging error models are discussed in Section II. For this reason, these models are not presented in this subsection. The nominal raw code ranging errors related to the other sources of pseudo range measurement error are modelled with order Gauss-Markov processes characterized by a standard deviation and by an autocorrelation time presented in Table 5.

Source of pseudo range measurement error

Standard deviation [m]

Auto-correlation time [s]

GNSS receiver noise Derived from [START_REF] Betz | Extended Theory of Early-Late Code Tracking for a Band limited GPS Receiver[END_REF] 1 (due to a DLL bandwidth of 1Hz) Tropospheric delay estimation error Derived from Section J.5 of [START_REF]Minimum operational performance standards for Global Positioning System / Aircrafts Based Augmentation System airborne equipment[END_REF], [START_REF]Minimum operational performance specification for airborne open service Galileo satellite receiving equipment[END_REF] 1800 [START_REF]Minimum operational performance standards for Global Positioning System / Aircrafts Based Augmentation System airborne equipment[END_REF] [START_REF]Minimum operational performance specification for airborne open service Galileo satellite receiving equipment[END_REF] Satellite clock error and ephemeris error Galileo: 0.85 [START_REF]Minimum operational performance specification for airborne open service Galileo satellite receiving equipment[END_REF] GPS: 0.85 (expected value for future GPS constellation [START_REF]Integrity monitoring applied to the reception of GNSS signals in urban environments[END_REF]) 3600 [8] [12] Table 5: Characteristics of the GNSS raw code pseudo range measurements errors models Note that the standard deviation of the GNSS receiver noise ranging error depends on the C/N0 ratio of the related GNSS signal. The impact of multipath on the C/N0 ratio of each GNSS signal is left as future work. Hence, in this paper, the impact of multipath on the GNSS receiver noise error standard deviation is not taken into account.

III.2. Accuracy performance

The goal of this subsection is to obtain the accuracy performance of the positioning algorithm presented in Figure 1 under nominal conditions. The following methodology has been used. An aircraft is simulated and located in Toulouse Blagnac airport, France, during a time period of 48hr. During this time period, the satellites geometries are simulated with a time step of 15min. For each satellite geometry:

-Horizontal positioning errors [ ] are computed over few hundred of seconds thanks to the positioning error simulator presented in Figure 1. The errors models presented in Section III.1 are used in these computations. In addition, in nominal conditions, raw code pseudo range measurements are affected by multipath from the airport surface and aircraft structure described Section II.1.1. In order to eliminate large (dm level) multipath ranging errors that appear at low elevation angles (see Figure 3), a GPS and Galileo elevation mask angle of 15° has been chosen. -A large number of independent horizontal positioning errors are then extracted from the computed errors.

179490 independent horizontal positioning errors have been simulated over 48hr. Data are then used to compute the percentage of the time when | | exceeds the 95% accuracy bound provided in Table 1. The results are shown in Table 6. Note that Table 6 shows results for both static and dynamic case since the dynamic of the system influences the performance of the positioning algorithm. 6, the positioning algorithm used in this paper allows meeting the accuracy requirements for the static case and for the dynamic case with a typical speed of the aircraft in the airport of 2m/s. Figures 9 and 10 have been computed under the following conditions:

Operation

-The selected satellites geometry and selected time instant are detailed in Section III.3.2.1. -The pseudo range measurement affected by the abnormal multipath error is selected in a way so as to maximize

( | | ( )).
- The raw code ranging errors is a multipath fault when (

) is located on or outside from the blue lines plotted in Figure 11 for the control -taxi operation and in Figure 12 for the controlparking on the apron operation. Finally, the HAL for the apron operation is almost twice the HAL for the taxi operation. This explains why some pairs (such as (

)) are considered as faults for the taxi operation and not for the apron operation.

CONCLUSION AND FUTURE WORKS

In this paper, we have analyzed the impact of raw code ranging errors due to multipath from the aircraft structure, from the airport surface and from obstacles surrounding the airborne antenna on the horizontal position error in both static and dynamic configurations. Based on this analysis, we have identified which multipath ranging errors are considered as multipath faults. In the following, the obtained results are summarized and future works are proposed.

The impact of the multipath ranging errors in the horizontal positioning domain depends on the navigation algorithm used to estimate the aircraft position. Here we have considered a GNSS+INS tight coupling Kalman filter as navigation algorithm. In order to estimate the horizontal aircraft position, the Kalman filter hybridizes Galileo and GPS iono free measurements with the position, velocity and attitude angles estimated by the INS. Simulations show that this algorithm enables meeting the accuracy requirements of the control function for both taxi on the taxiway and parking on the apron operations.

After the selection of the navigation algorithm, the impact of the raw code multipath ranging errors on the horizontal position error has been analyzed. Multipath ranging errors are classified as follows. Ranging errors due to multipath from the airport surface and from the aircraft structure are considered as "nominal" errors. Ranging errors due to multipath from the airport surface, from the aircraft structure and from obstacles are considered as errors "inconsistent with the nominal error distribution", also called "abnormal" errors in this paper. In the static case, nominal and abnormal multipath ranging errors are biases. In the horizontal positioning domain, these ranging biases induce 2-dimension (North and East) deterministic horizontal position errors that converge to roughly constant values. Nominal multipath ranging biases lead to few cm deterministic positioning errors in the North and East directions. Abnormal multipath ranging biases of few m on one pseudo range measurement may occur in airport environments [START_REF] Montloin | GNSS Multipath Error Model for Surface Operations[END_REF] and may lead to few dm deterministic positioning errors in the North and East directions. Abnormal multipath ranging biases combined with nominal ranging errors may lead to horizontal positioning errors larger than the HAL for the control function that is meter level. In the dynamic case, nominal raw code multipath ranging errors remain bias errors while each abnormal multipath ranging error follows a stationary Gaussian distribution characterized by a mean and a standard deviation . In the horizontal positioning domain, are up to few m in airport environments [START_REF] Montloin | GNSS Multipath Error Model for Surface Operations[END_REF] and induces 2-dimension (North and East) deterministic horizontal position errors that may reach few dm up to 1 meter. induces an inflation of the covariance matrix of the horizontal position error. A multipath ranging error characterized by a standard deviation of few m on one pseudo range measurement may occur in airport environments [START_REF] Montloin | GNSS Multipath Error Model for Surface Operations[END_REF] and may generate an increase of the standard deviation of positioning error in the North and East directions. More precisely, the standard deviation of the positioning error under nominal conditions is roughly 30cm and may be doubled in the presence of a stochastic multipath ranging error characterized with a standard deviation of 3m. Finally, abnormal multipath errors influence both the deterministic positioning errors and variance of positioning errors. This may lead to horizontal positioning errors larger than the HAL for the control function that is meter level.

Based on the analysis of the impact of the multipath ranging errors on the horizontal position error, multipath faults have been identified. A multipath single fault has been defined as a raw code multipath ranging error that is inconsistent with the nominal distribution error and that may induce a horizontal position error larger than the HAL with a probability that is non-negligible with respect to the integrity risk. Similarly, multipath multiple faults are multiple abnormal multipath errors on multiple ranging measurements that may induce a horizontal position error larger than the HAL with a probability that is nonnegligible with respect to the integrity risk. Hence, the multipath faults identification depends on the pair function/operation. In the static case, and for the control function during taxi operations, simulations results show that:

a single multipath fault is an abnormal ranging bias whose absolute value exceeds roughly 5.5m. a double multipath fault is a pair of abnormal ranging biases in a way so as the sum of the absolute value of each bias of the pair exceeds roughly 5.5m.

Since the aircraft is not likely to stop during parking on the apron operations, single faults modes during apron operations in static case have not been identified. In the dynamic case, a single multipath faults are characterized by a mean and by a standard deviation that have been identified for the control function during taxi operations and during parking on the apron operations. Simulations results show that: if , the standard deviation of single multipath faults exceeds 2m for the taxi operation and 4.4m for the apron operation.

if , the absolute value of the mean of single multipath faults exceeds 3.2m for the taxi operation and 8.5m for the apron operation.

This paper proposes an identification of the single and double multipath faults for the static configuration. It also proposes an identification of the single multipath faults for the dynamic configuration. Future works can be organized as follows. The identification of double multipath faults for the dynamic configuration has to be performed. Next, probabilities of occurrence of multipath faults modes are required in order to define an algorithm that properly detect the multipath faults. An understanding of the environmental parameters that influence the occurrence of multipath faults along a trajectory in a given airport is necessary to derive probabilities of occurrence of multipath faults. Finally, the identification of multipath faults modes and probabilities of occurrence will represent a basis to develop an integrity monitoring capable to properly detect and exclude multipath faults.

APPENDIX

Let's assume that, during a surface operation, GNSS pseudo range measurements are used in the Kalman filter to estimate the aircraft position. At time epoch , each GNSS pseudo range measurement " ", ⟦ ⟧, is affected by: a pseudo range measurement error that is the sum of the ranging error resulting from the receiver noise, the ranging error resulting from the residual satellite clock error and from ephemeris errors, the ranging error resulting from the tropospheric delay estimation error.

if , a pseudo range measurement error that can be decomposed into:

o A deterministic bias , o A stochastic error modeled as a first-order Gauss-Markov process with an auto-correlation time and a standard deviation :

{ where:  is the sampling period of the GNSS pseudo range measurement estimates,  is the noise term normally distributed with a standard deviation of and √ .

Let's compute the deterministic error vector and the covariance of the positioning error at the Kalman filter output in the horizontal domain at time . An important remark is that, in the following, a model of the true behavior of the position error is proposed assuming a sub-optimal Kalman filter. In other words, the filter does not contain the exact model of the measurement process. Indeed, the Kalman gain is calculated using a measurement error covariance matrix that does not contain the model of multipath errors. The In order to remove the highest correlated pseudo range measurement error, that is to say the ranging error resulting from the residual satellite clock error and from ephemeris errors, the state-augmented algorithm detailed in [START_REF] Wang | The practical approaches to Kalman filtering with time-correlated measurement errors[END_REF] is implemented in the Kalman filter bloc illustrated in Figure 1.

With the state-augmented algorithm implementation, the covariance matrix of the positioning error due to is denoted as . Note that, since the GNSS satellites are considered as static during a surface operation, the standard deviations of the of pseudo range measurement errors are roughly constant during this time period. As a consequence, remains roughly constant during a surface operation:

( [( ̂ )])
A.20

Conclusion:

Finally the following result has been demonstrated:

( [ ] [ ])
A.21

where:

 [ ] is given by Equation A.10,

 [ ] [ ] ,
[ ] is given by Equation A.19.

Figure 1 :

 1 Figure 1: Architecture of the position estimate error simulator

  , -The residual satellite clock and ephemeris errors after correction by Aircraft Based Augmentation System (ABAS) standardized correction models, -The residual tropospheric error after correction of the tropospheric delay by ABAS standardized correction models, -The GNSS airborne receiver noise error. The INS estimates the position, velocity and attitude angles of the aircraft based on measurements made by the INS sensors, namely the gyroscopes and the accelerometers. INS sensors measurements are affected by errors. Both GNSS and INS sensors measurements errors induce errors in the horizontal position estimated at the Kalman filter output. Horizontal position error is denoted as [ ] in Figure 1. The next section aims to propose models of adapted to airport surface operations.

Figure 3 :

 3 Figure 3: as a function of the elevation angle,

Figure 4

 4 Figure 4 represents the evolution of the multipath bias on the raw code pseudo range measurement between satellite "2" and the antenna (black line). In this graph, the transient period of the DLL is neglected and the multipath bias on the 2 nd GNSS measurement is assumed to converge immediately to at . The effect of the DLL transient state on the deterministic position error is not detailed in this paper and is left as future work. The North (blue line) and East (red line) components of the deterministic position estimate error vector [ ] in the presence of the multipath biases vector are computed based on Eq. 3 and plotted in Figure 4 as well. Note that the Kalman matrices , and needed to compute [ ] have been obtained by simulations using the position error simulator represented in Figure 1.

Figure 4 :

 4 Figure 4: Deterministic position error in the North and East directions As depicted in Figure 4, the deterministic position error in the North and East directions present few oscillations in the time domain and converge to final time-constant values. The amplitude of the oscillations can reach 2 times the value of the final time-constant value. The shape of the position error is interpreted as follows. From Eq.3, [ ] can be considered as the response of an equivalent filter to the raw code pseudo range biases

  ranging error covariance matrix at time ,  contains the correlation times of the stochastic multipath ranging errors at time ,  , , are defined in APPENDIX,  is the stochastic part of the multipath ranging error vector at time ,  , [ ] ,

Figure 5

 5 Figure 5 represents the evolution of the multipath error standard deviation on the raw code pseudo range

Figure 5 :

 5 Figure 5: Position error standard deviations in the North and East directions Note that, as for the deterministic position errors, the position error standard deviations in both North and East directions become constant after a transient state that follows the apparition of the stochastic ranging error on the pseudo range between satellite "2" and the airborne receiver. An interpretation of this behavior is proposed in Section II.1.2.

Figure 9 :Figure 10 :

 910 Figure 9: probability of positioning failure given a Gaussian ranging error ( ) on a single pseudo range measurementcontrol during taxi

Figure 11

 11 Figure 11: pairs ( ) that may lead to -control during taxi

Figure 12 :

 12 Figure 12: pairs ( ) that may lead to -control during parking on the apron

  Let's note the state vector at time and ̂ the updated state vector estimated by the Kalman filter at time . The first line of represents the position error at time in the Norh direction. The second component of represents the position error at time in the East direction:

  Table, HAL stands for Horizontal Alert Limit.

	Operation	Accuracy		Integrity requirement
		95% bound
		requirement	HAL Integrity risk	Time to
					Alert
	Taxi	1m	4,5m	1e-9/op	1s
	Parking on	1m	7,5m	1e-7/op	1s
	the apron

Table 2

 2

	: raw code multipath ranging errors models in
		static configuration
	Case (a)
	From Table 2, the raw code multipath ranging error due to
	multipath from the airport surface and from the aircraft
	structure	(Case	(a))	is	a	bias
	(	) in	the	DLL	steady	state.
	(	) mainly depends on the satellite
	elevation angle

Table 6 :

 6 Accuracy performance for the static casecontrol functionFrom Table

	% of the time |	| is below the
	95% accuracy bound
	Static aircraft	Dynamic aircraft

  state vector at time is given by the state propagation  ̂ is the a priori state vector estimated by the Kalman filter at time , and is obtained in the Kalman filter as:Coming back to the computation of the covariance of ̂ , we have from Equation A.11:

		(	)	(	)(		)			 Finally we get: [ ] [	] [	]
											[	]	[	]	A.19
												[	]
		[	̂ ]	[(	̂			)	]	A.12	̂ Equations A.2, A.3, A.4, A.5 in A.1 lead to: ̂ where:   [ ]	A.5
	Assuming that the noise process errors vectors and , and the error in the estimation of , the measurements the state vector ̂ are independent	 	̂	( [	] [ ]	)( )	( (	̂ ) )	A.6
	each other, we get:
		[	̂ ]	[(	̂		)]	A.13	Esperance of 	[	̂ :	]
					[	]					From Equation A.6:
	Let's compute	[	]:					[ Let's compute ̂ ] (	[(	)( ̂	[	̂ ] )]:	A.7
												[	])	[	]
					[	]				A.14
		[(				)			]		Assuming that the process noise vector is unbiased and remarking that [ ] since the [ ]
	For		can be written as follows:			elements of	are unbiased, Equation A.7
											becomes:
		[			]	[	]	[	]	A.15		[	̂ ] (	)(	A.8
		⏟				⏟		⏟				[	̂ ])
											where:
	Hence:									 [	̂ ]
	[ equation: where:  is the state transition matrix from epoch ] [( ) ( )] Since is independent of and  is the process noise vector at time . equivalent to:	A.16 A.2 , A.16 is to ,	To conclude we get: [ [ ] ((	]	)(	[	̂ ] [	̂ ])	A.9 A.10
	̂ The updated state vector is obtained in the Kalman filter as ̂ ( ̂ ) A.3 ] follows: ] [ ] [ [ ] [ ] [ A.17	where:  [	̂ ]	)
	where: where:
	 	is the Kalman gain at time ,					Covariance of	̂ :
		is the design matrix at time ,
	 is the measurement vector at time , and is given by Equation A.17 is equivalent to:	It can be demonstrated by recurrence that:
		the observation model: [ ] [	]		[ ]		A.17	̂	(	̂	)	A.11
	where: [		[	] ] can be computed by recurrence. A.4	where: 	̂	corresponds to the error in the
	 Indeed, using Equations A.15 and A.11 we get: [ ]		would have been , estimation of the state vector if the measurement vector
				[		]				A.18
		[			]			[	]

Leslie MONTLOIN has a Master degree in electronic from the ENAC (French

III.3. Multipath faults modes identification

III.3.1. Multipath faults definition

In the introductive part, a definition of the multipath faults is proposed. More specifically, it defines a multipath fault as a multipath ranging error inconsistent with the nominal error distribution that can lead to a positioning error larger than the HAL for a given function/operation. Following this definition, any multipath error due to signal reflection from the airport surface, the aircraft structure and obstacle(s) would be considered as fault. Indeed, any multipath error inconsistent with the nominal conditions and combined with nominal errors can potentially lead to a positioning error larger than the HAL with a probability that may be extremely small compared to the integrity risk. In this subsection, a criterion aiming to identify which multipath ranging errors are considered as faults is established. The integrity risk provided in Table 1 is the probability of providing a position that is out of tolerance without warning the user within the time-to-alert [START_REF]International Civil Aviation, Aeronautical Telecommunications, Volume I Radio Navigation Aids[END_REF]:

Eq.10

A positioning failure occurs when | | and a Misleading Information (MI) occurs when:

)) [START_REF]Minimum operational performance standards for Global Positioning System / Aircrafts Based Augmentation System airborne equipment[END_REF]. Hence, Eq.10 becomes:

Eq.11

where:  ( ) is the probability of MI knowing that the system is under nominal conditions,  ( ) is the probability that the system is under nominal conditions [/op],

 (

) Note that, in Eq.11, the probability of MI due to 3 or more simultaneous multipath ranging errors inconsistent with the nominal error distribution is neglected. Indeed, a preliminary analysis based on simulations shows that, in Toulouse Blagnac airport, this probability is not significant with respect to the allowed integrity risks. However, further investigations are needed to validate this remark. In this document, only single fault modes and double faults modes are considered and are defined below. Let's define a single multipath fault as follows. Let's denote a raw code multipath ranging error inconsistent with the nominal error distribution.

is a single multipath fault if the probability of MI due to the combination of to nominal errors is non-negligible with respect to the total allowed integrity risk. In the following, we consider that the probability of MI due to a particular condition is non-negligible with respect to the integrity risk if the probability of MI is equal to or above 10% of the integrity risk. Based on these definitions, we get: is a fault if may lead to:

Eq.12 Since ( ) is not available at this stage of the project, let's state the following conservative assumption:

Based on this previous assumption and considering no detection algorithms to detect the positioning failures, we get: is a fault if can lead to:

Eq.14

A similar methodology leads to:

Eq.15

III.3.2.1. Static case

The raw code ranging errors inconsistent with the nominal error distribution are errors due to multipath from the airport surface, the aircraft structure and obstacle(s). These errors are constant biases in the static case and are denoted as . From Eq.14, in static case, a multipath single ranging fault is a bias that may lead to:

Eq.16

Multipath double ranging faults are biases and that may lead to:

Eq.17 ). Results are shown in Figures 6 and7. The probabilities are provided for control function during taxi. Indeed, in this paper, we do not consider the case of a stop during the parking on the apron operation. Even if the aircraft can stop during apron operations, this situation is unlikely to happen and is not considered here. In these computations, ranging measurements that are not affected by abnormal multipath ranging errors are affected by nominal biases due to multipath from the airport surface and from the aircraft structure (see Section II.1.). In order to reject large nominal biases that appear at low elevation angles, a GPS and Galileo elevation mask angle of 15° has been chosen. In addition, all raw code pseudo range measurements are affected by nominal stochastic errors detailed in Section III.1.2.

is modeled by a 2 dimension Gaussian law proposed by Eq.3.

The covariance matrix and the Kalman matrices are evaluated by simulations thanks to the simulator presented in Figure 1. The deterministic position error vector [ ] is computed based on Eq.3.

and are finally computed from the technique detailed in [START_REF]Integrity prediction and monitoring of navigation systems[END_REF].

Note that the probabilities of positioning failure presented in Figure 6 have been computed under specific conditions. These conditions have been chosen in order to maximize the value of and and thus constitutes a worst case: -The selected satellites geometry is the geometry that maximizes the norm of the horizontal deterministic position error vector ) is located on or outside from the blue quadrilateral plotted in Figure 8. Note that the pseudo range measurements affected by the biases pair are not the same for all pairs. This creates small discontinuities on the blue line shown in Figure 8. ) that may lead to control function during taxi

III.3.2.2. Dynamic case

The raw code ranging errors inconsistent with the nominal error distribution are errors due to multipath from the airport surface, the aircraft structure and obstacle(s). These errors are stochastic errors in the dynamic case and follow a Gaussian distribution characterized by a mean and a standard deviation . From Eq.14, multipath single fault in the dynamic case is a multipath ranging error characterized by a mean and a standard deviation that may lead to:

Eq.18

Note that the identification of double multipath ranging faults for the dynamic case is not presented in this paper and is left as future work. is computed and plotted in Figure 9 for the controltaxi operation and in Figure 10 for the control-parking on the apron operation. In order to compute , the same methodology as the one presented in Section III.3.2.2 is used, knowing that is modeled by a 2 dimension Gaussian law proposed by Eq.7.

The probabilities of positioning failure presented in


Antoine Blais

Dr Christophe Macabiau

Olivier Julien

Matched Quantization and Band Separation in a Direct Sampling Dual Band GNSS Receiver for Civil Aviation

This paper studies the application of the Direct Sam-pling technique to GNSS receivers dedicated to Civil Aviation usage. After a description of the specific spectral environment to be withstood by such receivers, two previously proposed architectures are reviewed. In the first one the E5a/L5 band and the E1/L1 band are sampled by a unique ADC, providing full coherency between bands. In the second architecture each band is sampled separately. The minimum sampling frequencies are also recalled for each architecture. The quantization process is then analyzed and the required number of bits assessed in each case, taking into account the full range of the input signal, possibly tainted by strong interference. The high values found motivate the study of two low complexity non-linear functions dedicated to dynamic compression at the output of the ADCs, in order to decrease the computation workload of the following signal processing tasks. Finally the filters needed to separate the E5a/L5 and E1/L1 bands in the the first coherent sampling architecture are specified and the induced computation workload is estimated.

Introduction

Due to software's superior adaptability over hardware, software is being used increasingly often in new receiver designs, and especially in GNSS receivers. Each increase in computing capacity or advance in Analog to Digital Converter (ADC) technology brings the ADC closer to the antenna. In fact, analog correlators now no longer exist. Next to disappear will undoubtedly be Intermediate Frequency (IF) translation stages as sampling frequencies high enough to allow RF Direct Sampling (DS) are already available on the market. Finally the analog Automatic Gain Control (AGC) will give way to a digital one in light of the upcoming availability of a sufficient number of quantification bits required to linearly quantize the full range of input signal. The complete Software Defined Radio (SDR) seems to be in view: an antenna, an ADC and a processor. However, even if technology rapidly keeps its promises, and permits such a design for mass market GNSS receivers for which extreme operational robustness is not necessarily a key point, additional difficulties are expected for more demanding applications. In particular, for Civil Aviation purposes, GNSS receivers must meet the stringent requirements found in standards, for instance, the Galileo (EUROCAE, 2010) and GPS (RTCA, 2009), (RTCA, 2008b) or (RTCA, 2006) Minimum Operational Performance Specifications (MOPS) documents. Among the most severe requirements, are the interference levels the receiver must withstand without degradation of its operational capability. As an illustration, it is noted that at the input of the Civil Aviation receiver the dynamic range between useful signals and Carrier Wave (CW) interference can reach 50 dB for the L1/E1 band. For the L5/E5a band, the situation is no better because this dynamic range, with the cumulation of in-band Distance Measuring Equipment (DME) ground emissions, can be as high as 30 dB. Taking these requirements as design specifications, we proposed in previous papers (Bla, 2011) and (Bla, 2012) two different SDR DS GNSS receiver architectures for Civil Aviation, using both L5/E5 and L1/E1 bands. We targeted a final objective of "neither AGC nor IF". The first solution has one global channel leading to one ADC digitizing the input signal to provide as output a signal including both L5/E5 and L1/E1 bands very close to each other. The second solution has one channel per band each ending with an ADC. The aim of this article is to extend the study of DS SDR architectures for Civil Aviation GNSS receivers. More specifically we address two points. The first relates to the high binary throughput, which results from the digitization process and is necessary to reflect all interference levels. We propose methods to minimize this throughput in order to lower the workload of signal processing tasks, preserving the signal. The second point we study is specific to the first proposed architecture. It consists in determining a process to digitally separate L5/E5a and L1/E1 bands, subsampled in one block with the lowest sampling frequency and so closely aliased, whilst keeping out-of-band interference at a low power. In the first part of this paper we present the context of Civil Aviation GNSS receivers, focusing on requirements regarding robustness against interference. We display interference masks at the antenna port specified in the MOPS. They define the maximal power of the interfering signals below which all the minimum performance requirements must be met. Taking into account the minimum required selectivity of the on-board active antenna we are then able to deduce the second level interference masks, at the receiver input.

In a second section we review the two Direct Sampling architectures we previously elaborated from these second level masks, identifying the essential minimal RF hardware element and recalling the minimum values of sampling frequencies we calculated. The third part is dedicated to the calculation of the number of quantification bits required to linearly quantize the input signal over the range defined by the second level masks. We then present in a fourth section methods to minimize the bit rate immediately after the ADC. On average we would expect good data economy because most of the time the interference threat will be absent. In a fifth and last section we detail the digital separation process of L5/E5a and L1/E1 bands in the architecture where they are sampled together. That is we have to design sharp digital filters which not only select each band but also protect it from possible interference present in the other band.

GNSS Receivers for Civil Aviation

The interference threat a Civil Aviation GNSS receiver can face has been quantified by standardization committees and is summarized in three documents, (RTCA, 2008a) and (RTCA, 2004) for respectively the L1 and L5 GPS signals, and (EUROCAE, 2010) for both E1 and E5 Open Service Galileo signals. Considering that, from the spectral standpoint, E1 band equals L1 band and E5 band includes L5 band and that the MOPS for Airborne Open Service Galileo Satellite Receiving Equipment document (EUROCAE, 2010) is the only one which considers a dual band receiver in a unified way, (EUROCAE, 2010) will be used as a single reference in the rest of this paper. Anyway the levels of interference described in (RTCA, 2008a), (RTCA, 2004) and (EUROCAE, 2010) are comparable.

The requirements about robustness against interferences found in (EUROCAE, 2010) are defined at receiver antenna port. This port, as well as the receiver input are defined graphically in figure 1. These requirements can be represented, as proposed in figure 2, as masks which define the maximal power of the interfering signals below which all the minimum performance required for the receiver shall be achieved. The high dynamic range, from +30 dBm to -118 dBm, the receiver has to sustain can immediately be noted.

Interference Masks at Antenna Port

The receiver useful signal bands can also implicitly be defined from the figure 2 as the frequency ranges where the mask is at its lower values, that is B E5a = [1166.45, 1186.45]MHz for E5a/L5 and B E1 = [1565.42, 1585.42]MHz for E1/L1.

Antenna Selectivity

Standards also specify the requirements for active antenna to be used on board. The corresponding required minimum preamplifier selectivity in both E1 and E5 bands are represented in figure 3 extracted from (EU-ROCAE, 2010).

CW Interference Masks at Receiver Input

Using the minimum preamplifier selectivity curves it is possible to deduce the maximum interference levels at the receiver input as drawn in figure 4 if input interference is at the mask level. Narrow Band (NB) interference mask is not represented because, as it will be shown later on, Carrier Wave mask has higher power levels at frequencies of interest. Regarding Pulsed interferences with power above CW mask (which is, out- of-band, around 5 dB under the compression point of the preamplifier also specified in (EUROCAE, 2010)), they lead to saturation with unpredictable spectral effects. So we also propose to not consider their spectral content in the design of the RF Front-End and that is why they are not on the drawing as well.

3 Proposed Direct Sampling Architectures Direct Sampling architectures for multi-band GNSS receivers have already been studied in some other publications as (Psi, 2003), [START_REF] Alonso | A direct sampling digital receiver for multiple gnss signals[END_REF] and [START_REF] Parada | Design of a gps and galileo multi-frequency front-end[END_REF], but our approach includes the interference threat and goes further into hardware simplification by removing the need of AGC. That is, ideally, the structure of this kind of receiver could be as simple as the one represented in figure 5. Sampling directly the spectral content represented in figure 4 requires a sampling frequency greater or equal to the Shannon's Frequency, that is at least 2 times 2GHz here i.e 4GS/s, which is huge in comparison to the total bandwidth of the useful signals, 20MHz for each bands i.e. a total of 40MHz.

That is why undersampling was the chosen technique in (Bla, 2011) and(Bla, 2012) Defined Radio GNSS receiver. In the first architecture, called Coherent Sampling, the two bands E5/L5 and E1/L1 are sampled by the same unique ADC, by opposition to the second architecture where each band is sampled by its own ADC.

Coherent Sampling

In this first architecture, sketched in figure 6, both bands are digitized at the same time because only one ADC is used. This provides a perfect coherency between the two, as long as the differences in their different RF paths are known and compensated. In particular the sampling jitter is the same on each band. Some extra filters appear here because, as it is explained in (Bla, 2011), due to the interference threat, it is not possible to undersample the signal directly at the receiver input. Otherwise, aliasing due to the sampling operation could provoke the crushing of a useful band by a more powerful part of the interference mask. The task of the extra filters is then to attenuate the outband mask under the in-band mask so that out-band aliasing in-band is safe.

A representation of the magnitude of the ideal transfer functions of these extra filters is given in figure 7. They were calculated so that in-band aliasing of an out-band threat is attenuated to a level of 10dB less than the minimum in-band mask, -118dBm in the E1/L1 band.

Applying these ideal transfer functions to the CW interference mask 4 at the receiver input gives the new maximum spectral content to be considered at the input of the ADC. It is drawn in figure 8. It is then possible to calculate the minimum sampling frequency which allows the aliasing of the useful bands without overlapping. If the ideal transfer functions of the extra filters could not be reached for some reason, this minimum sampling frequency was also calculated for a supplementary transition bandwidth B around each side of the useful bands. The results are presented in figure 9. Two bound values are of interest. The first is the minimum minimorum sampling frequency, 88.079MS/s, which is approximately equal to twice the total bandwidth of the useful signals. The second is the maximum supplementary transition bandwidth, 28MHz, above which there is no sampling frequency which allows to undersample the signals.

Separate Sampling

The second proposed architecture does not provide coherency, but in return relaxes the need to perfectly compensate for the different RF paths. It is represented in figure 10. Again some extra filters are needed for the same reason than with Coherent Sampling. But the required minimum selectivity is different because, as each band is sampled separately, it does not matter if during the sampling of the E5a/L5 band the E1/L1 band A representation of the magnitude of the ideal transfer functions of these extra filters is given in figure 11. They were calculated, for each band, so that in-band aliasing of an out-band threat is attenuated to a level of 10 dB less than the minimum in-band mask, -103dBm in the E5a/L5 band and -118dBm in the E1/L1 band.

Applying these ideal transfer functions to the CW interference mask 4 at the receiver input gives the new maximum spectral content to be considered at the input of the ADCs. It is drawn in figure 12.

Again it is then possible to calculate the minimum sampling frequency, but for each band separately here, which allows the aliasing of the useful band without overlapping. If the ideal transfer functions of the extra filters could not be reached for some reason, this minimum sampling frequency was also calculated for a supplementary transition bandwidth B around each side of the useful band. The results are presented in figure 13. Only one bound value is of interest in this case: the minimum minimorum sampling frequency, 40.219MS/s for the E5a/L5 band and 40.137MS/s for the E1/L1

Quantization

After sampling, quantization is the second operation to address in order to completely characterize the digitization process which happens in a SDR GNSS receiver.

To scale the quantization stage of the ADC(s), two reference amplitude levels must be considered.

Low Reference Amplitude Level

This level sets the minimum number of bits k required to correctly digitize the minimum system equivalent thermal noise, in which the useful signals are completely buried. Usually k is between 1 and 5.

(EUROCAE, 2010) specifies the interference masks, but also assesses the system noise through the equivalent temperature of the noise at the antenna port, T sky = 100K and the actual cable temperature, As the proposed architectures aim at being aircraft installation independent, they must cope with the full range of preamplifier gain and cable losses. It means that k must be calculated for the lowest noise level, corresponding to N0 min . Under the classical assumption that this system noise can be modeled as a gaussian random process, the pertinent value is not the Power Spectrum Density (PSD) value N0 but its standard deviation σ ∈ [σ min , σ max ]. For the Coherent Sampling architecture σ 2 = N0 × (B E1 + B E5a ) = 2N0 × B E1 at the input of the unique ADC, whereas σ 2 = N0 × B E1 = N0 × B E5a ) at the input of each ADC for the Separate Sampling architecture.

Writing ∆ the quantifier step size and considering that the (useful signals +) maximum minimum noise amplitude is 3σ min (this value is proposed as the dimensioning element because the cumulative probability for the amplitude to lie in [-3σ min , +3σ min ] is above 99%), then we have the relation

2 k × ∆ ≥ 6σ min

High Reference Amplitude Level

This level sets the full scale of the ADC(s). It corresponds to the total number of bits N of the ADC(s). This high level is obviously a function of the interference mask at the input of the ADC(s), but also of the noise level as it is always present. Again, the proposed architectures aim at being aircraft installation independent. It means that this time this high reference level must be calculated with the maximum active antenna gain G max = 32.5dB and the minimum cable losses L min = -3dB to cover the maximum maximorum level of the signal at the input of the ADC(s). At this point we make the assumption that the CW aggression is limited to one interference at a time per band, as the tests imposed in (EUROCAE, 2010) suggest it. It means that for Coherent Sampling the maximum interference level A M to be considered is the sum of the maximum amplitudes found in the mask for E5a/L5 on the one hand and for E1/L1 on the other, whereas for Separate Sampling it is either the maximum amplitude A M 5a found in the mask for E5a/L5 for the corresponding ADC or the maximum amplitude A M 1 found in the mask for E1/L1 for the other ADC.

0 2 N-1 -2 N-1 2 k-1 -2 k
The high reference level is then the sum of the maximum interference level A M , A M 5a or A M 1 , multiplied by G max and L min , and of the highest noise level, corresponding to N0 max .

Writing A max the generic value for A M , A M 1 or A M 5a and considering again that the maximum maximum noise amplitude is 3σ max , this high reference level equals √ G max L min × A max + 3σ max . A graphical representation of the dimensioning values k and N can be found in figure 14.

The following relation can then be established

2 N × ∆ ≥ 2 G max L min × A max + 6σ max
A simple algebraic division gives the result

N -k ≥ log 2 ( √ G max L min × A max + 3σ max 3σ min ) (1)
Thus, setting k, N can be calculated as a function of A max only.

As it is detailed hereafter, this level A max is quite dependent of the real transfer functions of the extra filters required in both proposed architectures.

Quantization with ideal filters 4.3.1 Coherent Sampling

If the extra filters used before digitization meet the minimum selectivity drawn in figure 7 then the interference mask to be considered at the input of the ADC is the one represented in figure 8. It can be read that for E5a/L5 band the maximum threat is the composite ground DME signal maximum peak power at the European hotspot, -60dBm, and that it is the floor of its CW interference mask for the E1/L1 band, -118dBm.

It can be calculated then that Nk > 7.5: N = 9, 10, 11, 12, 13 for k = 1, 2, 3, 4, 5.

Separate Sampling

If the extra filters used before digitization meet the minimum selectivity drawn in figure 11 then the interference mask to be considered at the input of the ADCs is the one represented in figure 12. It can be read that A M 5a corresponds also to the composite ground DME signal maximum peak power at the European hotspot, -60dBm and that A M 1 is equal to the floor of the interference mask for the E1/L1 band, -118dBm too.

This gives

• on E5a/L5 Nk > 2.7: N = 4, 5, 6, 7, 8 for k = 1, 2, 3, 4, 5.

• on E1/L1 Nk > 8.0: N = 9, 10, 11, 12, 13 for k = 1, 2, 3, 4, 5.

Quantization after sub-optimal filters

If the extra filters are not as selective as required, they can pass the interference threat up to a very high level, depending of the width of their actual transition bandwidth.

Coherent Sampling

At the input of the ADC the situation can be as deteriorated as the one presented in figure 15, which corresponds to the maximum transition bandwidth B = 28MHz which allows coherent undersampling. In this worse situation, for the E5a/L5 band the maximum threat is a CW interference which can reach -30dBm at 1197.45MHz and for the E1/L1 band it is also a CW interference which can reach -57.26dBm at 1613.42MHz this time.

The new total number of bits N' of the ADC is then link to k by this relation : Nk > 12.5: N' = 18 for k = 5. At most 5 bits more than with ideal filters are needed.

Separate Sampling

This time the situation at the input of the ADCs can be as bad as the one presented in figure 16. It is to be noted that here a maximum transition bandwidth which allows coherent undersampling also exists, but is greater than 50MHz and so is not represented as undoubtedly the actual analog filters will exhibit a lower one. In addition we make the assumption that these analog filters will be selective enough so that we can consider the maximum threat for the E1/L1 band is a CW interference which can reach -40.5dBm at 1626.5MHz (it corresponds to a transition bandwidth greater than 40MHz). For the E5a/L5 band the maximum threat is as previously a CW interference which can reach -30dBm at 1197.45MHz. In these condi-tions, the new total number of bits N' of each ADC is given by the relation

• for the E5a/L5 band, Nk > 13.4: N' = 19 for k = 5. At most 6 bits more than with ideal filters are needed.

• for the E1/L1 band, Nk > 11.7: N' = 17 for k = 5. At most 9 bits more than with ideal filters are needed.

Binary Throughput Reduction

After digitization, the following process should be a filtering operation dedicated to the mitigation of the interference threat. As examples in the Separate Sampling case, the minimum attenuation which should be provided by these filters, versus frequency, is given in figure 17 for the E5a/L5 band and in figure 18 for the E1/L1 band.

The attenuation is calculated from figure 16 for each band respectively, so that in-band aliasing of an out-band threat is attenuated to the minimum inband mask level, -103dBm in the E5a/L5 band and -118dBm in the E1/L1 band.

If Finite Impulse Response (FIR) filters are to be used, figure 19 for the E5a/L5 band and figure 21 for the E1/L1 band give an estimation of the minimum order of the filter vs transition bandwidth. Figure 20 for the E5a/L5 band and figure 22 for the E1/L1 band propose an estimation of the induced calculation workload, versus transition bandwidth also.

The calculation workload is estimated here by the simple product of the filter order M and of the minimum sampling frequency, which is a function of the transition bandwidth as presented in section 3. So this estimated workload does not take into account the various optimization techniques which could be used in the actual implementation of these FIR filters. This workload can reach 2630MMAC/s for the E5a/L5 band and 4465MMAC/s for the E1/L1 band. The MAC unit corresponds to the computation of the product of two numbers and the addition of the result to an accumulator. This unit is used to quantify the performance of signal processors as for example the Analog Devices TigerSHARC ADSP-TS201S Digital Signal Processor (DSP) [START_REF] Ad | Analog devices tigersharc adsp-ts201s data sheet[END_REF], which can reach 4.8GMAC/s.

The following formula, from [START_REF] Herrmann | Practical design rules for optimum finite impulse response low-pass digital filters[END_REF], was used to estimate the filter order f p is the passband edge frequency, f s the stopband edge frequency, ∆f = (f s -f p ) is the transition bandwidth, δ p the passband ripple (10 log 10 (δ p ) = -10dB in the rest of this paper) and δ s is the stopband ripple.

M = D ∞ (δ p , δ s ) -f (δ p , δ s ) • (∆f ) 2 ∆f +
Considering these workload values, if we multiply them by the highest number of quantization bits, for example N = 17 (k = 5) for the E1/L1 band, this workload can then exceed 75Gbit/s which is really important. Therefore the question of decreasing the number bits before the filtering operation arises.

Three methods were envisaged

• Estimation of the actual antenna gain G and of the actual cable losses L once the receiver is put in the aircraft. As (1) shows it, the lower the product G × L the lower N. For G min and L max , with ideal analog filters we can throw away 3 MSBs in Coherent Sampling and 2 MSBs on each channel in Separate Sampling. With sub-optimal filters only 1 MSB can be thrown away, for Coherent or Separate Sampling. As it is not a general solution it was not more studied.

• Digital AGC, which can keep the number of bits to the minimum needed to correctly represent the signal, as a function of the signal power at the output of the ADC(s). This solution is well known and established so it was not studied here also.

• Dynamic Reduction with non-linear functions. This is the subject of the rest of this part about binary throughput reduction.

The main idea is to compress the dynamic of the input signal (which is supposed tainted by at most one CW interference) as much as possible so that a maximum number of the MSBs, useless after this operation, can be discarded. Of course the useful signal must be preserved as much as possible, especially when there is no interference. It means that linearity must be preserved as much as possible over the full range of the useful signal. And when the interference is present, the compression of the dynamic must not produce other interferences with a level higher than the mask.

It must be noted that, as our study is limited to one CW interference, it only applies to the Separate Sampling architecture.

Two non-linear functions with low computation complexity were considered.

The Linear-then-Log Function

This function F ll is represented in figure 23. Its main advantage is that it is fully linear before the corner value X0: if X0 is set greater than 3σ max , the dynamic compression operation is completely transparent for the useful signal in the nominal situation, i.e. when there is no interference.

The effect of this function F ll on a CW parametrized by its mean power P , its frequency f 0 and its initial phase φ 0 (set to 0 here without loss of generality) was evaluated through the Fourier coefficients of the output signal. Indeed, as the input CW interfering signal x(t) = sqrt2P sin(2πf 0 t) is periodic, the output y(t) = F ll (x(t)) is also periodic with the same period and can then be decomposed in a Fourier series

y(t) = +∞ n=-∞ C n e i2πnf 0 t
where

C n = f 0 1 2f 0 -1 2f 0 x(t)e -i2πnf 0 t dt, n ∈ Z
The single sided power spectrum of y is made of lines located at frequencies nf 0 with power 2(C n ) 2 . It must be noted that even if one harmonic is very far in frequency from the original CW interference, due to aliasing it is folded back in the sampled band.

To measure the performance of this linear-thenlog function, the interference masks represented in figure 16 for the E5a/L5 and the E1/L1 bands were sampled and for each sample the following numerical computations were done with a CW interference of corresponding power P mask 1. set X0 to √ 2P mask , 2. calculate by numerical integration the first Fourier coefficients, the more powerful, of y, 3. test if it exists one coefficient C n , n > 1 for which the power 2(C n ) 2 exceeds the in-band mask level,

• if yes it means that X0 is the compression limit under which dynamic reduction is dangerous,

• if not X0 can be lowered (of 1dB in our code) and the process can loop to step 1 Doing so, X0 is calculated as low as possible (better dynamic compression) but without reinjecting in-band secondary harmonics more powerful than the interference mask.

The results are presented in figures 24, 25 and 26 for the E5a/L5 band and in figures 27, 28 and 29 for the E1/L1 band. The first plots 24 and 27 superimpose the CW interference mask at the input of F ll and the corresponding power of the first harmonic 2(C 1 ) 2 at the output. The second plots 25 and 28 plot directly the From the first subplot we can define the full efficiency band of this function, the band over which after the dynamic reduction there is no more need of filtering because the level of the first harmonic lies under the inband mask. It is equal to [1162,1189]MHz for E5a/L5 and to [1555,1593]MHz for E1/L1. This result by itself is interesting. and what is more the third subplot shows that in absolute a quantization bit saving up to 2 bits is possible. Unfortunately the first plot also shows that in this full efficiency band the CW interference mask is around or under the linear quantization limit which is set to 3σ as stated before: the CW level to compress is the same or smaller than the noise in which the useful signal is buried. This function F ll is efficient where it can not be used. That is why a second function was considered.

The Pure Log Function

This function F pl is represented in figure 30. It is fully logarithmic and it has a continuous second order derivative, by opposition to F ll which has a discontinuous one at X0, property which should induce lower harmonics at the output. This is at the expense of linearity : the The effect of F lp on the CW interference masks was measured as previously for the F ll function. The results are presented in figures 33, 34 and 35 for the E5a/L5 band and in figures 36,37 and 38 for the E1/L1 band.

From the first plots 33 and 36 we can also define the full efficiency band of this function, it is equal to [1161,1189]MHz for E5a/L5 and to [1553,1594]MHz for E1/L1. As expected it is larger than for the linear-thenlog function, but slightly. And unfortunately, as in the case of the F ll function, in this full efficiency band the CW interference mask is around or under the linear quantization limit : the CW level to compress is here also the same or smaller than the noise in which the useful signal is buried. We reach the same conclusion than for the F pl function: F pl is efficient where it can not be used.

It is also the conclusion of this part of our study: unless a function with better performance, that is a larger efficiency band, is produced, dynamic compression is not a usable technique to decrease the binary throughput after the ADCs.

Band Separation in Coherent Sampling

This last part of the paper is dedicated to the digital separation of the two bands, E5a/L5 and E1/L1, which The situation to consider at the input of the ADC in structure 6 can be ideal as in figure 8 or as bad as the one represented in figure 16, depending on the selectivity of the analog filters put in front of the ADC. We decide here to study the worst case to bound the complexity of the filters needed to select from one hand the E5a/L5 band and on the other hand the E1/L1 band.

The figure 39 represents the situation of figure 16, aliased in the worst way after sampling at 322.710MSamples/s, which corresponds to the minimum sampling frequency for the maximum transition bandwidth of 28MHz. E5a appears to be the most difficult to separate, from itself, due to the high slope of its interference mask.

The minimum selectivity of the filters mentioned above is drawn in figure 40. The design objective was to allow the greatest decimation rate after the filtering operation. It means that, for each band, after the filter it must be possible to alias the band to itself as near as possible without suffering from an out-band threat reinjected in-band. The attenuation was then calculated so that any out-band level is attenuated to a level of 10dB less than the minimum in-band mask, -103dBm in the E5a/L5 band and -118dBm in the E1/L1 band.

If Finite Impulse Response (FIR) filters are to be used, their order can be estimated using the following formula, from (Kai 110 for the E1/L1 band. The previous formula, 2 from [START_REF] Herrmann | Practical design rules for optimum finite impulse response low-pass digital filters[END_REF], was not used here as it is not valid when the ratio δ p /∆f is less than 0.004 which is the case here. When applicable the formula from [START_REF] Herrmann | Practical design rules for optimum finite impulse response low-pass digital filters[END_REF]) is more accurate but 3 give results of the same order. The calculation workload can also be estimated here by the simple product of the order M and of the sampling frequency. It can reach 548GMAC/s for the E5a/L5 band. This figure should be compared to the performance of the nowadays processors to assess the feasibility of the Coherent Sampling solution with the less selective analog filters. For example the most powerful FPGA from Xilinx, the Virtex-7 XC7V2000T (Xilinx, 2013), which can process up to 5335GMAC/s, seems to offer a sufficient processing power.

Conclusion

In summary the paper has reviewed two previously proposed Direct Sampling SDR GNSS receiver architectures designed to take into account the very specific Civil Aviation requirements regarding robustness against interference. On this basis the number of quantification bits required to linearly quantize the input signal has been bounded, from the nominal situation with ideal analog RF filters to the worse situation with low selectivity filters. It appears that an important number of bits is needed to represent the full range of the input signal when interference is present. In view of the important computation workload induced in the following filtering processes, two low complexity nonlinear functions have been considered to compress the signal dynamic at the output of ADCs. It has been shown that even if efficient in absolute, these functions are not usable in the proposed architectures. Finally the filters required to digitally separate the E5a/L5 and E1/L1 bands in the Coherent Sampling architecture have been specified and the induced computation workload has been assessed. Future work will focus on the design and implementation of the analog RF filters needed before the ADC(s) in both proposed architectures.

Figure 2 .

 2 Figure 2. Interference masks at antenna port

  Figure 3. Minimum required selectivity of the active antenna

Figure 4 .

 4 Figure 4. CW interference mask at receiver input

Figure 7 .Figure 8 .

 78 Figure 7. Magnitude of the ideal transfer functions of the extra filters needed for Coherent Sampling

Figure 9 .

 9 Figure 9. Minimum coherent sampling frequency vs transition bandwidth B

Figure 11 .

 11 Figure 11. Magnitude of the ideal transfer functions of the extra filters needed for Separate Sampling

Figure 13 .

 13 Figure 13. Minimum separate sampling frequency vs transition bandwidth B

Figure 14 .

 14 Figure 14. Dimensioning values of the quantifier

Figure 17 .

 17 Figure 17. Minimum attenuation to be provided at the output of the ADC in the E5a/L5 band, Separate Sampling

Figure 18 .Figure 19 .Figure 20 .Figure 21 .

 18192021 Figure 18. Minimum attenuation to be provided at the output of the ADC in the E1/L1 band, Separate Sampling

Figure 22 .

 22 Figure 22. Estimated calculation workload on the E1/L1 band, Separate Sampling

Figure 23 .

 23 Figure 23. The linear-then-log function

Figure 24 .Figure 25 .

 2425 Figure 24. Compression effect of the linear-then-log function on the E5a/L5 CW interference mask

Figure 26 .Figure 27 .Figure 28 .

 262728 Figure 26. Quantification bit saving offered by the linearthen-log function on the E5a/L5 band

Figure 29 .yFigure 31 .

 2931 Figure 29. Quantification bit saving offered by the linearthen-log function on the E1/L1 band

Figure 32 .Figure 33 .Figure 34 .

 323334 Figure 32. Comparison of the two non-linear functions, close-up

Figure 35 .Figure 36 .Figure 37 .

 353637 Figure 35. Quantification bit saving offered by the pure log function on the E5a/L5 band

Figure 38 .Figure 39 .

 3839 Figure 38. Quantification bit saving offered by the pure log function on the E1/L1 band

Figure 40 .

 40 Figure 40. Minimum attenuation to be provided at the output of the ADC, Coherent Sampling

CW Interferences Mask on L1 at Receiver Input

		dBm		CW
		-17.75
		-20
		-25.5
							-28.5
		-30
		-33.75
	These levels do not include the preamplifier gain [26.5, 32.5] dB nor cable losses [3,13] dB	-103 -46.5 -42.75 -37.5 -41				1626.5	-42 -40.5 -40
							f MHz
		900 950 960	1000	1166.45	1186.45	1585.42	2000

Interferences Mask on E5a at Receiver Input

		950
		1000
	-27	Extrapolated segments	1197.45	1215
				1220	1400
		1150
		1100
			1250
	-45.61
		1120		1255
	-60			Composite Ground DME Signal Max Peak Power at	1495	-57.1
	-65			European Hotspot
	-74.37	1160
	-95
	-118
					1565.42

Digital Filter Ideal Transfer Function required to compensate for the Analog Sub-Optimal Filter on E5a

  1

	dB	1166.45	1186.45
	0		f MHz
	-28.63	1160
	-65.5
		1150
	-73
			1197.45

engineer for the French Civil Aviation Authority. He returned to the ENAC in 2001, where he obtained a Master Polytechnique from Toulouse in 2003.