Iman Khosravi

Farnaz Hosseini

Mahsa Khorshidifard

Mehdi Sahihi

Amiri Hadi

Rudbari

Synthesis, characterization, crystal structure and HSA binding of two new N,O,O-donor Schiff-base ligands derived from dihydroxybenzaldehyde and tert-butylamine

Keywords:

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

o-Hydroxy Schiff-bases compounds are important class of compounds from crystallography and biological activity points of view. From the crystallography point of view, o-hydroxy Schiff bases represent an important class of compounds that are of great interest to the investigators mostly because of the intramolecular H-bond. o-Hydroxy Schiff-bases compounds display two possible tautomeric forms, the phenol-imine (OH) and the keto-amine (NH) forms. Depending on the tautomers, two types of intramolecular hydrogen bonds are observed in Schiff bases: O H⋯N in phenol-imine and N H⋯O in keto-amine tautomers [1], [2], [3]. In the latter case, a proton transfer may cause a change in the electron structure: the electronic and protonic states are clearly coupled, and an intramolecular charge transfer may occur. Hence, a zwitterionic structure may appear, which is rarely seen for hydroxy derivatives. The characteristic property of this form is the presence of ionic N + -H⋯O -hydrogen bond [4], [5].

The previous research showed that the position of the proton in the OH⋯N-region dramatically affects the interactions and plays a very important role in determining the physical, chemical properties of o-hydroxy Schiff bases. In this regard, we can mention to photo-, thermo-, and solvatochromic properties [6] and a variety of their applications were revealed in both theoretical and experimental investigations [7], [8], [9], [10], [11], [START_REF] Hadjudis | Photochromism[END_REF]. From the biological activity perspective, the origin of biological activity, especially anticancer activity of Schiff bases has been attributed to their ability to cleave DNA. It has been proved that in Schiff bases compounds the HC N linkage is an essential structural requirement for biological activity. Since Schiff bases are potential anticancer drugs, it was of interest to obtain insight in to their modes of binding with the most abundant plasma carrier protein, Human Serum Albumin (HSA), using spectroscopic techniques and molecular modeling [START_REF] Kazemi | [END_REF], [14], [15], [16], [17], [18]. Fluorescence and UV-vis absorption spectroscopy are effective techniques to study the drugprotein interactions, because of their sensitivity, reproducibility and convenience. These approaches can reveal the binding affinity of small molecules with proteins and help to understand their binding mechanisms [START_REF] Kazemi | [END_REF], [14], [15], [16], [17], [18]. Furthermore, it was of interest to locate binding sites of drugs on HSA. It requires docking tools that are able to identify the pose of the molecule in the binding site and the binding affinity or a score representing the strength of binding [START_REF] Kazemi | [END_REF], [14], [15], [16], [17], [18]. As part of our ongoing investigation into designing new Schiff base ligands and complexes [START_REF] Kazemi | [END_REF], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], herein we present results of a detailed investigation on the synthesis and structural characterization of two new o-hydroxy Schiff bases ligands derived from the condensation of tert-butylamine with 2,3-dihydroxybenzaldehyde (L 1 ) and 2,4-dihydroxybenzaldehyde (L 2 ) using IR, NMR, elemental analysis and single-crystal X-ray diffraction. In addition, the interaction between Schiff base ligands and HSA has been investigated using spectroscopic methods-fluorescence, UV absorptionalong with the protein-ligand docking and molecular dynamics simulation techniques.

Experimental section

Chemicals and instrumentation

Human serum albumin (HSA) was purchased from Sigma-Aldrich. 2,3-dihydroxybenzaldehyde, 2,4-dihydroxybenzaldehyde and tert-butylamine were obtained from Merck Co. and used without further purification. All of the solutions were used freshly after preparation. The FT-IR spectra were recorded on a JASCO, FT/IR-6300 spectrometer (4000-400 cm -1 ) in KBr pellets. 1 H and 13 C NMR spectra for L 1 and L 2 were recorded on a Bruker Avance 400 spectrometer using CDCl3 and DMSO-d6 as solvent, respectively. The elemental analysis was performed on Leco, CHNS-932 and Perkin-Elmer 7300 DV elemental analyzers. The UV-vis spectra were recorded on a JASCO V-670 spectrophotometer. Fluorescence measurements were carried out on Shimadzu RF-5000 spectrofluorometer at room temperature.

Single crystal diffraction studies

The crystallographic data for Schiff base ligand, L 2 , was collected at room temperature with a Bruker APEX II CCD area detector diffractometer using Mo-Kα radiation (0.71073 Å). Data collections, cell refinements, data reductions and absorption corrections were performed using multiscan methods with Bruker software [START_REF] Cosmo | Version 1.60[END_REF]. The structures were solved by direct methods using SIR2004 [START_REF] Burla | [END_REF]. The non-hydrogen atoms were refined anisotropically by the full matrix least squares method on F 2 using SHELXL [START_REF] Sheldrick | SHELXL-97[END_REF].

Hydrogen atoms for CH and CH3 groups were positioned geometrically (C H = 0.93 Å for CH, C H = 0.96 Å for CH3) and refined using a riding model. H atoms of NH group was found in difference Fourier map and refined isotropically. Molecular graphics were prepared with the Olex2 program [START_REF] Dolomanov | OLEX2: A complete structure solution, refinement and analysis program[END_REF]. Crystallographic data for L 2 are listed in Table 1. Selected bond distances and angles are summarized in Table 2

Synthesis of Schiff-base ligands

Tert-butylamine (10 mmol) was dissolved in 30 mL of absolute methanol and added slowly to a stirred solution 2,3-dihydroxybenzaldehyde or 2,4-dihydroxybenzaldehyde (10 mmol) in 30 mL of absolute methanol in ambient temperature. The color immediately changed to yellow for L 1 and orange for L 2 . The mixture was then stirred for 2 h in ambient temperature. The reaction mixture was allowed to stand for slow evaporation of the solvent until the formation of precipitate. The precipitate was filtered off and purified by washing with 10 mL of cold methanol. For compound L 2 , the filtrate was allowed to stand at room temperature and orange crystals suitable for single crystal X-ray diffraction analysis were obtained after 1-2 days.

L

HSA binding experiments

A stock solution of HSA was prepared by dissolving the desired amount of HSA in 50 mM phosphate buffer (pH = 7). The HSA stock solution was stored at 4 °C in the dark and used within 2 h. HSA concentration was determined by UV-vis spectroscopy using the molar absorption coefficient 35,700 M -1 cm -1 at 278 nm [38]. The solution of the compounds was first prepared in DMSO as co-solvent, and then diluted with phosphate buffer. The final DMSO concentration never exceeded 0.5% v/v. All the solutions were used freshly after preparation. The interaction of HSA with the synthesized Schiff base ligands was investigated using absorption titration experiment at room temperature. The UV-vis absorption spectra of the solution of HSA (10 μM) in the absence and presence of various amounts of the ligands (100 μM) were recorded at room temperature and the changes in the HSA absorption were recorded after each addition. Moreover, fluorescence quenching experiment was also done to more investigation of the interaction between HSA and the Schiff base ligands. In this experiment 2 mL of HSA solution (5 μM) was placed into the cell and various amounts of the Schiff base ligands (50 μM) were added to the cell. The absorbance at excitation wavelengths was always less than 0.1, in order to avoid inner filter effects. The fluorescence intensity was measured with excitation wavelength at 295 nm and emission wavelength rang of 300-450 nm and the excitation and emission slits set at 5 and 10 nm, respectively. In each measurement the mixture was allowed to incubate for 2 min after addition of the ligand.

Molecular docking procedure

In this work, docking study was carried out to indicate the binding site for the synthesized ligands. The 3D structures of the L 2 ligand was generated using the CIF file of its X-ray crystal structure. The CIF file was converted to the PDB format by using the Mercury software (http://www.ccdc.cam.ac.uk/). The geometry of the L 1 ligand was optimized using Gaussian 03 [START_REF] Frisch | Gaussian Development Version, Revision B.01[END_REF] at the level of B3LYP/6-31G** [START_REF] Becke | [END_REF], [41]. The crystal structure of HSA (PDB ID: 1AO6) was taken from the Brookhaven Protein Data Bank (http://www.rcsb.org/pdb). The R-value and resolution of this file were 0.218 and 0.25 Å, respectively. Water molecules of the protein. pdb file were removed, polar hydrogen atoms and Gasteiger charges were added. Flexible-ligand docking was performed by AutoDock 4.2 molecular-docking program using the implemented empirical free energy function and the Lamarckian Genetic Algorithm [42]. The Autogrid was used to calculate Grids. A blind docking with 126 lattice points along X, Y, and Z axes was performed to find the active site of ligands on HSA. After determination of the active site, the dimensions of the grid map were selected 60 points with a grid point spacing of 0.375 Å, to allow the ligand to rotate freely. 200 docking runs with 25,000,000 energy evaluations for each run were performed.

Molecular dynamics simulation

The lowest binding free energy conformation of each complex was considered as the initial conformation for the MD studies. All MD studies were carried out using the GROMACS 4.5.6 (University of Groningen, Netherlands) package [43], [44] and the GROMOS96 43a1 force field [START_REF] Van Gunsteren | Biomolecular Simulation: the GROMOS96 Manual and User Guide[END_REF], [START_REF] Van Gunsteren | Encyclopedia of Computational Chemistry[END_REF]. The Dundee PRODRG2.5 server was used to generate the topology parameters of the ligands [START_REF] Schuttelkopf | [END_REF]. The partial atomic charges of ligands were calculated using Gaussian 03 [START_REF] Frisch | Gaussian Development Version, Revision B.01[END_REF] at the level of B3LYP/6-31G** [START_REF] Becke | [END_REF], [41]. The complex was located in the cubic box with the periodic boundary conditions. The box volume was 11.36381 × 11.36381 × 11.36381 nm 3 and the minimum distance between the protein surface and the box was 1.0 nm. The box, filled with extended simple point charges (SPC), water molecules [START_REF] Berendsen | Intermolecular Forces[END_REF], and the solvated systems, were neutralized by adding 15 sodium ions (Na + ). Energy minimization was done through using the steepest descent method. Then, the system was equilibrated for 100 ps at the temperature of 300 K. Finally, a 20 ns MD simulation was carried out at 1 bar and 300 K. A Berendsen thermostat [START_REF] Berendsen | [END_REF] at 300 K, the particle mesh Ewald (PME) method [50], [51] for long range electrostatics, and a 9 Å cut off for van der Walls interactions and Coulomb interactions were used. The equation of motions was integrated by the leap-frog algorithm with the 2 fs time steps. The atomic coordinates were recorded to the trajectory file every 0.5 ps for later analysis. Finally, an all-bond constraint was used to keep the ligand from drifting in the MD.

Results and discussion

The procedure synthesis of the Schiff base ligands, L 1 and L 2 , reported in Scheme 1. The L 1 and L 2 Schiff base ligands obtained by the self-condensation reaction between tertbutylamine with 2,3-dihydroxybenzaldehyde (L 1 ) and 2,4-dihydroxybenzaldehyde (L 2 ), respectively, in methanol at the ambient temperature. The Schiff base ligands are yellow (L 1 ) and orange (L 2 ) powder, stable in air and soluble in the most common organic solvents such as MeOH, EtOH, CHCl3, DMSO, DMF. The most characteristic feature in the IR spectra of the Schiff base ligands comes from the C N stretching vibrations. In IR spectra, these stretching vibrations are at 1636 cm -1 for L 1 and L 2 confirming the presence of the imine groups. The presence of several medium intensity bands in the range 3060-2740 cm -1 suggests the existence of C H stretching vibrations of aliphatic and aromatic protons. Another characteristic region of the Schiff bases derivate spectrum is 1100-1400 cm -1 , which is attributed to C O stretching vibrations. The C O stretching vibration was observed at 1185 cm -1 and 1204 cm -1 for L 1 and L 2 respectively, which confirms the presence of phenolate group in the compound. Also, The Schiff base ligands exhibit two broad IR-bands of medium intensity in the 3400-3200 cm -1 region, the higher frequency peak is due to the ν(OH) mode while the lower one can be assigned to the ν(N H) vibration [21]. The presence of the OH and NH hydrogens are also revealed in the X-ray crystal structure of the L 2 ligand. The 1 H NMR and 13 C NMR spectra of L 1 and L 2 were recorded using CDCl3 (L 1 ) and DMSO-d6 (for L 2 ) as the solvent and data are summarized in the experimental section and shown in Fig. 1, Fig. S1 and Fig. S2. The 1 H and 13 C NMR spectra were run immediately after solution in CDCl3 (L 1 ) and DMSO-d6 (for L 2 ) with TMS as internal standard and gave the expected simple spectra, indicating the integrity of the Schiff base ligands. The spectra obtained after 12, 24 and 120 h were similar to the initial spectra indicating that the Schiff base ligands are stable in solution. The 1 H NMR spectra of L 1 and L 2 Schiff base ligands exhibit -OHphenolic proton resonances at ∼14.70 ppm for two -OHphenolic protons as broad singlet peak (Fig. 1). The signal for the imine proton in the L 1 and L 2 Schiff base ligands appears at 8.10 and 8.39 ppm, respectively, as a singlet peak. Also, methyl protons appear as singlet at 1.47 and 1.34 ppm for L 1 and L 2 , respectively. 13 C NMR spectra of Schiff base ligands, L 1 and L 2 , show in the region corresponding to signals of aromatic rings (148.18-113.54 ppm for L 1 and 159.63-102.94 ppm for L 2 ) 5 peaks instead of 6 expected peaks are observed. It seems that one carbon in this region are overlapping to other carbons due to high similarity of carbons (See Figs. S1 andS2). Also, 13 C NMR spectra show the azomethine carbon peaks at 159.89 and 162.06 ppm, for L 1 and L 2 , respectively. The UV-vis electronic absorption spectra of the title compounds were recorded in MeOH solvent. The experimental diagrams of absorption spectra for two Schiff base compounds are shown in Fig. 2. The observed spectra showed four bands at 210, 231, 288 and 381 nm for L 1 and three bands at 221, 299 and 371 nm for L 2 . It should be noted that the electronic absorption spectra of 2-hydroxy Schiff bases which exist mainly as phenol-imine structure indicate the presence of a band at <400 nm, whereas compounds that existing in the keto-amine form show a new band, especially in polar and nonpolar solvents at >400 nm [52], [53], [54]. According to absence of bonds at >400 nm in UV-vis spectra of two compounds, we can say that the compounds exist in solution as phenol-imine form. This result also can be confirmed by 1 HNMR spectra (Fig 1). As a result, the observed bands at 210, 231, 288 and 381 nm for L 1 and three bands at 221, 299 and 371 nm for L 2 can be due to benzene π → π* and azomethine π → π* and n → π* transitions. These values are similar to those found in related compound [55], [56].

Crystal structure of L 2

The molecular structure of L 2 is shown in Fig. 3. Selected bond distances and bond angles are listed in Table 2. The molecule has adopted E-configuration about the C7 N1 double bond Intramolecular N H⋯O hydrogen bond between the NH + and the phenolate O¯ generates an S(6) ring motif (Fig. 3; Table 3) which help to stabilize the planarity of the molecule [58]. The bond distances are in normal ranges and comparable with those found in related structures [59], [60]. In the crystal packing, molecules are connected by O H⋯O hydrogen bonds, generating C(6) chains (Fig. 4) [58]. According to the hydrogen bonding classification [61], [START_REF] Desiraju | The Weak Hydrogen Bond in Structural Chemistry and Biology[END_REF], these intramolecular and intermolecular hydrogen-bonding are moderate (mostly electrostatic) and strong (strongly covalent) interactions, respectively.

UV-vis spectroscopy

UV-vis absorption spectroscopy is an effective and simple technique to study of the HSA-drug binding and to investigate the structural changes of protein upon its drug binding. The UV absorption spectra of HSA shows a strong absorption peak at about 220 nm resulted of the n→π* transition for the peptide bond of α-helix and a weak peak at 278 nm assigned to π→π* transition of the phenyl rings in aromatic acid residues (Trp, Tyr and Phe) [START_REF] Fu | [END_REF]. Through addition of various amount of the Schiff base ligands (10 × 10 -5 M), a hypochromic effect in spectrum of HSA is observed (Fig. 5). This observation indicates that the aromatic amino acid residues of HSA in a hydrophobic cavity were exposed to an aqueous upon binding to the compounds. Moreover, observation of hypochromic effect in UV-vis spectra of HSA with increase amount of the synthesized ligands reveals the π-π stacking interaction between the aromatic rings of the compounds and the phenyl rings of aromatic acid residues [START_REF] Fu | [END_REF]. To assess the binding ability of the Schiff base ligands with HSA, the intrinsic binding constant (Kb) was determined using Benesi-Hildebrand plot and the following equation [64], [65]:

where A0 and A are the absorbance of HSA in the absence and presence of drug, respectively. Amax is the obtained absorbance at saturation and [M] is the concentration of drug. The plot of 1/(A-A0) versus 1/[M] gives Kb as ratio of y-intercept to slop. The binding constants for L 1 and L 2 are equal to 1.31 × 10 4 and 3.16 × 10 4 , respectively (Fig. S3).

Fluorescence spectroscopy

Fluorescence quenching experiment has been performed to investigate the mechanism of interaction between HSA and the ligands. Although fluorescence of HSA protein is due to the presence of three amino acids, i.e. tryptophan (Trp), tyrosine (Tyr), and phenylalanine (Phe) residues, the intrinsic fluorescence of HSA comes from tryptophan [66].

The fluorescence intensity of protein was quenched through the addition of the synthesized ligands, revealing the interaction of the ligands and HSA takes place and leads to the changes of microenvironment around the Trp-214 residue in HSA [67], [68]. Fig. 6 represents the fluorescence quenching of 5 μM HSA at the presence of various amounts of the ligands (5 × 10 -5 M). In order to determine the binding ability between the ligands and HSA, the Stern-Volmer quenching plot was obtained by monitoring the fluorescence quenching of HSA with increasing the concentration of the compounds according to the Stern-Volmer equation [67]:

where, F0 and F are the fluorescence intensity of HSA in absence and presence of the compounds. Ksv is the Stern-Volmer quenching constant, kq is the quenching rate constant of HSA and τ is the average lifetime of HSA without quencher which is typically equal to 10 -8 s for biomacromolecules [38]. Ksv is determined from the plot of F0/F vs. [Q] (Fig. 6). In the present study, the values of Ksv were obtained and presented in Table 4. Fluorescence quenching is classified to two mechanisms: static quenching and dynamic quenching. In the static mechanism, the fluorophore and the quencher collide together in the ground state while fluorophore and quencher collide together in the excited state in dynamic mechanism. Linearity of the Stern-Volmer plot indicates that quenching fluorescence has only one mechanism, dynamic or static [38].

In this study, the value of kq was obtained and represented in Table 4 that is greater than limiting diffusion rate constant of the diffusional quenching for biopolymers (2 × 10 10 M -1 S -1 ). This observation supports that quenching fluorescence of HSA occurs by static mechanism [38].

In addition the changes in the absorption spectrum of HSA show the formation HSA-drug complex in the ground state, which is because of the change in the HSA structure. While dynamic quenching affects on the exited state and do not have impact on the absorption spectrum. Moreover, binding constant (Kb) of the ligands and HSA can be calculated by this experiment using following equation [38]:

where, F0 and F are the fluorescence intensity of HSA in absence and presence of Schiff base ligand, respectively. [Q] is the concentration of quencher that quencher is the synthesized ligand here. "Kb" is obtained from the plot of ln ((F0 F/F) versus Ln[Q]as a y-intercept. Furthermore, "n" which is the number of binding site per protein is slope of the plot. The value of n is nearly 1, indicating that the synthesized compounds bind to HSA with molar ratio of 1:1. The calculated results are shown in Table 4. The Kb value reveals that the L 2 -HSA complex is more stable than other one; in the other words, this is more available for drug-cell interaction. Moreover, this result is in good agreement with the UV-vis spectroscopy result. Table 4. The HSA binding constant (Kb), the number of binding sites (n), the Stern-Volmer constant (Ksv) of the compounds, the quenching rate constant of HSA (kq) and the obtained results from FRET theory for the compounds. In general, binding constant of a drug with a carrier protein such as HSA should be high enough to bind and transfer thought body. Moreover, in order to release a drug since arrival at its target, it should not be too high. With respect to above, HSA binding constants of these ligands are in a good range (2-10 × 10 4 ) [69].

Ligand

Energy transfer from HSA to ligands

Energy transfer between the ligands and HSA can provide valuable information about HSA-ligand binding. The fluorescence quenching of HSA upon the binding of ligands can be indicative of energy transfer between HSA and these compounds. This energy transfer can be explained by fluorescence resonance energy transfer (FRET) theory. FRET that also known as Förster , s resonance energy transfer is an interaction between the excited molecule and its adjacent molecule. Upon this interaction, energy absorbed by donor molecule is transferred to an acceptor [70].

According to this theory, three conditions are required to energy transfer: (1) the donor should have fluorescence, (2) the fluorescence emission spectrum of the donor and the UV-vis spectrum of the acceptor should have sufficient overlap (3) the small distance between donor and acceptor (<8 nm) [70]. The distance and efficiency of energy transfer (E) between tryptophan residue of protein (HSA) and ligand has been calculated using this theory through the following equation:

where F0 and F are fluorescence intensities of HSA in the absence and presence of ligands, respectively and R0 is the critical distance when the transfer efficiency is 50%; r is the distance between donor and acceptor. R0 can be calculated by Eq. ( 5) [71].

In the above equation, the term K 2 is the orientation factor of the dipoles; N is the refracted index of medium, J is the overlap integral of the fluorescence spectrum of the donor with absorption spectrum of the acceptor and φ is the fluorescence quantum yield of the donor. The value of J can be calculated by the following expression:

where, F(λ) is the fluorescence intensity of the donor in the absence of the acceptor at wavelength λ and ɛ is the molar absorption coefficient of the acceptor at λ. In the present case, K 2 = 2/3, N = 1.336 and φ = 0.15 for HSA. Therefore, according to Eqs. ( 4), ( 5), ( 6) the parameters for the synthesized ligands were calculated (Table 4). Fig. 7 represents the overlap of the fluorescence emission spectrum of HSA (5 μM) and the UV-vis spectrum of ligands (5 μM). The values of r for all the ligands are less than 8 nm and "0.5 R0 < r < 1.5 R0", suggesting energy transfer from HSA to the Schiff base ligands occurs with high probability. Furthermore, the higher values of r than R0, indicating the presence of a static quenching mechanism in the interaction of HSA and compounds [72].

Molecular docking studies

It has been confirmed that the major factor affecting on the distribution, metabolism, the free concentration and the elimination rate of drugs is the binding affinity of drugs to HSA. Moreover, the half-life of drugs depends on their HSA-binding that effects on the dosage of drugs [73].

Therefore, the study of interaction between HSA and drugs is essential to investigate the pharmacokinetics and availability of drugs in tissues. HSA as the most abundant carrier protein comprises three α-helical domains (I, II and III), each containing two subdomains (A and B) [74].

The previous crystal structure studies showed that many drugs bind to one of the three binding sites located in subdomains IIA, IIIA and IB. In this work, the synthesized ligands were docked to the crystal structure of HSA. The obtained results from molecular docking showed that two ligands bind to the IB subdomain of HSA. Also, the affinity of L 2 is higher than the L 1 . The standard Gibbs free energies for binding of ligands to HSA are -5.63 and -5.82 kcal mol -1 for L 1 and L 2 , respectively. The results indicated that TYR161 and ARG117 amino acid residues are involved in the hydrogen bond interaction in the HSA-L 1 system. Also, ASP108 and TYR148 amino acid residues are involved in the hydrogen bond interaction in the HSA-L 2 complex. The results of docking are presented in Fig. 8, Fig. 9. The docking results are in good agreement with the experimental results and reveal important interactions between Schiff base ligands and HSA. Thus it can be concluded that the interaction of L 1 and L 2 with HSA is not mainly hydrophobic and the phenolic OH groups in the structure of ligands have a significant role in the binding process.

Analysis of the dynamics trajectories

The beginning structures for the MD analyses were selected from the conformations with lowest docking energies. The trajectories were analyzed in terms of root mean square deviation (RMSD), radius of gyration (Rg), and root mean square fluctuation (RMSF) using the GROMACS routines.

Root mean square deviation

The stability of for HSA-ligands was examined using the RMSD of the backbone of HSA (Fig. 10). A glance at the analysis in Fig. 10 shows that the RMSD of all systems reached equilibrium and fluctuated around 0.44 and 0.39 nm for L 1 and L 2 , respectively at about 7500 ps. This means that HSA-L complexes are well equilibrated and RMSD has low fluctuation around its mean value. Fig. 10. Time dependence of RMSD. RMSD values for HSA-ligand complexes.

Radius of gyration

The Rgs for all systems were also determined and plotted against simulation time to study the protein compactness (Fig. 11) showed that all of the systems stabilized after about 8000 ps, indicating that the MD simulation equilibrated after 8000 ps. It can be clearly seen that the Rg was larger upon binding of the L 2 than for L 1 suggesting a less compact structure after the binding of L 2 Schiff base ligand to HSA. Also, Fig. 11 shows that the Rg value of HSA decreases upon the complexation with L 1 and L 2 . This indicates that the environment of HSA changes during its interaction with Schiff base ligands.

Root mean square fluctuation

The mobility of the HSA residues was evaluated by analysis of the RMSF of the Cα atoms of HSA in the presence of the Schiff base ligands (Fig. 12). The results indicate that the residues that were in contact with the ligands are the most stable and have low RMSF values.

Fig. 12. The RMSF values of HSA-ligand complexes were plotted against residue numbers.

In addition, the RMSF of the atomic positions of the Schiff base ligands was calculated to examine their conformational variations (Fig. 13). The results indicate that the ligands atoms showed limited fluctuations (<0.14 nm). Hence, it can be concluded that the interactions of HSA and the Schiff base ligands were stable during the simulation time.

Conclusions

In this study, two new o-hydroxy Schiff-bases compounds, L 1 and L 2 , have been synthesized from reaction of 2,3-dihydroxybenzaldehyde and 2,4-dihydroxybenzaldehyde with tert-butylamine. The synthesized compounds have been investigated by elemental analysis and FT-IR, 1 H and 13 C NMR spectroscopy. The structure of L 2 has been also confirmed by X-ray crystallography. The crystal structure of L 2 showed that the compound exists as a zwitterionic form in the solid state, with the H atom of the phenol group being transferred to the imine N atom. The HSA binding of the synthesized compounds were evaluated using experimental methods including fluorescence quenching and UV-vis spectroscopy. The computational methods (molecular docking and molecular dynamics simulation methods) were also carried out to investigate more details. The obtained results from both experimental and theoretical methods indicate that the compounds bind to HSA as L 2 > L 1 . The obtained results from UV-vis spectroscopy and fluorescence quenching represent that all the synthesized compounds bind to HSA with binding constants about 4 × 10 4 M -1 that is appropriate value for drugs. The results of fluorescence experiment as well as the changes in the absorption spectrum of HSA upon addition of the compounds show that the HSA-compound complexes formed in the ground state. The synthesized ligands bind to HSA with molar ratio of 1:1. Moreover the binding distance between the donor (HSA) and the acceptors (the ligands) was evaluated based on the FRET theory and the results indicated that the quenching of HSA by the synthesized compounds was static quenching processes.

In addition, the binding propensity of the compounds to HSA was investigated by computational methods (Molecular docking and molecular dynamics simulation methods). Molecular docking studies revealed that hydrogen bond and hydrophobic interactions have major role in the binding of these Schiff base ligands to HSA. The presence of the hydrogen bond gives additional stabilization to the HSA-ligands systems. Also, the details of HSA binding properties of the compounds on the molecular level were obtained by this method.

To investigate HSA-binding of the compounds more accurately, the molecular dynamics simulation method was carried out. MD simulation studies showed the stabilization of HSA-ligand complexes at around 8 ns It can be clearly seen that Rg decrease upon binding of Schiff base ligands, suggesting a more compact structure after the binding of these two ligands to HSA. In addition, the environment of HSA changes slightly during its interaction with L 1 and L 2 . Furthermore, the profiles of atomic fluctuations showed that the interactions of HSA and Schiff base ligands were stable during the simulation time.

Scheme 1 .

 1 Scheme 1. Synthesis of Schiff base ligands.

Fig. 1 .

 1 Fig. 1. 1 HNMR of (a) L 1 and (b) L 2 . A star marks in black color represents peak for CDCl3 (L 1 ) and DMSO-d6 (L 2 ).

Fig. 2 .

 2 Fig. 2. UV-Vis spectra of L 1 (a) and L 2 (b) in MeOH solution at 25 °C.

( 1 .

 1 2981(18) Å) with a C8 N1 C7 C1 torsion angle of 178.139°. The C11 N1 [1.300(4) A] and C2 O1 [1.304(4) A] bonds of the title compound are the most important indicators of the tautomeric type (Scheme 2). While the C2 O1 bond is of a double bond for the keto-amine tautomer, this bond displays single bond character in phenol-imine tautomer. In addition, the C11 N1 bond is also a double bond in phenol-imine tautomer and of single bond length in keto-amine tautomer [53], [57]. However, these bond distances have intermediate values between single and double C O (1.362 and 1.222 A, respectively) and C N (1.339 and 1.279 A, respectively) bond distance. The shortened C2 O1 bond and the slightly longer C11 N1 bond provide structural evidence for the zwitterionic form of the title compound.

Fig. 3 .

 3 Fig. 3. ORTEP representation of L 2 . Displacement ellipsoids are drawn at the 30% probability level and H atoms are shown as small spheres of arbitrary radii. Intramolecular H-bond is illustrated by dashed line.

Fig. 4 .

 4 Fig. 4. Intramolecular and intermolecular hydrogen bond geometries for L 2 (Å,°).

Fig. 5 .

 5 Fig. 5. Electronic absorption spectra of HSA (10 μM) in the absence and presence of various concentrations of (a) L 1 and (b) L 2 .

Fig. 6 .

 6 Fig. 6. Fluorescence quenching of HSA (5 μM) in the absence and presence of various concentrations of (a) L 1 and (b) L 2 .

Fig. 7 .

 7 Fig. 7. Spectral overlap of the HSA fluorescence with the (a) L 1 and (b) L 2 absorption spectrum. [HSA] = [L] = 5 μM, λex = 295 nm.

Fig. 8 .

 8 Fig. 8. L 1 and L 2 docked in HSA using AutoDock. Ligands, depicted as a surface model, and HSA, represented in cartoon ribbon (The red circles show the ligands).

Fig. 9 .

 9 Fig. 9. The docking poses of the HSA-ligand complexes. a) L 1 and b) L 2 . Hydrogen bond interactions showed as spheres.

Fig. 11 .

 11 Fig. 11. Time evolution of the radius of gyration (Rg) during 20,000 ps of MD simulation of HSAligand complexes.

Fig. 13 .

 13 Fig. 13. RMSF values of Schiff base ligands were plotted against atom numbers.

Table 1 .

 1 . Crystal data and structure refinement for L 2 .

	Empirical formula	C11H15NO2
	Formula weight	193.24
	Temperature (K)	298(2)
	Wavelength (Å)	0.71073
	Crystal system	Orthorhombic
	Space group	P212121
	Unit cell dimensions	a = 7.6122(15)
		b = 11.962(2)
		c = 12.631(3)
	Volume (Å 3 )	1150.1(4)

Table 3 .

 3 Hydrogen-bond geometry (Å, °) for L 2 .

	D-H⋯A	D-H	H⋯A	D⋯A	D-	Empty Cell Symmetry code
					H⋯A
		0.904(15) 1.875(15) 2.6233(17) 138.7	Intra
	N(1)
	H(1)⋯O(1)
		0.965(19) 1.593(18) 2.5429(15) 167.1	Inter	-1/2 + x,1/2-y,-z
	O(2)
	H(2)⋯O(1)
	Note: D donor, A acceptor.

Acknowledgment

Support for this research by the University of Isfahan and Islamic Azad University (Qeshm branch) is acknowledged.

Appendix A. Supplementary data

The following is the supplementary data related to this article: CCDC 1448114 contains the supplementary crystallographic data for L 2 . This data can be obtained free of charge via http://www.ccdc.cam.ac.uk/conts/retrieving.html, or from the Cambridge Crystallographic Data Centre, 12 Union Road, Cambridge CB2 1EZ, UK; fax: (+44) 1223-336-033; or e-mail: deposit@ccdc.cam.ac.uk.


Nicolas Gault

Axel Garcia-Pena

Alexandre Chabory

Christophe Macabiau

Impact of DME/TACAN on GNSS L5/E5a Receivers at Low Altitude Considering Multipath

Since 2007, he has been with ENAC where he is

INTRODUCTION

The nominal processing of GNSS received signals can be affected by noise as well as received additive signals such as multipath and Radio Frequency Interference (RFI). RFI sources are of various sorts and their impact depends on the user application. In the context of civil aviation, Global Navigation Satellite System (GNSS) L5/E5a interference environment is dominated by pulsed interferences such as DME/TACAN and JTIDS/MIDS. A long thread of activities led or are leading to the elaboration of various ICAO, RTCA and EUROCAE standards to determine and to bound the vulnerability of airborne GNSS receivers equipped with their relevant antenna in the L5/E5a band to these RFI sources. In these standards, the RFI impact on a GNSS receiver is well characterized as the 𝐶/𝑁 0 degradation observed at the receiver's correlator output, or equivalently, as an increase of the effective 𝑁 0 denoted as 𝑁 0,𝑒𝑓𝑓 .

In civil aviation, 𝑁 0,𝑒𝑓𝑓 is computed taking into account the presence of temporal blanker in the receiver Radio Frequency Front-End (RFFE) block, as depicted in the RTCA DO-292, (2004). The role of the blanker is to set to zeros the samples whenever the instantaneous power envelope of a received pulse is higher than a certain threshold. Depending on the DME/TACAN interfering scenarios (number of beacons, received power at the aircraft level, pulses time of arrival etc.), the value of bdc (percentage of samples set to zero by the blanker) and 𝑅 𝐼 (the below-blanker interfering-signal-to-thermal-noise ratio) can be determined and the 𝑁 0,𝑒𝑓𝑓 computed. A worst-case scenario was found in RTCA DO-292, (2004) at Harrisburg P.A, high altitude (FL400), where the 𝐶/𝑁 0 degradation due to DME/TACAN only is up to 5.99 dB. Note that this value is to be updated in the RTCA DO-292A since an interpretation error have been found in the RTCA, (2004) and confirmed by the RTCA SC-159 WG6.

However, none of the previous characterizations consider any multipath effect which is especially important at low altitude Flight Level (FL) since the airborne GNSS receiver could be significantly affected by echoes created by low altitude obstacles. Indeed, the echoed pulses would impact the receiver in the same manner as the direct pulses: echoed pulses could trigger the temporal blanker and thus increase the value of 𝑏𝑑𝑐 or could just increase the overall 𝑅 𝐼 value. As a result, new 𝑅 𝐼 and 𝑏𝑑𝑐 formulas have to be derived to correctly consider multipath in the 𝑁 0,𝑒𝑓𝑓 computation; these formulas must tackle the collision of echoes with the direct signal pulses as well as the collision of echoes generated by other obstacles, knowing that the collision can distort the original shape of the DME/TACAN and JTIDS/MIDS pulses, highly complexifying the analysis. Furthermore, these new formulas require the knowledge of the precise time of arrival of the echoed pulses as well as their received power at the aircraft level. Therefore, a propagation channel model specially adapted for DME/TACAN interference analysis had to be developed. However, the newly developed RFI-focuses propagation channel model should be as simple as possible (as opposite to highly resource consuming propagation channel models) to allow its application to complex scenarios (high number of obstacles, such as real 2D/3D maps) while still being representative from a RFI impact point of view.

The general objective of this paper is thus to first present new 𝑅 𝐼 and 𝑏𝑑𝑐 formulas considering multipath, second to present a DME/TACAN multipath interference propagation channel and third, to present two meaningful numerical examples of the proposed new formulas and propagation channel model. The first example is a comparison of the area necessary to increase 𝑏𝑑𝑐 by 50%, at low and at high altitude FL. The second example is the study of a potential low altitude FL US operational hot spot, JALTO, Philadelphia, determined during RTCA SC-159 WG6 activities.

Following this three-fold objective, the paper is organized as follows. In the first section, the DME/TACAN signal and system are presented. The second section presents the RTCA DO-292, (2004) analytical model for the multipath-free effective 𝐶/𝑁 0 calculation, as well as its application to the US worst-case scenario (US FL400 hot spot). In the third section, the 𝑏𝑑𝑐 and 𝑅 𝐼 mathematical models including multipath impact are introduced. The fourth section introduces the proposed propagation channel model adapted to interference calculations purposes. The sixth section presents the two numerical examples of the application of the new propagation channel model and the 𝑅 𝐼 and 𝑏𝑑𝑐 formulas.

DME/TACAN SIGNAL AND SYSTEM DESCRIPTION

DME, and its military equivalent, TACAN, are two systems used by aircrafts to determine their distance to a ground station, whose position is known. The systems operate as follows. The aircraft DME equipment, called interrogator, sends pulses to ground stations. Once the interrogation is detected, the station transponder replies to the interrogator. The distance is then determined by measuring the time elapsed between each pulse transmitted by the interrogator and the reception of its corresponding reply pulse from the transponder. This delay corresponds to twice the distance between the aircraft and the station, as well as a fixed processing time inside the ground station.

According to the RTCA DO-292, (2004) only the signals emitted in the band of interest of the study disturb GNSS receivers operations, the band of interest being the E5a/L5 one, equal to [1164 MHz;1191 MHz]. The aircraft's DME interrogators emitting their signals between 1025 and 1151 MHz or between 1191 MHz and 1215 MHz, they are ignored herein. The study focuses on DME ground stations, as they emit their signals between 962 and 1213 MHz, which includes the defined band of interest defined above.

The emitted signal of an individual DME station is composed of a pair of Gaussian pulses modulated by a cosine, which can be modelled as:

𝑠(𝑡) = √2𝑃 ∑ (𝑒 -𝛼(𝑡-𝑡 𝑘 ) 2 2 + 𝑒 -𝛼(𝑡-Δ 𝑡 -𝑡 𝑘 ) 2 2 ) cos (2𝜋𝑓 𝑖 𝑡 + 𝜃 𝑖 ) 𝐾 𝑘=1 (1)
Where:

• 𝑃 is the interference beacon transmitting peak envelope power (dBW),

• 𝛼 = 4.5 • 10 11 𝑠 -2 is a constant of pulse,

• Δ 𝑡 = 12 𝜇𝑠 is the inter pulse time separation,

• 𝑡 𝑘 is the emission time of the 𝑘 𝑡ℎ pulse pair,

• 𝑓 𝑖 is the carrier frequency of the DME/TACAN signal (Hz),

• 𝜃 𝑖 is the DME/TACAN signal initial carrier phase shift.

Note that Equation (1) has been corrected from previous publications of the authors to define 𝑃 as the Peak Envelope Power (PEP) instead of as an Instantaneous Peak Power (IPP). Definition of 𝑃 as peak envelope power leads to a more consistent definition since DME specifications are provided with a PEP values. From Equation (1), instantaneous power of a single pulse, 𝑝(𝑡), can be defined:

𝑝(𝑡) = 𝑃𝑒 -𝛼𝑡² (2)
Where

• 𝑃 is the peak envelope power (PEP) of the pulsed interfering source at the antenna output. In this section, the general analytical multipath-free 𝐶/𝑁 0 degradation model, and the mathematical multipath-free 𝑏𝑑𝑐 and 𝑅 𝐼 models are presented. Using these models, the value of the 𝐶/𝑁 0 degradation due to DME/TACAN only at the US FL400 hot spot without considering multipath is derived.

3.A. GENERAL ANALYTICAL MODEL

In order to understand the DO-292 degradation analytical model, a generic civil aviation GNSS receiver structure as well as the behavior and effect of its components on the received signals are described. The receiver structure is presented in The RFFE block is also responsible for digitizing the filtered signals with the application of the ADC (Analog-Digital-Converter) coupled with the AGC (Automatic-Gain-Control). For a civil aviation receiver, the temporal blanker is introduced at that point. The temporal blanker is a device which blanks (put to 0s) the time samples of the incoming signal (mix of signals) which are over a threshold; the digitized and blanked signal is found at point C. In the RTCA, ( 2004), the defined blanker is an instantaneous blanker. This blanking method removes all the time samples of the incoming signal which have an envelope power over a given threshold. For an optimal functioning, the blanker should also be coupled with the ADC/AGC blocks: to ensure that high-power signals are not saturating the ADC/AGC and that the blanked signal spans the ADC quantization range. Figure 4 presents an example of the RTCA, (2004) instantaneous blanker. Although the blanking method is going to affect the power of the useful signal (part of the signal is removed and its power is decreased), RTCA DO-292 proposes to model the 𝐶/𝑁 0,𝑒𝑓𝑓 by defining an equivalent 𝑁 0,𝑒𝑓𝑓 . 𝑁 0,𝑒𝑓𝑓 represents the effective noise power spectrum density that a receiver will observe at the correlator output if the receiver captures a useful signal with power 𝐶 at the correlator output and no RFI signals are present. In other words, RTCA DO-292 proposes a generic formula to compute the degradation of the 𝐶/𝑁 0 through the increase of the background noise due to pulsed and continuous RFI.

In order to mathematically model 𝑁 0,𝑒𝑓𝑓 , the following points about the impact of pulsed RFI signals on 𝑁 0,𝑒𝑓𝑓 must be considered:

• Part of the signal is removed due to the blanker. Since the impact on the useful signal power (1 -𝑏𝑑𝑐)² is higher than the impact on the power of the noise(1 -𝑏𝑑𝑐), the equivalent 𝑁 0,𝑒𝑓𝑓 is increased by a factor of 1/(1 -𝑏𝑑𝑐). 𝑏𝑑𝑐 is the blanker duty cycle, or in other words, the percentage of time the incoming signal is blanked (𝑏𝑑𝑐 ∈ [0,1]). • Not all the RFI signal samples have a power above the threshold; therefore, there is a part of the RFI signal that is not removed, and its influence must be added to the below-threshold interfering-signal-to-thermal-noise ratio, 𝑅 𝐼 .

With these considerations, the mathematical model proposed by the RTCA DO-292 for 𝑁 0,𝑒𝑓𝑓 is:

𝑁 0,𝑒𝑓𝑓 = 𝑁 𝑂 (1 -𝑏𝑑𝑐) • (1 + 𝐼 0,𝑊𝐵 𝑁 0 + 𝑅 𝐼 ) (3)
Where:

• 𝑁 0 is the thermal noise power spectrum density level generated by the RFFE,

• 𝑏𝑑𝑐 is the blanker duty cycle,

• 𝐼 0,𝑊𝐵 is the contribution of wideband (non-pulsed) signals to 𝑁 0,𝑒𝑓𝑓 ,

• 𝑅 𝐼 is the below-threshold interfering-signal-to-thermal-noise ratio. The objective of this paper is to assess the impact of multipath considering DME/TACAN; therefore, for simplification purposes it is assumed that 𝐼 0,𝑊𝐵 = 0, which yields:

𝑁 0,𝑒𝑓𝑓 = 𝑁 𝑂 (1 -𝑏𝑑𝑐) • (1 + 𝑅 𝐼 ) (4)
And finally, ( 𝐶 𝑁 0 ) 𝑑𝑒𝑔 = 10 log 10 ( 𝑁 0 𝑁 0,𝑒𝑓𝑓 ) = 10 log 10 ( 1 -𝑏𝑑𝑐 1 + 𝑅 𝐼 )

(5)

3.B. MULTIPATH-FREE DME/TACAN 𝒃𝒅𝒄 MATHEMATICAL MODEL

In this subsection, the example of a single pulse is presented to introduce the mathematical notations necessary to define the (RTCA, 2004) multipath-free DME/TACAN 𝑏𝑑𝑐 mathematical model. In particular, the definition of the above threshold pulse width 𝑝𝑤 is especially detailed since this factor will be later impacted by the introduction of multipath.

Figure 5 provides an example of a single above-threshold pulse.

From Figure 5, three definitions are introduced:

• -𝑤 and 𝑤 are the instants of time where the pulse starts (stops) to be blanked. The mathematical expression of 𝑤 is obtained by simply equalizing the pulse instantaneous power 𝑝(𝑡) (defined Equation ( 2)) to blanker threshold 𝑇 ℎ :

𝑝(𝑤) = 𝑇 𝐻 ⇔ 𝑤 = √ ln ( 𝑃 𝑇 ℎ ⁄ ) 𝛼 (6)
• 𝐼 𝑏𝑑𝑐 is the total blanked interval. From the definition of 𝑤, its mathematical expression is simply given by:

𝐼 𝑏𝑑𝑐 = [-𝑤, 𝑤] (7)
• 𝒑𝒘 is the above threshold pulse width. From the expression of 𝐼 𝑏𝑑𝑐 , the mathematical expression of 𝑝𝑤 is as follows:

𝑝𝑤 = 𝑙𝑒𝑛𝑔𝑡ℎ(𝐼 𝑏𝑑𝑐 ) = 2𝑤 (8)
Note that the definition of 𝑝𝑤 is especially important since it will be impacted by the introduction of multipath.

Finally, from these definitions, RTCA DO-292 proposes a model for 𝑏𝑑𝑐 specially adapted for DME/TACAN, derived from queuing theory, accounting for 𝑁 pulses:

𝑏𝑑𝑐 = 1 -𝑒 -2 ∑ 𝒑𝒘 𝒊 •𝑃𝑅𝐹 𝑖 𝑁 𝑖=1 (9)
Where:

• 𝑁 is the total number of sources, 𝑁 = 𝑁 𝐷 + 𝑁 𝑇 with 𝑁 𝐷 and 𝑁 𝑇 being the number of DME and TACAN respectively,

• 𝑃𝑅𝐹 𝑖 is the beacon pulse repetition frequency of source 𝑖, 2700 for DME and 3600 for TACAN

3.C. MULTIPATH-FREE DME/TACAN 𝑹 𝑰 MATHEMATICAL MODEL

In this subsection, first the multipath-free DME/TACAN mathematical model proposed by the (RTCA, 2004) is introduced. In particular, the definition of the below threshold equivalent pulse width 𝑃𝑊 is provided in detail for below and above blanker pulses, since its value will be directly impacted later by the introduction of multipath. Finally, a more accurate model for the multipath-free 𝑅 𝑖 mathematical model, proposed by Garcia-Pena, Julien, et al., (2020) is introduced.

From Equation (1), the multipath-free mathematical model of 𝑅 𝐼 proposed by RTCA DO-292 is introduced:

𝑅 𝐼 = ∑ 𝑃 𝑖 𝑷𝑾 𝒊 𝑃𝑅𝐹 𝑖 𝑁 0 𝐵𝑊 𝑁 𝑖=1 (10)
Where:

• 𝐵𝑊 is the RFFE filter bandwidth,

• 𝑁 0 is the thermal noise power spectrum density level generated by the RFFE,

• 𝑷𝑾 𝒊 is the below threshold equivalent pulse width of a pair of pulse, such that:

𝑃𝑊 𝑖 = 2 • 𝑃𝑊 𝑖,𝑝𝑢𝑙𝑠𝑒 (11)
• 𝑃𝑊 𝑖,𝑝𝑢𝑙𝑠𝑒 is the below threshold equivalent pulse width of a single pulse.

𝑃𝑊 𝑖,𝑝𝑢𝑙𝑠𝑒 can be seen as the width of an equivalent (in energy) rectangular pulse to the original RFI pulse (in this case, DME or TACAN) having the same peak envelope power 𝑃 as the initial pulse. Equalizing the energy expressions of this rectangular equivalent pulse and the initial pulse yields:

𝑃𝑊 𝑖,𝑝𝑢𝑙𝑠𝑒 = 1 𝑃 𝑖 ∫ 𝑝(𝑡)𝑑𝑡 ∞ -∞ (12)
Where 𝑝(𝑡) is the instantaneous power of one pulse as expressed in Equation (2). Figure 6 provides an example for 𝑃𝑊 𝑖 𝑝𝑢𝑙𝑠𝑒 for both a DME/TACAN below and above threshold pulse.

From Figure 6, two different cases must be distinguished to derive 𝑃𝑊 𝑖,𝑝𝑢𝑙𝑠𝑒 :

• Case (a): the pulse is below the threshold. By applying the Gaussian integral, we obtain:

𝑃𝑊 𝑖,𝑝𝑢𝑙𝑠𝑒 = 1 𝑃 𝑖 ∫ 𝑝(𝑡)𝑑𝑡 ∞ -∞ = ∫ 𝑒 -𝛼𝑡 2 𝑑𝑡 ∞ -∞ = √𝜋/𝛼 (13)
• Case (b): the pulse is only partially below the threshold. In that case, the blanked interval must be excluded from the derivation, and from 𝑢 = √𝛼𝑡 change of variable:

𝑃𝑊 𝑖,𝑝𝑢𝑙𝑠𝑒 = ∫ 𝑒 -𝛼𝑡 2 𝑑𝑡 -𝑤 -∞ + ∫ 𝑒 -𝛼𝑡 2 𝑑𝑡 ∞ 𝑤 = √ 𝜋 𝛼 𝑒𝑟𝑓𝑐(√𝛼𝑤) (14)
Where

𝑒𝑟𝑓𝑐(𝑥) = 2 √𝜋 ∫ 𝑒 -𝑢² 𝑑𝑢 +∞ 𝑥
is the complementary error function.

It must be noted that the 𝑅 𝐼 model of Equation ( 10) assumes that the post-blanker RFI signal power spectrum density (PSD) is completely spread which is a loose approximation. A more accurate model has been recently proposed by (Garcia-Pena, Julien, et al., 2020) to better model the post-blanker pulsed RFI signal PSD since simulations showed that the blanked signal is far from having a perfectly spread PSD. The proposed model is as follows:

𝑅 𝐼 = ∑ 𝑃 𝑖 𝑷𝑾 𝒊 𝑃𝑅𝐹 𝑖 𝑁 0 𝛽 0 • 𝑆𝑆𝐶(Δ𝑓 𝑖 ) 𝑁 𝑖=1 (15)
Where:

• 𝛽 0 is the thermal noise power degradation due to RFFE filter and correlator,

𝛽 0 = ∫ |𝐻 𝑅𝐹 (𝑓)|²𝑆 ̅ 𝑐𝑚 (𝑓)𝑑𝑓 ∞ -∞ (16)
• 𝐻 𝑅𝐹 (𝑓) is the baseband transfer function of the equivalent RFFE plus antenna filter, • 𝑆𝑆𝐶(Δ𝑓 𝑖 ) is the Spectral Separation Coefficient of the pulsed interfering source 𝑖,

𝑆𝑆𝐶(Δ𝑓 𝑖 ) = ∫ 𝑆 ̅ 𝑖,𝑃𝐵𝑓 (𝑓 -Δ𝑓 𝑖 )𝑆 ̅ 𝑐𝑚 (𝑓)𝑑𝑓 ∞ -∞ (17)
• 𝑆 ̅ 𝑖,𝑃𝐵𝑓 is the 𝑖 𝑡ℎ baseband post-blanker filtered normalized interfering signal PSD, • 𝑆 ̅ 𝑐𝑚 is the normalized local replica PSD of the 𝑚 𝑡ℎ satellite, • Δ𝑓 𝑖 is the difference between L5/E5a central frequency 𝑓 𝐿5 and carrier frequency of interfering source 𝑖.

Note that Equation ( 10) and (15). will be affected in the same manner by the introduction of multipath, through the computation of 𝑃𝑊 𝑖 , which is one of the most important factors of the 𝑅 𝐼 derivation. Hence, for simplification purposes, we propose to only keep equation ( 15) for the rest of the article.

3.D. APPLICATION TO RTCA DO-292 US HOT-SPOT

Applying the multipath-free model for 𝑅 𝐼 (Equation ( 10)), 𝑏𝑑𝑐 (Equation ( 9)) and for the 𝐶/𝑁 0 degradation (Equation ( 5)), RTCA, (2004) members determined in 2004 a worst-case scenario concerning the 𝐶/𝑁 0 degradation due to DME/TACAN only, at Harrisburg, Pennsylvania, at high-altitude (FL400). However, as mentioned in the introduction, the value of the 𝐶/𝑁 0 degradation found in the RTCA DO-292 needs to be updated since an interpretation error of the 𝑅 𝐼 model (Equation ( 10)) have been reported during the RTCA SC-159 WG6 activities. Applying the proposed model (Equation ( 15)) at the exact same spot, the value obtained for 𝑏𝑑𝑐 and 𝑅 𝐼 are 0.59 and 0.61. From these two values, the 𝐶/𝑁 0 degradation at this hot spot due to DME/TACAN only is Figure 7: Representation of a direct DME/TACAN pulse followed by a created echo precisely 7.34 dB. This value is important to remember and will be compared to a potential low altitude hot-spot at JALTO, Philadelphia, in section 6 of this article.

PROPOSED 𝒃𝒅𝒄 AND 𝑹 𝑰 MATHEMATICAL MODELS CONSIDERING MULTIPATH

Multipath is impacting the value of the 𝐶/𝑁 0 degradation (Equation ( 5)) by modifying the values of 𝑏𝑑𝑐 (Equation ( 9)) and 𝑅 𝐼 (Equation ( 15)), through the modification of the above threshold pulse width 𝑝𝑤 (Equation ( 8)) and the below threshold equivalent pulse width 𝑃𝑊 (Equation ( 11)) respectively. The objective of this section is to define the 𝑝𝑤 and 𝑃𝑊 formulas considering multipath.

In this section, the assumptions concerning the pulse collisions and the new notations necessary to the definition of the new 𝑝𝑤 and 𝑃𝑊 formulas are presented. Then, the new formula for 𝑝𝑤 and 𝑃𝑊 are introduced. Finally, an important remark on the necessary parameters to be obtained from the propagation channel model to derive the 𝑝𝑤 and 𝑃𝑊 formulas is highlighted.

Figure 7 provides a graphical example of the multipath impact on 𝑃𝑊 and 𝑝𝑤, where an echoed pulse (in green) collides on the right of direct pulse (in red).

4.A. PULSE COLLISION ASSUMPTIONS

As mentioned in the Introduction, the new formulas for 𝑅 𝐼 and 𝑏𝑑𝑐 must tackle the collision of echoes with the direct signal pulses as well as the collision of echoes generated by other obstacles, knowing that the collision can distort the original shape of the DME/TACAN and JTIDS/MIDS pulses, highly complexifying the analysis. In this subsection, two assumptions concerning the pulse collisions, on the phase difference between pulses, and on the impact of the echoes generated by the first pulse on the second pulse of a pair of pulse, are presented to simplify the overall derivation.

Phase difference:

The phase difference between two echoed pulses (or an echoed pulse and the LOS pulse) is modeled as a random uniform variable on [0,2𝜋]. With this assumption, it can be shown that in average, the contribution to 𝑅 𝐼 and 𝑏𝑑𝑐 of multiple pulses is equal to the sum of the contribution to 𝑅 𝐼 and 𝑏𝑑𝑐 for each individual pulse.

Impact of the echoes generated by the first pulse on the second pulse of a pair: Since the DME/TACAN beacon emits pair of pulses, echoes generated by the first pulse could collide with the second pulse of the pair. This would be particularly problematic if the blanked interval generated by an echo directly collides with the blanked interval of the second pulse of the pair (impact on 𝑝𝑤) or is nearer to the blanked interval of the second pulse than the blanked interval of the first pulse (impact of 𝑃𝑊). However, this would require to have echoes triggering the blanker after a long additional travel time generating an additional delay 𝜏 𝑖 𝑗 > 6 𝜇𝑠 since the time interval between the two pulses of a pair is 12 𝜇𝑠. During this long additional travel time, the echoed signal is very attenuated and it seems very unlikely to have an echo triggering the threshold with this additional delay. Therefore, this scenario is excluded from the 𝑅 𝐼 and 𝑏𝑑𝑐 derivation considering multipath.

4.B. UPDATED NOTATION

In this subsections, a new notation concerning the instants of time where the pulse starts and stops to be blanked, is introduced.

Since LOS pulse is chosen to be centered at 0, echoed pulses are not. Therefore, the instants of time where a echo triggers the blanker (blanker activation interval limits) must be defined since Equation ( 6) can no longer be directly used. Let's consider an echo arriving at the receiver with an additional delay 𝜏 𝑖 𝑗 with respect with the LOS. From Equation (2), instantaneous power 𝑝(𝑡)

of the echo is defined as:

𝑝(𝑡) = 𝑃 𝑖 𝑗 𝑒 -𝛼(𝑡-𝜏 𝑖 𝑗 )²
(18) Where:

• 𝑃 𝑖 𝑗 is the PEP of the echo at the antenna output, To obtain the instants of time where an echo triggers the blanker, 𝑝(𝑡) is equalized to the blanking threshold 𝑇 ℎ . The resolution of the equation yields two solutions:

𝑝(𝑡) = 𝑇 𝐻 ⇔ { 𝑤 𝑖,𝑗 𝑙 = 𝜏 𝑖 𝑗 - √ ln ( 𝑃 𝑖 𝑗 𝑇 ℎ ⁄ ) 𝛼 𝑤 𝑖,𝑗 𝑟 = 𝜏 𝑖 𝑗 + √ ln ( 𝑃 𝑖 𝑗 𝑇 ℎ ⁄ ) 𝛼 (19)
Where 𝑤 𝑖,𝑗 𝑙 is the instant of time the echo starts to trigger the threshold (the upper index "𝑙" stands for left) and 𝑤 𝑖,𝑗 𝑟 is the instant of time the echo stops to trigger the threshold (the upper index "𝑟" stands for right).

4.C. 𝒃𝒅𝒄 MATHEMATICAL MODEL CONSIDERING MULTIPATH

In this section, first a simple example, one LOS signal and one echo, from Figure 7, is presented to introduce the mathematical definitions, notions and elements necessary to define an updated 𝑝𝑤 and 𝑏𝑑𝑐 formula considering multipath. Afterwards, these formulas are generalized to consider any potential combination of LOS and number of echoes.

Application to a simple case: From Figure 7, the total blanked interval 𝐼 𝑖 𝑏𝑑𝑐 is no longer one interval created by the direct pulse but the combination of the blanked intervals generated by the direct and the echoed pulse. The blanked intervals generated by the direct and the echoed pulse are denoted 𝐼 𝑖,1 𝑏𝑑𝑐 and 𝐼 𝑖,2 𝑏𝑑𝑐 respectively. With these new notations, 𝐼 𝑖 𝑏𝑑𝑐 is derived from:

𝐼 𝑖 𝑏𝑑𝑐 = 𝐼 𝑖,1 𝑏𝑑𝑐 𝑈𝐼 𝑖,2 𝑏𝑑𝑐 (20)
Where:

{ 𝐼 𝑖,1 𝑏𝑑𝑐 = [𝑤 𝑖,1 𝑙 , 𝑤 𝑖,1 𝑟 ] 𝐼 𝑖,2 𝑏𝑑𝑐 = [𝑤 𝑖,2 𝑙 , 𝑤 𝑖,2 𝑟 ] (21)
𝐼 𝑖 𝑏𝑑𝑐 can also be seen at the union of all the disjoint blanked intervals 𝐵 𝑖 𝑘 . In the simple case presented Figure 7, they are two disjoint blanked intervals, which are directly equals to 𝐼 𝑖,1 𝑏𝑑𝑐 and 𝐼 𝑖,2 𝑏𝑑𝑐 . With this new definition, 𝐼 𝑖 𝑏𝑑𝑐 is derived from:

𝐼 𝑖 𝑏𝑑𝑐 = 𝐵 𝑖 1 𝑈𝐵 𝑖 2 (22)
The above threshold pulse width 𝑝𝑤 is then simply retrieved by applying Equation ( 8):

𝑝𝑤 = 𝑙𝑒𝑛𝑔𝑡ℎ(𝐼 𝑖 𝑏𝑑𝑐 ) = 𝑙𝑒𝑛𝑔𝑡ℎ(𝐵 𝑖 1 𝑈𝐵 𝑖 2 ) (23)
Generalization to 𝑱 𝒊 echoes: Generalizing to a number 𝐽 𝑖 of echoes, the updated expression for 𝐼 𝑖 𝑏𝑑𝑐 is:

𝐼 𝑖 𝑏𝑑𝑐 = ⋃ 𝐼 𝑖,𝑗 𝑏𝑑𝑐 𝐽 𝑖 𝑗=1 = ⋃ 𝐵 𝑖 𝑘 𝐾 𝑖 𝑘=1 (24)
Where:

• 𝐽 𝑖 is the number of echoes generated from source 𝑖,

• 𝐼 𝑖,𝑗 𝑏𝑑𝑐 is the blanked intervals generated by the echo 𝑗 of source 𝑖,

• 𝐾 𝑖 if the number of disjoint blanked intervals for source 𝑖, 𝐾 𝑖 ≤ 𝐽 𝑖 • 𝐵 𝑖 𝑘 are the disjoint blanked intervals, retrieved from the list of 𝐼 𝑖,𝑗

𝑏𝑑𝑐

From the new definition of 𝐼 𝑖 𝑏𝑑𝑐 , the new formula for 𝑝𝑤 considering multipath is:

𝑝𝑤 = 𝑙𝑒𝑛𝑔𝑡ℎ(𝐼 𝑖 𝑏𝑑𝑐 ) = 𝑙𝑒𝑛𝑔𝑡ℎ (⋃ 𝐵 𝑖 𝑘 𝐾 𝑖 𝑘=1 ) (25)
Which finally gives, from Equation ( 9), the 𝑏𝑑𝑐 mathematical model now considering multipath:

𝑏𝑑𝑐 = 1 -𝑒 -2 ∑ 𝑙𝑒𝑛𝑔𝑡ℎ(⋃ 𝐵 𝑖 𝑘 𝐾 𝑖 𝑘=1 )•𝑃𝑅𝐹 𝑖 𝑁 𝑖=1 (26)

4.D. 𝑹 𝑰 MATHEMATICAL MODEL CONSIDERING MULTIPATH

In this section, first a simple example, one LOS signal and one echo, from Figure 7 is presented to introduce the mathematical definitions, notions and elements necessary to define an updated 𝑃𝑊 and 𝑅 𝐼 formula considering multipath. Afterwards, these formulas are generalized to consider any potential combination of LOS and number of echoes.

Application to a simple case: From Figure 7, there are two 𝑃𝑊 𝑖,𝑝𝑢𝑙𝑠𝑒 that need to be derived. The one from the direct pulse, 𝑃𝑊 𝑖,𝑝𝑢𝑙𝑠𝑒 1 , and the one for the echoed pulse, 𝑃𝑊 𝑖,𝑝𝑢𝑙𝑠𝑒 2 . Both of these pulses are generating a blanked disjoint interval 𝐵 𝑖 1 and 𝐵 𝑖 2 , which blanks part of their tails. Taking the example of 𝑃𝑊 𝑖,𝑝𝑢𝑙𝑠𝑒 1 , applying Equation ( 14) and stating that 𝜏 𝑖 0 = 0 (the direct pulse has no additional delay) 𝑃𝑊 𝑖 1 is obtained as:

𝑃𝑊 𝑖,𝑝𝑢𝑙𝑠𝑒 1 = ∫ 𝑒 -𝛼(𝑡-𝜏 𝑖 0 ) 2 𝑑𝑡 -𝑤 𝑖,1 𝑙 -∞ + ∫ 𝑒 -𝛼(𝑡-𝜏 𝑖 0 ) 2 𝑑𝑡 𝑤 𝑖,2 𝑙 𝑤 𝑖,1 𝑟 + ∫ 𝑒 -𝛼(𝑡-𝜏 𝑖 0 ) 2 𝑑𝑡 +∞ 𝑤 𝑖,2 𝑟 (27)
Now, in order to keep the same formalism from Equation ( 14), it is possible to rewrite this expression as:

𝑃𝑊 𝑖,𝑝𝑢𝑙𝑠𝑒 1 = ∫ 𝑒 -𝛼(𝑡-𝜏 𝑖 0 ) 2 𝑑𝑡 -𝑤 𝑖,1 𝑙 -∞ + ∫ 𝑒 -𝛼(𝑡-𝜏 𝑖 0 ) 2 𝑑𝑡 -∫ 𝑒 -𝛼(𝑡-𝜏 𝑖 0 ) 2 𝑑𝑡 +∞ 𝑤 𝑖,2 𝑙 +∞ 𝑤 𝑖,1 𝑟 + ∫ 𝑒 -𝛼(𝑡-𝜏 𝑖 0 ) 2 𝑑𝑡 +∞ 𝑤 𝑖,2 𝑟 (28)
Which yields, applying 𝑢 = √𝛼(𝑡 -𝜏 𝑖 0 ) change of variable:

𝑃𝑊 𝑖,𝑝𝑢𝑙𝑠𝑒 1 = √𝜋/𝛼 2 [𝑒𝑟𝑓𝑐(√𝛼|𝑤 𝑖,1 𝑙 -𝜏 𝑖 0 |) + 𝑒𝑟𝑓𝑐(√𝛼|𝑤 𝑖,1 𝑟 -𝜏 𝑖 0 |) -(𝑒𝑟𝑓𝑐(√𝛼|𝑤 𝑖,2 𝑙 -𝜏 𝑖 0 |) -𝑒𝑟𝑓𝑐(√𝛼|𝑤 𝑖,2 𝑟 -𝜏 𝑖 0 |))] (29)
Which is the 𝑃𝑊 𝑖,𝑝𝑢𝑙𝑠𝑒 obtained without any multipath, with a removed part due to the blanked interval generated by the echoed pulse. Applying the exact same methodology for 𝑃𝑊 𝑖 2 , we obtain:

𝑃𝑊 𝑖,𝑝𝑢𝑙𝑠𝑒 2 = √𝜋/𝛼 2 [𝑒𝑟𝑓𝑐(√𝛼|𝑤 𝑖,2 𝑙 -𝜏 𝑖 1 |) + 𝑒𝑟𝑓𝑐(√𝛼|𝑤 𝑖,2 𝑟 -𝜏 𝑖 1 |) -(𝑒𝑟𝑓𝑐(√𝛼|𝑤 𝑖,1 𝑟 -𝜏 𝑖 1 |) -𝑒𝑟𝑓𝑐(√𝛼|𝑤 𝑖,1 𝑙 -𝜏 𝑖 1 |))] (30)
An interesting comment to make here is that the first 𝑒𝑟𝑓𝑐 removed in Equation ( 30) is the one starting at 𝑤 𝑖,1 𝑟 (right part of the blanked interval generated by the direct pulse) while the one added is the one starting at 𝑤 𝑖,1 𝑙 (left part of the blanked interval generated by the direct pulse). This is the opposite in Equation ( 29).

Therefore, from Equation ( 29) and Equation ( 30), two rules can be stated:

• The addition of a blanked interval 𝐵 𝑖 𝑘 adds a double sum of 𝑒𝑟𝑓𝑐 functions to the 𝑃𝑊 derivation

• The sign of both these added 𝑒𝑟𝑓𝑐 functions depends on the position of the added blanked interval relatively to the pulse considered Generalization to 𝑱 𝒊 echoes: Generalizing to a number 𝐽 𝑖 of echoes, the updated expression for 𝑃𝑊 is, from Equation ( 11):

𝑃𝑊 𝑖 = √𝜋/𝛼 ∑ 𝑃 𝑖 𝑗 𝐽 𝑖 𝑗=0 [2 (1 -1 𝐼 𝑖 𝑏𝑑𝑐 (𝜏 𝑖 𝑗 )) + ∑ 𝑠𝑖𝑔𝑛(𝜏 𝑖 𝑗 -𝐵 𝑖 𝑘,𝑙 ) 1-1 𝐵 𝑘 (𝜏 𝑖 𝑗 ) 𝑒𝑟𝑓𝑐(√𝛼|𝐵 𝑖 𝑘,𝑙 -𝜏 𝑖 𝑗 |) + +𝑠𝑖𝑔𝑛(𝐵 𝑖 𝑘,𝑟 -𝜏 𝑖 𝑗 ) 1-1 𝐵 𝑘 (𝜏 𝑖 𝑗 ) 𝑒𝑟𝑓𝑐(√𝛼|𝐵 𝑖 𝑘,𝑟 -𝜏 𝑖 𝑗 |) 𝐾 𝑖 𝑘=1 ] (31)
Where:

• 𝐽 𝑖 is the number of echoes generated by source 𝑖 • 𝐾 𝑖 is the number of disjoint blanked intervals generated by the 𝐽 𝑖 echoes of source 𝑖,

• 1 𝐴 (𝑥) is the indicator function of the ensemble 𝐴, 1 𝐴 (𝑥) = { 1 𝑖𝑓 𝑥 ∈ 𝐴 0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 ,

• 𝑃 𝑖 𝑗 is the received power generated by the obstacle 𝑗,

• 𝜏 𝑖 𝑗 is the additional delay of echo j with respect to the direct pulse generated by source 𝑖.

Which finally gives, from Equation ( 15), the 𝑅 𝐼 mathematical model now considering multipath:

𝑅 𝐼 = ∑ 𝑆𝑆𝐶(𝛥𝑓 𝑖 )𝑃𝑅𝐹 𝑖 √ 𝜋 𝛼 ⁄ 𝑁 0 𝛽 0 𝑁 𝑖=1 • (∑ 𝑃 𝑖 𝑗 𝐽 𝑖 𝑗=0 [2 (1 -1 𝐼 𝑖 𝑏𝑑𝑐 (𝜏 𝑖 𝑗 )) + ∑ 𝑠𝑖𝑔𝑛(𝜏 𝑖 𝑗 -𝐵 𝑖 𝑘,𝑙 ) 1-1 𝐵 𝑖 𝑘 (𝜏 𝑖 𝑗 ) 𝑒𝑟𝑓𝑐(√𝛼|𝐵 𝑖 𝑘,𝑙 -𝜏 𝑖 𝑗 |) + +𝑠𝑖𝑔𝑛(𝐵 𝑖 𝑘,𝑟 -𝜏 𝑖 𝑗 ) 1-1 𝐵 𝑖 𝑘 (𝜏 𝑖 𝑗 ) 𝑒𝑟𝑓𝑐(√𝛼|𝐵 𝑖 𝑘,𝑟 -𝜏 𝑖 𝑗 |) 𝐾 𝑖 𝑘=1 ]) (32)

4.E. IMPORTANT REMARK

A very important remark from both 𝑏𝑑𝑐 and 𝑅 𝐼 mathematical models considering multipath is that to compute 𝑏𝑑𝑐, the length of the 𝐵 𝑖 𝑘 must be derived, which implies the knowledge of the 𝑤 𝑖,𝑗 𝑙 and 𝑤 𝑖,𝑗 𝑟 and so 𝜏 𝑖 𝑗 and 𝑃 𝑖 𝑗 (Equation ( 19)). 𝜏 𝑖 𝑗 and 𝑃 𝑖 𝑗 are also directly used in the 𝑅 𝐼 computation. Therefore, to compute the 𝐶/𝑁 0 degradation, the precise knowledge of 𝑷 𝒊 𝒋 and 𝝉 𝒊 𝒋 is necessary. These values are often determined using a channel propagation model. The presentation of the proposed propagation channel model, specially adapted for the DME/TACAN interference analysis, is precisely the object of the next section.

PROPOSED PROPAGATION CHANNEL MODEL

As mentioned in the previous section, to be able to apply the 𝐶/𝑁 0 degradation analytical model considering multipath, the precise knowledge of the received power 𝑃 𝑖 𝑗 and the additional delay 𝜏 𝑖 𝑗 of the echoed pulse with respect to the direct ray is required to calculate the values of 𝑏𝑑𝑐 and 𝑅 𝐼 . The objective of the propagation channel is thus to obtain these values from few information about the obstacles. The scenarios investigated in this article and in future standards are scenarios where the aircraft is at a certain flight level (likely ≥ 2100 feet) which means that a high number of obstacles around many DME/TACAN stations must be considered. Therefore, on one hand, the propagation channel model proposed in this work is as simple as possible in order to deal with many obstacles without being too computationally expensive. On the other hand, the proposed channel model should be complex enough to provide a realistic impact of the multipath. The solutions and assumptions considered for the propagation channel derivation in order to reach the desired trade-off between simplicity and representativity are presented in the next subsections.

5.A. PRESENCE OF MULTIPATH

The first set of assumptions, namely the Radio Line Of Sight (RLOS), the shadowing phenomenon and the ground effect, concerns the determination of whether an obstacle present in the scenario reflects the direct signal or not. Indeed, considering the high number of DME/TACAN beacons potentially communicating with the aircraft, it is important to limit the number of obstacles reflecting the direct signal around these beacons while ensuring that the most threatening ones are not overlooked. These assumptions are presented in this subsection.

Radio Line Of Sight (RLOS):

Only the obstacles that are in the Radio Line Of Sight (RLOS) horizon of the DME/TACAN beacons are kept as potential candidates to reflect the direct signal. Using an effective earth radius of 4/3𝑅 𝑇 , where 𝑅 𝑇 is the actual earth radius as depicted in ITU-R P834-9, (2017), the RLOS is formulated as:

𝑅𝐿𝑂𝑆 = √(4/3𝑅 𝑇 + 𝐻 𝐷𝑀𝐸 )² -(4/3𝑅 𝑇 )² + √(4/3𝑅 𝑇 +𝐻)² -(4/3𝑅 𝑇 )² (33)
Where:

• 𝑅 𝑇 is the earth radius, 𝑅 𝑇 = 6378,14 𝑘𝑚

• 𝐻 𝐷𝑀𝐸 is the height of the DME/TACAN beacon antenna,

• 𝐻 is the height of the obstacle.

In the case of DME/TACAN, the height above ground of the beacon's antenna is usually 10 meters (𝐻 𝐷𝑀𝐸 = 10 𝑚). For example, with an obstacle's height of 10 meters (𝐻 = 10 𝑚), the RLOS is about 25 𝑘𝑚.

Shadowing:

A shadowing-phenomenon is also assumed in the model. This means that obstacles in the shadow of any other obstacles are not considered to reflect the direct signal (since it is assumed that the direct signal does not reach them). Figure 8 provides an example of this phenomenon, where the red obstacle does not reflect the direct signal because of the shadow created by the green obstacle.

Ground reflection: From simplification purpose, the ground reflection is not considered.

5.B. ASSUMPTIONS FOR THE ADDITIONAL DELAY 𝝉 𝒊 𝒋 CALCULATION

In this subsection, the assumptions used for the additional delay 𝜏 𝑖 𝑗 calculation, namely the first order of reflection and the earth flat assumption, are presented.

First order of reflection:

To simplify the analysis, only the first order of reflection are considered, meaning that the obstacles can only reflect the ray coming from the DME/TACAN beacon's antenna.

Earth flat: For simplification purpose, the earth is considered flat in our study. Similarly, the earth relief is not considered.

5.C. EXPRESSION OF THE ADDITIONAL DELAY 𝝉 𝒊 𝒋

Considering the previous assumptions, the formula of the additional delay 𝜏 𝑖 𝑗 of the echoes with respect with the LOS ray is the following:

𝜏 𝑖 𝑗 = 𝑅 1 + 𝑅 2 -𝐷 𝑐 (34)
Where:

• 𝑅 1 is the distance between the DME/TACAN beacon and the obstacle,

• 𝑅 2 is the distance between the obstacle and the aircraft,

• 𝐷 is the distance between the DME/TACAN beacon and the aircraft (direct path),

• 𝑐 is the speed of light.

5.D. ASSUMPTIONS FOR THE REFLECTED POWER 𝑷 𝒊 𝒋 DERIVATION

In this subsection, the assumptions used for the reflected power 𝑃 𝑖 𝑗 , namely the far-field propagation model and the obstacle material, are presented.

Far-field propagation model: The reflected power 𝑃 𝑖 𝑗 derivation is made assuming a far-field propagation model, meaning that the obstacle must be in the far-field of the DME/TACAN beacon antenna and that the aircraft must be in the far-field of the obstacle. This was deemed necessary since the resolution of the Stratton-Chu integrals is much easier in that case, as depicted in Chen, (2011). An object 𝐴 is in the far-field of an object 𝐵 if:

𝑅 ≥ 2𝐿 2 𝜆 (35)
Where:

• 𝑅 is the distance between 𝐴 and 𝐵,

• 𝐿 is the largest dimension of object 𝐴,

• 𝜆 is the wavelength associated to the DME/TACAN carrier frequency by 𝜆 = 𝑐/𝑓 𝑖 .

In the case of the link between the DME/TACAN antenna and the obstacle, the condition is always respected. Indeed, the maximum dimension 𝐿 of the DME/TACAN is the antenna size. Indeed, in that case, 𝐿 = 5m, 𝜆 = 25.5𝑐𝑚 (considering DME/TACAN emitting at exactly the L5 frequency) resulting in 𝑅 ≥ 200m, and a distance shorter than 200m between the DME and an obstacle is unlikely since no obstacle should be found in a circle of radius 300m around the DME/TACAN beacon according the guidance of ICAO, (2015).

Concerning the link between the obstacle's center and the aircraft, the condition is more difficult to enforce since the maximum dimension 𝐿 is now the dimension of the obstacle (that can reach a few hundred of meters). Therefore, to ensure that the far-field condition is always respected, not-far-field-complying obstacles are divided in smaller obstacles portions as much as needed such that every obstacle portion respects the far-field condition. The scattered electric field is then derived from every portion and the reflected power 𝑃 𝑖 𝑗 is retrieved from the coherent sum of the electric field of each individual portion. Figure 9 provides an example of that process, where the initial obstacle is divided three times. 1).

5.E. EXPRESSION OF THE REFLECTED POWER 𝑷 𝒊 𝒋

In this subsection, considering the previous assumptions, the derivation of the reflected power 𝑃 𝑖 𝑗 in the two following cases: when the obstacles are perfectly smooth, and when roughness is added to the surface, is conducted. Note that in this paper, the roughness term is used to characterize details such as windows, with dimension ≥ 𝜆, added to the obstacles. From Figure 10, four useful definitions can be introduced:

• 𝑠 𝑖 ̂ is the direction of incidence, 𝑠 𝑖 ̂= 𝐴𝑀 ⃗⃗⃗⃗⃗⃗ ||𝐴𝑀 ⃗⃗⃗⃗⃗⃗ ||
, where the DME/TACAN beacon antenna center is modelized 𝐴,

• 𝑠 𝑟 ̂ is the direction of observation, 𝑠 𝑟 ̂= 𝐵𝑀 ⃗⃗⃗⃗⃗⃗⃗

||𝐵𝑀 ⃗⃗⃗⃗⃗⃗⃗ ||

, where the aircraft antenna center is modelized by 𝐵,

• 𝑛 ̂ is the normal vector to the obstacle,

• 𝑢 ̂ and 𝑣 ̂ are the unit vectors completing the orthonormal base.

In the most general case, the calculation of the Stratton-Chu integrals gives the following equation for the reflected power 𝑃 𝑖 𝑗 :

𝑃 𝑖 𝑗 = 𝑃 𝑇𝑋 𝐺 𝑇𝑋 (𝑒𝑙 1 )𝐺 𝑅𝑋 (𝑒𝑙 2 ) (4𝜋) 2 𝑅 1 2 𝑅 2 2
|𝑅 𝑇𝐸 |²(𝐿𝐻)² 𝑓(𝑉, 𝐻)𝑔(𝑈, 𝐿) cos(𝜙 𝑝 -𝜙 1 )² sin(𝜃 2 )² (36) Where:

• 𝑅 𝑇𝐸 is the reflection coefficient (which depends on the material). The rationale to derive 𝑅 𝑇𝐸 is presented in [START_REF] Chabory | Modélisation Electromagnétique des radômes par des techniques basées sur les faisceaux gaussiens[END_REF].

• 𝑃 𝑇𝑋 is the transmitted power of the DME/TACAN beacon,

• 𝐺 𝑇𝑋 is the gain of DME/TACAN beacon's transmitting antenna,

• 𝐺 𝑅𝑋 is the gain of the receiving aircraft antenna,

• 𝑒𝑙 1 the elevation angle between the DME and the position of the obstacle center,

• 𝑒𝑙 2 is the elevation angle between the obstacle and the center of the aircraft antenna,

• 𝐿 and 𝐻 are respectively the length and the height of the obstacle,

• 𝑉 = (𝑠 𝑟 ̂-𝑠 𝑖 ̂). 𝑣 ̂, • 𝑈 = (𝑠 𝑟 ̂-𝑠 𝑖 ̂). 𝑢 ̂,
• 𝜙 1 is the azimuth angle between the DME/TACAN beacon and the obstacle, • 𝜙 𝑝 is the angle between 𝑥 ̂ and 𝑛 ̂, • 𝜃 2 is the 𝜃 angle from the spherical coordinate system between the obstacle and the aircraft,

• 𝒇 and 𝒈 are two functions that will be customized for the two cases presented in this subsection: when the obstacles are perfectly smooth, and when roughness is added to the surface.

Case when the obstacles are perfectly smooth: From Figure 10, and following the calculation of the Stratton-Chu integrals in the case of a perfectly smooth obstacle in the far-field condition conducted in Chen, (2011), the function 𝑓 and 𝑔 are perfectly solved as 𝑠𝑖𝑛𝑐² functions, such that the reflected power 𝑃 𝑖 𝑗 in that case is:

𝑃 𝑖 𝑗 = 𝑃 𝑇𝑋 𝐺 𝑇𝑋 (𝑒𝑙 1 )𝐺 𝑅𝑋 (𝑒𝑙 2 ) (4𝜋) 2 𝑅 1 2 𝑅 2 2 |𝑅 𝑇𝐸 |²(𝐿𝐻)² 𝑠𝑖𝑛𝑐² (𝑘 0 𝑉𝐻 2 ) 𝑠𝑖𝑛𝑐²(𝑘 0 𝑈𝐿 2 ) cos(𝜙 𝑝 -𝜙 1 )² sin(𝜃 2 )² (37)
Where:

• 𝑘 0 is the wave number, 𝑘 0 = 2𝜋 𝜆 .

Case when roughness is added to the obstacle:

To improve the realism of the propagation channel model, the inclusion of windows to the facades of the obstacles is necessary since most of them represent houses or other non-flat/non-smooth buildings.

Generally speaking, roughness is included in most propagation channel model thanks to the precise knowledge of the facades' roughness and by meshing the obstacles with rectangles in the order of 𝜆. However, no such knowledge is available for the high number of obstacles used in the analyzed scenarios . Moreover, the computational power required to produce such a precise meshing on all these obstacles is too high and will upset simplicity objective set for the modelling of this propagation channel.

To cope with these difficulties, it is proposed to find a customization of 𝑓 and 𝑔 thanks to the study of a unique facade, representative of a middle-tall sized building. The analysis is conducted in FRME (Fortran Range Multipath Error), a propagation channel model specially created for GNSS multipath analysis, Chen, (2011). Since a unique facade is studied, the facade details are perfectly introduced. Figure 11 represents the studied facade, where the core material can be customized to be metal, concrete, brick or wood. The objective of the study is to find two simple functions 𝑓 and 𝑔 such that the two sinc² functions from Equation ( 37) could be replaced to accurately introduce the details of the facade. From the observation of FMRE simulation results, it was observed that 𝑓 and 𝑔 were two sinc look-alike functions with: 1. Their highest peaks reduced 2. Never reaching zero 3. Having additional peaks that are Grating lobes created by the spatial periodicity of the windows 4. All previous observations depended on the obstacle material.

From these observations, function 𝑓 and 𝑔 are modelled as 𝑠𝑖𝑛𝑐² functions multiplied by a coefficient (< 1 except for wood), with the zeros removed (zeroes are replaced by the value of the nearest peak) and with a minimum value fixed at the maximum value of the periodicity peaks. 𝑓 and 𝑔 customized functions are called 𝑠𝑖𝑛𝑐 𝑒𝑞 𝑣𝑒𝑟𝑡 and 𝑠𝑖𝑛𝑐 𝑒𝑞 ℎ𝑜𝑟 . Their mathematical models are:

{ 𝑓(𝑈, 𝐿) = 𝑠𝑖𝑛𝑐 𝑒𝑞 ℎ𝑜𝑟 (𝑘 0 𝑈𝐿 2 ) 2 = 𝐾 • max (𝑠𝑖𝑛𝑐 𝑤0 (𝑘 0 𝑈𝐿 2 ) , ( 𝑀 𝑝𝑒𝑎𝑘 ℎ𝑜𝑟 𝐾 ) ) ² 𝑔(𝑉, 𝐻) = 𝑠𝑖𝑛𝑐 𝑒𝑞 𝑣𝑒𝑟𝑡 (𝑘 0 𝑉𝐻 2 ) 2 = 𝐾 • max (𝑠𝑖𝑛𝑐 𝑤0 (𝑘 0 𝑉𝐻 2 ) , ( 𝑀 𝑝𝑒𝑎𝑘 𝑣𝑒𝑟𝑡 𝐾 ) ) ² (38)
Where:

• 𝐾 is the maximum of the electric scattered field obtained from the FMRE simulations,

• 𝑀 𝑝𝑒𝑎𝑘 𝑣𝑒𝑟𝑡 and 𝑀 𝑝𝑒𝑎𝑘 ℎ𝑜𝑟 are the maximum of the Grating lobes found in the electric scattered field from the FMRE simulations,

• 𝑠𝑖𝑛𝑐 𝑤0 is defined as:

𝑠𝑖𝑛𝑐 𝑤0 (𝑥) = max (𝑠𝑖𝑛𝑐(𝑥), 𝑠𝑖𝑛𝑐(𝜁(𝑥))) (39)
Where: From the new 𝑓 and 𝑔 functions, the new formula for Equation ( 36) considering the roughness of the obstacle is:

𝑃 𝑖 𝑗 = 𝑃 𝑇𝑋 𝐺 𝑇𝑋 (𝑒𝑙 1 )𝐺 𝑅𝑋 (𝑒𝑙 2 ) (4𝜋) 2 𝑅 1 2 𝑅 2 2 |𝑅 𝑇𝐸 |²(𝐿𝐻) 2 𝑠𝑖𝑛𝑐 𝑒𝑞 𝑣𝑒𝑟𝑡 (𝑘 0 𝑉𝐻 2 ) 2 𝑠𝑖𝑛𝑐 𝑒𝑞 ℎ𝑜𝑟 (𝑘 0 𝑈𝐿 2 )² cos(𝜙 𝑝 -𝜙 1 )² sin(𝜃 2 )² (41)
Where all the useful parameters to derive 𝑠𝑖𝑛𝑐 𝑒𝑞 𝑣𝑒𝑟𝑡 and 𝑠𝑖𝑛𝑐 𝑒𝑞 ℎ𝑜𝑟 for each of the material are provided in Table 1.

Table 1: Parameters summary for 𝑠𝑖𝑛𝑐 𝑒𝑞 functions

5.F. REMARK

Four obstacles' parameters are necessary to apply ( 41) and (37) reflected power formula:

1. The position of the center of the obstacles, that are needed to derive 𝑅 1 , 𝑅 2 and many other parameters, 2. The dimensions 𝐿 and 𝐻 of the obstacles, that are directly used in the equations, 3. The orientation of the obstacles, which is used to derive 𝜙 𝑝 , 4. The composition of the material, which is needed for the derivation of 𝑅 𝑇𝐸 and both 𝑠𝑖𝑛𝑐 𝑒𝑞 𝑣𝑒𝑟𝑡 and 𝑠𝑖𝑛𝑐 𝑒𝑞 ℎ𝑜𝑟 .

Knowing these four parameters, the value of the additional delay 𝜏 𝑖 𝑗 (Equation ( 34)) and the received power 𝑃 𝑖 𝑗 (Equation ( 37) or ( 41)) can be retrieved for any obstacle. From 𝜏 𝑖 𝑗 and 𝑃 𝑖 𝑗 , the value of 𝑅 𝐼 and 𝑏𝑑𝑐 can be determined (Equation ( 32) and ( 26)) and finally the value of the 𝐶/𝑁 0 computed (Equation ( 5)).

NUMERICAL APPLICATIONS

Now that the different models to determine the 𝐶/𝑁 0 degradation in presence of multipath has been introduced, two numerical applications are introduced: A. A comparison of the impact of multipath at low and at high altitude.

B. An application of the 𝐶/𝑁 0 degradation model at a potential low-altitude US hot-spot, JALTO.

Material

6.A. COMPARISON OF THE IMPACT OF MULTIPATH AT LOW AND HIGH ALTITUDE

In this subsection, the first numerical application of the propagation channel model and the 𝐶/𝑁 0 degradation due to DME/TACAN only in presence of multipath is presented. After a presentation of the objective of the analysis, the methodology is introduced and further detailed. Finally, the results are presented and commented.

The objective of the first numerical application is to compare the impact of multipath at low and at high altitude flight levels. To achieve this objective, a single perfectly smooth metallic obstacle is considered for simplicity purposes. We are interested in the minimum area that is necessary to increase the above threshold pulse width 𝑝𝑤 by 50% (Equation ( 25)). Therefore, it is proposed to compare this minimum area at two Flight Levels (FL), FL21 and FL400 through six different scenarios. Note that this analysis is by no means exhaustive (other materials, flight levels and 𝑝𝑤 augmentations could have been investigated) but should provide a tendency on how the multipath impacts both low altitude and high altitude flight levels scenarios.

The 6 analyzed are defined as follows:

• The PEP of the DME/TACAN beacon, 𝑃 𝑇𝑋 , is either 20 or 30 dBW, to accurately model approach-used DME/TACAN beacons (𝑃 𝑇𝑋 = 20 dBW) or en-route used DME/TACAN beacon (𝑃 𝑇𝑋 = 30 dBW). • The Line Of Sight (LOS) received power PEP, 𝑃 𝑅𝑋 is chosen to be either -90,-100 or -110 dBW, to accurately model direct pulses being largely, partially and moderately above the blanking threshold (-120dBW). • The height 𝐻 of the obstacles is always set to 10 𝑚 • The blanking threshold is set to -120 𝑑𝐵𝑊 A summary of the 6 scenarios is provided Table 2.

Table 2: Summary table of the 6 defined scenarios

The proposed methodology to obtain the minimum area for the 6 scenarios at the two FLs is as follows:

1. Determination of the distance 𝐷 between the DME/TACAN beacon and the aircraft from 𝑃 𝑇𝑋 and 𝑃 𝑅𝑋 and 𝐹𝐿 2. Determination of the objective reflected peak envelope power 𝑃 𝑖 𝑗,𝑜𝑏𝑗 necessary to reach the objective of having a 𝑝𝑤 increased by 50% for every additional delay 𝜏 𝑖 𝑗 , where 𝜏 𝑖 𝑗 varies from 0.264 𝜇𝑠 to 7,92 𝜇𝑠 (3 times the rectangular equivalent width of a below-blanker pulse). Therefore, an initial candidate list of additional delays 𝜏 𝑖 𝑗 (values between 0.264 𝜇𝑠 to 7,92 𝜇𝑠) is defined and later processed in this step. This choice of values for 𝜏 𝑖 𝑗 is made to limit the obstacles' area obtained. 3. Determination of the obstacles' positions that generates 𝜏 𝑖 𝑗 values 4. Determination of the length 𝐿 of the obstacles to reach 𝑃 𝑖 𝑗 = 𝑃 𝑖 𝑗,𝑜𝑏𝑗 from 𝑃 𝑇𝑋 , the obstacles, DME and aircraft positions.

Note that the multiplication of L by 𝐻, which has been arbitrary chosen to 10m for this study, gives the different areas needed to reach a value of 𝑝𝑤 increased by 50%.

Determination of the distance 𝑫:

The distance 𝐷 between the DME/TACAN beacon and the aircraft is chosen such that LOS received power 𝑃 𝑅𝑋 is either -90,-100 or -110 dBW, given the scenario. 𝐷 is retrieved from the free-space losses equation:

𝑃 𝑅𝑋 = 𝑃 𝑇𝑋 𝐺 𝑇𝑋 (𝑒𝑙)𝐺 𝑅𝑋 (-𝑒𝑙). ( 𝜆 4𝜋𝐷 2 ) (42)
Where: • 𝑒𝑙 is the elevation angle between the DME/TACAN beacon and the aircraft Determination of the objective reflected power 𝑷 𝒊 𝒋,𝒐𝒃𝒋 : The objective power 𝑃 𝑖 𝑗,𝑜𝑏𝑗 needed to increase 𝑝𝑤 by 50% is found for every possible delay 𝜏 𝑖 𝑗 thanks to an algorithm testing every possible value of 𝑃 𝑖 𝑗,𝑜𝑏𝑗 starting at the blanking threshold 𝑇 𝐻 as its minimum value (𝑝𝑤 is not increased if the echoed pulse does not trigger the blanker and thus has a power above the threshold) and ending at LOS received power 𝑃 𝑅𝑋 as its maximum value (an echoed pulse cannot generate a power above the LOS power, unless for very specific cases that are outside the scope of this article).

𝑃 𝑇𝑋 (𝑑𝐵𝑊) 𝑃 𝑅𝑋 (𝑑𝐵𝑊) 𝐻 (
For every tested 𝑃 𝑖 𝑗,𝑜𝑏𝑗 , the values of the blanked instant of times 𝑤 𝑖,2 𝑙 and 𝑤 𝑖,2 𝑟 are determined from Equation ( 19). From 𝑤 𝑖,2 𝑙 and 𝑤 𝑖,2 𝑟 the blanked interval of the echoed pulse 𝐼 2 𝑏𝑑𝑐 is determined. From 𝐼 2 𝑏𝑑𝑐 and 𝐼 1 𝑏𝑑𝑐 , the disjoint blanked intervals can be determined and so 𝑝𝑤 (From Equation ( 25)). If for a specific additional delay 𝜏 𝑖 𝑗 , no objective power 𝑃 𝑖 𝑗,𝑜𝑏𝑗 valid value to increase 𝑝𝑤 by 50% is found (tested 𝑃 𝑖 𝑗,𝑜𝑏𝑗 ≤ 𝑃 𝑅𝑋 ), then this specific additional delay is removed from the candidate list of additional delay to investigate in the remaining steps of the methodology. Notes that the smaller the additional delay 𝜏 𝑖 𝑗 , the more difficult it is for the echoed pulse to increase 𝑝𝑤 by 50%. Determination of the obstacles' positions: All the obstacles' positions are defined such that an obstacle generates at least one of the targeted additional delays, 𝜏 𝑖 𝑗 , of the candidates list of additional delay generated in step 2. Three constraints must be additionally respected (if these 3 constraints are not respected, the obstacle position is not valid and discarded):

1. The obstacles must have a height 𝐻 = 10𝑚 (entry of the analysis).

2. The obstacles must respect the DME servitude.

3. The obstacles must respect the aircraft servitude.

For each individual delay 𝜏 𝑖 𝑗 , the obstacles' generating an additional delay 𝜏 𝑖 𝑗 are perfectly determined by an ellipsoid, with the following equation: Where:

𝑅 1 + 𝑅 2 = 𝐷 + 𝑐 • 𝜏 𝑖 𝑗 (43)
• 𝑅 1 is the distance between the DME/TACAN beacon and the obstacle,

• 𝑅 2 is the distance between the obstacle and the aircraft,

• 𝐷 is the distance between the DME/TACAN beacon and the aircraft (direct path),

• 𝑐 is the speed of light.

To fulfill the first constraint, the intersection of the ellipsoid and the plan 𝑧 = 𝐻/2 is determined. To respect the second constraint, the obstacles within a cylinder of radius 300𝑚 centered at the DME/TACAN beacons are excluded from the analysis (ICAO, 2015). To respect the third constraint, obstacles within a cylinder centered at the aircraft of radius 0.1 𝑛𝑚 (=185.2 𝑚) starting 150m above earth are excluded from the analysis.

Figure 14 shows (a) an example of an ellipsoid associated to an additional delay, with the two zones of servitude (in yellow) and the plan 𝑧 = 𝐻/2 and (b) the resulting intersection of the ellipsoid with the plan.

Determination of the length 𝑳:

The last step of the methodology is to find the lengths 𝐿 of the obstacles that generate a reflected power 𝑃 𝑖 𝑗 equal to the objective power 𝑃 𝑖 𝑗,𝑜𝑏𝑗 necessary to increase 𝑝𝑤 by 50%, where 𝑃 𝑖 𝑗 is obtained from Equation (37)

(obstacles are assumed perfectly smooth).

Recall that to be able to use Equation ( 37), 4 parameters are needed:

• The position of the obstacles (which is known from the additional delay 𝜏 𝑖 𝑗 )

• The dimensions 𝐿 and 𝐻 of the obstacle. 𝐻 = 10𝑚 is known since it is on of the entries of the analysis.

• The composition of the obstacle which is known since the obstacles have been assumed to be perfectly metallic.

• The orientation of the obstacle.

Since the horizontal orientation 𝜙 𝑝 of the obstacle is unknown, it is proposed to customize Equation ( 37) by computing the Cumulative Density Function (CDF) of 𝜓(𝜙 𝑝 ) where 𝜓 is the product of the two terms from Equation (37) that depends on the horizontal orientation:

𝜓(𝜙 𝑝 ) = 𝑠𝑖𝑛𝑐 (𝑘 0 𝑈𝐿 2 ) ²cos (𝜙 𝑝 -𝜙 1 )² (44)
The 99 𝑡ℎ quantile of that CDF, 𝜓 99 , is then chosen in order to model a realistic worst-case scenario, excluding the case where the obstacle always generate a specular reflection (unrealistic worst-case scenario), but still having 𝜓(𝜙 𝑝 )< 𝜓 99 99% of the time.

The expression of the reflected power 𝑃 𝑖 𝑗,99 that depends on 𝐿 can thus be expressed as: Multiple values of 𝐿, from 10 to 1000, are then tested to see if it is possible in this range of value to fulfil 𝑃 𝑖 𝑗,99 = 𝑃 𝑖 𝑗,𝑜𝑏𝑗 . If it is the case, the value of 𝐿 is multiplied by 𝐻 to obtain, for a given position, the area needed to increase 𝑝𝑤 by 50%. Figure 15 provides an example of the results obtained for the scenario 4 at FL21 where an upper view of the scene is represented. For each possible position, the value of the necessary area is represented by a color. The position is left blank if pw has not increased by 50% when 𝐿 = 1𝑘𝑚. The minimum area is surrounded by a red circle. Note that the discontinuities in the middle of the figure are only due to the step increase to speed up the process since it was noticed that less possible obstacles verifying 𝐿 < 1000 𝑚 to increase 𝑝𝑤 by 50% were found in this area.

𝑃 𝑖 𝑗,
The results of the minimum area for the 6 scenarios applied to FL21 and FL400 are presented in Table 3.

Table 3: Results on the minimum area needed to increase 𝑝𝑤 by 50% for the 6 different scenarios

From these results, it can be seen that it is more difficult to increase 𝑝𝑤 by 50% at high altitude (FL400) compared to low altitude (FL21), since the minimum area found at high-altitude is almost 4 times higher for every scenario. Therefore, it can be said, at least within the scope of this special application, that the impact of multipath is most significant at low-altitude FL.

6.B. APPLICATION TO JALTO, PHILADELPHIA

From a standardization point of view, the main concern brought by multipath impact is that the 𝐶/𝑁 0 degradation generated by a low-altitude hot-spot exceeds the 𝐶/𝑁 0 degradation already observed at Harrisburg, FL400. Indeed, the FL400 hotspot is determined as the location where the 𝐶/𝑁0 𝑒𝑓𝑓 is the lowest one (as well as the location where the link budget margin is the smallest one) and thus the FL400 hotspot is the one driving the RFI compatibility mask (Garcia-Pena, Macabiau, et al., 2020). However, the 𝐶/𝑁0 𝑒𝑓𝑓 calculation and specially the DME/TACAN 𝑏𝑑𝑐 and 𝑅 𝐼 values were calculated without considering multipath impact.

The objective of the next application is precisely to tackle this question through the study of a potential low-altitude hot-spot, JALTO. JALTO has been chosen for two main reasons: have been digitalized using images that vary in date (from 2012 to 2021). In this database, the coordinates of the corners of each buildings are expressed in the WGS84 Coordinate Reference Systems (CRS). They are later expressed in the ENU coordinate system of all the visible DME/TACAN beacons for this special analysis. Figure 17 presents an example of a building footprint extracted in that database.

𝑃 𝑅𝑋 =
From the building footprints only, 3 important parameters needed to use the propagation channel model are retrieved: the position of the center, the length 𝐿 and the horizontal orientation 𝜙 𝑝 . Let's denote 𝐸 and 𝐹 the coordinates of two consecutive corners of a building:

• The position 𝑀 of the center of the obstacles is simply retrieved by:

𝑀 = (𝐹 -𝐸)/2 (46)
• The length 𝐿 of the obstacles is simply given by:

𝐿 = ||𝐹 -𝐸||² (47)
• The horizontal orientation 𝜙 𝑝 is the angle between 𝑥 and 𝑛 ⃗ . Since 𝑥 and 𝑛 ⃗ are unit vectors:

𝜙 𝑝 = { acos(𝑥 ⃗ ⃗ . 𝑛 ⃗ ⃗ ) 𝑖𝑓 𝑛 ⃗ ⃗ . 𝑦 ⃗ ⃗ ≥ 0 2𝜋 -acos(𝑥 ⃗ ⃗ . 𝑛 ⃗ ⃗ ) 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 (48)

From the positions, lengths and orientations of all the obstacles within the RLOS of all the DME/TACAN beacons, the shadowing phenomenon is applied resulting as a results in an exclusion of some of the obstacles inside the DME/TACAN RLOS. Figure 18 shows an example of the process of extracting the building footprint from the data base for the 11 th DME/TACAN beacon with an upper view of (a) the Google Maps position of the beacon, (b) the extraction of the center of every obstacle within the RLOS where the color of each point represents the length of the obstacle and (c) the exact same plot after having applied the shadowing phenomenon.

Since the height 𝐻 of the buildings were not publicly available from this database, it was arbitrarily decided to take 𝐻 = 8𝑚 for this analysis, which completes the dimension parameter for the application of the 𝐶/𝑁 0 degradation.

Estimation of obstacles' composition through five different scenarios:

Since the composition of the materials was not directly given by the Microsoft database, in this analysis the composition is estimated through 5 different scenarios derived from the observation of the DME/TACAN beacons surroundings in Google Maps.

A scenario is defined by the distribution (or percentage) of the obstacles' materials. Table 4 provides the distribution proposed for the first 4 scenarios.

Table 4: Material distributions for the 4 first scenarios

In scenarios 1 and 2, there is no difference between the distribution for small (𝐿 ≤ 50𝑚) and large (𝐿 > 50𝑚) obstacles. They are the simplest scenarios and the ones where the most metal obstacles are assumed to be present. Therefore, they should also be the worst-case scenarios. A small change is made on the metal composition distribution between these two scenarios.

In scenarios 3 and 4, a difference is made between small and large obstacles for the compositions of the materials. This was deemed necessary since it was found that the largest obstacles were made of concrete and metals (commercial centers, hospital, penitential center, etc.) whereas smaller obstacles were primarily made of a mix of concrete and wood (residential houses, small farms, etc.). A small change is made on the metal composition distribution between these two scenarios.

The scenario 5 is the most evolved one. It relies of the definition of 4 different zones:

• Zone 1 is associated with large cities such as Philadelphia and Wilmington. Presence of large buildings. Dominance of concrete and brick. • Zone 2 is associated with normal-sized cities. Presence of many buildings. Dominance of wood and brick for small buildings and concrete and brick for large. • Zone 3 is associated with more rural areas such as allotment. Dominance of brick and wood.

• Zone 4 is associated with coastal areas. Huge dominance of wood.

Table 5: Summary of the distribution for the 4 different zones for the scenario 5

In this scenario, each of the DME/TACAN beacons are assigned to a particular zone after a close observation of their surroundings. Figure 19 presents the 19 DME/TACAN beacons associated with their particular zone, zone 1 being represented in red, zone 2 in yellow, zone 3 in green and zone 4 in blue. The scenario 5 is the result of our finest estimation of the composition of the obstacles. It is the most realistic scenario of the 5 five scenarios defined despite still lacking the realism provided by an accurate database. Indeed, it must be noted that this scenario are best estimation efforts without any backing data except for detailed visual observations on Google maps.

Length

Application of the 𝑪/𝑵 𝟎 degradation model, results and conclusions:

From the scenarios previously defined, we draw a material for each obstacles according to the distribution associated with the scenario. This allows to have the four necessary parameters to derive the additional delay 𝜏 𝑖 𝑗 and the reflected power 𝑃 𝑖 𝑗 of all the obstacles in the RLOS of the 19 DME/TACAN beacons. From the additional delay 𝜏 𝑖 𝑗 and the reflected power 𝑃 𝑖 𝑗 , a value for 𝑅 𝐼 , 𝑏𝑑𝑐 and then the 𝐶/𝑁 0 are determined. Since an aleatory component has been added to the model with the introduction of the scenarios, this process is repeated 1000 times for each scenario and the mean and standard deviation (𝑠𝑡𝑑) of 𝑅 𝐼 , 𝑏𝑑𝑐 and the 𝐶/𝑁 0 degradation are determined. The results are presented in Table 6.

Table 6: Results summary of the 𝐶/𝑁 0 degradation at JALTO for the 5 different scenarios

From Table 6 two conclusions can be made: 1. Even for the scenario 1, which is the scenario with the most metal obstacles (worst case scenario), the 𝐶/𝑁 0 degradation due to DME/TACAN only found at JALTO including multipath does not exceed the original US hot spot found at Harrisburg, high altitude. Therefore, from these results, it seems unlikely that a new low altitude hot-spot replaces the original US hot-spot, even with the inclusion of multipath. 2. The 𝐶/𝑁 0 degradation due to DME/TACAN only have been increased by at least 2.67 dB from the multipath-free theory in the 5 different scenarios. Therefore, multipath seem to have a significant impact on the 𝐶/𝑁 0 degradation computation.

CONCLUSIONS

In the context of civil aviation, the RFI impact on a GNSS L5/E5a receiver is well characterized as the 𝐶/𝑁 0 degradation observed at the receiver's correlator output, or equivalently, as an increase of the effective 𝑁 0 denoted as 𝑁 0,𝑒𝑓𝑓 . In presence of a temporal blanker, 𝑁 0,𝑒𝑓𝑓 is derived from the blanker duty cycle 𝑏𝑑𝑐 and the below-blanker interfering-signal-to-thermal-noise ratio 𝑅 𝐼 .

However, the model proposed RTCA, ( 2004) to derive both 𝑅 𝐼 and 𝑏𝑑𝑐 does not include any consideration for multipath which could be problematic at low altitude. In this paper, the complete 𝑅 𝐼 , 𝑏𝑑𝑐 and 𝐶/𝑁 0 formulas have been reviewed and a new model considering multipath has been proposed. This new model requires the knowledge of precise time of arrival and received power of the echoed pulse. Therefore, a propagation model specifically adapted for the DME/TACAN interference analysis have also been proposed in this work.

This model has been developed to RFI-focused and to target the following trade-off: to be simple enough to be applied to a large number of scenarios with a high number of obstacles but to be complex enough to provide realistic estimations.

Two applications of the propagation channel model and the 𝐶/𝑁 0 degradation computation considering multipath have been presented. The first one consists of a comparison of the impact of the inclusion of multipath in the 𝐶/𝑁 0 degradation at high and at low altitude. The results shows that, for a single perfectly metallic obstacle, the impact at low altitude (FL21) is four time more significant than at high altitude (FL400). The second one is the study of a potential US low-altitude hot spot at JALTO, Philadelphia. Although the results tend to imply that it would be unlikely to find a new US hot-spot at low altitude due to the multipath, the impact of multipath is deemed significant since even the lowest 𝐶/𝑁 0 degradation found considering multipath (4.96 𝑑𝐵) is 2.67 𝑑𝐵 higher than the multipath-free theory (2.29 𝑑𝐵)

The main limitations to this work is the lack of real data containing DME/TACAN interferences as well as multipath, and the lack of information from the Microsoft database concerning multiple parameters of the obstacles (namely the height and the composition). This would need to be tackled in future works. Furthermore, an analysis on the 𝐶/𝑁 0 degradation due to DME/TCAN only considering multipath of a potential European hot-spot, REDGO, Frankfurt, is to be conducted.

Figure 1 (

 1 Figure 1 (left part) represents a normalized DME/TACAN pulse pair, modulated at 14 𝑀𝐻𝑧. Figure 1 (right part) represent the normalized pulse complex envelope of a DME/TACAN signal.

Figure 1 :

 1 Figure 1: DME signal pattern

Figure 2 .

 2 First, the antenna is the element responsible of capturing the incoming signal: at the antenna port (point A), there is a mix of all incoming signals; useful signals, GNSS and SBAS signals, and RFI signals such as DME/TACAN, JTIDS/MIDS, etc. Once the signals have been captured by the antenna, they are passed to the Radio-Frequency Front End (RFFE) block. This block amplifies the received signals, shifts them from their received signal frequency carrier to the intermediate frequency and filters them. The filtered signals are modelled in Figure 2 at the RF (Radio-Frequency) and IF (Intermediate Frequency) filters output at point B. The joint effect of these two filters and the antenna filtering are defined by RTCA, (2004)with an equivalent filter whose transfer function is provided Figure 3.

Figure 2 :Figure 3 :Figure 4 :

 234 Figure 2: Generic civil aviation GNSS receiver block scheme

Figure 5 :

 5 Figure 5: Representation of an above-threshold DME/TACAN normalized pulse

Figure 6 :

 6 Figure 6: Representation of DME/TACAN (a) weak pulse and (b) strong pulse

Figure 8 :

 8 Figure 8: Representation of shadowing-phenomenon, upper view

Figure 9 :Figure 10 :

 910 Figure 9: Example of the proposed methodology to ensure that the far-field condition is always respected (three divisions)

  General scheme: Figure 10 shows (a) the DME/TACAN beacon (represented by 𝐴), obstacle (represented by 𝑀) and aircraft (represented by 𝐵) scene in the spheric coordinate system and (b) a zoom on a perfectly smooth obstacle with center 𝑀, length 𝐿 and height 𝐻.

Figure 11 :

 11 Figure 11: Representation of the studied facade

Figure 12 :

 12 Figure 12: Representation of 𝑠𝑖𝑛𝑐 𝑒𝑞 ℎ𝑜𝑟 function for metal

  𝜁(𝑥) = { 𝑚𝑖 𝑛(𝑥 𝑛 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡 𝑥 𝑛 ≥ |𝑥|) ∀𝑛 ∈ [1, 𝑁] 𝑖𝑓 |𝑥| ≤ 𝑥 𝑁 𝑥 𝑁 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 (40) With (𝑥 1 , … . , 𝑥 𝑁 ) the abscissa of the 𝑁 first peaks of the 𝑠𝑖𝑛𝑐 function.In total, eight 𝑠𝑖𝑛𝑐 𝑒𝑞 functions are determined, two per material. Figure12provides the example of the 𝑠𝑖𝑛𝑐 𝑒𝑞 ℎ𝑜𝑟 (in blue) compared with the true 𝑠𝑖𝑛𝑐 function (in orange) and the true 𝑓 function (from FMRE) when windows are added (in green) for metal:

Figure 13 :

 13 Figure 13: Graphical example of (a) the objective power 𝑃 𝑖 𝑗,𝑜𝑏𝑗 for 𝑃 𝑅𝑋 = -117 𝑑𝐵𝑊 needed to increase 𝑝𝑤 by 50% and (b) all the 𝑃 𝑖 𝑗,𝑜𝑏𝑗 for every possible delay 𝜏 𝑖 𝑗 considering scenario 4 at FL21

Figure 13

 13 Figure 13 (a) presents a graphical example of the objective power 𝑃 𝑖 𝑗,𝑜𝑏𝑗 for 𝑃 𝑅𝑋 = -117 𝑑𝐵𝑊 (3dB above threshold) needed to increase 𝑝𝑤 by 50% with 𝜏 𝑖 𝑗 = 1.5 𝜇𝑠. Figure 13 (b) presents an upper view of all 𝑃 𝑖 𝑗,𝑜𝑏𝑗 for every possible delay 𝜏 𝑖 𝑗 considering scenario 4 at FL21. Note that the blanked spots on the left of the DME/TACAN beacon and on the right of the aircraft are just aggregates due to the additional delay steps.

Figure 14 :

 14 Figure 14: Representation of (a) an ellipsoid and the two zones of servitude and (b) the intersection of this ellipsoid and the plan 𝑧 = 𝐻/2

Figure 15 :

 15 Figure 15:Upper view of the results for scenario FL21, 𝑃 𝑇𝑋 = 30 𝑑𝐵𝑊, 𝑃 𝑅𝑋 = -90 𝑑𝐵𝑊

Figure 16 :

 16 Figure 16: Map of JALTO and the DME/TACAN beacons in view

Figure 17 :Figure 18 :

 1718 Figure 17: Example of building footprint extracted from the Microsoft Maps database

Figure 19 :

 19 Figure 19: Representation of the 19 visible DME/TACAN beacons at JALTO within their different zones

ACKNOWLEDGMENTS

The authors of this paper would like to deeply thank Davis Huerta C. for his contribution to this work.