Koffi Samuel Koulekpa
email: koffi.koulekpa@ensam.eu

Michael Deligant

Hélène Elias-Birembaux

Frédéric Rossi

Gérard Poulachon

Analysis of CO 2 Jet's Temperature by BOS Method for Applications in Cryogenic Machining

Keywords: Supercritical Carbon Dioxide, Background Oriented Schlieren method, Jet expansion, Cutting fluids, Tool life

In order to face the challenges in machining difficult-to-cut materials as nickel and titanium alloys, many cooling methods have been developed through the years, among which is the usage of supercritical carbon dioxide (scCO2). By combining the cooling provided by the expansion of the CO2 jet, and the scCO2 solvent properties (for dissolving lubricants), it is possible to reach higher tool life and productivity, with respect to the other classical cooling methods. The purpose of this study is to compare the Background Oriented Schlieren (BOS) method with thermocouple measurements in the context of CO2 jets, whether in liquid or supercritical state. A typical BOS setup using an enlightened background pattern and a high-speed camera was used to determine the temperature of CO2 jets. While the trend of BOS centerline temperature aligns with that of the thermocouple measurements, there exist noticeable disparities between them. These discrepancies average around 20°C in favourable instances and can reach up to 60°C in the most extreme cases. These findings underscore the need to continue the investigation to identify areas for improvement. This pursuit is essential to achieve accurate results for using this method in the development of future temperature models for CO2 jets.

Introduction -State of art

Assisted-machining

The control of cutting temperature when machining difficult-to-cut materials, such as titanium and nickel-based alloys, has a considerable impact on tool-life, wear mechanisms, surface integrity, production time, and chip flow [START_REF] Laforce | CETIM_Etude comparative du fraisage assisté par cryogénie -Version française et anglaise[END_REF]...This has been achieved through the years by using flood coolants, which provide the needed lubrication and cooling at the tool/part interface [START_REF] Yang | Application of coolants during tool-based machining -A review[END_REF]. But the cost and health risks linked to their usage have led to the development of other techniques such as Minimum Quantity Lubrication (MQL), liquid Nitrogen (LN2), liquid CO2 and supercritical CO2 (scCO2) [START_REF] Jawahir | Cryogenic manufacturing processes[END_REF]. This latter has demonstrated a great interest due to its influence on the process.

Supercritical CO2 in machining processes

When raised to the supercritical state (above 74 bars and 31°C), CO2 can dissolve some particles of oils or lubricants given that scCO2 is widely recognized as an environmentally friendly solvent [START_REF] Hyatt | Liquid and supercritical carbon dioxide as organic solvents[END_REF]. The combination of this lubrication and Electronic copy available at: https://ssrn.com/abstract=4657853 the cooling effect produced by the expansion (Joule-Thompson effect [START_REF] Proud | A Review of CO2 Coolants for Sustainable Machining[END_REF]), when delivered to the cutting zone, allows the decrease of the temperature produced in the process [START_REF] Mulyana | The influence of cryogenic supercritical carbon dioxide cooling on tool wear during machining high thermal conductivity steel[END_REF][START_REF] Rahim | Experimental Investigation of Supercritical Carbon Dioxide (SCCO2) Performance as a Sustainable Cooling Technique[END_REF], leading to longer tool life and good surface integrity [START_REF] Tapoglou | Milling of aerospace alloys using supercritical CO2 assisted machining[END_REF][START_REF] Khosravi | High-speed milling of Ti6Al4V under a supercritical CO2 + MQL hybrid cooling system[END_REF], often better than traditional MQL or flood coolants.

Cooling ability of CO2 jets

In order to analyze the CO2 jet properties, Mulyana et al. [START_REF] Mulyana | The influence of cryogenic supercritical carbon dioxide cooling on tool wear during machining high thermal conductivity steel[END_REF], studied the cooling rate for scCO2 and scCO2 + MQL jets, by placing a K-thermocouple in a heated carbide rod, for the pressure values of 78.8, 59.6 and 104 bars. A maximum cooling rate of 13.6°C/s was observed for scCO2 + MQL at 104 bars.

Gross et al. [START_REF] Gross | Investigation on the productivity of milling ti6al4v with cryogenic minimum quantity lubrication[END_REF] placed a thermocouple in different positions along the axis of a CO2 jet and recorded the centreline profile temperature for nozzles of 0.2, 0.3 and 0.5 mm diameter. They noticed that increasing the diameter did not lower the minimum temperature but rather enlarged the cold zone.

Pursell et al. [START_REF] Pursell | Experimental investigation of high pressure liquid CO2 release behaviour[END_REF] placed a thermocouple rack in liquid and gaseous jets of CO2, using 2 mm and 4 mm orifices. They recorded a decrease of temperature near the nozzle output (down to -80 °C), before the warming of the jet as the downstream distance increases. Khalil et al. [START_REF] Khalil | The structure of supercritical fluid free-jet expansions[END_REF] modelled the density contours after expansion of scCO2 flows into air, and compared the results to shadowgraph measurements (wave structures) and impact pressures on a plate placed in front of the jet.

Few works studied the properties of the whole jet (density, phase and temperature). Accessing and establishing a whole jet temperature model could be useful for modelling CO2 jets and their impacts on cutting processes. This latter temperature model could be paired to a heat transfer coefficient model, to properly simulate scCO2-assisted machining operations. This preliminary work aims then to study the temperature during the expansion of the jet using Background Oriented Schlieren method (BOS). The results were compared to a distinct thermocouple rack measurement taken under identical conditions for validation purposes. This comparative analysis serves the larger purpose of facilitating the future development of a comprehensive jet temperature model.

Principles of the BOS method

The Background Oriented Schlieren method is a nonintrusive technique used to characterise the density and temperature gradients in inhomogeneous fluids [START_REF] Dalziel | Whole-field density measurements by 'synthetic schlieren[END_REF][START_REF] Venkatakrishnan | Density Measurements in an Axisymmetric Underexpanded Jet by Background-Oriented Schlieren Technique[END_REF][START_REF] Raffel | Background-oriented schlieren (BOS) techniques[END_REF]. It consists in recording with a camera, focused on a background pattern enlightened from behind (figure 1), two images corresponding to the scene without, and with the fluid. The displacement field obtained by cross-correlation algorithms is the image of the gradients contained within the fluid. This displacement field is used to calculate the refractive index field by different approaches:

• The Poisson equation in the case of 2D plane fluids, when the width of the medium is known [START_REF] Jensen | Optical density and velocity measurements in cryogenic gas flows[END_REF],

• Abel Transform for axisymmetric fluids using the vertical displacement [START_REF] Kaganovich | Benchmarking background oriented schlieren against interferometric measurement using open source tools[END_REF][START_REF] Sipkens | New transform to project axisymmetric deflection fields along arbitrary rays[END_REF], • Other tomographic reconstruction algorithms [START_REF] Grauer | Instantaneous 3D flame imaging by background-oriented schlieren tomography[END_REF].

In this work, the axisymmetric theory is used, as the type of jets studied here are axisymmetric due the rounded jets nozzle used.

The notation described below refer to figure 1. The equations of the BOS are well established in the literature, so only the key steps are listed here. Using the principle of Fermat and the small refraction angles principles [START_REF]Principles of Optics[END_REF][START_REF] Walsh | Tomographic reconstruction of laser speckle photography for axisymmetric flame temperature measurement[END_REF], the vertical deflection can be expressed as a function of the index of refraction [START_REF] Kaganovich | Benchmarking background oriented schlieren against interferometric measurement using open source tools[END_REF][START_REF] Moumen | A new approach for the reconstruction of axisymmetric refractive index fields from background-oriented schlieren measurements[END_REF] :

𝜀(y)= ∫ rdr √r 2 -y 2 ( y r ∂n(x,r) ∂r ) ∞ |y| = ∆h(x,y) MZ 3 (1)
Where ε(y) is the vertical angular deflection, n(x,r) is the spatial axisymmetric distribution of the index of the refractive index with r=√y 2 +z², y the vertical component on the jet plane, and Δh the vertical displacement recorded by the sensor of the camera. Δh is linked to the displacement ΔY on the background plane, by the distance Z3 and the magnification factor M of the camera. The factor M is not taken into account in the following calculations because it is already included in the calibration of the camera during the experiments [START_REF] Kaganovich | Benchmarking background oriented schlieren against interferometric measurement using open source tools[END_REF].

Using the inverse Abel transform, the index refraction field is extracted from equation 1 to equation 2:

n(r,x) = n 0 + 1 πZ 3 ∫ dr ' r' ∞ r ∫ dh √ h 2 -r' 2 ∂ ∂h ( ∆h(x,h) h ) ∞ r' ( 2
)
n 0 being the refractive index of the ambient air. Using the ideal gas law and the Gladstone-Dale [START_REF] Vasilev | Schlieren methods[END_REF] relation, in equation 3 (with 𝜌 being the density, p the pressure, R the gas constant and T the temperature),

𝑛 = 1 + 𝐺𝜌 = 1 + 𝐺 𝑝 𝑅𝑇 (3)
the density and temperature gradients can be calculated by equations 3 and 4 as: Electronic copy available at: https://ssrn.com/abstract=4657853 n(r,x)-1=(n 0 -1) ρ(r,x) ρ 0 [START_REF] Hyatt | Liquid and supercritical carbon dioxide as organic solvents[END_REF] With 𝜌 0 the density of ambient air;

T(r,x) T 0 = [( n(r,x) n 0 -1) ( RT 0 GP 0 +1) +1] -1 ( 5
)
With 𝐺 = 2.26 × 10 -4 𝑚 3 /𝑘𝑔 being the Gladstone-Dale coefficient of CO2 [START_REF] Grauer | Instantaneous 3D flame imaging by background-oriented schlieren tomography[END_REF], T 0 and P 0 , the ambient temperature and pressure respectively.

Experimental setup

BOS setup

A FlowSense Cx 5M-124 camera with a 2448x2048 pixel sensor was placed on a tripod and focused on a printed background pattern generated by the Dantec Dynamics software (DynamicStudio 8.1). This pattern was placed on a transparent plexiglas plate enlightened from behind by a simple lamp as shown on figure 2. The area of interest was set to 40x40 mm because, the tool-nozzle distance in machining operations is rarely over 40mm. This zone of interest was vertically centered in relation to the nozzle axis, aiming to achieve symmetry in the captured displacement images as much as possible.

Round jet sapphire nozzles with diameters of 0.1 mm and 0.25 mm, sourced from TESCHKE, were employed for these preliminary tests. The conditions should be limited in pressure and temperature to avoid an opaque jet (with ice) leading to poor displacements measurements. Three conditions are presented in Table 1, to investigate the feasibility of studying CO2 jets temperature (liquid and supercritical) by the BOS method. In Table 1, D stands for diameter, P and T, for the upstream pressure and temperature, defining the CO2 state before the nozzle (liquid or supercritical). Validating this method would subsequently contribute to the development of a jet temperature model T = f(D, P, T) with the aforementioned parameters serving as inputs.

The main distances on the experimental setup are illustrated on figure 1. The recorded displacements calculated in DynamicStudio, were postprocessed in Python using the pyAbel [START_REF] Gibson | [END_REF] library to perform the inverse Abel transform and calculate the temperature distribution. A mean value is computed from the data collected by each individual thermocouple. In the case of non-stable jets, when the formation of dry ice occurs, the stable measured zone is used for calculations. The thermocouple measurements conditions, conducted under the same conditions are outlined in Table 1 will serve as a benchmark for evaluating the results obtained through the BOS method.

ScCO2 production device

The machine employed for the experiments, shown on figure 4, is a custom-built system designed for delivering liquid and supercritical CO2 [START_REF] Koulekpa | Etude expérimentale de l'usinage du Ti6Al4V sous assistance CO2 supercritique[END_REF]. The mix of liquid and gaseous CO2, taken from the CO2 bottles is cooled down at 10°C to ensure it is in a liquid state before compression and heating, to reach the supercritical state. The scCO2 is stocked in a pressure vessel before delivered throughout the nozzle. The pressure and temperature conditions can be provided to Electronic copy available at: https://ssrn.com/abstract=4657853

operate with either liquid or supercritical CO2. These conditions are measured by pressure sensors and thermocouples placed before the nozzle, and regulated by a controller unit.

Results

The figures 5a and 5b illustrate respectively an example of the jet and vertical displacement recorded for the supercritical condition. The dashed rectangle zone illustrates an example of the analyzed zone. After applying the previously described BOS method to the axisymmetric displacement distribution, the resulting temperature maps are depicted in figure 6.

The radial distributions extracted at x = 10 mm and the centerline temperatures are plotted on figure 7 for each condition. The corresponding thermocouples measurements are added on figure 7b, c andd.

Similar to the findings of Gross et al. [START_REF] Gross | Investigation on the productivity of milling ti6al4v with cryogenic minimum quantity lubrication[END_REF], the core jet dimension is larger for D = 0.25 mm (extending up to 15 mm in the axial direction on Figure 6c), while it is smaller for 6a and 6b, corresponding to a diameter of D = 0.1 mm. The decrease of the centerline temperature (figure 7c and7d) recorded near the output of the nozzle before the warming with downstream distance, is similar to the behavior modeled by Pursell et al. [START_REF] Pursell | Experimental investigation of high pressure liquid CO2 release behaviour[END_REF], which correspond to the expansion zone. But for the condition D0.1P60T30 (figure 7b), the BOS temperature values in the expansion zone are unexpectedly far above the thermocouple's measurements (A 60°C gap at x=0). Knowing that changing the diameter should not lower that much the cold temperature values [START_REF] Gross | Investigation on the productivity of milling ti6al4v with cryogenic minimum quantity lubrication[END_REF], there is a need of repetition to state about possible source of error.

The gap between the BOS measurement and the thermocouple measurement (realized in the same conditions) could be explained either by the bias of ideal gas law used in the BOS method which suppose a constant pressure in the jet core, or the fact that there could be slight changes in the jet from one experiment to another (BOS to thermocouple), even though the upstream conditions are the same. However, the strength of the BOS technique lies in its ability to capture both radial and axial density and temperature gradient distributions, thereby offering more comprehensive and valuable insights compared to discrete thermocouple measurements.

A difficulty noticed in the application of the BOS method is the formation of dry-ice for higher supercritical condition or higher diameter values, which lead to poor results because of the jet's opacity. The conditions to be used in future applications of the BOS method on CO2 jets, should be carefully considered to prevent such occurrences. For instance, with a nozzle diameter of 0.25 mm, pressures higher than 100 bars often leads to an opaque jet with ice. Electronic copy available at: https://ssrn.com/abstract=4657853

Conclusion and perspectives

In the present work, the Background Oriented Schlieren method has been applied to liquid an supercritical CO2 and benchmarked to thermocouple measurements. The expansion behavior described by Pursell et al. [START_REF] Pursell | Experimental investigation of high pressure liquid CO2 release behaviour[END_REF], was observed, resulting in a near-nozzle temperature decrease reaching as low as -80°C. Discrepancies were identified between the BOS results and those obtained through thermocouples:

• Regular error of about 20°C for condition D0.25P80T40 and D0.1P150T30, which could be reduced with careful and meticulous experiments execution, • Unexpected gap of 60°C in the expansion zone for D0.1P60T30 which will need repetition to uncover the source of such gap.

Nevertheless, this method has shown potentialities to establish a model of temperature describing the spatial axisymmetric distribution of CO2 jets, based on existing models [START_REF] Joshi | Variation of axial temperature in thermal plasma jets[END_REF]. Future works will focus on implementation of careful and repeated design of experiment to create a robust database for model development. This model will subsequently find application in simulations of liquid and supercritical CO2-assisted cutting processes.

Fig. 1

 1 Fig. 1 Illustration of ray path in the BOS experiment

D

  was designed to record the centerline-temperature in the CO2 jets for liquid and supercritical conditions, as illustrated on figure3. It consists in thirteen (13) K+ and K-thermocouple wires welded and fixed to springs, spaced by 5 mm, a part from the last two which are spaced by 15 mm. The springs attached to the frame allow the alignment of the welding points on the same line. The acquisition set Ni9205 +cDAQ-9171 is used to record the temperature during the experiments, with a 15kHz frequency.

Fig. 3 Fig. 2 .

 32 Fig. 3 BOS setup with enlightened pattern

Fig. 6 Fig. 5 .Fig. 4

 654 Fig. 6 Liquid and supercritical CO2 delivery machine

Fig. 7

 7 Fig. 7. a. Radial temperature distribution extracted at x = 10 mm BOS centreline temperatures against thermocouples measurements, for (b)D0.1P60T30 (c)D0.1P150T30 and (d) D0.25P80T40

Table 1 :

 1 Conditions used in the experiments

Electronic copy available at: https://ssrn.com/abstract=4657853

Acknowledgements

The authors would like to thank the funders of this research: INSTITUT CARNOT ARTS, CETIM and Bourgogne Franche-Comté council.


P Develter

J Bosse

O Rabaste

P Forster

J.-P Ovarlez

ON THE FALSE ALARM PROBABILITY OF THE NORMALIZED MATCHED FILTER FOR OFF-GRID TARGET DETECTION

Keywords: Radar detection, Off-Grid, GLRT, P F Athreshold relationship, Theory of Tubes

Off-grid targets are known to induce a mismatch that dramatically impacts the detection probability of the popular Normalized Matched Filter. To overcome this problem, the unknown target parameter is usually estimated through a Maximum Likelihood strategy resulting in a GLRT detection scheme. While the test statistic for the null hypothesis is well known in the on-grid case, the off-grid scenario is more involved and, to the best of our knowledge, no such theoretical result is available. This paper fills this gap by proposing such an expression under circular compound Gaussian noise with known covariance matrix thanks to a geometrical approach.

INTRODUCTION

Classically, detection of signals with unknown parameters are addressed with a Generalized Likelihood Ratio test (GLRT) that replaces the unknown parameters with their Maximum Likelihood estimators (MLE) in the Likelihood Ratio detection test [START_REF] Scharf | Statistical signal processing: detection, estimation, and time series analysis[END_REF]. When analytical MLE solutions are not available for signal parameters of interest, most detection strategies assume for ease of implementation that those parameters lie over a discrete set, called the grid. However, parameters have no reason to fall precisely on the grid, since they are distributed over a continuous range. This induces a mismatch between the tested parameters and the true target parameters that deteriorates the detection performance of most state-ofthe-art tests made under the on-grid assumption. In this paper, we illustrate this problem in the radar context where unknown parameters can include Doppler shift, distance, or direction.

The off-grid impact is particularly dramatic for detection schemes like the Normalized Matched Filter (NMF) test [START_REF] Rabaste | Off-grid target detection with Normalized Matched Subspace Filter[END_REF]. This test is used under Gaussian noise of unknown level. It is also widely used for adaptive radar in non-Gaussian contexts [START_REF] Pascal | On a SIRV-CFAR detector with radar experimentations in impulsive noise[END_REF] for example, when the noise is distributed according to a Complex Elliptically Symmetric (CES) distribution [START_REF] Ollila | Complex Elliptically Symmetric distributions: Survey, new results and applications[END_REF]. In some cases, the detection probability may vanish to 0 even for high Signal to Noise ratio (SNR) [START_REF] Rabaste | Off-grid target detection with Normalized Matched Subspace Filter[END_REF], especially for low P F A , familiar to Radar context. To overcome this problem, the most obvious solution consists in testing over the whole continuous parameter support, not just the grid: this is the true off-grid GLRT. However, to the best of our knowledge, the analytical expression of the null hypothesis statistic and the related P F A is unknown in the literature for this GLRT.

Some works provide optimal analytical solutions for other particular mismatches [START_REF] Besson | Detection of a signal in linear subspace with bounded mismatch[END_REF][START_REF] Bandiera | Advanced Radar Detection Schemes Under Mismatched Signal Models[END_REF][START_REF] De Maio | Fractional QCQP with applications in ML steering direction estimation for radar detection[END_REF], as such they are not optimal for our specific mismatch. Other approaches include approximating the off-grid GLRT, either with a subspace approach [START_REF] Rabaste | Off-grid target detection with Normalized Matched Subspace Filter[END_REF] or a Monopulse-inspired method [START_REF] Develter | Off-grid radar target detection with the normalized matched filter: A monopulse-based detection scheme[END_REF]. Those solutions are suboptimal: they are designed for practical conditions where the use of a refined, precise approximation of the GLRT is unrealistic because its lack of closed-form can lead to high computational cost. In this paper, we present a derivation of a P F A -threshold relationship for the off-grid NMF. It allows an insightful geometrical analysis of the problem. It is valid for low P F A of interest in common applications.

Section 2 presents the signal model, the off-grid problem and the true GLRT formulation. Section 3 gives the derivation of the P F A -threshold relationship. In Section 4 we check the validity of our derivation by comparing the theoretical thresholds to Monte-Carlo simulations.

Notations: Matrices are in bold and capital, vectors in bold. For any matrix A or vector, A T is the transpose of A and A H is the Hermitian transpose of A. I is the N × N identity matrix and CN (µ, Γ) is the circular complex Normal distribution of mean µ and covariance matrix Γ. S n-1 is the unit sphere in R n . The real part operator of a complex number is denoted by Re(.). The operator u is the angle of a complex number u.

PROBLEM FORMULATION

In radar detection, the main problem consists in detecting a complex signal d ∈ C N corrupted by an additive noise n (clutter, thermal noise, etc.). This problem can be stated as the following binary hypothesis test:

H 0 : r = n , H 1 : r = α d(θ) + n , ( 1
)
where r is the complex N -vector of the sampled received signal, α is an unknown complex target amplitude and d(θ) stands for a generally known steering vector characterized by unknown target parameters θ (time-delay, Doppler or angle).

In the sequel, we will assume n is a zero-mean complex circular Gaussian noise vector with unknown variance σ2 i.e. n ∼ CN (0, σ 2 Γ). This context is known as a partially homogeneous Gaussian environment. Without loss of generality, we will here assume θ to be the scalar Doppler shift of the target and we here investigate only the normalized Doppler steering vector d(θ):

d(θ) = 1 √ N 1, e 2iπθ , . . . , e 2iπ(N -1)θ T . (2)
This model of steering vector is often encountered in radar Range-Doppler detection schemes where the problem consists in estimating a complex sinusoid embedded in noise after range Matched Filter processing.

For unknown parameters {λ i } i∈[0,1] depending on each hypothesis {H i } i∈[0,1] (either parameters of interest and/or nuisance parameters), the usual procedure relies on the Generalized Likelihood Ratio (GLR) statistic, namely the ratio Λ(r) between the Probability Density Function (PDF) f H1 (.) of the data under H 1 and the PDF f H0 (.) under H 0 where the unknown parameters are replaced by their ML estimate and where w 2 is the detection threshold:

Λ(r) = max λ1 f H1 (r) max λ0 f H0 (r) H1 ≷ H0 w 2 .
( When λ 1 = {α, σ} and λ 0 = {σ} with θ known, the corresponding GLRT is known as the NMF (Normalized Matched Filter) [START_REF] Scharf | Signal detection in Gaussian noise of unknown level: an invariance application[END_REF][START_REF] Scharf | Matched subspace detectors[END_REF]:

d(θ) H Γ -1 r 2 d(θ) H Γ -1 d(θ) r H Γ -1 r H1 ≷ H0 w 2 . ( 4
)
This test is also widely used for adaptive radar in non-Gaussian contexts [START_REF] Pascal | On a SIRV-CFAR detector with radar experimentations in impulsive noise[END_REF][START_REF] Conte | Asymptotically optimum radar detection in compound-Gaussian clutter[END_REF] for example when the noise is distributed according to a Complex Elliptically Symmetric (CES) distribution [START_REF] Ollila | Complex Elliptically Symmetric distributions: Survey, new results and applications[END_REF]. Its statistic in this case is the same as in the Gaussian case. Equivalently, Eq. ( 4) can be rewritten with normalized whitened vectors:

s(θ) H u Γ -1/2 d(θ) and u = Γ -1/2 r Γ -1/2 r
.

The corresponding P F A -threshold relationship is well known:

P F A = 1 -w 2 N -1 . (6)
When the point θ where the NMF is tested is different from the true parameter θ 0 of the target, the target is said to be off-grid. This induces a mismatch (θ = θ 0 ) between the real target steering vector s(θ) and the steering vector s(θ 0 ) under test. Unfortunately, it was shown in [START_REF] Rabaste | Geometrical design of radar detectors in moderately impulsive noise[END_REF] that the NMF detector is very sensitive to steering vector mismatch, potentially leading to a dramatic deterioration of the detection performance: in particular for a mismatch larger than the detection threshold, the asymptotic detection probability tends to 0 at high SNR: this phenomenon occurs [START_REF] Rabaste | Off-grid target detection with Normalized Matched Subspace Filter[END_REF] for P F A as high as 10 -3 in the chosen resolution cell of width 1/N . This can be seen in Figure 1 which displays the NMF response without any noise as a function of the mismatch: this response falls below the detection thresholds at the edge of the cell. To correct this issue, θ must be estimated.

When θ is unknown, the natural GLRT procedure leads to

GLRT (u, D) = max θ∈D s(θ) H u 2 H1 ≷ H0 w 2 , ( 7
)
where D is the search domain relative to the unknown parameter θ. This detector corrects the off-grid issue of test (5) pointed out in Figure 1. Unfortunately, as of now, no analytical P F A -threshold relationship is known, the difficulty consisting here to evaluate the statistics of the maximum of a continuum of non-independent random variables.

In the next section, we propose to fill this gap.

AN ANALYTICAL P F A -THRESHOLD RELATIONSHIP WITH A GEOMETRICAL INTERPRETATION

Through geometrical considerations, Hotelling [START_REF] Hotelling | Tubes and spheres in n-spaces, and a class of statistical problems[END_REF] derived a methodology to study statistical tests over the real sphere.

Remarkably, it can be used to evaluate the P F A of the GLRT in the real case: his insightful approach is presented in Section 3.1. Unfortunately, as will be explained in Section 3.2, his derivation cannot be directly transposed to the complex case expressed in (5).

Hotelling's geometrical approach in the real case

Following Hotelling, let us see how finding a threshold guaranteeing a certain P F A reduces to a simple geometrical problem in the real case, that is, when replacing C with R in the previous section. First, note that the NMF expression (5) has a simple geometrical interpretation. It is indeed the squared cosine of the angle between the normalized steering vector under test s (θ) and the normalized whitened received signal u. The threshold can be seen as the squared cosine of the angle cos -1 w. When the vectors angle is below this limit angle, a target is detected. Since u has been whitened, it is uniformly distributed over the unit N -sphere under the null hypothesis.

Thanks to this distribution, we can compute the P F A of the NMF test given in Eq. ( 5). Let us consider the spherical cap SC + θ defined by all vectors u on the sphere verifying u T s (θ) < w and its symmetric version SC - θ satisfying -u T s (θ) < w. Let us define SC θ = SC + θ SC - θ as the union of the two previous spherical caps. A false alarm occurs as soon as u ∈ SC θ . Therefore, the P F A is the ratio between the spherical caps surface and the surface of the unit sphere.

Let us now consider the GLRT in Eq. [START_REF] De Maio | Fractional QCQP with applications in ML steering direction estimation for radar detection[END_REF]. A false alarm occurs when u ∈ T = θ∈D SC θ . Therefore the P F A is the ratio of the surface of T and the surface of the unit sphere. An example in R 3 is represented in Figure 2 on S 2 . Note that T is the union of two symmetric tubes embedded in the sphere, with geodesic radius cos -1 w around the two curves s(θ) and -s(θ) with added semi-spherical caps at the ends. T can thus be written into two equivalent forms [START_REF] Johnstone | On Hotelling's formula for the volume of tubes and Naiman's inequality[END_REF]:

T = u ∈ S N -1 : max θ∈D s(θ) T u > w = β=±1 u ∈ S N -1 : min θ∈D βs(θ) -u < 2(1 -w) .
With this form, computing the P F A amounts to finding the surface of the set of points in S N -1 that lie at a Euclidean distance 2(1 -w) of the curves.

In [START_REF] Hotelling | Tubes and spheres in n-spaces, and a class of statistical problems[END_REF], Hotelling derives an analytical formula for the surface of tubes on the real n-dimensional sphere S n-1 . Theorem 3.1 [START_REF] Hotelling | Tubes and spheres in n-spaces, and a class of statistical problems[END_REF] The surface enclosed by a tube of geodesic radius ψ around a curve on the real unit sphere S n-1 is the product of the length of the axial curve by

π(n -2) 2 Γ n 2 sin n-2 (ψ) . ( 8
)
It can be noted that this formula only holds when the considered tubes do not overlap [START_REF] Hotelling | Tubes and spheres in n-spaces, and a class of statistical problems[END_REF]. Overlap phenomenons can happen when the tubes draw back into themselves, or when their curvature becomes too high relative to their radius (local overlap). Non-overlap is locally guaranteed when ψ = Fig. 2. T embedded on the unit sphere S 2 in R 3 composed of a tube (in violet) around a one dimensional manifold θ∈D s(θ), two end semi-spherical caps (in green) and their opposite whose outline is dashed. SC θ is drawn in blue. [START_REF] Hohenwarter | GeoGebra -ein Softwaresystem für dynamische Geometrie und Algebra der Ebene[END_REF] cos -1 w becomes low enough. In case of overlap, Eq. ( 8) becomes an upper bound. By adding the surface of the end caps to [START_REF] Develter | Off-grid radar target detection with the normalized matched filter: A monopulse-based detection scheme[END_REF], one can find the surface of T in the real case.

Extending Hotelling's approach to the complex case

Unfortunately Hotelling's result is not immediately applicable to the considered GLRT. As we will see, when transposing our complex problem in R 2N the resulting manifold becomes now two-dimensional since the absolute value in the right hand term of the GLRT given in Eq. ( 7) then implies a search of the maximum on the phase as well.

Indeed, for any real scalar α ∈ [0, 2π], let us remark that Re(s (θ) H ue -iα ) ≤ s (θ) H u , those two quantities being equal for α = s (θ) H u. We then have, decomposing s (θ) = s r (θ) + is i (θ) and u = u r + iu i into real and complex parts:

Re(s (θ) H ue -iα ) = s r (θ) T u r + s i (θ) T u i cos α + s r (θ) T u i -s i (θ) T u r sin α = γ 1 (θ) T u cos α + γ 2 (θ) T u sin α , where γ 1 (θ) = s r (θ) s i (θ) , γ 2 (θ) = -s i (θ) s r (θ)
and u = u r u i is a 2N -real valued noise vector drawn uniformly on S 2N -1 under H 0 . The GLRT [START_REF] De Maio | Fractional QCQP with applications in ML steering direction estimation for radar detection[END_REF] can then be written as:

max α,θ γ 1 (θ) T u cos α + γ 2 (θ) T u sin α H1 ≷ H0 w . (9)
The two-dimensional nature of the so-defined real manifold to inspect is directly apparent with this formulation. Indeed, by denoting γ(α, θ) = γ 1 (θ) cos α + γ 2 (θ) sin α, the acceptance region in the complex case is a new tube T around the two-dimensional manifold γ(θ, α):

T = u ∈ S 2N -1 : max θ∈D,α γ(θ, α) T u > w = u ∈ S 2N -1 : min θ∈D,α γ(θ, α) -u < 2(1 -w) .
Hotelling's result does not cover this multi-dimensional manifold case as it gives the surface of a tube around a curve. However, in [START_REF] Johnstone | On Hotelling's formula for the volume of tubes and Naiman's inequality[END_REF] and [START_REF] Knowles | On Hotelling's approach to testing for a nonlinear parameter in regression[END_REF], this result is extended to a special case of two-dimensional manifolds which is of interest to us: Theorem 3.2 [START_REF] Johnstone | On Hotelling's formula for the volume of tubes and Naiman's inequality[END_REF] For i ∈ [START_REF] Scharf | Statistical signal processing: detection, estimation, and time series analysis[END_REF][START_REF] Rabaste | Off-grid target detection with Normalized Matched Subspace Filter[END_REF], let γ i : [0, t 0 ] → S n-1 be regular curves. Assume γ 1 (t) T γ 2 (t) = 0 for all t. Let

Z(t) = γ 1 (t) T u 2 + γ 2 (t) T u 2 1/2
where u is uniformly distributed on S n-1 . Then for 0 < w < 1, we have:

P max 0≤t≤t0 Z (t) > w ≤ (1 -w 2 ) (n-2)/2 + Γ n 2 w (1 -w 2 ) (n-3)/2 2π 3/2 Γ n -1 2 × t0 0 2π 0 γ1 (t) cos ω + γ2 (t) sin ω 2 - γ1 (t) T γ 2 (t) 2 1/2 dω dt , (10)
where γi (t) is the derivative of γ i (t) with respect to t. When there is no overlap, this inequality becomes an equality.

It turns out that we can reformulate our problem in order to fulfill the assumptions of the above theorem. With our notations, it is easy to check that

s (θ) H u 2 = γ 1 (θ) T u 2 + γ 2 (θ) T u 2 , (11)
so that Theorem 3.2 gives us the desired P F A (when the equality holds). Follows our main result:

Corollary 3.2.1 In the absence of overlap (low P F A regimes), the P F A for the GLRT (7) for a search interval D = [θ 1 , θ 2 ] with the steering vector d (θ) defined in (2) is given by:

• Under white noise (Γ = σ 2 I):

P F A = (1 -w 2 ) N -1 + (12) π 3 Γ(N ) w (1 -w 2 ) N -3 2 Γ N -1 2 N 2 -1 1 2 (θ 2 -θ 1 ) .
• Under colored noise (Γ = σ 2 I), the integral in [START_REF] Scharf | Matched subspace detectors[END_REF] can be evaluated numerically.

The first term in [START_REF] Rabaste | Geometrical design of radar detectors in moderately impulsive noise[END_REF] represents the surface of the two semi-spherical caps at the extremities of the tube. As such, it is equal to the known P F A of the NMF expressed in Eq. ( 6). The second term shows the influence of the manifold induced Here, θ 2 -θ 1 plays the role of the manifold length, and the rest of the rightmost term is the surface of the cross-section.

NUMERICAL RESULTS

Let us check the validity of the formula [START_REF] Rabaste | Geometrical design of radar detectors in moderately impulsive noise[END_REF]. Figure 3 presents the P F A -threshold relationship given by Eq. ( 12) and empirically computed thresholds using 10 8 complex circular white Gaussian noise samples for a steering vector size of N = 10. The continuous research over the domain D is replaced by a discrete search using 30 tests, where D = [0, 1/N ] is one of the usual Fourier resolution cells.

The formula fits perfectly when the P F A is low enough (or, equivalently, if the threshold is high enough). The formula is not valid for P F A close to 1 because of overlap (it even exceeds 1). However, such high P F A have no practical interest for common applications. A detailed analysis of overlap phenomena will be given in a subsequent paper.

CONCLUSION

In this paper, we addressed the off-grid detection problem using the NMF-GLRT by finding an analytical P F A -threshold relationship, valid for most common applications. In future works, we will mathematically investigate the conditions guaranteeing that no overlap is present for a given P F A value. This will enable us to provide the precise domain and the conditions under which our relationship is valid.

)Fig. 1 .

 1 Fig. 1. Comparison of the NMF response with various thresholds w 2 = 1 -P 1/(N -1) F A when N = 10, in cell D = [-0.5/N, 0.5/N ].

Fig. 3 .

 3 Fig. 3. Comparison between the theoretical P F A -threshold given in (12) and the empirical Monte Carlo P F A -threshold relationships for N = 10. The on-grid relation (6) is also drawn for comparison purposes. Tests in the cell [0/N , 1/N].

H1 ≷ H0 w 2 ,(5)where s(θ) = Γ -1/2 d(θ)


Sonia Guelton

Roelof Verhage

Public Value Capture of Increasing Property Values across Europe

France

Sonia Guelton and Roelof Verhage

Local authorities and planning system

France is known as a highly centralized country. In fact, the centralist approach applied to the field of urban planning after the Second World War until the end of the twentieth century. However, in 1982 and1983, a set of decentralization laws changed the situation. With these laws, numerous prerogatives were transferred from the State level to the local and supra local level, including controls and responsibilities in urban planning. As a result, local and supra local authorities became the main actors in planning. In particular, they became responsible for the elaboration of the legally binding local land use plan. Still, these local land use plans must be consistent with national laws and obligations dealing with risk and environmental issues or social housing provision.

Since the sixties, intermunicipal bodies were created to rationalise the delivery and management of public services at a larger territorial scale and coordinate all municipal planning activities. Known as Communauté Urbaine, the State created the first four of these intermunicipal bodies in 1969. The decentralisation laws transferred urban planning empowerment to the intermunicipal level. Since then, diverse intermunicipal bodies have been created with diverse powers: Communauté de communes in 1992, Communauté d'agglomération in 1999, and Métropole since 2014. In 2021, there are 1253 intermunicipal bodies in France and every municipality is a member of at least one intermunicipal authority. Strategic structure plans (SCOT: Schémas de Cohérence Territoriale) are elaborated at the intermunicipal level. They aim at assuring territorial coherence on a supra local scale in a country where 85% of the municipalities have less than 2000 inhabitants1 . Local land use plans (PLU: Plan Local d'Urbanisme) have to be compatible with the strategic structure plans.

In order to better adjust local land use plans to territorial dynamics, the State has gradually reinforced the role of the intermunicipal level, resulting in 2014 in the Law n°2014-366. This is known as the ALUR law (Accès au Logement et Urbanisme Rénové -Access to housing and remodelled urban planning policy). The ALUR law transferred the elaboration of the local land use plans to the intermunicipal level at the horizon of 2021.

Despite taking a significant step towards decentralization, it has to be noted that the State nevertheless retains a certain level control. In particular, only the State can determine the application, directives and objectives of French planning law (code de l'urbanisme). This includes instruments of value capture that are defined and framed at the central level. However, the municipalities or intermunicipal cooperation structures, in charge of the local land use plans, have a certain amount of freedom in their concrete application.

Recurring forms of public value capture

In France, recurring forms of public value capture can take the form of annual property taxes, real estate transfer taxes and capital gain taxes.

Recurring forms (annual payments)

The property taxes

France has implemented property taxes long ago (before the 1789 revolution), which leads to call them "les 4 vieilles" ("the four old women"): a property tax on unbuilt land, a property tax on built land, a housing tax and a corporate tax. Only three of them survive: the property taxes on built and unbuilt lands and the corporate tax. Since 2010 the corporate tax includes two parts: a part based on business property (the corporate property tax -CFE: cotisation foncière des entreprises) and another part (the corporate value added tax -CVAE: cotisation sur la valeur ajoutée des entreprises) calculated according to the added value created by companies. This last part of the tax is no longer related to land or property values.

These taxes are "local taxes" as they are raised by local authorities and benefit to them: property tax on built land is fixed by municipalities, intermunicipalities and départements (an intermediate level of government between the regional and the local level) in an independent but additional process. Property tax on unbuilt land and corporate property tax are fixed by municipalities only.

These taxes (CFE included) are due by the land or property owner every year, considering the situation on January 1 st . The tax revenues supply the local budgets and can be spent for any operating expenditures. In 2020, the annual revenues collected were 68.4 billion euros, 33.2% of the total receipts of the local authorities (source Direction Générale des Collectivités locales, 2021).

Each Property tax liability is calculated by multiplying the rental value of the property by the property tax rate. Following the French tradition of local taxes on land and property, the tax rate is fixed by local institutions (municipality, intermunicipal level or départements, depending on the tax). According to the law (code général des impôts -Art. 1379 to 1518 F ), the assessment of the rental value of the propertieswhich serves as the tax base -is carried out by national budget services. This assessment is supposed to reflect the annual market rental value of the property considering the quality, use, and localisation. In reality, the assessments done in 1975 have not been re-evaluated locally since then. As a consequence, several distortions appear with the present market property value2 .

The government regularly tries to re-evaluate the assessments to follow the current market price index. Up to now, a national index price is applied but it does not consider the evolutions in local markets. To be able to capture the current property value increase at a local level, a re-evaluation has been done in 2017, for the first time since 1975, but only for commercial and business properties. The revaluation is supposed to be effective for residential properties in 2026. In parallel, several tax exemptions are proposed to support social amenities and social housing situations. For example, a 15 years' property tax exemption is allowed for dwellings being built by social housing companies3 .

An additional taxation on unbuilt properties is expressly supposed to capture the potential land value due to increased building rights in the local land use plan. The tax has been introduced by national law in 2000 as an option. Municipalities can decide whether or not they want to levy this tax. When they do, it becomes part of their income (Art. 1396 Code general des impôts). The tax concerns properties which are designated as "buildable" in the local land use plan, but which remain unbuilt-upon. The landowner has to pay an additional charge fixed by municipalities as a maximum of 0.75€/m² additional value to the assessment of the rental value of these unbuilt properties. In 2014, an attempt to add a mandatory 5€/m² to the rental value assessment of identified unbuilt properties has been experimented4 . However, in 2017, due to landowner's recrimination, the government had to come back to an optional increase freely fixed by municipalities, from 0 € to 3€/m². About 400 municipalities out of 35,360 French municipalities have adopted the additional charge in 20185 .

In case of sales/purchase

Real estate transfer taxes

The real estate transfer tax is a legal transaction tax raised by départements and municipalities on all property purchases. It is not levied on a legal property transfer by heritage or donation. The tax rate is determined nationally by the law at 1.20% of the purchase price for the municipal part, and from 1.2% to 4.5% for the departmental part. In addition, the State levies a maximum of 2,37 % of the departmental part to compensate for the costs of collection of the tax. New buildings are exempted from this tax, regardless whether they are subject to VAT. In 2020, the annual revenues collected were 16.4 billion euros (source DGCL -Direction Générale des Collectivités locales, 2021).

Capital gain taxes

A national tax aims at transferring part of the capital gain of any purchase of real property to the State. The capital gain is reduced according to the length of possession6 and the tax rate is fixed by the State. In 2020, the rate is 36.2% of the capital gain calculated on the basis of a book value. This tax is considered as unstable as the taxation rate can vary in time, depending on the policy issues. In the past, some governments have decided to reduce it in order to boost real estate markets while others have decided to increase it in order to generate incomes for the public budgets.

In addition, a national tax concerns the sale of land of which the designation as agricultural land has been changed into a designation as "buildable" land in the latest revision of the local land use plan. Only the first transaction of the land after the change of designation is concerned by this tax. As such, the tax aims at capturing part of the value increase of the land due to its new status in the local land use plan (planning gain). The taxation is progressive with the level of capital gain: tax rate of 0% if the gain counts for less than 10 times the purchase price, tax rate of 5% if the gain is between 10 to 30 times the purchase price, and tax rate of 10% if the gain is over 30 times of purchase value. These tax revenues are allocated to a national agency which funds supports farmers by enabling their access to land and by supporting innovation.

Both national taxes aim at capturing the land value increase to the benefit of the State (i.e. the national budget).

In addition, municipalities can also tax capital gains generated from a planning decision (planning gain). The planning gain tax is levied on the first sale of unbuilt land after its designation as "buildable" in the local land use plan. The tax base is calculated as the difference between the current selling price, and the declared purchase cost at the time of the prior acquisition of the land (or declared value in case of heritage for instance), actualized by an official real estate index. If the historical purchase cost is unknown, the assessed capital gain is 1/3 of the current selling price. The tax rate is 10%. The tax contributes to the financing of the municipal development strategy, but the annual revenues remain low. In 2020, only 6,500 municipalities of the 35,360 municipalities have decided to implement the planning gain tax (fiscal statistics from ministry of Finance7 ).

Non recurring forms of public value capture

Non-recurring forms (focussing on one factor of value increase)

Since 1965, the law enables two types of instruments to capture land value increases induced by new public infrastructures. Local authorities can either use a development tax (taxe d'aménagement) or two forms of contractual participations (or developer obligations)8 .

Development tax (Taxe d'aménagement)

The development tax is levied by local authorities and assigned to the local budget to finance public infrastructure. It is paid on the entire municipal perimeter by the recipient of a building permit (for new constructions as well as for parking spaces). The tax is levied on the number of newly produced square meters of floor space, stated by local authorities in the building permit. The tax base is fixed at the national level, currently it is 870 €/m² of newly produced floor space in the Paris region, and 767 €/m² in other regions (as by the law in 2020). The tax rate is fixed by municipalities, between 1% and 5%. An additional 2.5% taxe rate is levied by the départements. Municipalities have the possibility to increase the tax rate to 20% on an identified area when the infrastructure cost is massive. The revenues can be spent on infrastructure of all kinds.

The tax is commonly used in France, as 80% of local authorities collect the development tax9 . In 2020, the annual revenues accounted for 1.1 billion euros (DGCL -Direction Générale des Collectivités locales, 2021).

Contractual participation (developer obligation)

Two instruments jfor contractual participation of developers to public infrastructure exist : the ZAC: Zone d'Aménagement Concerté (Comprehensive Development Zone), and the PUP: Projet Urbain Partenarial (Partnership Urban Project). Both these instruments are quite common to support local authorities investments in identified areas. They are designed as financial partnerships and very often include additional negotiations on the urban quality of the development and the building program10 .

Comprehensive Development Zone -Zone d'Aménagement Concerté

The Comprehensive Development Zone (Zone d'Aménagement Concerté -ZAC) has been initiated in 1967. It applies to development projects initiated by the public sector, and gives a lot of power to the land developer (expropriation rights), under strict public conditions, requirements, and controls. The land developer is required to finance part of the public investment in the area via "developer contributions". The range of developer contributions is wide but it has to comply with two requirements: (1) Causality: a direct link should be made between facilities imposed to the developers and the real needs related to the new buildings; (2) Proportionnality: fees charged to the developers should be proportional to the use made of them by (occupiers of) the new buildings. The ZAC process is transparent (that is, publicly advised) and strictly controlled by the law. The counterpart of the developers contributions is a solid guarantee of the achievement of the project.

Traditionnally, all the land included in the perimeter of a ZAC is temporarily owned by the land developer, who services the land and then sells the remaining area as serviced building plots. Recently, new ZAC contracts can include private landowners in the area, who wish to keep the land for their proper use. They still have to pay the same contributions as those required in the developed area proportionally to their land surface.

Partnership Urban Project -Projet Urbain Partenarial

The Partnership Urban Project (Projet Urbain Partenarial -PUP) is a contract introduced after 2014 which enables local authorities to charge private landowners for participation to public infrastructure in a private project, initiated and developped by the private developer. The contribution scheme is the same as in a ZAC, but results from a negociated process. Even though negotiations depend on the local context, they have to comply with the same requirements of "causality" and "proportionnality" as in a ZAC.

3.3.2 Non-recurring forms (focussing on more than one factor of value increase)

France experiments tools to transfer land value increase to social achievement without a direct contribution to public budget. During the previous decades, a focus was made on the provision of social housing. Public authorities use two kinds of instruments to include social housing in construction programs: firstly, a legislative obligation provided by the State and, secondly, an instrument allowing long-term land leases.

Social mix obligations -Servitudes de mixité sociale

Since 2000, the law « Solidarité et Renouvellement Urbain » -Art. 55 makes it compulsory for large cities to reach 20% of social housing of the total housing stock. In 2014, the rate was increased to 25% when a tense market pressure occurred. Every year, a statement is made to identify the municipalities where the ratio is not met. In 2019, 1,100 municipalities did not reach the requirement. If they remain in default after three years, the State can impose penalties and decide to replace the municipal authority in the control of urbanization processes. This might imply various interventions without the approval of the local authority: through the use of preemption rights, through the delivery of building permits, and through the development of new social housing projects. In 2020, 36 municipalities were in this situation, while the others succeeded in convincing developers to build enough social housing. In addition to large publicly led projects, municipalities use regulations in the local land use plans and negotiations to require a mix of private and social housing in new programs of housing development by private developers. This is called "servitudes de mixité sociale" or social mix obligations in English.

Interim landownership

In 2014 and 2015, the French government introduced a new instrument of value capture based on longterm land leases, with the objective to secure social home ownership. The system mixes two instruments 11 . A non-profit organisation (organisme de foncier solidaire (OFS), ALUR law, 2014) has been designed to buy land plots, build dwellings and sell them without the land to low or middle-income households. The land remains property of the OFS, who leases it to the home owners. A new long-term land lease called bail réel solidaire (BRS) (the Macron Law of 2015) was created to enable a kind of tenure in which homeownership remains affordable. Under this system, the future homeowner can buy the house from the OFS at a lower cost without paying the land price. The land lease that the homeowners have to pay to the OFS is set at such a rate that access to housing under these conditions is much cheaper as in the free market. A specificity of the type of land lease that is used (the BRS) is that when the dwelling is sold, anti-speculations clauses apply, and the land lease is renewed. This system enables the OFS to capture and freeze the land value in the long term for homeownership benefits. Since 2014, 9,400 housings have been launched through 71 OFS, most of them being supported by local authorities (source Foncier Solidaire France association, nov. 2021). A public report suggested opening of the system to a profitable organisation, but the proposal has been rejected by the Parliament.

Interim conclusion for France

11 See [START_REF] Guelton | The French system of Long-term land leases[END_REF] While there is a political involvement in France to implement recurrent tools of value capture, the attempts remain weak. Property taxes are not based on the market value of the property and capital value taxes do not apply to owner-occupiers. This situation reflects a political reluctancy to tax property, which can be related to the importance that is assigned to propert rights in France (considered, following art. 17 of the declaration of human rights of 1789 as "inviolable et sacré", inviolable and sacred). State interference in property rights is considered justified only if there is a demonstrated public necessity, and a just compensation for the property owner.

Nevertheless, a number of indirect instruments succeed in reallocating part of the value increase induced by urban planning. Contractual « tailor-made solutions » via PUP and ZAC are efficient in buoyant markets. Other instruments, like development tax or municipal capital gain tax, have a reduced impact as they concern very few cities and limited places. They are using the local land-use plan to impose some reallocation of land value increase for collective use, like the realisation of social housing (through « servitudes de mixité sociale ») or environmental preservation.

Figure 1 :

 1 Figure 1: Value capture in France

DGCL -Direction Générale des Collectivités locales,

Brémond, 2012

French Social housing companies benefit from a special status when they sign a contractual agreement with the State. The agreement includes maximal rental prices for low incomed households, between others.

[START_REF] Guelton | La sur-taxation des terrains constructibles non bâtis[END_REF]

Pouillaude, 2019, p39

In 2020, the tax is not due after 22 years of possession

https://www.data.gouv.fr/fr/datasets/impots-locaux/

See Booth, 2012

For a full description see[START_REF] Guelton | Development Tax in France as a Tool for Land Value Capture and Social Equity[END_REF]

See Guelton, 2018


Yangyang Qu

François-Xavier Sauvage

Gilles Clavier

Fabien Miomandre

Pierre Audebert
email: audebert@ppsm.ens-cachan.fr

Novel Metal-Free Synthetic Approach to 3-Monosubstituted Unsymmetrical 1,2,4,5-Tetrazines Useful for Bio-orthogonal Reactions

A facile, efficient and metal-free synthetic approach to 3-monosubstituted unsymmetrical 1,2,4,5-tetrazines is presented. Dichloromethane (DCM) is for the first time recognized as a novel reagent in the synthetic chemistry of tetrazines. Eleven 3-aryl/alkyl 1,2,4,5-tetrazines were prepared in excellent yields (up to 75%) through this novel approach. The mechanism of this new reaction, including the role of DCM in the tetrazine ring formation, has been investigated by DCM13C labeling, and is also presented and discussed as well as the photophysical and electrochemical properties.

Among other bio-orthogonal reactions, tetrazine click chemistry with strained alkenes or alkynes (also called inverse Electron Demand Diels Alder (iEDDA) reaction) has recently emerged as a powerful tool for biological imaging and detection, cancer targeting, drug delivery and biomaterials science. [1] The synthesis of reactive tetrazine reagents has recently attracted significant interest since the inception of this field. [2] Moreover, as a well-known electron-deficient heterocycle, [3] 1,2,4,5-tetrazines have already seen significant use for decades in various application fields such as organic electronics (e.g., OPVs, OFETs), [4] energetic materials, [5] coordination chemistry, [6] electrofluorochromism [7] and total synthesis of natural products. [8] Despite the promising application potential of 1,2,4,5-tetrazines, the lack of practical and convenient tetrazine synthetic approaches has hampered their development, especially for click chemistry related applications. [3] Therefore, developing synthetic routes that improve access to tetrazine molecules has recently become very useful, as it not only eases existing tetrazine applications, but also facilitates the emergence of new applications.

Herein we report a facile and efficient synthetic approach to 3-monosubstituted unsymmetrical 1,2,4,5tetrazines, which are well-known to be the most reactive tetrazines for room temperature (RT) click reactions. With our new approach, 3-monosubstituted unsymmetrical tetrazines are conveniently prepared in mild conditions from aromatic/alkyl nitrile precursors and dichloromethane (DCM). 11 tetrazine examples were prepared with excellent yields (up to 75%) through this novel approach. To the best of our knowledge, this is the first example utilizing DCM as a reactant in the chemistry of tetrazine synthesis. This approach gives improved access to both 3-aryl and 3-alkyl unsymmetrical 1,2,4,5-tetrazines in high yields, which are expected to be widely utilized in various applications, particularly in bioorthogonal click chemistry.

The classical approach to prepare 1,2,4,5-tetrazines is a two-step procedure starting from the addition of hydrazine to nitrile precursors, followed by oxidation of the resulting 1,2-dihydrotetrazine. [9] However, this approach is usually not efficient for unsymmetrical tetrazines. 3-monosubstituted unsymmetrical 1,2,4,5tetrazines were conventionally prepared from aromatic or alkyl nitrile precursors and formamidine salts, but yields are typically low, between 10% to 20%. [10] Recently, Devaraj et al. demonstrated an effective metal-catalyzed one-pot procedure to prepare 3-monosubstituted unsymmetrical 1,2,4,5-tetrazines. [11] This approach is currently intensively employed by scientists to prepare tetrazines useful for iEDDA reactions. However, the limitations of this approach are also obvious because it requires excess anhydrous hydrazine as the solvent (usually 50 equivalents to the nitrile precursors). Anhydrous hydrazine is an oily, highly toxic and hazardous reagent, which is commercially limited in several parts of the world, including Europe and China due to safety regulations. As a result, this synthetic procedure is limited to certain labs or companies. In addition, it also requires excess formamidine salts (usually 10 equiv.) and a nonneglectable amount of expensive metal catalysts (0.5 equiv.) (Therefore explaining the high prices of commercial clickable tetrazines, usually in the 1,000 $/g range). Moreover, the harsh conditions are not compatible with several functional groups such as carbonyls and alkyl halides, which are susceptible to give either nucleophilic addition or reduction. [12]

Synthesis

We envisioned developing a simple but also efficient synthetic approach to prepare 3-monosubstituted unsymmetrical 1,2,4,5-tetrazines. In our synthetic approach, DCM was selected as a novel starting material instead of formamidine salts, and we found out it exhibits a surprisingly good reactivity and selectivity (Scheme 1). Because of this excellent reactivity and selectivity, only mild reaction conditions are required. The dangerous anhydrous hydrazine can be replaced by the much more stable hydrazine hydrate as a reactant, and ethanol (EtOH) as the solvent, along with 2 equivalents of sulfur. Only 1 equiv. of DCM is needed rather than excess amount (10 equiv.) of formamidine salts. More importantly, no metal catalyst is required in this novel approach. All these factors make our approach simple, economical and efficient.

Scheme 1: Synthetic approach for 3-monosubstituted unsymmetrical 1,2,4,5-tetrazines.

We optimized the reaction conditions using 4-cyanophenylacetic acid as nitrile precursor to prepare tetrazine T1, which is a well-known tetrazine for bio-orthogonal reactions (Table 1). The reaction temperature is extremely important to the outcome of this reaction. We used microwave heating throughout, but we have verified that the microwave irradiation was not necessary, and heating in a sealed tube worked as well, though minor changes to the yields did happen. At 80 °C, the reaction can be finished within 6 hours (most of the nitrile precursor consumed, Entry 1). However, considerable amount of side products were also generated, making the purification tedious, along with a moderate yield of 38%. The yield of the reaction is significantly increased by decreasing the temperature to 50 °C (Entry 3). Using these conditions, an excellent yield of 72% was obtained. However, further decreasing the temperature decreases the yield again (Entry 4). In addition, (as historically noticed in Pinner reaction [13] ) sulfur addition was also found important in this reaction. As an inducer of the reaction, 1 equiv. of sulfur already leads to a quite reasonable yield, while slight excesses of sulfur can still improve the yield and 2 equiv. of sulfur gives the best yield. The importance of sulfur addition is also important to elucidate the mechanism of the reaction (see later on). When no sulfur is used, most of the nitrile precursor was recovered and only a small amount of product could be isolated (3%, Entry 5). However, further excess sulfur (4 equiv.) lowers the yield of the reaction probably because of unwanted side reactions (Entry 7). 50 °C, 24 h 4 equiv. 55 (a) All reactions were carried out on a 0.5 mmol scale in 0.5 ml EtOH. 1 equiv. DCM, 1 equiv. nitrile precursor and 8 equiv. hydrazine hydrate (MW = microwave, equiv. = equivalents). (b) Yields (isolated) based on the nitrile precursor.

To extend the scope of this synthetic approach, a series of 3-monosubstituted unsymmetrical 1,2,4,5tetrazines were prepared employing this approach (Table 2). Several useful unsymmetrical tetrazines T1, T2, T3 and T4 were successfully prepared in good to high yields. Interestingly, when using 4-(aminomethyl)benzonitrile hydrochloride as the nitrile precursor, the desired tetrazine with amine functional group was not obtained, instead tetrazine T3 with hydroxy group was obtained in high yield (71%, Scheme 2), due to the reaction of nitrous acid (HNO2) with primary amine during the oxidation step. The yield is even higher compared to using 4-(hydroxymethyl)benzonitrile as the nitrile precursor which is commercially-unfriendly. This reaction offers a more efficient and economical pathway to tetrazine T3.

Table 2: Synthesis of 3-monosubstituted unsymmetrical tetrazines T1-T11 (a) .

(a) Reactions were carried out on a 1 mmol scale in 1 ml EtOH. Yields (isolated) based on the starting nitrile. 1 equiv. DCM, 1 equiv. nitrile precursor, 2 equiv. sulfur and 8 equiv. hydrazine hydrate (except T11). (b) T3 was alternatively prepared using 4-(aminomethyl)benzonitrile hydrochloride as the nitrile precursor. (c) 0.2 ml DMF as co-solvent. (d) 2 equiv. DCM, 1 equiv. nitrile precursor, 4 equiv. sulfur and 16 equiv. hydrazine hydrate.

Tetrazines T5, T6, T7 and T8 were also easily prepared in moderate to high yields as expected, among which T5 and T6 are useful for further preparation of new photo and/or electroactive molecular materials through coupling reactions, an outcome which is currently under investigation in our group. Importantly, unsymmetrical alkyl tetrazines could also be prepared in high yield by this approach. T9 was directly prepared from the alkyl nitrile precursor benzyl cyanide and DCM in a moderate yield (45%).

Remarkably, when using 1,4-dicyanobenzene as the nitrile precursor, the two cyano groups can be sequentially converted to tetrazine groups. Therefore, tetrazine T10 and T11 were readily prepared in high yields. Notably, T11 is a di-tetrazine molecule which has two functional sites adapted for click reactions, which could potentially serve as a bifunctional bio-orthogonal tetrazine reagent.

Scheme 2: Synthesis of T3 using 4-(aminomethyl)benzonitrile hydrochloride as the nitrile precursor.

In order to explore the mechanism of the reaction, the participation of CH2Cl2 in the tetrazine ring formation was firstly confirmed by using 13 CH2Cl2 as the reactant (Scheme 3). The doublet peak at 10.17 ppm ( 1 JCH = 212.04 Hz) of 1 H NMR and the extraordinary high intensity of singlet peak at 157.73 ppm of 13 C NMR can be ascribed to the proton and carbon of H-13 C group, respectively (Figure 1). In addition, the C-3 carbon shows a doublet peak at 166.61 ppm ( 3 JCC = 29.20 Hz) in 13 C NMR (Figure 1 The formation of 1,2,4,5-tetrazine is commonly agreed as a two-step procedure starting from the nucleophilic attack of nitrile precursor by sulfur activated hydrazine, followed by oxidation of the resulting 1,2-dihydrotetrazine. [3] Since no standard "Pinner" product is observed, with as little as 1 equiv. of DCM added, the first step is very likely addition of the sulfur-activated hydrazine on DCM, followed by addition on the nitrile. Since a noticeable amount of product is formed with as little as 1 equiv. of sulfur (despite 2 equiv. improve the yields), and that the dihydrotetrazine is formed before oxidant addition, it is likely that hydrazine alone reacts in the following step, with cyclization kicking out the less reactive second chlorine in the end. Following this reasoning, a possible mechanism for tetrazine ring formation using CH2Cl2 as a reagent is summarized in Scheme 4. Scheme 4: Proposed mechanism for tetrazine ring formation using CH2Cl2 as a reagent.

To support more strongly our mechanism, and lead by the idea that actually, the activated hydrazine was implicated in the first addition step, and not further, we have investigated the influence of intermediate amounts of sulfur. Actually, 1.2 equiv. of sulfur already leads to quite good yieds, and 1.5 equiv. give almost the same high yield than 2 equiv. This demonstrates that sulfur in moderate excess to 1 equiv. is sufficient to the reaction, the excess probably serving to compensate for a non-quantitative formation of the sulfurhydrazine activated nucleophile (due to the heterogenous reaction). This strongly suggests that only 1 equiv. of activated nucleophile is used at the beginning of the reaction process, 1 equiv. of hydrazine alone being involved afterwards in one of the following steps, as proposed in Scheme 4.

It is worth mentioning, however, that the reaction did not work in a few trials with the more hindered dichloroethane, and seems probably limited to DCM as a synthon, despite our attempts were limited to this dichloroalkane only.

Photophysics

The photophysical data for the T1-T9 series are gathered in Table 3. All compounds present similar features, typical of tetrazine derivatives, namely an absorption band in the visible with a maximum located near 535 nm and a rather low extinction coefficient. Unfortunately, a very weak emission in the 575-595 nm range has been recorded for all the series (only T4 exhibits a luminescence quantum yield over 1%). These poor fluorescence properties were expected since it is known that only tetrazines substituted with heteroatoms are highly emissive. [3] Table 3: Absorption and emission maximum wavelengths Absorption wavelength (a) Emission wavelength Quantum yield (c) T1 536 (300) (b)

Electrochemistry

The electrochemistry of compounds T1-T9 was studied in solution in acetonitrile. Two types of cyclic voltammetries (CV) are observed:

-For non-acidic compounds a nicely reversible CV is observed (Figure 2, green curve).

-For acidic compounds (T1-T2), a fully irreversible reduction peak can be seen (Figure 2, red curve)

All CVs have been compared to a standard (chloroethoxytetrazine) to assess the number of electrons exchanged on one hand and the standard potential on the other hand. Using a standard allows one to get rid of electrode surface issues for the relation between peak current and number of electrons and of reference drift for the determination of standard potential.

The results are summarized in Table 4. All compounds behaving reversibly shows a one-electron reduction as expected, leading to the anion radical which is thus stable at the timescale of cyclic voltammetry. Moreover, the standard potentials for the redox couple formed by tetrazine and its anion radical all lie around -1.35 V vs. Fc which is similar to tetrazines substituted by donating aromatic groups like pyrrole or thiophene. [3] It can be noticed that T5-T6 have a more positive reduction potential of ca. +0.1 V in agreement with the withdrawing character of the halogen substituents born by the tetrazine core. This demonstrates that the redox potentials of these tetrazine derivatives are actually sensitive to the electronic properties of the substituent. The two compounds substituted with acidic functions show a non-reversible electrochemical behavior assigned to the rapid protonation of the anion radical. This leads to more positive reduction potential (typically -1.14 V for the peak reduction potential of T1) in agreement with an 'EC' mechanism. Besides the number of electrons exchanged which has been found close to 2 is in agreement with a '2H + + 2e -' mechanism similar to quinone reduction. Typically after the one-electron first reduction and fast protonation, the generated radical is further reduced and the anion protonated.

Table 4: Electrochemical data for T1-T9 (a) Standard potential /V vs. Fc + /Fc Exp. det. number of electrons exchanged T1 n.d. (b) 2.25 In summary, we have discovered a facile, efficient and metal-free synthetic approach to 3monosubstituted unsymmetrical 1,2,4,5-tetrazines. DCM has been demonstrated to be a novel alternative reagent to formamidine salts which exhibits excellent reactivity and selectivity during the tetrazine ring formation, resulting in high yields and mild reaction conditions in the tetrazine synthesis. 3monosubstituted unsymmetrical 1,2,4,5-tetrazines were thus prepared which are among the most useful reagents in bio-orthogonal click chemistry as well as other application fields (e.g., photo and/or electroactive molecular materials).

Scheme 3 :

 3 Scheme 3: Synthesis of T12.

Figure 1 :

 1 Figure 1: 1 H NMR ((left, 400 MHz, CDCl3) and 13 C NMR (right, 100 MHz, CDCl3) spectra of T12. Only downfield parts of the spectra are shown here to show the characteristic signals clearly (see NMR spectra T12 in supporting information for full spectra).

  (a) Values in nm. (b) The values in brackets are extinction coefficients. (c) Luminescence quantum yields of solutions of compounds T1-T9 in acetonitrile.

Figure 2 :

 2 Figure 2: CVs of T7 (full green line) and T1 (dotted red line) in acetonitrile (+TBAPF6) on glassy carbon electrode at 0.1 V/s. Currents are normalized vs. the peak cathodic value for clarity sake.

  In acetonitrile (+TBAPF6) on glassy carbon electrode. (b) n.d. = not defined.

Table 1 :

 1 Optimization of the reaction conditions(a) .

	Entry	Temp., time	S	Yield [%] (b)
	1	80 °C, 6 h	2 equiv.	38
	2	60°C, 24 h	2 equiv.	61
	3	50 °C, 24 h	2 equiv.	72
	4	40 °C, 24 h	2 equiv.	40
	5	50 °C, 24 h	0 equiv.	3
	6	50 °C, 24 h	1 equiv.	45
	7	50 °C, 24 h	1.2 equiv.	62
	8	50 °C, 24 h	1.5 equiv.	70
	9

Acknowledgements

The research leading to these results has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 674990 (EXCILIGHT).

COMMUNICATION

A facile, efficient and metal-free synthetic approach to 3-monosubstituted unsymmetrical 1,2,4,5-tetrazines is presented. Dichloromethane (DCM) is for the first time recognized as a novel reagent in the chemistry of tetrazine synthesis. 11 tetrazine examples were prepared with excellent yields (up to 75%) through this novel approach. The role of DCM in the tetrazine ring formation, has been investigated by DCM 13


P Develter

J Bosse

O Rabaste

P Forster

J.-P Ovarlez

OFF-GRID RADAR TARGET DETECTION WITH THE NORMALIZED MATCHED FILTER: A MONOPULSE-BASED DETECTION SCHEME

Keywords: Radar, Detection, Off-grid, Monopulse, Normalized Matched Filter

In this paper, we tackle the problem of off-grid radar target detection based on the Normalized Matched Filter. State-ofthe-art solutions to this problem include the use of the Generalized Likelihood Ratio test (GLRT), whose implementation can be costly, subspace detectors, or oversampling. In this paper, we introduce a new solution for improving detection of an off-grid target. It is inspired by monopulse angle estimation methods. Using simulations, we show that our solution is comparable to the GLRT under Gaussian noise hypothesis with a known covariance matrix in terms of detection probability. As such, its detection performance appears to be often better than those of detectors in the same computational cost range, and significantly better in some cases.

INTRODUCTION

The goal of radar systems is to detect the presence of targets with unknown parameters in unknown environment. Classically, in presence of unknown parameters, the commonly used detection strategy is the Generalized Likelihood Ratio test (GLRT) that replaces the unknown parameters by their Maximum Likelihood estimators (MLE) in the detection test. Unfortunately, analytical MLE solutions are not available for some target parameters of interest (Doppler shift, distance, and direction for example).

Therefore, for ease of implementation, most detection strategies assume that target parameters lie over a discrete set, called the grid. However, target parameters have no reason to fall exactly on the grid, since they are distributed over a continuous range. This mismatch between the tested parameters and the real target parameters deteriorates the response of most state-of-the-art tests. For example, in homogeneous Gaussian environment, it is well known that the Matched Filter (MF) test can approximately suffer from a maximum of 3dB loss for common grid sampling strategies (e.g. at the resolution cell). This tends to deteriorate the detection probability. The off-grid impact has also been shown to be particularly dramatic for other detection schemes like the Normalized Matched Filter (NMF) test. In some cases, the detection probability may vanish to 0 even for high Signal to Noise ratio (SNR) [START_REF] Rabaste | Off-grid target detection with Normalized Matched Subspace Filter[END_REF], especially for common low probability of false alarm (P F A ).

To overcome this problem, the most obvious solution is to approximate the GLRT on a refined parameter grid, but this can be costly in terms of computations. Other works have addressed this issue in adaptive context with a steering vector mismatch (bounded angle mismatch) [START_REF] Besson | Detection of a signal in linear subspace with bounded mismatch[END_REF], but they are not yet suited for low P F A radar contexts [START_REF] Bosse | Adaptive subspace detectors for off-grid mismatched targets[END_REF]. The authors in [START_REF] Bandiera | Advanced Radar Detection Schemes Under Mismatched Signal Models[END_REF] proposed a two-stage detector to reduce the loss in performance. Those solutions do not cure the asymptotic SNR loss under grid mismatch [START_REF] Rabaste | Geometrical design of radar detectors in moderately impulsive noise[END_REF]. Some authors have proposed to model the mismatch through relevant subspaces [START_REF] Bosse | Subspace rejection for matching pursuit in the presence of unresolved targets[END_REF] using subspace detection framework.

In this paper, we propose a new method that solves the off-grid problem. It enables to achieve a detection probability of 1 at high SNR, even for arbitrary low P F A . Moreover, this is achieved at a low computational cost.

Section 2 presents the signal model and the off-grid detection problem. We present our method in Section 3. Finally, its performance is evaluated in Section 4 through simulations.

Notations: Matrices are in bold and capital, vectors in bold. For any matrix A or vector, A T is the transpose of A and A H is the Hermitian transpose of A. I is the N × N identity matrix and CN (µ, Γ) is the circular complex Normal distribution of mean µ and covariance matrix Γ.

THE OFF-GRID RADAR DETECTION PROBLEM

In radar detection, the main problem consists in detecting a complex signal s ∈ C N corrupted by an additive noise n (clutter, thermal noise, etc.). This problem can be stated as the following binary hypothesis test:

H 0 : r = n , H 1 : r = α s(θ) + n , ( 1
)
where r is the complex N -vector of the received signal, α is an unknown complex target amplitude and s(θ) stands for a generally known steering vector characterized by unknown target parameters θ (time-delay, Doppler or angle). In the sequel, we will assume n is a zero-mean complex circular Gaussian noise vector with known covariance matrix Γ but with unknown variance σ 2 i.e. n ∼ CN (0, σ 2 Γ). This context is known as a partially homogeneous Gaussian environment. Without loss of generality, we will here assume θ to be the scalar Doppler shift of the target and we here investigate only the normalized Doppler steering vector s(θ):

s(θ) = 1 √ N 1, e 2iπθ , . . . , e 2iπ(N -1)θ T . (2)
This model of steering vector is often encountered in radar Range-Doppler detection schemes where the problem consists in estimating a complex sinusoid embedded in noise after range Matched Filter processing.

For unknown parameters {λ} i∈[0,1] depending on each hypothesis {H i } i∈[0,1] (either parameters of interest and/or nuisance parameters), the usual procedure to derive the best strategy relies on the Generalized Likelihood Ratio (GLR) statistic, namely the ratio Λ(r) between the Probability Density Function (PDF) f H1 (.) of the data under H 1 and the PDF f H0 (.) under H 0 where the unknown parameters are replaced by their MLE estimate:

Λ(r) = max λ1 f H1 (r) max λ0 f H0 (r) H1 ≷ H0 η. (3)
When λ 1 = {α, σ} and λ 0 = {σ} with θ = θ 0 known, the corresponding GLRT is known as the NMF (Normalized Matched Filter) [START_REF] Scharf | Signal detection in Gaussian noise of unknown level: an invariance application[END_REF][START_REF] Scharf | Matched subspace detectors[END_REF]:

t Γ (r, θ 0 ) = s(θ 0 ) H Γ -1 r 2 s(θ 0 ) H Γ -1 s(θ 0 ) r H Γ -1 r H1 ≷ H0 η. ( 4
)
This test is also widely used for adaptive radar in non-Gaussian contexts [START_REF] Conte | Asymptotically optimum radar detection in compound-Gaussian clutter[END_REF][START_REF] Pascal | On a SIRV-CFAR detector with radar experimentations in impulsive noise[END_REF] for example when the noise is distributed according to a Complex Elliptically Symmetric (CES) distribution [START_REF] Ollila | Complex elliptically symmetric distributions: Survey, new results and applications[END_REF]. When θ is unknown, what is done in practice is to run a variety of tests. The set of the parameters Θ used for testing is called the grid. For a parameter θ 0 on the grid, we will define the cell of width ∆ associated with this parameter as

ν θ0 = [θ 0 -∆/2, θ 0 + ∆/2] . (5)
The grid step size ∆ is classically taken such that s(θ 0 ) and s(θ 0 + ∆) are orthogonal. In our case, ∆ = 1 N i.e. the grid samples are separated by the Doppler resolution.

Each test is run with the assumption that the target parameters are equal to those under test. If a target is detected for any of those tests, it will be considered as detected. However, target parameters θ are never exactly equal to the parameters under test, as target parameters are continuous.

If the point θ 0 where the NMF is tested is different from the parameter θ of the target, the target is said to be off-grid. This induces a mismatch (θ = θ 0 ) between the real target steering vector s(θ) and the steering vector s(θ 0 ) under test. Unfortunately, it was shown in [START_REF] Rabaste | Geometrical design of radar detectors in moderately impulsive noise[END_REF] that the NMF detector is very sensitive to steering vector mismatch, potentially leading to a dramatic deterioration of the detection performance: in particular for a mismatch larger than the detection threshold, it was shown that the asymptotic detection probability tends to 0 at high SNR: this phenomenon occurs [START_REF] Rabaste | Off-grid target detection with Normalized Matched Subspace Filter[END_REF] for P F A as high as 10 -3 . Indeed, the NMF value can be interpreted as a squared cosine between Γ -1/2 s(θ 0 ) and Γ -1/2 r [START_REF] Rabaste | Geometrical design of radar detectors in moderately impulsive noise[END_REF]. As soon as the mismatch causes this angle to lie outside the detection cone defined by the detection threshold η in (4), no detection occurs even without any noise. This can be seen in Figure 1, which displays the NMF response without any noise as a function of the mismatch: this response falls below the detection thresholds at the edge of the cell. This pathological behavior motivates the search for robust detection schemes.

The most natural solution is to consider the GLRT with unknown θ ∈ ν θ0 :

GLRT (r, θ 0 ) = max θ∈ν θ 0 t Γ (r, θ) H1 ≷ H0 η. (6)
Unfortunately, a closed-form expression of this test is unknown, it has to be approximated using possibly computationally-heavy methods.

PROPOSED MONOPULSE-INSPIRED SCHEME

In this section, we introduce a new method for coping with the mismatch. We took inspiration from monopulse methods [START_REF] Mosca | Angle estimation in amplitude comparison monopulse systems[END_REF], and consider the following function:

h Γ,θ0 (r) = t Γ r, θ 0 - ∆ 2 -t Γ r, θ 0 + ∆ 2 t Γ r, θ 0 -∆ 2 + t Γ r, θ 0 + ∆ 2 . ( 7
)
-0.5/N 0 0.5/N -1 0 1 Fig. 2. Candidate g(.) functions for N=10. Γ(ρ) refers to the covariance matrix model in [START_REF] Pascal | On a SIRV-CFAR detector with radar experimentations in impulsive noise[END_REF] The principle is the following: without noise, knowing two test points of the NMF, namely θ 0 + ∆ 2 and θ 0 -∆ 2 , in the main lobe (here, the resolution cell) is enough to provide an estimate of θ, as the distribution of energy between those points is directly linked to the mismatch:

δ = θ -θ 0 . (8)
The quantity [START_REF] Scharf | Signal detection in Gaussian noise of unknown level: an invariance application[END_REF] known as the monopulse ratio can be seen as an approximation of the MLE [START_REF] Mosca | Angle estimation in amplitude comparison monopulse systems[END_REF]. The link between δ and h Γ,θ0 (r) is made thanks to the noiseless g Γ,θ0 function defined as follows:

g Γ,θ0 (δ) = h Γ,θ0 (s(θ 0 + δ)) (9)
Our goal is to find an estimate θ = δ + θ 0 of θ, to plug it in the NMF in order to approximate the GLRT in [START_REF] Bosse | Subspace rejection for matching pursuit in the presence of unresolved targets[END_REF]. For that purpose, δ is obtained by computing δ = g -1 Γ,θ0 (h I,θ0 (r)). Of course, this only makes sense if g Γ,θ0 is invertible. As can be seen in Figure 2, this is not always the case, depending on Γ and θ 0 . For the particular case Γ = I, the functions g Γ,θ0 are invertible. This is why we will use these functions, even in colored noise. Moreover, they do not depend on θ 0 and will be simply denoted by:

g(δ) = g I,θ0 (δ) = sin (πN (δ + ∆)) sin (π(δ + ∆)) 2 - sin(πN (δ -∆)) sin(π(δ -∆)) 2 sin(πN (δ + ∆)) sin(π(δ + ∆)) 2 + sin(πN (δ -∆)) sin(π(δ -∆)) 2 .
The function g takes its values between -1 (at the right edge of the cell) and 1 (at the left edge of the cell).

To summarize, our proposed method is as follows: for every θ 0 on the grid:

1. compute t I r, θ 0 -∆ 2 and t I r, θ 0 + ∆ 2 ; 2. compute δ = g -1 (h I,θ0 (r));

run the final tests t

Γ r, δ + θ 0 H1 ≷ H0 η g .
Finding η g analytically is much more involved than for the NMF, because the true statistic of t Γ r, δ + θ 0 is related to the non-independent random variables r and δ = g -1 Γ,θ0 (h I,θ0 (r)). This is why η g was computed offline resorting to Monte Carlo methods.

Let us evaluate the computational load of this scheme. The method has to be applied in each of the N resolution cells. In total, t Γ needs to be computed 2N times. Consequently, the overall computational load is equivalent to that of approximating GLRT using 2 points per cell, neglecting the cost of inverting g, which is minimal since g can be tabulated offline. Finding δ by computing g -1 then reduces to a simple lookup table operation. It turns out that our approach outperforms existing methods with equivalent computational cost.

Finally, let us describe some properties of this approach. Firstly, when the SNR tends to infinity, r is equivalent to αs(θ) and our procedure yields the true parameter value i.e. θ = θ. This ensures that the probability of detection tends to 1. Secondly, when Γ = I, θ is an approximate MLE [START_REF] Mosca | Angle estimation in amplitude comparison monopulse systems[END_REF] and our test is an approximate GLRT. Thirdly, simulations show that our procedure is still close to the GLRT even when Γ = I.

NUMERICAL RESULTS

In this section, the performance of our detection scheme is illustrated using Monte Carlo simulations. In each simulation, target parameter θ is drawn at random, uniformly in a given cell. For a given cell under test, we compared the performance of different detectors as a function of SNR. The detectors we evaluated are the DPSS NSMF with subspaces of dimension Fig. 4. P D versus SNR with ρ = 0.9 at cell θ 0 = 0 for a P F A of 10 -6 , N = 10.

2 introduced in [1], our scheme under the name "Proposed" in the figures, an approximation of the GLRT done computing 50 tests per cell, the classical and mismatched NMF under the name "NMF" and a crude approximation of the GLRT exploiting the maximum of two tests per cell under the name "Oversampling x2". Performance for known θ is also provided as "Oracle". Oracle and the GLRT are given as benchmarks. Other detectors ("DPSS NSMF","Oversampling x2") are in the same computational range as our proposed method. Thresholds are computed for each detector, using 20 P -1

F A
Monte Carlo simulations. For each SNR, 20000 Monte Carlo simulations are run, considering steering vectors of length N = 10 and setting P F A = 10 -6 . Figure 3 first compares the performance of our method with the other detection schemes under additive white noise (Γ = I). Note that in this case, all the cells are equivalent in terms of detection probability. It can be seen that our detector is very close to the GLRT even though its computational cost is lower. As such, it does better than the detectors it is compared to, at a similar cost.

Our proposed detection scheme was also tested against colored Gaussian noise. Its covariance matrix was generated according to the following well-known model:

Γ(ρ) = T 1 ρ . . . ρ N -1 , (10)
where T (.) is the Toeplitz operator. We here tested all the schemes against colored noise of covariance matrice Γ(0.9).

It should be noted that when Γ = I, tests behave differently for each resolution cell. In Figure 4, the target is located in the cell centered around θ 0 = 0/N , and in Figure 5, θ 0 = 5/N . The relationship between |α| 2 and the SNR in dB is : Note that for the oracle detector, the detection probability depends only on the parameter |α| 2 σ 2 s(θ) H Γ -1 s(θ) that is the non-centrality parameter characterizing the ratio between the target power and the noise power in the cell under test. It partly explains why the performance depends on the choice of the cell under test.

SNR = 10 log 10 |α| 2 σ 2 . (11)
In cell ν 0 , the shape of the whitened steering vectors variety makes it so that both DPSS NSMF and oversampling by a factor of two do not approach a detection rate of 1 as SNR becomes very high. However our scheme performs only a bit worse than GLRT, and as expected its detection rate increases towards 1 when the SNR is high. In cell ν 5/N , all detectors perform much better than in cell ν 0/N and than in white noise, which was to be expected because of the noncentrality parameter. In this case, the shape of the whitened steering vectors variety also makes it possible for the detectors to detect targets asymptotically. As a result, the probability of detection of DPSS NSMF and oversampling by a factor of two tends toward 1 as SNR increases. In this context, our scheme does approximately as well as the DPSS NSMF detector asymptotically, which makes it slightly worse than GLRT, as it was before.

CONCLUSION

This paper is motivated by detector performance degradation in the case of off-grid targets. The proposed solution is based on monopulse technique and can be considered as an approximate GLRT for off-grid targets in Gaussian noise. It also overcomes the problem of the off-grid NMF not reaching a probability of detection equal to 1 when SNR tends to infinity. It outperforms existing detectors of the same computational cost order and provides detection performance close to the true GLRT, whatever the noise covariance matrix.

Future works will investigate the derivation of the P F Athreshold relationships, and the adaptive case as well as non-Gaussian additive noise environments.

8 Fig. 1 .

 81 Fig. 1. Comparison of the NMF response with various thresholds η = 1 -P 1/(N -1) F A when Γ = I, N = 10.

Fig. 3 .

 3 Fig. 3. P D of the detectors in white Gaussian noise, for a P F A of 10 -6 , N = 10.

Fig. 5 .

 5 Fig.5. P D versus SNR with ρ = 0.9 at cell θ 0 = 5 for a P F A of 10 -6 , N = 10.