S V Endruweit

S Lomov

P Advani

A X H Yong

A Aktas

D May

A Endruweit

S V Lomov Dg

S Advani

P Hubert

S G Abaimov

D Abliz

I Akhatov

M A Ali

S Allaoui

T Allen

D C Berg

S Bickerton

B Caglar

P Causse

A Chiminelli

S Comas-Cardona

M Danzi

J Dittmann

C Dransfeld

P Ermanni

E Fauster

A George

J Gillibert

Q Govignon

R Graupner

V Grishaev

A Guilloux

M A Kabachi

A Keller

K Kind

D Large

M Laspalas

O V Lebedev

M Lizaranzu

A C Long C C López

K Masania

V Michaud

P Middendorf

P Mitschang

S Van Oosterom

R Schubnel

N Sharp

P Sousa

F Trochu

R Umer

J Valette

J H Wang Aa

Experimental characterisation of textile compaction response: a benchmark exercise

Keywords: A. Fabric/textiles, D. Mechanical testing, Compressibility 1

HAL is

Introduction

During the processing of fibre-reinforced composites, the fibre network is subjected to through-thickness compressive forces. These compressive forces are applied at different stages of impregnation with a liquid matrix system, depending on the processing method applied, and dictate the volume fraction of the finished component. In some processes, the reinforcement is compacted after impregnation to obtain the desired fibre volume fraction and to minimise the volume of voids in the polymer matrix. An example is filament winding, where individual fibre tows are compressed by the tension induced during the fibre deposition process on a mandrel. In pultrusion, impregnated fibre tows are compressed as they are pulled through a die. In autoclave or out-of-autoclave processing of prepregs, the impregnated fibre bed reacts to the pressure in the autoclave or atmospheric pressure, which is applied to a vacuum bag enclosing the lay-up. On the other hand, there are processes where the dry reinforcement is compressed before it is impregnated with the matrix. In Resin Transfer Moulding, where rigid mould tools are used and the tool cavity height is given, the preform compaction response determines the reaction force on the tooling surfaces, which indicates if a given fibre volume fraction can be achieved. In vacuum infusion, where the compaction pressure is given (atmospheric pressure), the compaction response allows the component thickness and the resulting fibre volume fraction to be predicted. As high values of the fibre volume fraction, V f , are sought after in the manufacture of structural composites to obtain high mechanical properties of the finished component, characterisation of the relation between compaction pressure and V f of the reinforcement is vital to achieving the desired mechanical properties. Changes in V f during preform compression also result in a reduction in its permeability, which impacts the flow of liquid resin during the subsequent preform impregnation.

From compaction experiments on carbon fibre beds impregnated with silicone oil, Gutowski et al. [START_REF] Gutowski | Experiments in composites consolidation: fiber deformation[END_REF][START_REF] Gutowski | Resin flow/fiber deformation experiments[END_REF] found that at high fibre volume fraction (V f > 0.5) the fibres carry a gradually increasing portion of the applied load, defined as the effective stress, . This behaviour is caused by multiple fibre-to-fibre contacts 𝜎 resulting from the fibre waviness. Assuming the fibres to be curved beams in bending, the following expression was derived to express the effective stress in the fibre bed as a function of the fibre volume fraction:

𝜎(𝑉 𝑓 ) = 3𝜋𝐸 𝑓 𝛽 4
( 1 -𝑉 𝑓 𝑉 𝑓0 ) ( 𝑉 𝑎 𝑉 𝑓0 -1 ) 4 (Equation 1)

where E f is the flexural modulus of the fibre, β is the ratio of arc length and arc height in wavy fibres (related to crimp), V f0 is the initial fibre volume fraction in the uncompressed preform, and V a is the maximum achievable fibre volume fraction. The fibre bed compaction is measured by compressing a fabric stack, typically using a universal test machine (UTM) which can measure the force-displacement relationship. This type of test has been widely used to study different aspects of fibre bed compaction, and several loadvolume fraction relationships have been proposed based on these studies [START_REF] Robitaille | Compaction of textile reinforcements for composites manufacturing. I: Review of experimental results[END_REF][START_REF] Chen | A nonlinear compaction model for fibrous preforms[END_REF][START_REF] Comas-Cardona | Unidirectional compression of fibre reinforcements. Part 1: A non-linear elastic-plastic behaviour[END_REF][START_REF] Pearce | The compressibility of a reinforcement fabric[END_REF][START_REF] Bickerton | The viscoelastic compression behavior of liquid composite molding preforms[END_REF]. Empirical or semi-empirical models derived from compaction tests are typically expressed as a power-law relation as follows:

𝑉 𝑓 = 𝑎𝜎 𝑏 (Equation 2)
where a and b are fitting constants.

Although many aspects of fibre bed compaction have been extensively studied, there is currently no standard test procedure in place for this type of measurement. In 2016, an international survey of procedures for permeability measurement was carried out by the National Physical Laboratory (Teddington, UK) and the National Composites Centre (Bristol, UK). Responses were obtained from 34 organisations across both industry and academia, which highlighted that there is a current need for standardisation of test methods for reinforcement processing properties, permeability and compaction response [START_REF] Aktas | NPL REPORT MAT 82: Survey of procedures in use for permeability measurements in liquid composite moulding processes[END_REF]. Measurement standards ensure that data is traceable and comparable. In the manufacture of composite materials, standard test procedures for measurement of material properties related to processing will speed up production times by providing confidence and quality in the data used for process design. Following three international benchmark exercises on reinforcement permeability testing carried out between 2011 and 2019 [START_REF] Arbter | Experimental determination of the permeability of textiles: a benchmark exercise[END_REF][START_REF] Vernet | Experimental determination of the permeability of engineering textiles: Benchmark II[END_REF][START_REF] May | In-plane permeability characterization of engineering textiles based on radial flow experiments: a benchmark exercise[END_REF], a new test standard for radial in-plane permeability measurement will be developed. However, the issue of compaction testing has not been addressed so far. This paper reports the results of an international benchmark exercise on measurement of textile preform compaction. In this exercise, various test configurations in current use for thickness-controlled compression measurement are compared. Previous observations indicate that wet samples are typically easier to compress than dry samples due to lubrication effects [START_REF] Kelly | Viscoelastic response of dry and wet fibrous materials during infusion processes[END_REF][START_REF] Kim | Compressibility and relaxation of fiber reinforcements during composite processing[END_REF], therefore the benchmark exercise was carried out for both wet and dry specimens. The reinforcement fabrics to be characterised as well as the test fluid, target thicknesses of the compressed material, and crosshead speed were initially prescribed for all users in order to isolate the effect of measurement set-up on the repeatability and reproducibility of the data.

Participants had to report if they followed the instructions but had the choice in other aspects of the test, such as the sample and testing fixture geometry and testing equipment instrumentation. The aim of this study was to obtain an overview of the different approaches presently used in order to move towards a recommended test procedure for fibre bed compaction measurements by identifying aspects of the test method critical to obtaining reliable results, and assessing the degree of scatter in compression data between organisations following current methods. Table 1 lists the participants of this study.

Materials and Methods

Materials

Two different fabrics were used in this benchmark; a glass fibre 2/2 twill weave (in the following referred to as WOVEN) with a nominal superficial density of 295 g/m 2 , supplied by Hexcel, and a biaxial (±45°) glass fibre non-crimp fabric (in the following referred to as NCF) with a nominal superficial density of 444 g/m 2 , supplied by Saertex. In both cases, the material sent to participants was taken from the same batch of fabric. Further details of these fabrics can be obtained in the report for a study on permeability testing in radial flow experiments [START_REF] May | In-plane permeability characterization of engineering textiles based on radial flow experiments: a benchmark exercise[END_REF] which ran in parallel to this study, using the same materials. For the wet compaction tests, specimens were saturated with Dow Corning Xiameter PMX-200 silicone fluid (100 cs).

Silicone oil was chosen as its viscosity is similar to that of liquid resin systems and it is often used as a test fluid in permeability measurement. This specific oil was also used in permeability benchmark exercises [START_REF] Vernet | Experimental determination of the permeability of engineering textiles: Benchmark II[END_REF][START_REF] May | In-plane permeability characterization of engineering textiles based on radial flow experiments: a benchmark exercise[END_REF]. The viscosity-temperature curve of the test fluid was characterised experimentally at TU München (a viscosity  = 97 mPa.s was measured at a temperature T = 20 C).

Sample preparation

Participants were required to carry out compression tests on both fabrics, testing each under two different conditions; wet and dry. Either 10 layers of the multiaxial NCF or 14 layers of the WOVEN were compressed to a minimum thickness of 3 mm in order to achieve maximum (target) fibre volume fractions of 58 % and 54 %, respectively, as shown in Table 2. For the relatively high numbers of fabric layers used in the specimens, laying up and aligning the layers accurately may be challenging. On the other hand, different configurations of nesting between layers would be expected to cause larger scatter in results for lower numbers of layers, while convergence, i.e. smaller scatter, would be expected to be obtained at these numbers of layers. Values for the minimum thickness and the corresponding fibre volume fractions were selected as they allow the specimens to be exposed to realistic ranges (for actual composite components) of thickness and V f during the compression tests. Each test was repeated a minimum of 5 times with a fresh specimen. Participants were asked to lay up the stacks with all layers placed in the same orientation, with the same surface facing upwards. While handling-induced fabric deformation may affect the material properties (e.g. through shear), no particular instructions were given on how to handle the fabrics to minimise this effect. To prepare specimens for the wet compaction tests, participants were asked to soak each fabric stack in a bath of test fluid for a minimum of 15 minutes to ensure complete wet-out of the fibre bundles. Excess fluid was then drained from the wetted fabric stacks by placing them on a mesh. This step was carried out in order to minimise any effect of excess fluid on the compaction test. Following the wetting procedure, the fabric stacks were tested in compression using the same protocol as for the dry samples.

Sample and compaction platen dimensions

No recommendation was given for the dimensions of samples or compaction platens. The choice of set-up of each participant for both sample and platen shape and dimensions are shown in Table 3. Sample dimensions ranged from 50 mm to 200 mm, while platen dimensions ranged from 50 mm to 250 mm. In order to assess the effect of these dimensions on the acquired data, participants were divided into three categories. Category 1: sample area < platen area, with square (or rectangular) samples and round platens (1a), square samples and square platens (1b) and round samples and round platens (1c). Category 2: sample area > platen area, with square samples and round platens (2a), square samples and square platens (2b) and round samples and round platens (2c). Category 3: sample area = platen area, with square samples and square platens (3a) and round samples and round platens (3b). Figure 1a) and b) shows that 81 % of participants used circular platens and 65 % used square samples, making a combination of circular platens and square samples the most commonly used one. Figure 1c) presents the distribution of specimen size relative to platen size. The majority of participants (50 %) used a sample area smaller than the platen area (Category 1), 31 % had larger samples compared to the platen size (Category 2) and 15 % used equal sample and platen size (Category 3). One participant (4 %) did not fall in any of these categories.

Participant test set-up

Table 4 gives the individual test set-up details for each participant, and Table 5 lists the data acquisition and displacement control methods. The load cells used to measure the compaction force during the tests ranged from 5 kN to 250 kN in capacity. The majority of participants used a typical compression test setup comprising of two solid stiff compaction platens, however participants 6, 8 and 9 used bespoke test rigs.

Participant 8 compressed their specimens using their radial permeability test rig, which has a central hole in the upper platen for the injection of resin. Participant 9 used their through-thickness permeability test rig to carry out the compression tests, therefore specimens were constrained between porous plates within a cylinder of the same internal diameter as the specimens. Participant 6 followed a completely different procedure using a vacuum bag to compress the specimens and measuring the specimen thickness using linear variable differential transformers (LVDTs) and the vacuum level to characterise the compaction pressure. Figure 1d) shows that 19 participants (73 %) used the UTM displacement to measure the sample thickness, while 7 participants (27 %) used LVDTs or other direct measurement techniques (laser, extensometer).

In a recent study on compression testing of a reference specimen [START_REF] May | A Reference Specimen for Compaction Tests of Fiber Reinforcements[END_REF], the reproducibility of results was found to be generally high for different experimental set-ups. The difference between results obtained on different UTMs was in the order of a few percent. This suggested that the influence of the test set-up on acquired pressure data was small. However, the reference specimen used in these tests had an uncompressed thickness of 68 mm and was compressed by more than 5 mm, which makes the measurements less sensitive to small uncertainties in thickness measurement. The effect of uncertainties in thickness can be expected to be more significant for compression measurement of thin specimens.

Compression profile

The participants were asked to carry out compression tests using the displacement-controlled loading profile depicted in Figure 2a). This profile required all participants to start the test with a 10 mm gap between compression platens, which ensured tests were started with zero load on the samples. The samples were compressed at a cross-head speed of 1 mm/min to a stack thickness of 3 mm and held for a period of 30 min to ensure a stable and relaxed state was captured in the results. The sample was then fully unloaded at a cross-head speed of 1 mm/min. The applied compaction force was recorded during the test. Participants were only required to carry out one compression cycle per specimen.

Machine compliance adjustment

As summarised by Sousa et al. [START_REF] Sousa | Methodology of dry and wet compressibility measurement[END_REF], it has been reported that machine compliance can impact the precision of the values measured during compression testing. Participants were asked to record the machine compliance prior to testing by first pressing the platens together, without a sample, at a speed of 1 mm/min and recording the force at the load cell as a function of the cross-head displacement. Participants were asked to use these results to correct data for the preform compaction response. Typically, a correction would be applied according to: 3) where t t is the true gap height between platens, t ch is the gap height according to the cross-head displacement reading, and t(F c ) is a height correction which depends on the applied compaction force, F c . t can be determined from a compliance curve as shown in Figure 3a), where the compaction force, F c , is plotted as a function of the apparent cross-head displacement, t. Here, a value of t = 0 corresponds to the point of contact between the compression plates, i.e. F c = 0. Any further increase in t beyond the point of plateplate contact is related to machine compliance at an applied load, F c . For the example shown here, the relation between applied compaction force and cross-head displacement (for t > 0) can be expressed as: 4)

𝑡 𝑡 = 𝑡 𝑐ℎ + ∆𝑡(𝐹 𝑐 ) (Equation
𝐹 𝑐 (∆𝑡) = 𝐶 ∆𝑡 (Equation
where C is a compliance constant. For this specific data set, a linear approximation fits the experimental data with good accuracy (coefficient of correlation R 2 = 0.999). It is to be noted that the compliance curves were acquired at increasing compaction load. Because of relaxation effects, a different curve would be acquired at decreasing compaction load. F c (t) can be inverted to find: 5)

∆𝑡(𝐹 𝑐 ) = 𝐹 𝑐 /𝐶 (Equation
It is to be noted that the relation between F c and t is in general not necessarily linear. The reported data for machine compliance displacement (Figure 3b) indicate that t is not negligible. The average value is t = 0.22 mm, which corresponds to the average maximum load in the compliance tests (F c = 2700 N) reported by the participants. The average machine compliance displacement is of the same order of magnitude as the thickness deviation from the mean (of all participants) at maximum compaction pressure and at 10 5 Pa (Figure 3c) in the fabric compaction tests. Machine compliance displacement should ideally be at least one order of magnitude smaller than the scatter in measured thickness values for the load range experienced during the test. Furthermore, machine compliance would be expected to be linear in the range of the test, which is not the case for many participants. The non-linearity of the curves indicates an instability in the compression caused by non-controlled elements in the testing fixtures used (e.g. friction, different materials, pins, bolts, etc.).

In tests where the specimen thickness was measured using the UTM reading, the machine compliance was accounted for by applying a correction as described above to improve the accuracy of the thickness values.

Most participants applied the correction to the recorded data after testing. However, some participants used the compliance curve for calibration of the UTM prior to testing. In tests where data for the gap height was collected via a direct thickness measurement method, e.g. using LVDTs or lasers, no compliance correction needed to be applied to the specimen compaction data.

Results and discussion

Evaluation of recorded data

Figure 2b) shows a typical compaction curve obtained from the entire test cycle. The following information was extracted from the compaction curve as quantitative descriptors of the results: 1) maximum compaction stress,  c,max , 2) thickness at maximum compaction stress, t max , 3) thickness at a compaction stress of 10 5

Pa, t 1 , and 4) compaction stress at the end of the hold,  c,hold . The average value per participant for the five test repeats was calculated for each of these descriptors. It is to be noted that not all participants reached pressure values of 10 5 Pa in all test series. This was a problem particularly for the wet specimens.

Although the tests were carried out under nominally the same conditions, there is a large scatter in the data between the participants. To quantify this, minimum values, maximum values, averages and coefficients of variation (c.v.) of the descriptors described above were determined for each test series (Table 6). The percentage drop in load from the maximum compaction stress to the end of hold, F c , and its c.v. is also given in Table 6.

As a measure for the scatter in measured data for each participant individually, values for the c.v. of data for each participant are listed in Table 7. Typically, c.v. is in the order of 10 % for  c,max , and  c,hold .

However, there are a few outliers with very high values of c.v., particularly for the woven fabric. The c.v.

for t max and t 1 is typically in the order of 1 %, indicating that the repeatability of thickness measurement for each participant is generally high. Importantly, the typical scatter in data for each participant is significantly smaller than the scatter between participants. This indicates that the scatter in data between participants is related to differences in experimental set-ups.

Table 6 shows that the thickness at a compaction stress of 10 5 Pa and the compaction stress at the end of the hold have similar coefficients of variation as t max and  c,max . Therefore, the discussion will focus on t max and  c,max , as the initial compression phase to minimum thickness and maximum pressure provides sufficient information to compare compaction curves among participants. It is to be noted that the data acquired by participant 6 are not included in Tables 6 and7 and in Figs. 4 to 6, as a different compression protocol (pressure-controlled compaction) was followed.

The coefficient of variation in maximum compaction stress (scatter between participants) for the dry tests for both fabrics was 38 %. The c.v. in the wet tests was 40 % for the NCF and 50 % for the WOVEN. Figure 4 reports the average measured maximum compaction stress and corresponding sample average thickness for both materials in dry and wet conditions. The data shows a large scatter in both stress and thickness.

The range of values in compaction pressure at the sample target thickness of 3 mm is particularly large. In dry condition, the measured values range from 152 kPa to 695 kPa (average: 370 kPa) and from 63 kPa to 224 kPa (average: 114 kPa) for the NCF and WOVEN, respectively (Figure 4a) and Figure 4c)). For the wet condition, the measured maximum compaction stress is in the range from 59 kPa to 551 kPa (average: 317 kPa) and from 19 kPa to 196 kPa (average: 73 kPa) for the NCF and WOVEN, respectively (Figure 4b) and 4d)). In general, the measured compaction stress tends to be higher for the NCF than for WOVEN.

This difference is related to the higher fibre volume fraction in the NCF at the target thickness. It is also a result of the difference in fabric architectures which affects the fibre mobility in fabric compression. Effects of nesting can be significant in woven fabrics, while they tend to be insignificant in NCFs. Also, there is a trend for the maximum compaction stress to be lower for wet fabric than for dry fabric, which is related to fibre lubrication. The effect of lubrication is less significant for the NCF than for WOVEN, as the fibre fixation is generally stronger in the NCF and there is a lower degree of fibre reordering.

The same trends were observed by most participants individually. However, there were a few outliers (e.g. maximum observed compaction stress higher for wet than for dry fabric), which may be related to measurement errors.

Effect of thickness measurement method

As the recorded compaction pressure near the minimum specimen thickness is very sensitive to changes in specimen thickness (Figure 2), small uncertainties in thickness can result in large variations in the maximum value of the compaction stress. In order to examine the source of the variability observed in the compaction stress, the dataset was divided into two groups based on the method used to record the sample thickness.

Figure 5 shows the data from participants using the testing machine displacement sensor (UTM) and Figure 6 shows the data for participants using calibrated LVDTs or other direct measurement methods (excluding participant 6). The results listed in Table 8 and plotted in Figs. 4 and5 show that a similar variability in the measured compaction stress as for all participants (c.v. is 38 % to 50 %) is generally observed for the UTM group (38 % to 58 %), while the average values of  c,max are comparable. A lower variability (10 % to 39 %) is typically observed when LVDTs or direct methods are used to measure the sample thickness (Figure 6 compared to Figure 4), while average stress values are similar (with the exception of NCF wet). However, it is to be considered that the size of the LVDT group is much smaller (7 participants) than that of the UTM group (19 participants). Carrying out a single-factor ANOVA test on the mean values obtained by each participant for  c,max and t max did not show a statistically significant effect of the thickness measurement method on the scatter in results (i.e. p-value was greater than 5%).

In general, characterisation of the machine compliance showed thickness variations in the same order of magnitude as the scatter in measured thickness values (Fig. 3c). Applying a correction to the measured thickness data to account for the compliance (in the UTM group) should eliminate the compliance as a source of scatter. However, if there is a high level of uncertainty on the compliance data (as seems to be the case for some compliance curves in Fig. 3b), there is a risk that applying a compliance correction to measured thickness data may not necessarily improve the accuracy. In these cases, the compliance may contribute significantly to the scatter in results.

To assess the compaction response independently of the large scatter in  c,max , a curve fit was applied to the raw data for the specimen thickness, t, as a function of the compaction pressure, p, acquired during the compression phase of the tests (i.e. for 0 mm to 7 mm cross-head displacement, corresponding to an experiment time from 0 min to 7 min). A power-law function was used as proposed by Robitaille et al. [START_REF] Robitaille | Compaction of textile reinforcements for composites manufacturing. I: Review of experimental results[END_REF]: 6)

𝑡 = 𝐴𝑝 𝐵 𝑛 (Equation
where:

𝑝 𝑛 = 𝑝 1 kPa (Equation 7)
Here, p n is a dimensionless compaction pressure, A is the specimen thickness at p n = 1, i.e. at a compaction pressure of 1 kPa, and the exponent, B, describes the shape of the curve. In some cases, curve fitting caused problems because of the long (vertical) tail in the acquired data for t at p n ≈ 0, which reflects the phase of the experiment where the compression platen is not yet in contact with the specimen. To overcome this difficulty, data points were removed from the experimental curves starting at the smallest values of p n , until a fit was found such that the coefficient of correlation, R 2 , between the fit curve and the original data was greater than 0.995 (Figure 7). Average values for the constant and exponent values, A and B, respectively, are summarised in Table 9. Cases where R 2 > 0.995 could not be achieved are highlighted in the table. In general, the confidence in the acquired data is the higher, the more regular the curves and the better the fit.

Curves of specimen thickness as a function of compaction pressure acquired by participant 10 showed strongly irregular shapes at low values of p. As a result, values for A and B derived from curve fitting using Eq. ( 6) are outliers. However, this appears not to have affected the maximum recorded compaction stress or the recorded minimum specimen thickness. Aiming at high accuracy in specimen thickness measurement, participant 10 used a video extensometer to track the movement of markers through a transparent Perspex frame, which enclosed the specimens. Interactions between the frame and the specimens and build-up of fluid pressure in the frame (in wet compaction) have affected the acquired data.

Hence, use of this specific type of set-up is not recommended. For the remaining participants, the coefficient of variation of the constant, A, was 6 % to 8 % and ranged from 9 % to 14 % for the exponent, B. This includes one participant who carried out the compression using a vacuum bag. For each participant individually, the c.v. was typically in the range between 0 % and 4 % for A and between 0 % and 9 % for B (with only few outliers).

Figures 8 and9 compared the fitted compaction curves between participants using the UTM and LVDT/direct displacement readings. The curves for participants using LVDT/direct sample thickness measurement appear to show less scatter compared to the UTM measurement method. This would be consistent with the latter being more prone to variability, although the machine compliance curve was applied in the reduction of the thickness data. However, as in evaluation of  c,max and t max , it is to be considered that the number of curves based on direct thickness measurement methods is smaller than the number of curves generated using UTM thickness measurement. Hence, the difference is not statistically significant. In terms of the c.v. of the parameters A and B, there appear to be trends for the c.v. of A to be higher for UTM and for the c.v. of B to be higher for LVDT/direct (Table 10). It is difficult to clearly see any difference in variability between UTM and LVDT/direct methods based on the fit parameters.

Effect of areal density

All participants determined the mass of each fabric stack, i.e. each specimen, prior to testing. This allowed the areal density of the fabrics to be calculated as an indicator for the specimen quality. Using data from the participants excluding unexplained outliers, the determined areal densities varied by 1 % for the WOVEN and 2 % for NCF. This may reflect actual local variations in material properties. Based on Equation ( 2), it can be estimated that this small variability in areal density, which translates into an equally small variability in fibre volume fraction, cannot be the source of the very significant variations in  c,max .

Effect of specimen and platen dimensions

For both fabrics, the dimensions of the unit cells were smaller than 10 mm, implying that even for the smallest compression platens used here (square, 50 mm × 50 mm), a representative area of the fabrics was tested. It can be argued that the dimensions of the specimens relative to the dimensions of the compression platens may have an effect on the measured pressure, as  the specimen in-plane dimensions increase as a result of fibre straightening in the compressed reinforcement (if sample area < platen area);  shear stresses occur in the compressed specimen around the edge of the platens (if sample area > platen area).

However, when the data are separated into groups as defined in Section 2.3 (Table 3), it is hard to identify any trends for the variability, i.e. the c.v., of the maximum pressure for each group (Table 8) at generally comparable average values (with a few outliers). This is confirmed by single-factor ANOVA analysis, which does not indicate a statistically significant effect of the specimen dimension relative to the platen dimension on the scatter in results. The c.v. of  c,max appears to be consistently higher for circular platens than for square platens. However, there is no obvious physical explanation why the shape of the platens would have any effect on the results, particularly for set-ups where the platens are larger or equal in size to the specimens (which is the case for 69 % of participants). This apparent effect would need to be verified in a more detailed study. Table 10 shows no clear effect of shape and size of specimens and platens on the c.v. of A and B. Therefore, no conclusion can be drawn with regards to the effect of the geometry of the specimens and platens and their relative size on the measured pressure.

Effect of specimen wetting

A number of participants used a different test set-up for the wet tests to that used in the dry, which may have affected the results. However, a significant difference in c.v. was seen between the non-crimp fabric and the woven fabric. The coefficient of variation in  c,max for WOVEN was higher in the wet tests than in the dry tests, while the c.v. was unaffected for the NCF. This is likely due to the architecture of the woven fabric, which reportedly caused difficulties during specimen preparation. The participants reported notable difficulties in handling the woven fabric, resulting in a higher degree of fraying and deformation during handling (as it has low shear resistance). The specimen wetting time for these tests was prescribed.

However, a number of participants did not follow this procedure. Unlike the wetting time, the draining time for wetted specimens prior to testing was not described in the set test procedure for this benchmark.

Therefore, this varied between all participants. As a result, the fluid content in the specimens at the beginning of the tests may have been variable, which may have affected the measured compaction pressure.

Some participants modified the compression setup for the wet tests. One participant used a porous sinter metal structure above and below the specimens through which the oil could drain. One participant sealed the specimen in a bag with a drainage tube to allow the oil to flow away. Two participants used a perforated bottom platen and another participant placed a limiting rubber circle around the specimen, with a gap through which the oil could flow.

It is to be noted that, for wet specimens, the compaction pressure is related not only the properties of the reinforcement, but also to flow of the test fluid, which is squeezed out of the reinforcement as the specimen thickness is reduced. This effect was utilised by Buntain and Bickerton [START_REF] Buntain | Compression flow permeability measurement: a continuous technique[END_REF] and Comas-Cardona et al. [START_REF] Comas-Cardona | Unidirectional compression of fibre reinforcements. Part 2: A continuous permeability tensor measurement[END_REF] for continuous characterisation of the reinforcement permeability in compaction tests. Because of the relation between compaction and flow, the measured compaction pressure may depend on the speed of compaction, the reinforcement permeability (which decreases with increasing level of compaction, implying that the fluid pressure will increase during a compression test), the fluid viscosity (i.e. test temperature), and also on the specimen size, which affects the length of fluid flow paths. All participants carried out the compaction tests at room temperature, which may have varied from lab to lab. However, as the change in viscosity with temperature is relatively small in this temperature range (between  = 0.11

Pa.s at T = 15 C and  = 0.09 Pa.s at T = 25 C), it can be assumed that the viscosity of the test fluid was similar for all participants. Hence, different test temperatures cannot explain the significant scatter in results. On the other hand, the dependence on specimen size may explain the larger scatter in data for the wetted specimens. It is worth pointing out that this does not contradict the observation from Section 3.2, which implied that the specimen size relative to the compaction platens does generally not have a clear effect on the measured pressure. In general, effects related to fluid flow can be minimised if the compaction speed is minimised.

Parallelism effects

Pressure calculations during compression tests assume that two parallel platens close and apply a uniformly distributed load across the compression area. If the compression platens are not parallel, the closing platens will make initial contact with the specimen on one edge, leading to an increase in apparent pressure reading while the specimen is not loaded uniformly. Similarly, lack of parallelism will affect the accuracy of zeroing the gap height between the platens. As a result, recorded curves as shown in Figure 2b) will be inaccurate.

A lack of parallelism may also be a contributing factor to the irregular behaviour seen in the machine compliance tests shown in Figure 3b).

To minimise parallelism effects, participants employed a number of different strategies. Some participants tried to minimise parallelism effects during the test, others used feeler gauges to adjust the platens. One participant employed pressure sensors to check for differences in pressure distribution across the test area while one participant used a self-aligning pivot beneath the bottom platen.

To avoid lack of parallelism, it is generally helpful to use platens with a spherical seating which allows platen alignment to be adjusted. Relatively cheap pressure-sensitive film can be used to check if the pressure distribution between the platens is uniform.

Effect of load cell capacity

The load values recorded during these tests were used to calculate the compaction pressure exerted on the fabrics, therefore it was important for the load to be measured and recorded precisely. The load cell capacity for each participant varied significantly, with a range of 5 kN to 250 kN. The fraction of the load cell capacity of the recorded peak loads ranged from 0.1% to 53.8%. The accuracy of the load measurement would be influenced by how the test machine was calibrated. The calibration of each load cell is not known, and no conclusion can be drawn. However, it is a possibility that this is a major source of variation in the data.

It is also to be considered that the total force applied to compress a specimen to a given thickness depends on the specimen area. This means that, in theory, there may be an optimum load cell capacity for each specimen size. However, in reality, the accuracy of a properly calibrated load cell is typically below 1 % of the load cell reading. In addition, linearity and repeatability are typically high, even at very low percentages of load cell capacity (typically from 0.1 % and up). Hence, the effect of the combination of load cell capacity and specimen size on the recorded data can be expected to be small.

Conclusions

From this benchmark exercise on characterisation of preform compaction, a number of factors which may potentially influence the outcome were identified from the participants' choices of test fixture, load and displacement measurement methods and sample geometry. Analysis of the data shows that there is a large scatter in results between participants, with coefficients of variation of maximum recorded stresses ranging from 38% to 50% for a prescribed displacement of 3 mm. This scatter is more significant than the typical scatter for each participant individually, which indicates that it is related to differences in experimental methods used.

In statistical analysis of the collected data, the large number of factors potentially affecting the measurement combined with the comparatively small size of some of the analysed sub-groups meant that it was not possible to isolate main effects on scatter from confounding factors. Nevertheless, the data suggests that the machine compliance is high for some participants, which may cause problems in terms of accuracy of results of compaction tests. Although the difference in scatter between participants who used LDVTs or other direct methods to measure the thickness and participants who used the UTM reading and applied a compliance correction was not statistically significant, the thickness measurement method may have an effect on the consistency of obtained data. Use of a power-law model to describe the compaction behaviour may be a useful way to compare results between different participants, as it considers the entire compaction curve rather than just one point on the curve. No conclusion can be drawn with regards to the effect of the geometry of the specimens and platens and their relative size on the measured pressure.

Although it was difficult to see clear trends in the data due to the number of confounding factors, three main variables were recognised in the approach to testing and analysis between organisations; thickness measurement, approach to compliance correction and parallelism, and specimen saturation in wet compression tests. It is therefore recommended that a second benchmarking exercise be carried out in which these main parameters are defined, in order to reduce the scatter between participants, which will be necessary for the development of a standard for the measurement and characterization of compaction curve for fabrics.

Figure 1 .Figure 2 .Figure 3 .Figure 4 .Figure 5 .Figure 6 .

 123456 Figure 3. Machine compliance characterisation; a) Example of a compliance curve for a compression plate set-up on a universal testing machine; compaction force, F c , is a linear function of apparent crosshead displacement, t. b) Compliance curves measured by different participants show a wide variety of dependencies of F c on t. c) Specimen thickness (maximum and at 10 5 Pa of pressure) deviation from mean for dry and wet conditions compared to the average compliance reported by the participants. The dotted line box represents the average compliance t = 0.22 mm for an average compaction load F c =

Figure 7 .Figure 8 .Figure 9 .

 789 Figure 7. Specimen thickness, t, as a function of normalised compaction pressure, p n ; example of processed data allowing a power-curve to be fitted with R 2 = 0.999.

Table 1 :

 1 List of participants.

		Zwick	100 kN	Steel	Free to flow
	12	Zwick	0 -10 kN	Steel	Free to flow
	13	Instron	250 kN	Steel	Opening in limiting rubber circle
	14	Zwick	100 kN	Steel	Free to flow
	15	Walter + Bai	10 kN	Steel	Free to flow
	16	Zwick	100 kN	Steel	Immersed
	17	Zwick	100 kN	Steel	Free to flow
	18	Instron	50 kN	Steel	Free to flow
	19	Instron	100 kN	Steel	Free to flow
	20	Reger	30 kN	Steel	Perforated and grooved bottom platen
	21	Instron	Auto Ranging	Aluminium & glass	Free to flow
	22	Instron	5 kN	Steel	Free to flow
	23	MTS	5 kN	Aluminium	Sealed with drainage tube
	24	Instron	100 kN	Steel	Free to flow
	25	Instron	5 kN	Aluminium	Free to flow
	26	MTS	22 kN	Steel	Free to flow

Table 5 :

 5 Methods for displacement control and thickness measurement for each participant. Where thickness was measured using UTM readings, a compliance correction was typically applied to the data after testing. Some participants used a compliance correction for machine calibration prior to compression testing. These are labelled with an asterisk (*).

	ID	Closing control	Thickness measurement
	1	LVDT	LVDT
	2	UTM	UTM
	3	UTM	UTM
	4	UTM	UTM
	5	UTM	UTM
	6	LVDT	LVDT
	7	UTM	UTM
	8	UTM	Laser
	9	UTM*	UTM
	10	UTM	Video extensometer
	11	UTM*	UTM
	12	UTM	UTM
	13	Strain gauge extensometer	Strain gauge extensometer
	14	UTM	UTM
	15	UTM	LVDT
	16	UTM	UTM
	17	UTM	UTM
	18	LVDT	LVDT
	19	UTM	UTM
	20	UTM	UTM
	21	UTM*	UTM
	22	UTM	UTM
	23	UTM	UTM
	24	UTM	UTM
	25	UTM	UTM
	26	UTM	UTM

Table 6 :

 6 Measured values for maximum compaction stress,  c,max , thickness at maximum compaction stress, t max , thickness at a compaction stress of 10 5 Pa, t 1 , and compaction stress at the end of the hold,  c,hold . Scatter of values between participants. Minimum value, maximum value, average and coefficient of variation are given based on the average values for each participant. The average percentage drop of load from maximum to end of hold, F c , and its coefficient of variation are also given.

	Test series			 c,max			t max				t 1					 c,hold		F c
		min	max	average	c.v.	min	max	average	c.v.	min	max	average	c.v.	min	max	average	c.v.	average	c.v.
		[kPa]	[kPa]	[kPa]		[mm]	[mm]	[mm]		[mm]	[mm]	[mm]		[kPa]	[kPa]	[kPa]
	NCF (dry)	152	695	370	38 %	2.96	3.31	3.04	3 %	3.20	3.60	3.45	3 %	125	505	230	37 %	36 %	23 %
	NCF (wet)	59	551	317	40 %	2.98	3.20	3.03	2 %	3.00	3.54	3.39	4 %	33	311	196	38 %	37 %	22 %
	WOVEN	63	224	114	38 %	2.95	3.14	3.02	2 %	3.02	3.25	3.12	2 %	52	182	91	37 %	20 %	22 %
	(dry)
	WOVEN	19	196	73	50 %	2.97	3.19	3.01	1 %	3.04	3.26	3.11	3 %	14	146	50	58 %	33 %	23 %
	(wet)

Table 7 :

 7 Measured values for maximum compaction stress,  c,max , thickness at maximum compaction stress, t max , thickness at a compaction stress of 10 5 Pa, t 1 , and compaction stress at the end of the hold,  c,hold . Scatter in measured values for each participant. Minimum, maximum and average values of coefficient of variation for each participant are given.

	Test	 c,max			t max			t 1			 c,hold
	series
	mi	ma	averag	mi	ma	averag	mi	ma	averag	mi	ma	averag
	n	x	e c.v.	n	x	e c.v.	n	x	e c.v.	n	x	e c.v.
	c.v.	c.v.		c.v.	c.v.		c.v.	c.v.		c.v.	c.v.
	NCF
	(dry)

Acknowledgements

The authors thank Hexcel Corporation and Saertex GmbH for supplying the materials used in this exercise free of charge. For their contribution towards the experimental work, the authors would also like to thank Jan Ivens (KU Leuven), Mohsen Orouji (Ecole Polytechnique Montreal), Damiano Salvatori and Amaël Cohades (Ecole Polytechnique Federale de Lausanne), Lucie Riffard (McGill) and Bradley Buttars and Collin Childs (Brigham Young University). Finally, the authors thank Louise Wright (National Physical Laboratory) for supporting with the statistical analysis and Argyro Martidi (National Physical Laboratory) for her contribution towards processing the data.

Credit Author Statement

This paper was produced following the efforts of several authors. The details of each author's efforts are detailed below.

The benchmark exercise was planned and organised by: A. X. H. Yong: writing (original draft) and writing (review and editing), formal analysis, visualization, project administration A. Aktas: conceptualization, methodology, formal analysis, project administration, investigation D. May: conceptualization, methodology, formal analysis, investigation

The organisers were supported by an extended Steering Committee of:

A. Endruweit: investigation, formal analysis, writing (original draft) and writing (review and editing), visualization S. V. Lomov: investigation, formal analysis, visualization P. Hubert: investigation, formal analysis, writing (original draft), visualization S. Advani: writing (original draft)

Viscosity measurements for all participants were carried out by K. Kind (investigation), who also participated in the benchmark.

Declaration of interests

☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:


Marc Oudin

Jean-Pierre P Delmas

ASYMPTOTIC GENERALIZED EIGENVALUE DISTRIBUTION OF TOEPLITZ BLOCK TOEPLITZ MATRICES

Keywords: Toeplitz block Toeplitz matrix, Szegö's theorem, generalized eigenvalues

come

INTRODUCTION

2-D discrete random processes appear in many signal processing applications. A typical example is imagery, where observation data are modelized by such processes. Furthermore, in certain imagery applications, the 2-D discrete random processes are second order stationary (e.g., see [1, chap. 5]). Other examples can be found in different application areas, such as for instance in magnetic recording systems, where the magnetic medium can be modelled by a 2-D second order stationary process (e.g., see [START_REF] Voois | Upper Bounds on Achievable Storage Density: A Two-Dimensional Approach[END_REF]), where the two dimensions correspond to the linear and radial coordinates. In that case, the covariance matrix of the random process is structured. More specifically, when a 2-D n 1 × n 2 block of samples from a second order stationary random process is grouped into a vector by stacking its columns, the vector's covariance matrix has a Toeplitz block Toeplitz (TBT) structure (e.g., see [START_REF] Davis | Circulant Matrices, Pure and Applied Mathematics[END_REF]). In many detection and estimation problems associated with Thanks to the French MOD (DGA) for funding.

processing of such 2-D random processes, for which the main performance criterion is the Signal to Noise Ratio (SNR), the data are the sum of two zero-mean second order stationary processes: the process of interest and the noise process. For instance, in [START_REF] Voois | Upper Bounds on Achievable Storage Density: A Two-Dimensional Approach[END_REF], the data are the sum of a channel input and a noise refered as a "flux noise". In that case, after linear filtering, the optimal SNR corresponds to the maximal value of a Rayleigh quotient which can be interpreted as the largest generalized eigenvalue of the covariance matrices associated with the signal and noise processes [START_REF] Oudin | Asymptotic generalized eigenvalue distribution of block Toeplitz matrices and application to spacetime beamforming[END_REF].

In this paper, we study the problem of the influence of the size of the samples vector along with the different dimensions, on the generalized eigenvalues of TBT matrices. More specifically, we assume that the number of samples along with both dimensions tends to infinity and analyze the asymptotic generalized eigenvalue distribution of such matrices.

The problem of the asymptotic eigenvalue distribution of Toeplitz matrices has first been analyzed by Grenander and Szegö, whose famous result asserts that the eigenvalues of a sequence of Hermitian Toeplitz matrices asymptotically behave like the samples of the Fourier transform of its entries [START_REF] Grenander | Toeplitz Forms and Their Applications[END_REF]. However, this analysis has been performed by use of sophisticated mathematics under the general hypothesis that the Toeplitz matrix is generated by measurable and bounded spectrum. Then, Gray has proposed a simpler proof of this result for banded Toeplitz matrices [START_REF] Gray | On the asymptotic eigenvalue distribution of Toeplitz matrices[END_REF] (which are called finiteorder Toeplitz matrices) and then for infinite-order Toeplitz matrices, under the assumption of absolutely summable elements, based on the asymptotic equivalence between two matrices [START_REF] Gray | Toeplitz and circulant matrices: a review[END_REF]. Following this approach and under the same assumption, the Grenander and Szegö result has been later extended to the eigenvalues of block Toeplitz with Toeplitz block matrices where both the size and the number of blocks tend to infinity [START_REF] Voois | A Theorem on the Asymptotic Eigenvalue Distribution of Toeplitz-Block-Toeplitz Matrices[END_REF], then to the eigenvalues and generalized eigenvalues of block Toeplitz with non-Toeplitz blocks where only the number of blocks tends to infinity in [START_REF] Gazzah | Asymptotic eigenvalue distribution of block Toeplitz matrices and applications to blind SIMO channel identification[END_REF] and [START_REF] Oudin | Asymptotic generalized eigenvalue distribution of block Toeplitz matrices and application to spacetime beamforming[END_REF], respectively. Let us note that extension of the celebrated Grenander and Szegö result has been also considered in the mathematical literature under the general assumption that the involved Toeplitz, block Toeplitz matrices are generated by in-tegrable spectra (e.g., see [START_REF] Tilli | Asymptotic spectral distributions of Toeplitz-related matrices[END_REF]) for the asymptotic eigenvalue distribution. However, to the best of our knowledge, the extension to generalized eigenvalues of TBT matrices has not been studied until now. In the present work, we propose an extension of Szegö's theorem to the generalized eigenvalues of TBT matrices, under the hypothesis of absolutely summable elements, which relies on an extension of the notion of asymptotic equivalence between matrix sequences established by Gray in [START_REF] Gray | On the asymptotic eigenvalue distribution of Toeplitz matrices[END_REF].

This paper is organized as follows. In Section 2, we give an interpretation of the generalized eigenvalues of Hermitian matrices from the point of view of SNR after linear filtering. Then, in Section 3, the notation of TBT matrices and Circulant Block Circulant (CBC) matrices are introduced and preliminary results about asymptotic equivalence between block matrix sequences are given. Finally, a generalized eigenvalue distribution theorem of TBT matrices is proved in Section 4.

SNR AFTER LINEAR FILTERING OF BIDIMENSIONAL PROCESSES

In this Section, we give an interpretation of the generalized eigenvalues of two Hermitian matrices. Thus, let x be a data vector where the 2-D data are arranged in alphabetical order of dimension n 1 × n 2 composed by the sum of s, the signal of interest and n, the noise process. Both processes are modelled by second-order stationary processes with

n 1 n 2 × n 1 n 2 Hermitian covariance matrices R s = E{ss H } and R n =
E{nn H }, respectively. Since the processes are assumed to be second-order stationary, the covariance matrices are TBT structured. Moreover, due to the presence of background noise in n, R n is almost always positive definite. After linear filtering with n 1 × n 2 -dimensional filter w of the data x, the SNR is given by the Rayleigh quotient

SNR = w H R s w w H R n w
whose stationary points are given by the generalized eigenvalues of (R s , R n ) (e.g., see [START_REF] Oudin | Asymptotic generalized eigenvalue distribution of block Toeplitz matrices and application to spacetime beamforming[END_REF]), denoted λ k (R s , R n ) which are real-valued and strictly positive. Moreover, since R n is positive definite and Hermitian, these generalized eigenvalues are given by the eigenvalues 1

of R -1 n R s , λ k (R s , R n ) = λ k (R -1 n R s ).
In particular, the maximum of this SNR is given by the maximum generalized eigenvalue of (R s , R n ). To prove our main result (theorem 1), we need some notations and background.

NOTATIONS AND PRELIMINARY RESULTS

We first recall the definition of the structured TBT and CBC matrices and extend the asymptotic equivalence of matrix se- 1 Note that these generalized eigenvalues are also given by the eigenvalues of the Hermitian matrix R

-1/2 n R s R -1/2 n
, but theorem 1 is more directly proved by using the eigenvalues of R -1 n R s .

quences to block matrix sequences. Then, we give some preliminary lemmas necessary to prove our main result in Section 4.

Definition 1 (TBT matrix):

Given an index (n 1 , n 2 ), a TBT matrix A n 1 ,n 2 is defined as:

A n1,n2 def =       A 0 A -1 • • • A -(n1-1) A 1 . . . . . . . . . . . . . . . . . . A -1 A n 1 -1 • • • A 1 A 0       and each block A m1 , m 1 = -(n 1 -1), ..., n 1 -2, n 1 -1 is a (n 2 × n 2 ) Toeplitz matrix, A m 1 def =       a m 1 ,0 a m 1 ,-1 • • • a m1,-(n2-1) a m1,1 . . . . . . . . . . . . . . . . . . a m 1 ,-1 a m1,(n2-1) • • • a m 1 ,1 a m 1 ,0      
The approach of this paper is to relate the generalized eigenvalues of TBT matrices to those of simpler structured associated CBC matrices that we now formalize.

Definition 2 (CBC matrix):

Given an index (n 1 , n 2 ), a CBC matrix C n1,n2 is defined as a TBT matrix previously defined, where Toeplitz blocks are replaced by circulant blocks. These CBC matrices have an eigenvalue decomposition (EVD) that extends those of circulant matrices (e.g., see [START_REF] Davis | Circulant Matrices, Pure and Applied Mathematics[END_REF]Th. 5.8.1]). More specifically, a n 1 n 2 × n 1 n 2 CBC matrix C n1,n2 is diagonalizable by the unitary matrix

U n1,n2 = U n1 ⊗ U n2 (1)
where (U n p ) p=1,2 are the unitary discrete Fourier transform (DFT) matrices of terms

(U n p ) k,l = 1 √ n p e -j2π (k-1)(l-1) n p
, of size n p × n p , where the associated eigenvalues are the 2-D DFT of its first row

λ k 1 ,k 2 = n 1 -1 m1=0 n 2 -1 m2=0 c m 1 ,m 2 e -j2π m 1 k 1 n 1 + m 2 k 2 n 2
. Now, we define 2-D asymptotic equivalence.

Definition 3 (2-D Asymptotic Equivalence):

Let {A n 1 ,n 2 } and {B n 1 ,n 2 } be sequences of n 1 n 2 × n 1 n 2 matrices. These matrices are said to be 2-D asymptotically equivalent and noted A n1,n2 ∼ B n1,n2 if the following conditions hold:

• A n1,n2 ≤ M < ∞ • B n 1 ,n 2 ≤ M < ∞ • lim n 1 ,n 2 →∞ |A n 1 ,n 2 -B n 1 ,n 2 | = 0
where . is the spectral norm and |.| is a normalized Frobenius norm defined by

|A n 1 ,n 2 | 2 = 1 n 1 n 2 n 1 m 1 =1 n 2 m 2 =1 |a m1,m2 | 2 .
Then, we give the following lemma about the asymptotic eigenvalue distribution of 2-D asymptotically equivalent matrices.

Lemma 1

Let {A n1,n2 } and {B n1,n2 } be sequences of 2-D asymptotically equivalent matrices of dimension n 1 n 2 × n 1 n 2 with eigenvalues λ k (A n1,n2 ) and λ k (B n1,n2 ) for k = 0...(n 1 n 2 -1), respectively, then for any positive integer s

lim n 1 ,n 2 →∞ 1 n 1 n 2 n1n2-1 k=0 (λ s k (A n1,n2 ) -λ s k (B n1,n2
)) = 0 and hence if either limit exists individually,

lim n 1 ,n 2 →∞ 1 n 1 n 2 n1n2-1 k=0 λ s k (A n 1 ,n 2 ) = lim n 1 ,n 2 →∞ 1 n 1 n 2 n1n2-1 k=0 λ s k (B n 1 ,n 2 ).
Proof: The proof is a mere extension of the proof of [7, Th. 2] to the 2-D case.

From now on, we consider Hermitian TBT matrices only, for which a -i 1 ,-i 2 = a * i 1 ,i 2 is equivalent to a real-valued Fourier transform a(ω 1 , ω 2 ) = i1,i2 a i 1 ,i 2 e -j P 2 p=1 ωpip . To construct a sequence of CBC matrices that are 2-D asymptotically equivalent to {A n1,n2 } and whose eigenvalues are the samples of a(ω 1 , ω 2 ), we define the sequence

c n 1 ,n 2 m1,m2 (a) def = 1 n 1 n 2 n 1 -1 k 1 =0 n 2 -1 k 2 =0 a(2π k 1 n 1 , 2π k 2 n 2 )e -j2π P 2 p=1 m p k p np
(2) and {C n 1 ,n2 (a)}, the sequence of CBC 2 matrices induced by {c i 1 ,i 2 (a)}. We note that C n 1 ,n 2 (a) may be written more compactly as

C n1,n2 (a) = U H n 1 ,n 2 ∆ n 1 ,n 2 (a)U n 1 ,n 2 (3)
where U n 1 ,n 2 is defined as in [START_REF] Pratt | Digital Image Processing[END_REF] and

∆ n 1 ,n 2 (a) is the n 1 n 2 × n 1 n 2 diagonal matrix of elements a(2π k 1 n 1 , 2π k 2 n 2 )
, arranged in alphabetical order. We are now ready to state the following lemma, proved in [8, Lemma 1].

Lemma 2

Let {a i1,i2 } be a Hermitian, absolutely summable sequence with Fourier transform a(ω 1 , ω 2 ). Let {c i 1 ,i 2 (a)} be defined by [START_REF] Voois | Upper Bounds on Achievable Storage Density: A Two-Dimensional Approach[END_REF]. Then, the induced sequences of matrices {A n 1 ,n 2 } and {C n1,n2 (a)} are 2-D asymptotically equivalent.

TOEPLITZ BLOCK TOEPLITZ MATRICES GENERALIZED EIGENVALUE DISTRIBUTION THEOREM

The aim of this Section is to extend Szegö's theorem to the case of the generalized eigenvalues of Hermitian TBT matrices, under the assumption that the elements generating the matrices are absolutely summable. We proceed as in [START_REF] Oudin | Asymptotic generalized eigenvalue distribution of block Toeplitz matrices and application to spacetime beamforming[END_REF] and prove three lemmas used for the proof of Theorem 1. More precisely, we first prove in Lemma 3 that the eigenvalues of Hermitian TBT matrices generated from an absolutely summable sequence are bounded by the minimum and maximum values of the 2-D Fourier transform of the sequence. Then, this lemma is used for the proof of the 2-D asymptotic equivalence between the inverse of a positive definite Hermitian TBT matrix and the inverse of its 2-D asymptotically equivalent CBC matrix, given by Lemma 4. Indeed, Lemma 3 shows that the spectral norm of the inverse of a positive definite Hermitian TBT matrix is bounded [START_REF] Oudin | Asymptotic generalized eigenvalue distribution of block Toeplitz matrices and application to spacetime beamforming[END_REF]. Furthermore, Lemma 5 shows that the product of the inverse of a positive definite Hermitian TBT matrix by a Hermitian BTB matrix is 2-D asymptotically equivalent to the product of the inverse of the Hermitian CBC matrix by a Hermitian CBC, both derived from (2). Finally, using this 2-D asymptotic equivalence and Lemma 1, we straightforwardly obtain Theorem 1.

Lemma 3

Let {a i 1 ,i 2 } be a Hermitian, absolutely summable sequence with Fourier transform a(ω 1 , ω 2 ). Then, for all eigenvalues λ(A n 1 ,n 2 ) of the induced sequences of matrices {A n 1 ,n 2 }, we have

m a = min ω1,ω2 a(ω 1 , ω 2 ) ≤ λ(A n 1 ,n 2 ) ≤ max ω1,ω2 a(ω 1 , ω 2 ) = M a .
Proof: This lemma is proved in the first step of the proof of [8, Lemma 1].

Lemma 4

Let B n1,n2 be a positive definite Hermitian, TBT matrix generated from the absolutely summable sequence {b i1,i2 } with Fourier transform b(ω 1 , ω 2 ) and the associated 2-D asymptotically equivalent CBC matrix

C n 1 ,n 2 (b) as defined in Lemma 2. If min ω 1 ,ω 2 b(ω 1 , ω 2 ) = m b > 0, then (B n 1 ,n 2 ) -1 ∼ (C n 1 ,n 2 (b)) -1 .
Proof: Using Lemma 3, the proof is similar to that of Lemma 3 in [START_REF] Oudin | Asymptotic generalized eigenvalue distribution of block Toeplitz matrices and application to spacetime beamforming[END_REF].

Lemma 5

With the assumptions of Lemma 4, if A n 1 ,n 2 is a Hermitian TBT matrix generated by an absolutely summable sequence {a i 1 ,i 2 }, the associated CBC matrices C n 1 ,n 2 (a) and C n 1 ,n 2 (b) given by ( 2) satisfy

(B n1,n2 ) -1 A n1,n2 ∼ (C n1,n2 (b)) -1 C n1,n2 (a).
Proof: The proof is the same as for Lemma 4 in [START_REF] Oudin | Asymptotic generalized eigenvalue distribution of block Toeplitz matrices and application to spacetime beamforming[END_REF].

We now introduce the interval

I ω = [ min ω 1 ,ω 2 b -1 (ω 1 , ω 2 )a(ω 1 , ω 2 );max ω 1 ,ω 2 b -1 (ω 1 , ω 2 )a(ω 1 , ω 2 )]
and give a theorem about the asymptotic distribution of the generalized eigenvalues of Hermitian TBT matrices.

Theorem 1

Let A n1,n2 and B n1,n2 be two Hermitian TBT matrices, such that B n1,n2 is positive definite, and generated by absolutely summable sequences {a i1,i2 } and {b i1,i2 }, respectively, with

min ω 1 ,ω 2 b(ω 1 , ω 2 ) = m b > 0.
Then for all continuous functions

F on I ω lim n 1 ,n 2 →∞ 1 n 1 n 2 n1n2-1 k=0 F (λ k (A n1,n2 , B n1,n2 )) = 1 (2π) 2 π -π π -π F (b -1 (ω 1 , ω 2 )a(ω 1 , ω 2 ))dω 1 dω 2 . Proof: Using Lemma 5, (B n 1 ,n 2 ) -1 A n 1 ,n 2 is 2-D asymptoti- cally equivalent to (C n1,n2 (b)) -1 C n1,n2
(a) and thanks to the similarity of C n1,n2 (b) and C n1,n2 (a) to the diagonal matrices ∆ n1,n2 (b) and ∆ n1,n2 (a) with the same unitary matrix U n 1 ,n 2 (3), we have using Lemma 1, for arbitrary integer s,

lim n 1 ,n 2 →∞ 1 n 1 n 2 n1n2-1 k=0 [λ s k (A n 1 ,n 2 , B n 1 ,n 2 ) - λ s k (∆ -1 n 1 ,n 2 (b)∆ n 1 ,n 2 (a))] = 0 with lim n1,n2→∞ 1 n 1 n 2 n 1 n 2 -1 k=0 λ s k (∆ -1 n 1 ,n 2 (b)∆ n1,n2 (a))] = lim n1,n2→∞ 1 n 1 n 2 n 1 -1 k 1 =0 n 2 -1 k 2 =0 b -s (2π k 1 n 1 ,2π k 2 n 2 )a s (2π k 1 n 1 ,2π k 2 n 2 ) = 1 (2π) 2 π -π π -π b -s (ω 1 , ω 2 )a s (ω 1 , ω 2 )dω 1 dω 2 ,
where the continuity of the Fourier transforms b(ω 1 , ω 2 ) and a(ω 1 , ω 2 ) guarantees the existence of the integral.

Finally, extending this result to any polynomial and after invoking the Stone-Weierstrass approximation theorem, Theorem 1 is proved.

As shown in [START_REF] Gray | On the asymptotic eigenvalue distribution of Toeplitz matrices[END_REF][START_REF] Gray | Toeplitz and circulant matrices: a review[END_REF], and combined with the fact that for all vectors n 1 , n 2 , the eigenvalues of (B n1,n2 ) -1 A n1,n2 lie in I ω , Theorem 1 leads to the following corollary:

Corollary 1

For any positive integer l, the smallest and the largest l generalized eigenvalues of (A n 1 ,n 2 , B n 1 ,n 2 ) are convergent in n 1 , n 2 and lim n1,n2→∞

λ n 1 n 2 -l+1 (A n1,n2 ,B n1,n2 ) =min where the eigenvalues are ranked in decreasing order.

In particular, the asymptotic optimal SNR after linear filtering is given by

lim n 1 ,n 2 →∞ SNR = max ω 1 ,ω 2 b -1 (ω 1 , ω 2 )a(ω 1 , ω 2 ).

CONCLUSION

In this paper, we have given an extension of Szegö's theorem to the generalized eigenvalues of Hermitian TBT matrices using a simple proof under the hypothesis of absolutely summable elements, based on the notion of 2-D asymptotic equivalence between 2-D block matrix sequences.

b - 1

 1 (ω 1 ,ω 2 )a(ω 1 ,ω 2 )and limn1,n2→∞ λ l (A n 1 ,n 2 , B n 1 ,n 2 ) = max ω1,ω2 b -1 (ω 1 , ω 2 )a(ω 1 , ω 2 )

The CBC matrix structure is easily shown by noticing that the sequence defined by (2) is periodic.


Marc Oudin
email: marc.oudin@fr.thalesgroup.com

Jean-Pierre P Delmas
email: jean-pierre.delmas@it-sudparis.eu

Frédéric Barbaresco
email: frederic.barbaresco@thalesgroup.com

Siegel distance-based covariance matrix selection for Space-Time Adaptive Processing

Keywords: STAP, Siegel distance, Generalized Inner Product

published or not. The documents may come

I. INTRODUCTION

STAP is a well-known technique used for clutter mitigation in radar applications among any others. For a cell under test (CUT), STAP algorithms are based on the estimation of the covariance matrix of clutter and background noise. In an ideal situation, there exist training samples which are independent and identically distributed (i.i.d.) and share the same covariance matrix. In that case, the true covariance matrix is estimated by its maximum-likelihood covariance matrix (the sample covariance matrix (SCM) under the assumption of Gaussian training samples) and the estimation improves when the number of training samples increases. Consequently, the performance of adaptive filters increases with the number of training samples. For instance, when the sample matrix inversion (SMI) STAP processor [START_REF] Reed | Rapid convergence rate in adaptive arrays[END_REF] is used, the average SINR performance becomes within 3 dB of the optimal performance when more than 2N (where N is the number of space-time degrees of freedom) samples are used. However, in practical applications, i.i.d. training samples are scarce since the environments are very frequently heterogeneous (e.g., see [START_REF] Melvin | Space-time adaptive radar performance in heterogeneous clutter[END_REF]). Since the presence of heterogeneous samples (outliers) in the training data, leads to a biased covariance matrix estimate, the performance of adaptive algorithms degrade. To take this problem into account, preprocessing schemes of the training data have been developed. An important class of solutions are based on non homogeneity detection (NHD). The principle of NHD techniques is to detect and excise nonhomogeneous training snapshots to avoid a bias in the covariance matrix estimation and thus keep good SINR performance for STAP algorithms. For instance, the GIP [START_REF] Chen | Screening Among Multivariate Normal Data[END_REF], [START_REF] Rangaswamy | Statistical Analysis of the Nonhomogeneity Detector[END_REF] is a commonly-used NHD technique which aims at detecting outliers in the training snapshots. For each sample x of the training data, it consists in computing a scalar value equal to x H R -1 x, where R is the SCM derived from all the training data. Then, the resulting value is compared to a threshold and the sample is removed from the training samples if the value exceeds the threshold. Other techniques also exist, based for instance on power tests or on more sophisticated techniques like the projection statistics test (PS) [START_REF] Schoenig | Improved Detection of Strong Nonhomogeneities for STAP via Projection Statistics[END_REF].

In this paper, we consider the context of a forward-looking radar where data heterogeneity is mostly due to the presence/absence of clutter in the training data and to the fact that clutter characteristics depend on physical parameters and vary with range and Doppler [START_REF] Klemm | Principles of Space-Time Adaptive Processing[END_REF]. Other factors can also generate heterogeneities such as for instance complex scattering environments. In that context, heterogeneous training data cannot only be modelled by discrete outliers, but other models must be developed [START_REF] Bidon | A Bayesian approach to adaptive detection in non-homogeneous environments[END_REF]. In that condition, the use of NHD techniques may not be the most appropriate solution to deal with it. Moreover, although these techniques can be efficient to detect one or more outliers in the data, they can have an high computational complexity since they need to process each of the different samples in the training data. The number of operations is therefore proportional to that number of training samples.

Here, we propose another approach which aims at having good SINR performance on powerful clutter areas, without degrading the performance with respect to conventional processing on other areas, and by minimizing the number of detection tests. It consists in computing a SCM with a few training samples in the neighborhood of the CUT (so that we limit clutter heterogeneity which extends with the distance to the CUT). Then, a test on the resulting SCM is performed to decide whether powerful clutter is present in the training data or not and whether there is a need to implement an adaptive filter or not. This test is significant since when a small number of samples are used to compute an adaptive filter, estimation errors occur, which limit the algorithm SINR performance. Choosing a non-adaptive filter on background noise therefore avoids those SINR performance losses. Then, let us note that compared to NHD techniques, this algorithm does not need any detection for each training sample, but only for the SCM computed for each CUT.

To test whether using a STAP filter based on the SCM is useful, we compute the distance between the background noise covariance matrix (which is assumed to be known in this paper) and the SCM. The chosen distance is the Siegel distance [START_REF] Siegel | Symplectic Geometry[END_REF], which is defined on the space of Hermitian positive matrices.

This paper is organized as follows. In Section II, data model and problem statement are presented. Then, in Section III, the expression of Siegel distance is given and its PDF in the case of an SCM estimated with one training snapshot is derived. Finally, simulations are performed in Section IV to illustrate the SINR performance of the proposed scheme.

II. PROBLEM STATEMENT

A. Data model

Let us consider an airborne radar with an arbitrary array composed of N sensors. The radar emits an M -pulse waveform. Then, let us suppose that the environment be composed of clutter, background noise and a moving target. For STAP processing, the data is divided into two sets of data called primary and secondary data. The primary data consists of the samples to be filtered and is composed of background noise, and possibly clutter and signal. The secondary data is the training data, and is supposed to be only composed of background noise and possibly clutter. We denote K the number of secondary samples used to compute adaptive filters and assume that for each adaptive filter, the K secondary samples used for its computation share the same distribution. The background noise is modelled by a zero-mean i.i.d. complex process not necessarily spatially white, with covariance matrix R. Then, the signal is considered as deterministic, with unknown power, but known direction and speed. Finally, the clutter is modelled by elementary reflectors. Those ones are supposed to be motionless, spatially and from pulse to pulse correlated but white from sample to sample during a pulse. They are assumed to be zero-mean, with power depending on their direction of arrival (DOA) and distance.

B. Problem statement

STAP filtering consists in computing, for each sample of the primary data, an adaptive filter based on the inverse of an estimated covariance matrix. Let us denote z i , the primary sample to be filtered, R i its covariance matrix and z k (i) k=1..K the secondary samples in the neighborhood of z i which are used for the estimation of the covariance matrix R i . This one will be computed by:

R i = 1 K K k=1 z k (i)z k (i) H .
In the case where the secondary samples are composed of background noise alone Gaussian distributed, the estimated covariance will be distributed with the following Wishart distribution:

R i ∼ W(N, K, R).
Before filtering the primary sample, our problem is first to decide whether R i is the SCM of R (hypothesis H 0 ) or not. Then, to optimize the SINR, we apply the filter R -1 φ based on R under hypothesis H 0 or the adaptive filter R -1 i φ based on R i otherwise, where φ is the spatio-temporal steering vector of the target.

III. USE OF SIEGEL DISTANCE BETWEEN HERMITIAN POSITIVE DEFINITE MATRICES

A. Test based on Siegel distance

To test the hypothesis H 0 , we compute the distance between R i and R. We choose the Siegel distance, since it is a distance on the space of Hermitian positive definite matrices, with good invariance properties [START_REF] Barbaresco | Innovative Tools for Radar Signal Processing Based on Cartan's Geometry of SPD Matrices and Information Geometry[END_REF], [START_REF] Barbaresco | Interactions between Symmetric Cones and Information Geometrics: Bruhat-Tits and Siegel Spaces Models for High Resolution Autoregressive Doppler Imagery[END_REF]. This distance is defined by:

d(R, R i ) 2 = log(R -1/2 R i R -1/2 ) 2 = N n=1 log 2 (λ n )
where

(λ n ) n=1...N are the generalized eigenvalues of ( R i , R), equal to the eigenvalues of R -1 R i or R -1/2 R i R -1/2
where R -1/2 is an arbitrary square root of R. The test consists in comparing d(R, R i ) with a threshold s. If the distance is below the threshold s, we decide to implement processing using the a priori matrix R, whereas in the contrary case, we implement STAP algorithm based on the SCM R i . It is clear that since R -1/2 R i R -1/2 ∼ W(N, K, I), under hypothesis H 0 , where I is the identity matrix of dimension N , the test based on this distance is constant false alarm rate (CFAR). Indeed, the joint density of the eigenvalues of

R -1/2 R i R -1/2
and consequently that of d(R, .) only depends on K and N and not on the background covariance matrix.

B. Statistical analysis of Siegel distance

Now, to fix the threshold s, we need to analyze the statistical behavior of Siegel distance. (A supprimer: First, we consider the particular case K = 1). But deriving the theoretical PDF of the Siegel distance for arbitrary K is rather challenging except for the specific case K = 1. In this case, although d(R, .) is not defined ( R i is singular), the Siegel distance PDF can be derived and compared to that of the GIP, using a regularization. For instance, let us add the known covariance matrice R and consider R i + R.

In that case, we have:

d(R, R i + R) 2 = N n=1 log 2 (λ n (R -1 R i + I)).
Since R -1 R i is rank-one, it has a single non-zero eigenvalue which is equal to

λ = Tr(R -1 z 1 (i)z 1 (i) H ) = Tr(z 1 (i) H R -1 z 1 (i)) = z 1 (i) H R -1 z 1 (i)
which is equal to the GIP measure denoted Q. The largest eigenvalue of R -1 R i + I is therefore equal to Q+ , whereas the other eigenvalues are equal to . In that case, the Siegel distance is then a function of the GIP measure, given by:

d(R, R i ) = log(Q + ) 2 + (N -1)log( ) 2 . ( 1
)
When the data are composed of background noise alone, the PDF of z k (i) is known, and the corresponding PDF of Q can be computed. For instance, when z k (i) are Gaussian distributed, it is well known that the PDF of Q follows a Chi-Squared distribution with N degrees of freedom [START_REF] Rangaswamy | Statistical Analysis of the Nonhomogeneity Detector[END_REF], given by:

f Q (q) = q N -1 Γ(N ) e -q 0 ≤ q < ∞ (2)
where Γ(N ) is the Euler-Gamma function. Therefore, the PDF of the Siegel distance given by (1) can be easily deduced after a straightforward change of variable. It is given by:

f D (d) = 1 Γ(N ) [(e √ d 2 -a -) N -1 e -e √ d 2 -a + √ d 2 -a+ + (e - √ d 2 -a -) N -1 e -e - √ d 2 -a - √ d 2 -a+ ] d √ d 2 -a (3)
for d > √ a where a = (N -1)log( ) 2 , which reduces to In the simulation, we consider white noise, with unitary power, so that R = I. We note a goof fit between the theoretical and empirical PDF of the Siegel distance. Moreover, we observe that the mean of D tends to infinity whereas its variance converges to zero when tends to zero. That figure shows that the choice of , which is arbitrary, defines different schemes associated with different thresholds s, each threshold being chosen to have a given type-I error, and therefore very dependent on the value of . In the following, we will make the choice = 1 for simplicity. Then, for comparison, we also plot the empirical and theoretical PDF of Q obtained by [START_REF] Melvin | Space-time adaptive radar performance in heterogeneous clutter[END_REF]. We note that whatever the value of , the variance of Q is much larger than that of D which results from the application of the logarithm function in the definition of Siegel distance.

f D (d) = (e d -1) (N -1) Γ(N ) e d+1-e d 0 ≤ d < ∞ for = 1.
Then, we consider the case where K > 1. Since, Ri is not positive definite for K < N , we still consider the regularized matrice. In that case, the derivation of the theoretical PDF of Siegel distance is challenging. Therefore, we only consider the mean and variance of d K = d(R, Ri + R) for K > 1. For both moments, we compute asymptotic values w.r.t. K. We have:

E{d K } = E{ N n=1 log 2 (λ n (R -1 R i + I))} and E{d 2 K } is equal to E{ N n,n =1 log 2 (λ n (R -1 R i + I)) log 2 (λ n (R -1 R i + I))}.
For computing the limiting values of the mean and variance of d K , when K → ∞, we use the continuity of the eigenvalue function. Thus, we obtain and

lim K→∞ E{d K } = N n=1 log 2 (λ n ((1 + )I)) = √ N log(1 + ) 0 0.5 1 1.5 2
lim K→∞ E{d 2 K } = N n=1 N n =1 log 2 (λ n ((1 + )I)) log 2 (λ n ((1 + )I)) = lim K→∞ E{d K } 2
which shows that the variance converges to zero when K increases to infinity 1 . In Fig. 2, we plot the empirical PDF for Siegel distance, for different values of K for N = 4, with = 1 and white noise of unit power. We see that the variance of d K converges to zero very rapidly. Then, in Tab.I, we compute the empirical mean of d K as a function of K. We see that the asymptotic value √ N log(2) ≈ 1.386 is obtained very rapidly as well.

IV. SIMULATIONS

We now illustrate the proposed scheme with simulations. Thus, let us consider the context of an airborne radar with rectangular array composed of N = 4 subarrays. More, precisely, we consider the case of a forward-looking radar, for which clutter is distributed along an ellipse in the angle-Doppler plane which varies with distance [START_REF] Klemm | Principles of Space-Time Adaptive Processing[END_REF]. This naturally leads to clutter heterogeneity. Therefore, for each primary sample to filter, the number of secondary samples K sharing the same distribution as the CUT is limited. Consequently, errors occur in the estimation of the noise covariance matrix, what leads to performance degradation. In Fig. 3, we illustrate clutter heterogeneities by showing a clutter + noise power map. 1 Let us note that this is a natural result since b R i is a consistent estimate of R. The number of pulses is equal to M = 256 and the number of range cells is equal to 60. The objective of the proposed processing is to attain the maximal STAP SINR performance.

In the illustrations, we plot the normalized SINR, i.e., the ratio between the SINR and its optimal value (obtained in exact statistics with background noise alone). First, we compare in terms of SINR performance, the standard STAP processing based on the estimation of the noise covariance matrix, to the processing based on a priori background noise covariance matrix (that will be called in the following a priori processing), which is optimal with background noise alone. Thus, we plot in Fig. 4 the normalized SINR for both STAP processing (computed with the SMI algorithm with diagonal loading) and a priori processing, at a given (peut être préciser cette Doppler cell pour pouvoir mieux s'appuyer sur la Fig. 3) Doppler cell and with a number of training samples equal to K = 20. Both SINR must be compared to the optimal SINR which is reached under assumption of a known clutter + noise covariance matrix. Depending on the range cell, 3 areas must be distinguished. From range cell 0 to range cell 15, we observe that a priori processing is sufficient to attain optimal performance, since at those range cells, clutter is less powerful than background noise as can be seen from Fig. 3. By comparison, STAP with K = 20 samples leads to poor SINR performance. From range cell 15 to range cell 30, clutter is powerful and performance of a priori processing significantly degrades. In that case, STAP is very useful since it allows to increase the SINR up to about 10 dB. Note that at range cell 17, the clutter that is present in the SCM estimated from the cells around it, is very strong (as shown by the power map in Fig. 3), whereas it is not present in the a priori covariance matrix. Finally, the 3rd area is between range cell 30 to 60, where the performance of a priori processing becomes better than those of STAP, though clutter is more powerful than background noise, as shown by power map in Fig. 3.

Then, we implement the proposed scheme. Depending on the value of Siegel distance, it consists in choosing between a priori and estimated from training data covariance matrix, for space-time processing. In Fig. 5, we compare the resulting normalized SINR of the proposed scheme to the optimal SINR. The threshold s is chosen by use of Monte Carlo simulations with background noise alone, so that the type-I error is less than 10 -6 .

We observe that this scheme makes a good choice between conventional STAP and a priori filter since it allows to attain good SINR performance. It allows both to improve the performance of conventional STAP processing on the areas of background noise alone (except in the close vicinity of clutter discontinuities) and to avoid significant SINR losses, resulting from the presence of clutter, when using a priori processing. Finally, let us note that the use of this scheme can be extended to the case where several a priori matrices are available, for instance for knowledge-aided STAP.

V. CONCLUSIONS

In this paper, we have considered the use of Siegel distance, in radar array processing. We have proposed to compute the distance between the SCM of training data and an a priori covariance matrix before STAP processing, to decide what kind of filter to use. After having analyzed the statistical behavior of the Siegel distance, we have performed simulations to illustrate the interest of that preprocessing in terms of STAP SINR performance.

Fig. 1

 1 presents a comparison of the PDF of D obtained from Monte-Carlo realizations using simulated data with the theoretical value obtained by[START_REF] Chen | Screening Among Multivariate Normal Data[END_REF] for N = 4 with 4 values of .

Fig. 1 .

 1 Fig. 1. Theoretical (-) and empirical PDF for Siegel distance (××) and GIP (+ +).

Fig. 2 .

 2 Fig. 2. Empirical (-+) PDF of Siegel distance for five values of K with = 1.

  K } 1.53 1.513 1.466 1.408 1.39 1.386 Tab.I: Empirical mean of d K as a function of K

Fig. 3 .

 3 Fig. 3. Power map vs range and Doppler cell.

Fig. 4 .

 4 Fig.[START_REF] Rangaswamy | Statistical Analysis of the Nonhomogeneity Detector[END_REF]. Normalized SINR vs range cell, at Doppler cell 100, for optimal (-), a priori (+-+) and STAP algorithm (---).

Fig. 5 .

 5 Fig.5. Normalized SINR vs range cell, at Doppler cell 100, for optimal processing (-), and the proposed scheme (*).