Thomas Gicquel
email: thomas.gicquel@chu-rennes.fr

Chloé Hugbart

Françoise Lohézic-Le Dévéhat

Sylvie Lepage

Alain Baert

Renaud Bouvet

Isabelle Morel

Françoise Le Devehat

Death related to consumption of Rauvolfia sp. powder mislabeled as Tabernanthe iboga

Keywords: poisoning, iboga, Rauvolfia, ajmaline, yohimbine, reserpine

Powdered roots of iboga (Tabernanthe iboga) contain ibogaine, an alkaloid considered by some authors, notably on the basis of experiments conducted in rats [START_REF] Glick | Effects and after effects of ibogaine on morphine self-administration in rats[END_REF][START_REF] Dzoljic | Effect of ibogaine on naloxone precipitated withdrawal syndrome in chronic morphine-dependent rats[END_REF][START_REF] Cappendijk | Inhibitory effects of ibogaine on cocaine selfadministration in rats[END_REF][START_REF] Rezvani | Attenuation of alcohol intake by ibogaine in three strains of alcohol preferring rats[END_REF], to be a treatment for addiction [START_REF] Brown | Ibogaine in the treatment of substance dependence[END_REF][START_REF] Alper | Treatment of acute opioid withdrawal with ibogaine[END_REF]. This hypothesis was put forward as early as 1985 but it has not been scientifically established [START_REF] Lotsof | Rapid method for interrupting the narcotic addiction syndrome[END_REF]. Nevertheless, use of iboga has increased due to its online availability. We report the case of a heroin addict who died after oral ingestion of a powder labeled Tabernanthe iboga bought online. We set about identifying this powder, and developed a specific method for quantification in blood and bile samples of alkaloids identified in the powder. The relationship between exposure to the powder and death is discussed in relation to the other results of forensic investigations.

Case report

Case history

A 30-year-old white woman, with a history of drug abuse and receiving methadone treatment, was found dead in the caravan where she was living by a friend. Foam was present at her mouth. Cannabis, methadone (one empty bottle with no indication of concentration), oxazepam and a small bag of powder labeled "Top quality Tabernanthe iboga 50g Gabon, Africa" were found in her caravan. This powder would have been consumed orally, mixed with juice or as an infusion, to treat opiate withdrawal symptoms. A friend of the victim said that she had already used iboga in the past as an indication for opiate withdrawal and that she had stopped taking opiates two days before her death, describing loss of appetite, diarrhea and severe fatigue. Her medical history was unremarkable except for anorexia.

Autopsy findings

An autopsy was performed four days post-mortem. External examination found no lesions other than non-specific injuries on the legs. The autopsy showed congestion of internal organs and pulmonary edema. No organs were injured, and although the cause and manner of death were not established, trauma was definitely ruled out. No previous pathological state was found. Samples of femoral blood, bile and hair were taken for toxicological analysis. Urine was not available (empty bladder). Histopathological analysis of the heart showed no macroscopic or microscopic abnormalities.

A c c e p t e d M a n u s c r i p t

Analytical methods

Chemicals: Tabernanthe iboga was generously donated to us by Dr Alexandre Maciuk (Paris Sud University, France). Rauvolfia vomitoria powder and the alkaloids of Rauvolfia serpentina (ajmaline and reserpine) were available in the Pharmacognosy's laboratory of Faculty of Pharmacy, Rennes, France. Ajmaline (Gilurytmal®) was purchased from Carinopharm GmbH (Elze, Germany). Formic acid, ammonium acetate, yohimbine, reserpine and flurazepam, as the internal standard (IS), were purchased from Sigma Aldrich (St Louis, MO, USA). Methanol, acetonitrile and water were purchased from Fisher Scientific UK (Loughborough, Leicestershire, UK). All chemicals, reagents and solvents were of LC-MS grade quality. Stock solutions of ajmaline, reserpine, yohimbine and IS were prepared in methanol (1.0 g/L) and then stored at 4°C.

Analysis of powders

Powders of Rauvolfia vomitoria, Tabernanthe iboga and the unidentified powder (found in the caravan) were examined under a microscope (Olympus CX41) equipped with a camera (Olympus DP20) in bidistilled water.

Two extractions were carried out, one by maceration in methanol under ultrasonic shaking and a second one based on the specific protocol for extracting alkaloids [START_REF] Delourmé-Houdé | Contribution à l'étude de l'iboga (Tabernanthe iboga H.Bn)[END_REF]. Briefly free lipophilic alkaloids were extracted by diethylether in alkaline solution. Then, liquid-liquid extraction was performed after series of acidification and alkalinisation to remove fat soluble impurities and to purify the alkaloids.

The Tabernanthe iboga and Rauvolfia vomitoria extracts and the standards of yohimbine, ajmaline, reserpine, and alkaloids from Rauvolfia serpentina were used to identify compounds in the unknown powder. An automated sample applicator, CAMAG ATS-4, was used for spotting High Performance Thin Layer Chromatography (HPTLC) plates and a densitometric UV scanner, CAMAG Scanner type 3, was used for plate analysis (CAMAG, Muttenz, Switzerland). Both instruments were controlled by Wincats software version 1.4.3. The HPTLC analysis for all extracts was carried out on silica gel plates (Merck silica gel 60F 254 ) with a mobile phase consisting of diethylether/acetone/diethylamine (85/8/7). Ten microliters of extracts (1 mg.mL -1 ) were spotted on TLC plates. After development, plates were scanned using the scanner at 6 wavelengths: 220, 260, 290, 310, 340 and 370 nm. Spot identification

Mass spectrometry analysis

Screening analyses

Powders and biological samples screening analyses were performed using two chromatography systems: GC-MS (Gas Chromatography Mass Spectrometry) and LC-HR-MS (Liquid Chromatography High Resolution Mass Spectrometry) on a Thermo Scientific Finnigan Trace GC Ultra DSQ II (Thermo Scientific, San Jose, USA) and on a Thermo Scientific Q Exactive (San Jose, USA) mass spectrometer, respectively.

Quantitative analyses of biological samples

After decontamination with dichloromethane, hair (16 cm) was segmented into 2 cm segments, extracted and analyzed as per Society of Hair Testing guidelines [START_REF] Cooper | Society of Hair Testing guidelines for drug testing in hair[END_REF]. Blood was extracted according to specific methods to detect various type of analytes (i.e. psychotropic, narcotic or opioids withdrawal medication). Briefly samples were extracted by liquid-liquid extraction (LLE) or solid phase extraction (SPE) on Bond Elut Agilent C18 (200 mg, 3 mL) columns before analysis under different chromatographic conditions with a LC-MS-MS system (Thermo Scientific, San Jose, USA) including an Accela pump and a tandem triple quadrupole mass spectrometer (TSQ Quantum Ultra) equipped with an electrospray ionization (ESI) source. Data acquisition, peak integration and calibration were performed using Xcalibur 2.1 software. for confirmation purposes (Table 1). Seven-point calibration curves were obtained by fortifying drug-free human blood with working solution of ajmaline, reserpine, yohimbine at final concentrations of 0, 10, 50, 100, 500, 1000 and 5000 ng/mL. Standard curves corresponded to peak area ratios of each analyte to IS using weighted linear least-squares regression (1/x²) and were linear in the range of 10 to 5000 ng/mL (r² >0.99 for ajmaline, reserpine, yohimbine). Mass Spectrometry (LC-HR-MS) screening revealed the presence of ajmaline, yohimbine and reserpine in blood and bile samples, and the absence of ibogaine, as in the unknown powder (Figure 1). A specific LC-MS/MS method developed to quantify ajmaline, yohimbine and reserpine showed respective concentrations of 109.1 ng/mL, 98.2 ng/mL and 30.8 ng/mL in blood, and 1528.2 ng/mL, 914.2 ng/mL and 561.2 ng/mL in bile.

Quantification of alkaloids

Results

Powders of

Hair analysis revealed the presence of morphine, 6-monoacetylmorphine, amphetamine, MDMA (3,4-methylenedioxy-methamphetamine), cocaine, benzoylecgonine, methadone, 2ethylidene-1,5-dimethyl-3,3-diphenylpyrrolidine (EDDP), buprenorphine, norbuprenorphine, acetaminophen and ketamine in all hair segments (Table 2). Alkaloids present in the powder were not detected in hair samples when using the specific analytical method.

Discussion

Microscopic and toxicological analyses of the powder labeled Tabernanthe iboga demonstrated the absence of ibogaine and the presence of ajmaline, yohimbine and reserpine.

Allowing that the powder was not Tabernanthe iboga, as the labelled on the package. The powder was rather obtained from Rauvolfia sp., a shrub of the more common and less expensive Apocynaceae family [START_REF] Hamet | Le problème du vrai et des faux Iboga[END_REF]. Rauvolfia sp. can easily be confused with Tabernanthe Alkaloids identified in the powder have also been identified in blood of the deceased, so it can be concluded that she had recently consumed this powder. They were also identified in the bile, at higher concentrations than in post-mortem blood, which is consistent with descriptions in the literature [START_REF] Vanbinst | Bile analysis of drugs in postmortem cases[END_REF]. Furthermore, cannabis and oxazepam were also found in the blood at very low concentrations, such that they can be excluded as contributors to death.

Toxicological analysis of the hair confirmed the addiction to narcotics.

Data on the toxic or lethal concentrations of these alkaloids are virtually non-existent.

According to literature data, the measured concentrations corresponded, at least for ajmaline and yohimbine, to concentrations observed after consumption of therapeutic doses (200 to 1000 ng/mL for ajmaline; 50 to 300 ng/mL for yohimbine; 1400 to 1800 ng/mL for reserpine), without symptoms of toxicity, and were far below lethal concentrations (above 5500 ng/mL for ajmaline; unknown for yohimbine and reserpine) [START_REF] Regenthal | Drug levels: therapeutic and toxic serum/plasma concentrations of common drugs[END_REF][START_REF] Maass | Studies on absorption, excretion, and metabolism of 3H-reserpine in man[END_REF].

The low blood concentrations must be interpreted with caution, taking into account certain limitations; although the body was discovered shortly after death, the autopsy was carried out 4 days later. In the absence of data on the stability of these compounds, one cannot rule out post-mortem degradation and hence underestimation of their concentration in the blood at the time of death. Moreover, degradation of ajmaline and yohimbine leads to formation of active metabolites such as ajmaline-N-oxide and hydroxy-yohimbine, which were not identified. The Rauvolfia sp. powder also possibly contained other xenobiotics that could not be identified using the laboratory's analytical techniques, and so their possible involvement in the death cannot be discussed.

There are few scientific data about the toxicity of Rauvolfia, which varies in accordance with the nature and concentration of alkaloids contained in the plant. The alkaloids are primarily concentrated in the roots and especially in the bark [START_REF] Rathi | Therapeutic characteristics of Rauvolfia serpentine[END_REF]. The concentration of alkaloids is affected by the Rauvolfia species, the geographical area where it grows, the age of the plant, the season and the harvesting technique [START_REF] Rathi | Therapeutic characteristics of Rauvolfia serpentine[END_REF][START_REF] Bronstein | Annual Report of the American Association of Poison Control Centers' National Poison Data System (NPDS): 27 th . Annual Report[END_REF]. The root extract is used, in south Asia, as the A c c e p t e d M a n u s c r i p t 10 main constituent of certain medications notably used to treat high blood pressure and certain mental illnesses [START_REF] Mehrotra | Hairy root biotechnology of Rauwolfia serpentina: a potent approach for the production of pharmaceutically important terpenoid indole alkaloids[END_REF].

Reserpine is described as the most potent alkaloid of the plant, responsible for the hypotensive activity of Rauvolfia [START_REF] Bronstein | Annual Report of the American Association of Poison Control Centers' National Poison Data System (NPDS): 27 th . Annual Report[END_REF]. It acts on receptors of the central and peripheral nervous system, lowering the level of endogenous bioamines (norepinephrine, serotonin and dopamine), by specifically inhibiting the uptake of these amines by storage vesicles [START_REF] Nammi | Reserpine-induced central effects: pharmacological evidence for the lack of central effects of reserpine methiodide[END_REF]. This mechanism induces depression of the central nervous system and a reduction in blood pressure [START_REF] Mehrotra | Hairy root biotechnology of Rauwolfia serpentina: a potent approach for the production of pharmaceutically important terpenoid indole alkaloids[END_REF][START_REF] Nammi | Reserpine-induced central effects: pharmacological evidence for the lack of central effects of reserpine methiodide[END_REF]. Reported cases of reserpine intoxication are rare. The observed toxic effects were gastrointestinal (hypersalivation, gastric hypersecretion, nausea, vomiting, diarrhea), skin flushes, nasal congestion, sedation or even coma, hypotension and bradycardia [START_REF]Poisons Information Monograph: Reserpine (PIM 467)[END_REF]. As with any substance acting on a tissue receptor, the blood concentration of the xenobiotic is not representative of its clinical effects, and if intoxication is suspected, the blood concentration is a priori uninformative for assessing clinical gravity. Moreover, a characteristic of the molecule is that a low proportion of administered reserpine binds specifically and irreversibly to its site of action [START_REF] Rudnick | Energetics of reserpine binding and occlusion by the chromaffin granule biogenic amine transporter[END_REF][START_REF] Hoffman | Adrenergic pharmalogy[END_REF] leading to long-lasting clinical signs. After the end of consumption of reserpine, these clinical signs only subside when storage of endogenous bioamines is restored, a process which requires several days or even weeks. A study was carried out to evaluate the efficacy of Rauvolfia serpentina in people with essential hypertension: the hypotensive effect induced by ingestion of R. serpentina persisted 2 to 4 weeks following discontinuation of the "treatment" [START_REF] Vakil | A clinical trial of Rauwolfia serpentina in essential hypertension[END_REF]. The fraction bound to tissue receptors, responsible for clinical signs, is not detectable in blood, while the free form (not bound to the receptor, quantifiable in the blood), representing the inactive portion, is metabolized and/or excreted [START_REF] Stitzel | The biological fate of reserpine[END_REF]. A clinical-biological discordance is therefore possible, in which clinical or even toxicity symptoms are present when reserpine is no longer detectable or is detected in small amounts in the blood [START_REF] Good | Rauwolfia serpentina toxicity in a dog[END_REF].

The potentially toxic nature of the other alkaloids identified in the powder must not be underestimated. Ajmaline is an inhibitor of -adrenergic receptors used for its antihypertensive and antiarrhythmic properties [START_REF] Baselt | Disposition of toxic drugs and chemicals in man[END_REF]. Its toxicity is primarily cardiac, by induction of rhythm and conduction disorders [START_REF] Rolf | The ajmaline challenge in Brugada syndrome: diagnostic impact, safety, and recommended protocol[END_REF][START_REF] Bermúdez | Spontaneous sustained monomorphic ventricular tachycardia after administration of ajmaline in a patient with Brugada syndrome[END_REF][START_REF] Jornod | Suicidal attempt by overdosage of ajmaline[END_REF][START_REF] Medina-Ravell | Nonfatal ajmaline cardiotoxicity[END_REF][START_REF] Wellens | Death after ajmaline administration[END_REF][START_REF] Köppel | Clinical course and outcome in class IC antiarrhythmic overdose[END_REF]. One of the metabolites of ajmaline, ajmaline-N-oxide, is pharmacologically active in animals [START_REF] Köppel | Clinical course and outcome in class IC antiarrhythmic overdose[END_REF]. ohimbine is a selective antagonist of 2adrenergic receptors that increases the release of norepinephrine and dopamine [START_REF] Anderson | Case study: two fatal case reports of acute yohimbine intoxication[END_REF]. One of its metabolites, 1-hydroxy-yohimbine, is pharmacologically active [START_REF] Le Corre | Biopharmaceutics and metabolism of yohimbine in humans[END_REF]. Among the rare reported cases of yohimbine intoxication, the adverse events included: increase in blood pressure, tachycardia, paresthesias, loss of coordination, headaches, gastrointestinal disorders, A c c e p t e d M a n u s c r i p t 11 agitation and even coma [START_REF] Baselt | Disposition of toxic drugs and chemicals in man[END_REF][START_REF] Linden | Yohimbine: a new street drug[END_REF]. A case of atrial fibrillation after ingestion of 350 mg yohimbine has been reported [START_REF] Varkey | Overdose of yohimbine[END_REF].

The toxicity of each of these alkaloids, considered separately, is partially known, but there are very few scientific articles concerning the toxicity of these alkaloids consumed concomitantly. There is one report in which medicinal plants were adulterated by alkaloids derived from R. serpentina (containing ajmaline and reserpine), in a dietary supplement of Italian origin: the people who ingested the powder experienced hypotensive problems [START_REF] Karioti | Combined HPLC-DAD-MS, HPLC-MS(n) and NMR spectroscopy for quality control of plant extracts: the case of a commercial blend sold as dietary supplement[END_REF].

To our knowledge, there are no cases of death in the literature related to simultaneous consumption of these three alkaloids. The symptoms reported by witnesses, nausea and diarrhea in combination with fatigue, are compatible with opiate withdrawal syndrome.

However, muscle pain, rhinorrhea, hyperhidrosis, increased gastrointestinal motility with diarrhea and vomiting have also been described in association with reserpine or yohimbine intoxication. It is therefore not possible, on the basis of symptomatology, to distinguish between the withdrawal syndrome and possible intoxication by Rauvolfia.

Conclusion

Post-mortem investigations did not find any active pathological processes at the time of death or a previous pathological state. Exposure to a powder containing three toxic alkaloids could be demonstrated. These data suggest that the death, which occurred violently after initiation of the "treatment", was due to the cardiovascular effects of the powder. The present case is therefore the first reported case of death following consumption of Rauvolfia sp. powder mislabeled as Tabernanthe iboga. It highlights the worldwide problem of counterfeit products available over the internet and the danger associated with their consumption.

A c c e p t e d M a n u s c r i p t 1 Highlights-

 1 An original case of a death after roots powder mislabeled -Alkaloids (ajmaline, yohimbine and reserpine) were quantified by an original LC-HR-MS method in blood and bile samples -Consumption of counterfeit products available over the internet represent a worldwide danger

  Rf value, on maximum absorbance wavelength and on visual evaluation after spraying Dragendorff reagent.

  Blood (100 μL) or bile(10 μL) samples were supplemented with 20 μL of 1 μg/mL flurazepam (IS) solution and extracted by basic liquid-liquid extraction (LLE) after adding 2.3 mL of a solvent mix (heptane : isopropanol : 1,2-dichloroethane : dichloromethane (33:17:25:25) (v/v)), 1.8 g of a powders mix (sodium chloride : sodium bicarbonate : sodium carbonate (84:8:8) (w/w)) and 4.5 mL of deionized water (pH=8.9) then mixing by gentle inversion for 10 min. After 10 min centrifugation at 3000 g, supernatants were evaporated to dryness at 50°C under a stream of nitrogen. Residues were dissolved in 500 μL of mobile phase at initial condition (90:10 A:B (v/v)) with solvent A corresponding to 10 mM ammonium acetate buffer containing 0.1% (v/v) formic acid and solvent B consisting of acetonitrile with 0.1% (v/v) formic acid. The analysis was performed by a LC-HR-MS A c c e p t e d M a n u s c r i p t 7 system equipped with a Thermo Scientific Q Exactive TM (San Jose, USA) mass spectrometer system. LC separation was performed by an Accela pump (Thermo Scientific, San Jose, USA) on a C18 Accucore column (100 mm x 2.1, 2.6 μm) at a flow rate of 500 μL/min using the following gradient elution : initial conditions of 90:10 (A:B) maintained for 1 min, increasing to 40:60 (A:B) for 4 minutes, and then 1 min for 100% B and return to initial conditions 90:10 (A:B) for equilibration. This corresponded to a total chromatographic run of 8 minutes. All samples were kept at 15°C in the autosampler until injection of 5 μL into the LC-HR-MS system (partial loop) in a thermostated column at 25°C. The MS conditions were as follows: HESI in positive mode, capillary temperature: 300°C; spray voltage: 4500 V; sheath and auxiliary gas (nitrogen) flow-rate: 35 and 15 (arbitrary units), respectively. Data were acquired in targeted MS² (t-MS²) mode, where MRM transitions for quantitative determination were 327.2060 144.0805 m/z for ajmaline, 355.2050 144.0805 m/z for yohimbine and 609.2796 195.0650 m/z for reserpine whereas second transitions were used

  Tabernanthe iboga, Rauvolfia vomitoria and the unknown powder were yellowbrownish. Microscopically, all of them showed lignified cork on surface view, starch grains with distinct hilum, isodiametrical and elongated sclereids. Tracheids and xylem vessels were clearly found only in R. vomitoria. The unknown powder contained isolated sclereids -not grouped as in iboga -but also simple or grouped starch grains like in R. vomitoria, therefore ruling out iboga. Moreover, many prismatic crystals of calcium oxalate were found in the unidentified powder as in R. vomitoria. In conclusion, this powder, purportedly derived from roots or barks of Tabernanthe iboga, was macroscopically similar to those of the Apocynaceae family and had microscopic findings closer to Rauvolfia species than to Tabernanthe iboga.

A c c e p t e d M a n u s c r i p t 8 Preliminary

 8 studies using chromatographic HPTLC analyses show that the methanolic extract and the purified fractions of the unknown powder were different from that of iboga powder (data not shown). Ajmaline and reserpine were only found in R. vomitoria and R. serpentina as major alkaloids. Screening of the powders by LC-HR-MS and GC-MS showed the presence of ajmaline, reserpine and yohimbine in R. serpentina, R. vomitoria and the unknown powder. Ibogaine was not found in these powders whereas analysis of authentic Tabernanthe iboga powder using the same screening methods confirmed the presence of ibogaine and ibogamine, two indole alkaloids usually contained in iboga roots.Toxicological analyses of blood samples reported the absence of alcohol and the presence of oxazepam (100.6 ng/mL), delta-9-tetrahydrocannabinol (THC) (3.01 ng/mL) and carboxytetrahydrocannabinol (THC-COOH) (105.2 ng/mL). Liquid Chromatography High Resolution

A c c e p t e d M a n u s c r i p t 9 iboga,

 9 since powders derived from the crushed roots of these plants were macroscopically similar. Rauvolfia genus contains about sixty plant species which are found in all tropical and subtropical regions. Their roots contain a high number of alkaloids including ajmaline, reserpine and yohimbine[START_REF] Rathi | Therapeutic characteristics of Rauvolfia serpentine[END_REF][START_REF] Kumari | phytochemical, pharmacological and therapeutic aspects[END_REF].

Figure 1 .

 1 Figure 1. Analysis of alkaloids in unknown powder (A) and blood samples (B) : Chromatograms of protonated ajmaline, yohimbine, and reserpine

Table 1 .

 1 Reaction monitoring parameters for ajmaline, yohimbine, flurazepam and reserpine. (RT = retention time, NCE = Normalized Collision Energy)

	Compound	Chemical formula	RT (min)	Precursor ion [M+H] + (m/z)	NCE (%)	Product ion (m/z)
	Ajmaline	C 20 H 26 N 2 O 2	3.0	327.2060	50 50	144.0805 158.0963
					50	212.1277
	Yohimbine	C 21 H 26 O 3 N 2	3.2	355.2050
					50	144.0805
	Flurazepam	C 21 H 23 ClFN 3 O	3.8	388.1586	50	315.0686
					50	174.0911
	Reserpine	C 33 H 40 N 2 O 9	4.6	609.2796	50	195.0650

Table 2 .

 2 Concentrations in hair segments. Segment 1 is the most recent and segment 8 is the oldest. (ND = not detected) (6-MAM = 6-MonoAcetylMorphine ; MDMA=3,4-methylenedioxy-methamphetamine ; EDDP=2-ethylidene-1,5-dimethyl-3,3-diphenylpyrrolidine)

	Concentration (ng/mg)	Segment 1 Segment 2 Segment 3 Segment 4 Segment 5 Segment 6 Segment 7 Segment 8
	Morphine	15.6	2.0	1.7	1.8	1.7	2.1	1.5	1.8
	6-MAM	0.1	0.2	0.2	1.2	0.2	0.3	0.2	0.2
	Amphetamine	7.0	6.1	3.5	2.2	1.8	1.7	1.7	1.7
	MDMA	4.2	3.8	6.0	4.1	2.3	2.2	2.1	2.5
	Cocaine	1.4	2.4	3.5	4.7	5.2	6.8	5.9	8.7
	Benzoylecgonine	0.4	0.6	1.0	1.3	1.7	3.2	2.7	4.4
	Methadone	6.5	8.4	6.7	4.8	3.4	3.6	2.9	2.1
	EDDP	0.7	1.1	0.8	0.6	0.2	0.6	0.4	0.3
	Buprenorphine	ND	ND	ND	ND	ND	ND	ND	ND
	Norbuprenorphine	ND	ND	ND	ND	ND	ND	ND	ND
	Acetaminophen	13.0	13.5	13.8	13.7	13.6	13.7	13.6	13.6
	ketamine	0.92	0.33	0.31	0.41	0.49	0.47	0.5	0.56


Sandro Franceschi
email: sandro.franceschi@upmc.fr

Kilian Raschel
email: kilian.raschel@lmpt.univ-tours.fr

TUTTE'S INVARIANT APPROACH FOR BROWNIAN MOTION REFLECTED IN THE QUADRANT

Keywords: Reflected Brownian motion in the quarter plane, Stationary distribution, Laplace transform, Tutte's invariant approach, Generalized Chebyshev polynomials. Version of July 10, 2017

Introduction and main results

1.1. Reflected Brownian motion in the quadrant. The object of study here is the reflected Brownian motion with drift in the quarter plane

Z(t) = Z 0 + W (t) + µt + RL(t), ∀t 0, (1)
associated to the triplet (Σ, µ, R), composed of a non-singular covariance matrix, a drift and a reflection matrix, see Figure 1:

Σ = σ 11 σ 12 σ 12 σ 22 , µ = µ 1 µ 2 , R = (R 1 , R 2 ) = r 11 r 12 r 21 r 22 .
In Equation (1), Z 0 is any initial point in R 2 + , the process (W (t)) t 0 is an unconstrained planar Brownian motion with covariance matrix Σ starting from the origin, and for i = 1, 2, L i (t) is a continuous non-decreasing process, that increases only at time t such that Z i (t) = 0, namely t 0 1 {Z i (s) =0} dL i (s) = 0, for all t 0. The columns R 1 and R 2 represent the directions in which the Brownian motion is pushed when the axes are reached. The reflected Brownian motion (Z(t)) t 0 associated with (Σ, µ, R) is well defined [START_REF] Harrison | Multidimensional reflected Brownian motions having exponential stationary distributions[END_REF][START_REF] Williams | Semimartingale reflecting Brownian motions in the orthant[END_REF], and is a fondamental stochastic process, from many respects. There is a large literature on reflected Brownian motion in the quadrant (and also in orthants, generalization to higher dimension of the quadrant). First, it serves as an approximation of large queuing networks [START_REF] Foddy | Analysis of Brownian motion with drift, confined to a quadrant by oblique reflection (diffusions, Riemann-Hilbert problem)[END_REF][START_REF] Baccelli | Analysis of models reducible to a class of diffusion processes in the positive quarter plane[END_REF]; this was the initial motivation for its study. In the same vein, it is the continuous counterpart of (random) walks in the quarter plane, which are an important combinatorial and probabilistic object, see [START_REF] Bousquet-Mélou | Walks with small steps in the quarter plane[END_REF][START_REF] Bernardi | Counting quadrant walks via Tutte's invariant method[END_REF]. In other directions, it is studied for its Lyapunov functions [START_REF] Dupuis | Lyapunov functions for semimartingale reflecting Brownian motions[END_REF], cone points of Brownian motion [START_REF] Gall | Mouvement brownien, cônes et processus stables[END_REF], intertwining relations and crossing probabilities [START_REF] Dubédat | Reflected planar Brownian motions, intertwining relations and crossing probabilities[END_REF], and of particular interest for us, for its recurrence or transience [START_REF] Hobson | Recurrence and transience of reflecting Brownian motion in the quadrant[END_REF]. Its stationary distribution exists and is unique if and only if the following (geometric) conditions are satisfied: r 11 > 0, r 22 > 0, r 11 r 22 -r 12 r 21 > 0, r 22 µ 1 -r 12 µ 2 < 0, r 11 µ 2 -r 21 µ 1 < 0.

(

(For orthogonal reflections, ( 2) is equivalent for the drift µ to have two negative coordinates.) Moreover, the asymptotics of the stationary distribution (when it exists) is now well known, see [START_REF] Harrison | Reflected Brownian motion in the quadrant: tail behavior of the stationary distribution[END_REF][START_REF] Dai | Reflecting Brownian motion in two dimensions: Exact asymptotics for the stationary distribution[END_REF][START_REF] Franceschi | Asymptotic expansion for the stationary distribution of a reflected Brownian motion in the quarter plane[END_REF]. There exist, however, very few results giving an exact expression for the stationary distribution, and the main contribution of this paper is precisely to propose a method (based on boundary value problems) for deriving an explicit formula for the (Laplace transform of the) stationary distribution. Our study constitutes one of the first attempts to apply these techniques to reflected Brownian motion, after [START_REF] Foschini | Equilibria for diffusion models of pairs of communicating computerssymmetric case[END_REF] (under the strong symmetry condition µ 1 = µ 2 , σ 11 = σ 22 and symmetric reflection vectors R 1 , R 2 in (1)), [START_REF] Foddy | Analysis of Brownian motion with drift, confined to a quadrant by oblique reflection (diffusions, Riemann-Hilbert problem)[END_REF] (with the identity covariance matrix Σ) and [START_REF] Baccelli | Analysis of models reducible to a class of diffusion processes in the positive quarter plane[END_REF] (with a diffusion having a quite special behavior on the boundary). We also refer to [START_REF] Burdzy | Obliquely reflected Brownian motion in non-smooth planar domains[END_REF] for the analysis of reflected Brownian motion in bounded domains by complex analysis techniques.

1.2. Laplace transform of the stationary distribution. Under assumption [START_REF] Bernardi | Counting quadrant walks via Tutte's invariant method[END_REF], that we shall do throughout the manuscript, the stationary distribution is absolutely continuous w.r.t. the Lebesgue measure, see [START_REF] Harrison | Multidimensional reflected Brownian motions having exponential stationary distributions[END_REF][START_REF] Dai | Steady-state analysis of reflected Brownian motions: Characterization, numerical methods and queueing applications[END_REF]. We denote its density by π(x) = π(x 1 , x 2 ). Let the moment generating function (or Laplace transform) of π be defined by

ϕ(θ) = E π [e θ|Z ] = R 2 + e θ|x π(x)dx.
The above integral converges at least for θ = (θ 1 , θ 2 ) ∈ C 2 such that θ 1 0 and θ 2 0. We further define two finite boundary measures ν 1 and ν 2 with support on the axes, by mean of the formula

ν i (B) = E π 1 0 1 {Z(t)∈B} dL i (t) .
The measure ν i is continuous w.r.t. the Lebesgue measure [START_REF] Harrison | Multidimensional reflected Brownian motions having exponential stationary distributions[END_REF], and may be viewed as the boundary invariant measure. We define the moment generating function of ν 1 and ν 2 by

ϕ 2 (θ 1 ) = R + e θ 1 x 1 ν 2 (x 1 )dx 1 , ϕ 1 (θ 2 ) = R + e θ 2 x 2 ν 1 (x 2 )dx 2 .
The functions ϕ 1 and ϕ 2 exist a priori for values of the argument with non-positive real parts.

There is a functional equation between the Laplace transforms ϕ, ϕ 1 and ϕ 2 , see (5) in Section 2, which is reminiscent of the functional equation counting (discrete) quadrant walks [START_REF] Bousquet-Mélou | Walks with small steps in the quarter plane[END_REF][START_REF] Bernardi | Counting quadrant walks via Tutte's invariant method[END_REF].

1.3. Main result. We derive an explicit expression for ϕ 1 , and therefore also for ϕ 2 and ϕ by the functional equation [START_REF] Dai | Steady-state analysis of reflected Brownian motions: Characterization, numerical methods and queueing applications[END_REF], in the particular case where R is the identity matrix, which means that the reflections are orthogonal (Figure 1, right). Define the generalized Chebyshev polynomial by (for a 0)

T a (x) = cos(a arccos(x)) = 1 2 x + x 2 -1 a + x -x 2 -1 a .
It admits an analytic continuation on C \ (-∞, -1), and even on C if a is a non-negative integer.

We also need to introduce

θ ± 2 = (µ 1 σ 12 -µ 2 σ 11 ) ± (µ 1 σ 12 -µ 2 σ 11 ) 2 + µ 2 1 det Σ det Σ (3)
(notice that the sign of θ ± 2 is ±, see Figure 2), as well as the angle (related to the correlation coefficient of the Brownian motion (W (t)) t 0 )

β = arccos - σ 12 √ σ 11 σ 22 . (4)
Theorem 1. Let R be the identity matrix in [START_REF] Baccelli | Analysis of models reducible to a class of diffusion processes in the positive quarter plane[END_REF]. The Laplace transform ϕ 1 is equal to

ϕ 1 (θ 2 ) = -µ 1 w (0) w(θ 2 ) -w(0) θ 2 ,
where the function w can be expressed in terms of the generalized Chebyshev polynomial T π β as follows:

w(θ 2 ) = T π β - 2θ 2 -(θ + 2 + θ - 2 ) θ + 2 -θ - 2 .
Accordingly, ϕ 1 can be continued meromorphically on the cut plane C \ (θ + 2 , ∞). There exists, of course, an analogous expression for ϕ 2 (θ 1 ), and the functional equation ( 5) finally gives a simple explicit formula for the bivariate Laplace transform ϕ.

Let us now give some comments around Theorem 1.

• It connects two a priori unrelated objects: the stationary distribution of reflected Brownian motion in the quadrant and a particular special function, viz, a generalized Chebyshev polynomial (which is a hypergeometric function). The expression that we obtain is quite tractable: as an example we will recover the well-known case of one-dimensional reflected Brownian motion (Section 3.5). • To prove Theorem 1, we apply a constructive (and combinatorial in nature) variation of the boundary value method of [START_REF] Fayolle | Random Walks in the Quarter-Plane[END_REF], recently introduced in [2] as Tutte's invariant approach [START_REF] Tutte | Chromatic sums revisited[END_REF], see our Section 3. This paper is one of the first attempts to apply boundary value techniques to (continuous) diffusions in the quadrant, after [START_REF] Foddy | Analysis of Brownian motion with drift, confined to a quadrant by oblique reflection (diffusions, Riemann-Hilbert problem)[END_REF] (which concerns very particular cases of the covariance matrix, essentially the identity matrix) and [START_REF] Baccelli | Analysis of models reducible to a class of diffusion processes in the positive quarter plane[END_REF] (on diffusions with completely different behavior on the boundary). • The recent paper [START_REF] Franceschi | Asymptotic expansion for the stationary distribution of a reflected Brownian motion in the quarter plane[END_REF] obtains the exact asymptotic behavior of the stationary distribution along any direction in the quadrant. The constant in that asymptotics involves both functions ϕ 1 and ϕ 2 (see [START_REF] Franceschi | Asymptotic expansion for the stationary distribution of a reflected Brownian motion in the quarter plane[END_REF]), and can thus be made explicit with our Theorem 1. • Theorem 1 is well suited for asymptotic analysis: we shall derive the asymptotics of the stationary distribution along the axes, by using classical arguments from singularity analysis of Laplace transforms (see the reference book [START_REF] Doetsch | Introduction to the theory and application of the Laplace transformation[END_REF]). See our Section 4. • Theorem 1 implies that the Laplace transform is algebraic if and only if a certain group (to be properly introduced later on) is finite, see Section 5.3. This result has an analogue in the discrete setting, see [START_REF] Bousquet-Mélou | Walks with small steps in the quarter plane[END_REF]. In the same vein, the authors of [START_REF] Dieker | Reflected Brownian motion in a wedge: sum-ofexponential stationary densities[END_REF] give necessary and sufficient conditions for the stationary density of two-dimensional reflected Brownian motion with negative drift in a wedge to have the form of a sum of exponentials. The intersection with our results is the polynomial case π β ∈ Z. • More that the Brownian motion in the quadrant, all results presented here (including Theorem 1) concern the Brownian motion in two-dimensional cones (by a simple linear transformation of the cones). This is a major and interesting difference of the continuous case in comparison with the discrete case, which also illustrates that the analytic approach is very well suited to that context.

Though being self-contained (this is one of the reasons why we focus on the case of orthogonal reflections), this paper is part of a larger project, dealing with any reflection matrix R (as in Figure 1, left). Let us finally mention that an extended abstract of this paper and of [START_REF] Franceschi | Asymptotic expansion for the stationary distribution of a reflected Brownian motion in the quarter plane[END_REF] may be found in [START_REF] Franceschi | Analytic approach for reflected Brownian motion in the quadrant[END_REF].

Acknowledgements. We thank Mireille Bousquet-Mélou, Irina Kurkova and Marni Mishna for interesting discussions. We also thank an anonymous referee for her/his careful reading and her/his suggestions. Finally, we acknowledge support from the projet Région Centre-Val de Loire (France) MADACA and from Simon Fraser University, Burnaby, BC (Canada).

Analytic preliminaries and continuation of the Laplace transforms

In this section we state the key functional equation (a kernel equation, see Section 2.1), which is the starting point of our entire analysis. We study the kernel (a second degree polynomial in two variables) in Section 2.2. Finally, we continue the Laplace transforms to larger domains (Section 2.3), which will be used in Section 3 to state a boundary value problem (BVP).

2.1.

A kernel functional equation. We have the following key functional equation between the Laplace transforms:

-γ(θ)ϕ(θ) = γ 1 (θ)ϕ 1 (θ 2 ) + γ 2 (θ)ϕ 2 (θ 1 ), (5)
where

     γ(θ) = 1 2 θ|σθ + θ|µ = 1 2 (σ 11 θ 2 1 + σ 22 θ 2 2 + 2σ 12 θ 1 θ 2 ) + µ 1 θ 1 + µ 2 θ 2 , γ 1 (θ) = R 1 |θ = r 11 θ 1 + r 21 θ 2 , γ 2 (θ) = R 2 |θ = r 12 θ 1 + r 22 θ 2 .
By definition of the Laplace transforms, this equation holds at least for any θ = (θ 1 , θ 2 ) with θ 1 0 and θ 2 0.

To prove (5) the main idea is to use an identity called basic adjoint relationship (first proved in [START_REF] Harrison | Brownian models of open queueing networks with homogeneous customer populations[END_REF] in some particular cases, then extended in [START_REF] Dai | Reflected Brownian motion in an orthant: numerical methods for steady-state analysis[END_REF]), which characterizes the stationary distribution, see [START_REF] Dai | Characterization of the stationary distribution for a semimartingale reflecting brownian motion in a convex polyhedron[END_REF][START_REF] Dai | Characterization of the stationary distribution for a reflecting brownian motion in a convex polyhedron[END_REF][START_REF] Dai | Nonnegativity of solutions to the basic adjoint relationship for some diffusion processes[END_REF]. (It is the continuous analogue of the equation πQ = 0, with π the stationary distribution of a recurrent continuous-time Markov chain having infinitesimal generator matrix Q.) This basic adjoint relationship connects the stationary distribution π and the corresponding boundary measures ν 1 and ν 2 . We refer to [START_REF] Foddy | Analysis of Brownian motion with drift, confined to a quadrant by oblique reflection (diffusions, Riemann-Hilbert problem)[END_REF][START_REF] Dai | Reflected Brownian motion in an orthant: numerical methods for steady-state analysis[END_REF][START_REF] Dai | Reflecting Brownian motion in two dimensions: Exact asymptotics for the stationary distribution[END_REF] for the details.

As stated in [6, Open problem 1], it is an open problem to prove that there is no finite signed measure which satisfies the basic adjoint relationship (by signed we mean a measure taking positive and negative values). Going back to our case, we shall see that the functional equation ( 5) admits a unique solution (the stationary distribution); in particular there is no solution of the functional equation with sign changes. 2.2. Kernel. By definition, the kernel of Equation ( 5) is the polynomial

γ(θ 1 , θ 2 ) = 1 2 (σ 11 θ 2 1 + 2σ 12 θ 1 θ 2 + σ 22 θ 2 2 ) + µ 1 θ 1 + µ 2 θ 2 .
It can be alternatively written as

γ(θ 1 , θ 2 ) = a(θ 2 )θ 2 1 + b(θ 2 )θ 1 + c(θ 2 ) = a(θ 1 )θ 2 2 + b(θ 1 )θ 2 + c(θ 1 ), where a(θ 2 ) = 1 2 σ 11 , b(θ 2 ) = σ 12 θ 2 + µ 1 , c(θ 2 ) = 1 2 σ 22 θ 2 2 + µ 2 θ 2 , a(θ 1 ) = 1 2 σ 22 , b(θ 1 ) = σ 12 θ 1 + µ 2 , c(θ 1 ) = 1 2 σ 11 θ 2 1 + µ 1 θ 1 . (6)
The equation γ(θ 1 , θ 2 ) = 0 defines a two-valued algebraic function Θ 1 (θ 2 ) such that γ(Θ 1 (θ 2 ), θ 2 ) = 0, and similarly Θ 2 (θ 1 ) such that γ(θ 1 , Θ 2 (θ 1 )) = 0. Their expressions are given by

Θ ± 1 (θ 2 ) = -b(θ 2 ) ± d(θ 2 ) 2 a(θ 2 ) , Θ ± 2 (θ 1 ) = -b(θ 1 ) ± d(θ 1 ) 2a(θ 1 ) ,
where d and d are the discriminants of the kernel:

d(θ 2 ) = θ 2 2 (σ 2 12 -σ 11 σ 22 ) + 2θ 2 (µ 1 σ 12 -µ 2 σ 11 ) + µ 2 1 , d(θ 1 ) = θ 2 1 (σ 2 12 -σ 11 σ 22 ) + 2θ 1 (µ 2 σ 12 -µ 1 σ 22 ) + µ 2 2 .
The polynomials d and d have two zeros, real and of opposite signs, they are denoted by θ ± 2 and θ ± 1 , respectively: θ ± 2 is introduced in (3) and

θ ± 1 = (µ 2 σ 12 -µ 1 σ 22 ) ± (µ 2 σ 12 -µ 1 σ 22 ) 2 + µ 2 2 det Σ det Σ .
Equivalently, θ ± 1 and θ ± 2 are the branch points of the algebraic functions Θ 2 and Θ 1 . Finally, notice that d is positive on

(θ - 1 , θ + 1 ) and negative on R \ [θ - 1 , θ + 1 ]
. Accordingly, the branches Θ ± 2 take real and complex conjugate values on these sets, respectively. A similar statement holds for Θ ± 1 .

2.3. Continuation of the Laplace transforms. In Section 3 we shall state a boundary condition for the functions ϕ 1 and ϕ 2 , on curves which lie outside their natural domains of definition (the half-plane with negative real-part). We therefore need to continue these functions, which is done in the result hereafter.

Lemma 2. We can continue meromorphically ϕ 1 (θ 2 ) to the open and simply connected set

{θ 2 ∈ C : θ 2 0 or Θ - 1 (θ 2 ) < 0}, (7)
by setting

ϕ 1 (θ 2 ) = γ 2 γ 1 (Θ - 1 (θ 2 ), θ 2 )ϕ 2 (Θ - 1 (θ 2 )).
Lemma 2 is immediate (one can find a refined version of it in [START_REF] Franceschi | Asymptotic expansion for the stationary distribution of a reflected Brownian motion in the quarter plane[END_REF]). A similar statement holds for the Laplace transform ϕ 1 . Anticipating slightly, we notice that the domain in [START_REF] Dai | Characterization of the stationary distribution for a reflecting brownian motion in a convex polyhedron[END_REF] contains the domain G R in Figure 2.

Statement of the boundary value problem

3.1. An important hyperbola. For further use, we need to introduce the curve

R = {θ 2 ∈ C : γ(θ 1 , θ 2 ) = 0 and θ 1 ∈ (-∞, θ - 1 )} = Θ ± 2 ((-∞, θ - 1 )). (8)
By the results of Section 2.2, d is negative on the interval (-∞, θ - 1 ), and thus the curve R is symmetrical w.r.t. the real axis, see Figure 2. Furthermore, it has a simple structure, as shown by the following elementary result, taken from [1, Lemma 9]: Lemma 3. The curve R in (8) is a (branch of a) hyperbola, given by the equation

σ 22 (σ 2 12 -σ 11 σ 22 )x 2 + σ 2 12 σ 22 y 2 -2σ 22 (σ 11 µ 2 -σ 12 µ 1 )x = µ 2 (σ 11 µ 2 -2σ 12 µ 1 ). (9)
Proof. We saw in Section 2.2 that whenever θ 1 ∈ (-∞, θ - 1 ), the branches Θ + 2 (θ 1 ) and Θ - 2 (θ 1 ) are complex conjugate. Define x and y by Θ + 2 (θ 1 ) = x + iy. Then from the identities

Θ + 2 + Θ - 2 = - b(θ 1 ) a(θ 1 ) = -2 σ 12 θ 1 + µ 2 σ 22 = 2x and Θ + 2 • Θ - 2 = c(θ 1 ) a(θ 1 ) = σ 11 θ 2 1 + 2µ 1 θ 1 σ 22 = x 2 + y 2 ,
we obtain that x and y are real solutions of [START_REF] Dai | Characterization of the stationary distribution for a semimartingale reflecting brownian motion in a convex polyhedron[END_REF].

We shall denote by G R the open domain of C bounded by R and containing 0, see Figure 3. 8) is symmetric w.r.t. the horizontal axis, and G R is the domain in green 3.2. BVP for orthogonal reflections. In the case of orthogonal reflections, R is the identity matrix in (1), and we have γ

Obviously G R , the closure of G R , is equal to G R ∪ R.
1 (θ 1 , θ 2 ) = θ 1 and γ 2 (θ 1 , θ 2 ) = θ 2 . We set ψ 1 (θ 2 ) = 1 θ 2 ϕ 1 (θ 2 ), ψ 2 (θ 1 ) = 1 θ 1 ϕ 2 (θ 1 ). (10)
Proposition 4. The function ψ 1 satisfies the following BVP: (i) ψ 1 is meromorphic on G R with a single pole at 0, of order 1 and residue ϕ 1 (0), and vanishes at infinity, (ii) ψ 1 is continuous on G R \ {0} and

ψ 1 (θ 2 ) = ψ 1 (θ 2 ), ∀θ 2 ∈ R. ( 11
)
Proof. Using the formula [START_REF] Dai | Reflecting Brownian motion in two dimensions: Exact asymptotics for the stationary distribution[END_REF], Point (i) is equivalent to proving that ϕ 1 is analytic in G R and is bounded at infinity. Both properties are obvious in the half-plane {θ 2 ∈ C : θ 2 < 0}, thanks to the definition of ϕ 1 as a Laplace transform. In the domain of G R where θ 2 0, one can use the continuation formula given in Lemma 2, and the conclusion follows.

Let us now prove (ii). Evaluating the (continued) functional equation ( 5) at (θ 1 , Θ ± 2 (θ 1 )), we obtain ψ 1 (Θ ± 2 (θ 1 )) + ψ 2 (θ 1 ) = 0, which immediately implies that

ψ 1 (Θ + 2 (θ 1 )) = ψ 1 (Θ - 2 (θ 1 )). (12)
Choosing θ 1 ∈ (-∞, θ - 1 ), the two quantities Θ + 2 (θ 1 ) and Θ - 2 (θ 1 ) are complex conjugate the one of the other, see Section 2. Equation ( 12) can then be reformulated as [START_REF] Dai | Stationary distribution of a two-dimensional SRBM: geometric views and boundary measures[END_REF], using the definition (8) of the curve R.

3.3.

Conformal gluing function and invariant theorem. The BVP stated in Proposition 4 is called a homogeneous BVP with shift (the shift stands here for the complex conjugation, but the theory applies to more general shifts, see [START_REF] Litvinchuk | Solvability Theory of Boundary Value Problems and Singular Integral Equations with Shift[END_REF]). Due to its particularly simple form, we can solve it in an explicit way, using the two following steps:

• Using a certain conformal mapping (to be introduced below), we can construct a particular solution to the BVP of Proposition 4.

• The solution to the BVP of Proposition 4 is unique, see Lemma 5 below (taken from Lemma 2 in [27, Section 10.2]). In other words, two different solutions must coincide, and the explicit solution constructed above must be the function ψ 1 .

Lemma 5 (Invariant lemma). The problem of finding functions f such that (i) f is analytic in G R and continuous in G R , (ii) f satisfies the boundary condition [START_REF] Dai | Stationary distribution of a two-dimensional SRBM: geometric views and boundary measures[END_REF], does not have non-trivial solutions in the class of functions f vanishing at infinity.

To construct a particular solution to the BVP of Proposition 4, we shall use the function

w(θ 2 ) = T π β - 2θ 2 -(θ + 2 + θ - 2 ) θ + 2 -θ - 2 (13)
introduced in Theorem 1. Let us first establish some of its properties.

Lemma 6. The function w in (13) is such that:

(i) w is analytic in G, continuous in G R and unbounded at infinity, (ii) w is injective in G (onto C \ (-∞, -1]), (iii) w(θ 2 ) = w(θ 2 ) for all θ 2 ∈ R.
The function w is called a conformal gluing function. The conformal property comes from (i) and (ii), and the gluing from (iii): w glues together the upper and lower parts of the hyperbola R. There are at least two ways for proving Lemma 6. First, it turns out that in the literature there exist expressions for conformal gluing functions for relatively simple curves: circles, ellipses, and also for hyperbolas, see [START_REF] Baccelli | Analysis of models reducible to a class of diffusion processes in the positive quarter plane[END_REF]Equation (4.6)]. Following this way, we could obtain the expression (13) for w given above and its different properties stated in Lemma 6. Instead, we would like to use the Riemann sphere given by

{(θ 1 , θ 2 ) ∈ C 2 : γ(θ 1 , θ 2 ) = 0}.
Indeed, as we shall see in Section 5, many (not to say all) technical aspects, in particular finding the conformal mapping, happen to be quite simpler on that surface. The proof of Lemma 6 is thus postponed to Section 5.

Under the symmetry condition (µ 1 = µ 2 , σ 11 = σ 22 and symmetric reflection vectors R 1 , R 2 in (1)), Foschini also obtained an expression for that conformal mapping, see [18, Figure 3].

3.4.

Proof of Theorem 1. We are now ready to prove our main result. Let us introduce the function

f (θ 2 ) = ψ 1 (θ 2 ) -ϕ 1 (0) w (0) w(θ 2 ) -w(0)
.

The key point is that f satisfies the assumptions of Lemma 5.

The first item (i) of Lemma 5 holds by construction: the only possible pole of f is at 0 (ψ 1 has a unique pole at 0, see Proposition 4, and since w is injective, see Lemma 6, the equation w(θ 2 ) -w(0) = 0 has only one solution, viz, θ 2 = 0). However, a series expansion shows that the residue of f at 0 is 0, in other words 0 is a removable singularity.

Point (ii) is also clear, since both ψ 1 and w satisfy the boundary condition [START_REF] Dai | Stationary distribution of a two-dimensional SRBM: geometric views and boundary measures[END_REF]. Furthermore, f vanishes at infinity, since on the one hand ψ 1 does, and on the other hand w goes to infinity at infinity. Using Lemma 5, we conclude that f = 0. It remains to show that ϕ 1 (0) = -µ 1 . For this it is enough to evaluate the functional equation ( 5), first at θ 2 = 0, and then at θ 1 = 0.

Diagonal covariance and dimension one.

Assuming that the covariance matrix Σ is diagonal, we have β = π 2 in (4), and T π β = T 2 is the second Chebyshev polynomial, given by T 2 (x) = 2x 2 -1. The formula [START_REF] Doetsch | Introduction to the theory and application of the Laplace transformation[END_REF] for w together with the expression (3) of θ ± 2 yields

w(θ 2 ) -w(0) = 8 µ 2 1 σ 11 σ 22 + µ 2 2 σ 2 22 θ 2 θ 2 + 2µ 2 σ 22 .
The formula of Theorem 1 then gives

ϕ 1 (θ 2 ) = -µ 1 w (0) w(θ 2 ) -w(0) θ 2 = - 2µ 1 µ 2 σ 22 1 θ 2 + 2µ 2 σ 22
, and finally, after some elementary computations and the use of functional equation ( 5), we obtain

ϕ(θ) = 2µ 1 /σ 11 (θ 1 + 2µ 1 /σ 11 ) 2µ 2 /σ 22 (θ 2 + 2µ 2 /σ 22 ) . ( 14
)
By inversion of the Laplace transform, we reach the conclusion that

ν 1 (x 2 ) = 2µ 1 µ 2 σ 22 e 2µ 2 σ 22 x 2 , ν 2 (x 1 ) = 2µ 1 µ 2 σ 11 e 2µ 1 σ 11 x 1 , π(x 1 , x 2 ) = 4µ 1 µ 2 σ 11 σ 22 e 2µ 1 σ 11 x 1 + 2µ 2 σ 22 x 2 .
Let us do two comments around [START_REF] Dubédat | Reflected planar Brownian motions, intertwining relations and crossing probabilities[END_REF]. Firstly, the product-form expression ( 14) is easily explained by the skew-symmetric condition (see [START_REF] Harrison | On the distribution of multidimensional reflected Brownian motion[END_REF]Equation (10.2)])

2Σ = R • diag(R) -1 • diag(Σ) + diag(Σ) • diag(R) -1 • R ,
which is obviously satisfied if R is the identity matrix and Σ is diagonal (above, diag(R) denotes the diagonal matrix with same diagonal entries as R and R is the transpose matrix of R). Secondly, our method works also to study the one-dimensional case. Let indeed

X t = X 0 + W t + µt + L 0 t
be a one-dimensional reflected Brownian motion with drift µ, where W is a Brownian motion of variance σ > 0 and L 0 t is the local time at 0. Applying Itô formula to (e θXt ) and taking expected value over the invariant measure π, we obtain, with ϕ(θ

) = E π [e θX 1 ], σ 2 θ + µ ϕ(θ) = -E π [L 0 1 ].
This is the functional equation in dimension one (compare with ( 5)). Evaluating this identity at 0 and remembering that ϕ(0) = 1 we find E π [L 0 1 ] = -µ, and then

ϕ(θ) = 2µ/σ θ + 2µ/σ .
This coincides with ( 14).

3.6. Statement of the BVP in the general case. We would like to close Section 3 by stating the BVP in the case of arbitrary reflections (non necessarily orthogonal). Let us define for

θ 2 ∈ R G(θ 2 ) = γ 1 γ 2 (Θ - 1 (θ 2 ), θ 2 ) γ 2 γ 1 (Θ - 1 (θ 2 ), θ 2 ).
Using the same line of arguments as in the proof of Proposition 4, we obtain the following result:

Proposition 7. The function ϕ 1 satisfies the following BVP: (i) ϕ 1 is meromorphic on G R with at most one pole p of order 1, and is bounded at infinity, (ii) ϕ 1 is continuous on G R \ {p} and

ϕ 1 (θ 2 ) = G(θ 2 )ϕ 1 (θ 2 ), ∀θ 2 ∈ R. ( 15
)
Due to the presence of the function G = 1 in (15), this BVP (still homogeneous with shift) is more complicated than the one encountered in Proposition 4, and cannot be solved thanks to an invariant lemma. Instead, the resolution is less combinatorial and far more technical, and the solution should be expressed in terms of Cauchy integrals and the conformal mapping w of Lemma 6. This will be achieved in a future work.

From the viewpoint of the general BVP of Proposition 7, we thus solved in this note a case where the variables θ 1 and θ 2 could be separated in the quantity (the right-hand side of the functional equation ( 5))

γ 1 (θ)ϕ 1 (θ 2 ) + γ 2 (θ)ϕ 2 (θ 1 ).
Let us say a few words on another case (the only other case, in fact) where the variables can also be separated, namely, when γ 1 = γ 2 . In this case the reflections are parallel, and the function ϕ 1 (instead of ψ 1 ) satisfies the BVP of Proposition 4, with no pole in G R . With Lemma 5, it has to be a constant. In other words, there is no invariant measure, which is in accordance with the fact that with parallel reflections the condition of ergodicity ( 2) is obviously not satisfied.

Singularity analysis

4.1. Statement of the result. In the literature, an important aspect of reflected Brownian motion in the quarter plane (and more generally in orthants) is the asymptotics of its stationary distribution, see indeed [START_REF] Dai | Reflecting Brownian motion in two dimensions: Exact asymptotics for the stationary distribution[END_REF][START_REF] Franceschi | Asymptotic expansion for the stationary distribution of a reflected Brownian motion in the quarter plane[END_REF] for the asymptotics of the interior measure, and [START_REF] Dai | Stationary distribution of a two-dimensional SRBM: geometric views and boundary measures[END_REF] for the boundary measures. With our Theorem 1 and a classical singularity analysis of Laplace transforms (our main reference for this is the book [START_REF] Doetsch | Introduction to the theory and application of the Laplace transformation[END_REF]), we can easily obtain such asymptotic expansions. Precisely, we identify three regimes, depending on the sign of Θ 1 (θ + 2 ). The fact that the latter quantity determines the asymptotics can be easily explained: as we shall see in Section 5, the pole

θ 2 ∈ C \ [θ +
2 , ∞) of ϕ 1 that is the closest to the origin is (provided it exists) a solution to Θ + 1 (θ 2 ) = 0. The relative locations of Θ + 1 (θ 2 ) and 0 will therefore decide which of that pole or of the algebraic singularity θ + 2 will be the first singularity of ϕ 1 . Define the constants

C 1 = -ϕ 1 (θ + 2 )2 π β sin( π β π) (w(θ + 2 ) -w(0)) θ + 2 -θ - 2 , C 2 = -µ 1 w (0)θ + 2 θ + 2 -θ - 2 2 π β sin( π β π) . (16)
Proposition 8. As x 2 → ∞, the asymptotics of ν 1 (x 2 ) is given by

ν 1 (x 2 ) =                x 2 -3 2 e -θ + 2 x 2 -C 1 2 √ π + o(1) if Θ 1 (θ + 2 ) < 0, x 2 -1 2 e -θ + 2 x 2 C 2 √ π + o(1) if Θ 1 (θ + 2 ) = 0, e 2 µ 2 σ 22 x 2 2µ 1 µ 2 σ 22 + o(1) if Θ 1 (θ + 2 ) > 0,
where the constants C 1 and C 2 are defined in [START_REF] Fayolle | Random Walks in the Quarter-Plane[END_REF].

Note that we obtain (for the particular case of orthogonal reflections) the asymptotic result of Dai and Miyazawa in [10, Theorem 6.1], with the additional information of the explicit expression for the constants.

Note that β = π/k for k 2 implies Θ 1 (θ + 2 ) > 0 and that k cannot be 1 due to the condition on Σ.

4.2.

Proof of Proposition 8. Proposition 8 is an easy consequence of the singularity analysis of ϕ 1 and of classical transfer theorems, as [START_REF] Doetsch | Introduction to the theory and application of the Laplace transformation[END_REF]Theorem 37.1]. Due to the expression of ϕ 1 in Theorem 1, there are two sources of singularities: the singularities of w and the points θ 2 where the denominator w(θ 2 ) -w(0) of ϕ 1 vanishes.

Let us first study the singularities of w. In fact, the function w can not only be analytically continued on G R as claimed in Lemma 6, but on the whole of the cut plane C \ [θ + 2 , ∞). Further, except if π/β ∈ Z, in which case w is a polynomial, w has an algebraic-type singularity at θ + 2 , given by the following result: Lemma 9. If π/β / ∈ Z, ϕ 1 has an algebraic-type singularity at θ + 2 , in the neighborhood of which it admits the expansion:

ϕ 1 (θ 2 ) =    ϕ 1 (θ + 2 ) + C 1 θ + 2 -θ 2 + O(θ 2 -θ + 2 ) if w(θ + 2 ) -w(0) = 0, C 2 √ θ + 2 -θ 2 + O(1) if w(θ + 2 ) -w(0) = 0,
where C 1 and C 2 are defined in [START_REF] Fayolle | Random Walks in the Quarter-Plane[END_REF].

Proof. Lemma 9 follows from doing an expansion of ϕ 1 at θ + 2 , using (the elementary) Lemma 10 below.

Lemma 10. If a is an integer, the generalized Chebyshev polynomial T a is the classical Chebyshev polynomial. If not, T a is not a polynomial, and admits an analytic extension on C \ (-∞, -1]. The point -1 is an algebraic-type singularity, and there is the expansion

T a (x) = cos(aπ) + a √ 2 sin(aπ) √ x + 1 + O(x + 1). (17)
Proof. The considerations on the algebraic nature of T a are clear, and [START_REF] Foddy | Analysis of Brownian motion with drift, confined to a quadrant by oblique reflection (diffusions, Riemann-Hilbert problem)[END_REF] comes from making an expansion of T a (x) = cos(a arccos(x)) in the neighborhood of -1.

We now turn to the singularities introduced by the denominator of ϕ 1 ; these singularities are poles.

Lemma 11. There are two cases concerning the poles of ϕ 1 in the cut plane

C \ [θ + 2 , ∞): • Θ 1 (θ + 2 ) 0, then ϕ 1 is analytic on C \ [θ + 2 , ∞), • Θ 1 (θ + 2 ) > 0, then ϕ 1 is meromorphic on C \ [θ + 2 , ∞)
, with poles on the segment [0, θ + 2 ] only. The closest pole to the origin is at -2 µ 2 σ 22 and has order one. The proof of Lemma 11 is postponed to Section 5, since the tools that we shall introduce there will simplify it. Then Proposition 8 is an easy consequence of Lemmas 9 and 11.

Riemann sphere and related facts

We have many objectives in this section, which all concern the set of zeros of the kernel

S = {(θ 1 , θ 2 ) ∈ C 2 : γ(θ 1 , θ 2 ) = 0}.
We add some complexity here, by using the framework of Riemann surfaces. In return, many technical aspects become more intrinsic and some key quantities admit nice and natural interpretations. We first (Section 5.1) study the structure of S, as a Riemann surface. Then we find a simple formula for the conformal mapping w (Section 5.2). We finally introduce the notion of group of the model (Section 5.3), similar to the notion of group of the walk in the discrete setting [START_REF] Malyšev | An analytic method in the theory of two-dimensional positive random walks[END_REF][START_REF] Fayolle | Random Walks in the Quarter-Plane[END_REF][START_REF] Bousquet-Mélou | Walks with small steps in the quarter plane[END_REF], and we prove that the algebraic nature of the solution is related to the finiteness of this group. 5.1. Uniformisation. Due to the degree of the kernel γ, the surface S has genus 0 and is a Riemann sphere C ∪ {∞}, see [START_REF] Franceschi | Asymptotic expansion for the stationary distribution of a reflected Brownian motion in the quarter plane[END_REF]. It thus admits a rational parametrization (or uniformisation) {(θ 1 (s), θ 2 (s)) : s ∈ S}, given by

         θ 1 (s) = θ + 1 + θ - 1 2 + θ + 1 -θ - 1 4 s + 1 s , θ 2 (s) = θ + 2 + θ - 2 2 + θ + 2 -θ - 2 4 s e iβ + e iβ s , (18)
with β as in [START_REF] Burdzy | Obliquely reflected Brownian motion in non-smooth planar domains[END_REF]. The equation γ(θ 1 (s), θ 2 (s)) = 0 is valid for any s ∈ S. We will often represent a point s ∈ S by the pair of coordinates (θ 1 (s), θ 2 (s)).

Any point θ 1 ∈ C has two images on S. More specifically, let s ∈ S be defined by θ 1 (s) = θ 1 . Then the two points are given by (θ 1 (s), θ 2 (s)) and (θ 1 (s), θ 2 (1/s)). They are always different, except if θ 1 is a branch point θ ± 1 . Similarly, any θ 2 ∈ C corresponds to two points on S, viz, (θ 1 (s), θ 2 (s)) and (θ 1 (e 2iβ /s), θ 2 (s)).

Any point or domain of C can be represented on the Riemann sphere, and there is the following correspondance (which is easily proved by using the formulas ( 18)), see Figure 3: • The branch points θ ± 1 and θ ± 2 are located at ±1 and ±e iβ ; • The point of coordinates (0, 0) is at s 0 , and ∞ is at 0 and ∞;

• The real points (θ 1 , θ 2 ) ⊂ R 2 such that γ(θ 1 , θ 2 ) = 0 form the unit circle;

• The curve R is sent to (-∞, 0), and if θ 2 ∈ R corresponds to s ∈ (-∞, 0), then θ 2 ∈ R corresponds to 1/s; the domain G R is the cone bounded by (0, -∞) and (0, -e iβ ∞).

F (s) = f (θ 2 (s)).
In particular, W (s) stands for the lifted conformal mapping w(θ 2 (s)). Reciprocally, for a function F (s) to define a univalent function f (θ 2 ), the condition F (s) = F (e 2iβ /s) needs to hold, due to the obvious identity θ 2 (s) = θ 2 (e 2iβ /s).

Proof of Lemma 6. We first translate on W the properties of Lemma 6 stated for w. First, W has to be analytic in the open cone G R (in S, this is the cone delimitated by (0, -∞) and (0, -e iβ ∞)) and continuous on the closed cone G R except at 0, see (i). Second, W has to be injective on G R , see (ii). Finally, the boundary condition (iii) has to be replaced by the pair of conditions

W (s) = W (1/s), ∀s ∈ R, W (s) = W (e 2iβ /s), ∀s ∈ (0, -e iβ ∞). (19)
We easily come up with the (at this point: conjectured) formula

W (s) = - 1 2 (-s) π β + (-s) -π β = - 1 2 e π β log(-s) + e -π β log(-s) , (20)
where we make use of the principal determination of the logarithm. (Since the conditions [START_REF] Franceschi | Asymptotic expansion for the stationary distribution of a reflected Brownian motion in the quarter plane[END_REF] are invariant under multiplication by a constant, we can choose the constant in front of the right-hand side of [START_REF] Franceschi | Analytic approach for reflected Brownian motion in the quadrant[END_REF]. We choose -1/2, so as to match the expression W (s) with w(θ 2 (s)), w being as in Theorem 1:

w(Θ 2 (θ - 1 )) = T π β (cos β) = cos π = -1 = W (-1).)
Let us briefly verify each property. First, the analyticity property (i) is clear from the properties of the logarithm. In order to show (ii), we first remark that W (s) = W (t) if and only if for some k ∈ Z, s = t ±1 e 2ikβ . (21) Since s and t both belong to the cone G R , we must have s = t, and therefore W is injective. Finally, (iii) is clear from the construction of W .

To obtain the expression of w in terms of the generalized Chebyshev polynomial, we use the fact that θ 2 (s) = θ 2 yields the formula

s = e iβ θ + 2 -θ - 2 2θ 2 -(θ + 2 + θ - 2 ) ± 2 (θ + 2 -θ 2 )(θ - 2 -θ 2 ) .
The proof is complete.

Thanks to the expression (20) of W derived above, we can now prove Lemma 11, concerning the (eventual) poles of ϕ 1 in the cut plane C \ [θ + 2 , ∞). Proof of Lemma 11. Using the expression of ϕ 1 obtained in Theorem 1, we conclude that the poles of ϕ 1 must be located at points θ 2 where w(θ 2 ) = w(0). Using the function W in (20), we reformulate the latter identity as W (s) = W (s 0 ), see Figure 3, and thanks to [START_REF] Harrison | Reflected Brownian motion in the quadrant: tail behavior of the stationary distribution[END_REF] we obtain s = (s 0 ) ±1 e 2ikβ . The closest pole to the origin s 0 is then 1/s 0 , which corresponds to the point

θ 2 = -2 µ 2 σ 22
such that Θ + 1 (θ 2 ) = 0, since θ 1 (s 0 ) = θ 1 (1/s 0 ) = 0. The sign of Θ 1 (θ + 2 ) then determines if 1/s 0 is before or after e iβ (which corresponds to θ + 2 ) on the unit circle, see Figure 3. The conditions given in terms of sign of Θ 1 (θ + 2 ) follows. Finally, let us notice that w(θ + 2 ) = w(0) if Θ 1 (θ + 2 ) = 0 and that w(θ + 2 ) = w(0) if Θ 1 (θ + 2 ) < 0. These transformations leave invariant θ 1 (s) and θ 2 (s), respectively, see [START_REF] Foschini | Equilibria for diffusion models of pairs of communicating computerssymmetric case[END_REF]. In particular, we have the following result (consequence of the Lemma 14), which connects the nature of the solution of the BVP to the finiteness of the group. Such a result holds for discrete walks, see [START_REF] Bousquet-Mélou | Walks with small steps in the quarter plane[END_REF][START_REF] Bernardi | Counting quadrant walks via Tutte's invariant method[END_REF].

Proposition 13. The solution ϕ 1 given in Theorem 1 is algebraic if and only if the group ζ, η is finite.

The proof of Proposition 13 builds on the following elementary result: Lemma 14. Let a 0. The generalized Chebyshev polynomial T a is

• rational if a ∈ Z,

• algebraic (and not polynomial) if a ∈ Q \ Z,

• not algebraic if a ∈ R \ Q.

However, even in the non-algebraic case, the Chebyshev polynomial T a always satisfies a linear differential equation with coefficients in R, since it can be written as the particular hypergeometric function 2 F 1 ([-a, a], [1/2], 1 -x), where (with (A) k = A(A + 1)

• • • (A + (k -1))) 2 F 1 ([A, B], [C], t) = ∞ k=0 (A) k (B) k (C) k t k n! .

Figure 1 .

 1 Figure 1. Drift µ and reflection vectors R 1 and R 2 in non-orthogonal and orthogonal cases

Figure 2 .

 2 Figure 2. The curve R in (8) is symmetric w.r.t. the horizontal axis, and G R is the domain in green

Figure 3 .it 5 . 2 .

 352 Figure 3. The set S viewed as the Riemann sphere, and important points and curves on it

5. 3 . 1 , θ 2 .

 312 Group of the model and nature of the solution. The notion of group of the model has been introduced by Malyshev[START_REF] Malyšev | An analytic method in the theory of two-dimensional positive random walks[END_REF] in the context of random walks in the quarter plane. It turns out to be an important characteristic of the model, in particular to decide whether generating functions or Laplace transforms are algebraic or D-finite functions, see[START_REF] Bousquet-Mélou | Walks with small steps in the quarter plane[END_REF].It can be introduced directly on the kernel γ: with the notation[START_REF] Dai | Nonnegativity of solutions to the basic adjoint relationship for some diffusion processes[END_REF], this is the group ζ, η generated by ζ and η, given byζ(θ 1 , θ 2 ) = θ 1 , c(θ 1 )By construction, the generators satisfy γ(ζ(θ 1 , θ 2 )) = γ(η(θ 1 , θ 2 )) = 0 as soon as γ(θ 1 , θ 2 ) = 0. In other words, there are (covering) automorphisms of the surface S. Sinceζ 2 = η 2 =1, the group ζ, η is a dihedral group, which is finite if and only if the element ζη (or ηζ) has finite order. With the above definition, it is not clear how to see if the group is finite, nor to see if this has any implication on the problem. In fact, we have: Proposition 12. The group ζ, η is finite if and only if π/β ∈ Q. The proof of Proposition 12 is simple, once the elements ζ and η have been reformulated on the sphere S:


Anna Guerra
email: anna.guerra3@unibo.it

Francesco Guidi
email: francesco.guidi@cea.fr

Antonio Clemente
email: antonio.clemente@cea.fr

Raffaele D'errico
email: raffaele.derrico@cea.fr

Davide Dardari
email: davide.dardari@unibo.it

Delay Spread Characterization of Millimeter-Wave Indoor Backscattering Channel

Keywords: Millimeter-waves channel measurements, massive antenna array, delay spread, personal radar, transmitarray

In this paper we present two indoor measurement campaigns performed using millimeter-waves massive arrays in a corridor and in an office room. In particular, we characterize the delay spread of the backscattered channel responses which can be exploited in future personal radars applications for indoor localization and mapping.

I. INTRODUCTION

The concept of smart personal radars has been recently proposed as a solution to map an environment exploiting the compactness and the beamforming potentialities of millimeterwave (mmW) massive arrays [START_REF] Guidi | Personal mobile radars with millimeter-wave massive arrays for indoor mapping[END_REF]. It is based on the idea of a massive array operating at mmW frequencies that, thanks to the narrow beam formed, is able to electronically scan the surrounding with an angular resolution comparable to that of laser-based radar (i.e. typically 6 • of half power beamwidth (HPBW) with 26 dBi of directivity) and with the additional advantage to operate even in non-line-of-sight (NLOS) and poor visibility conditions. From a technological point of view, the adoption of transmitarrays (TAs) for this kind of application has been investigated considering the impact of the antenna radiation properties on the map reconstruction accuracy [START_REF] Guerra | Application of transmitarray antennas for indoor mapping at millimeter-waves[END_REF]. The performance of the personal radar system depends on the ability of collecting the environmental backscattered response at the radar interrogation signal and secondly, on the exploitation of such measurements as an input for a mapping algorithm. As a consequence the characterization of the indoor backscattering mmW channel becomes an essential first step.

In this paper we aim at characterizing the delay spread of indoor backscattering mmW channel by exploiting measurements collected in the V-band adopting TAs in two typical office environments (i.e., an office room and a corridor) at CEA-Grenoble premises, reported in Fig. 1. At the best of authors' knowledge, there is not a literature available on this kind of channel, whereas many works deal with the one-way 60 GHz channel such as [START_REF] Smulders | Statistical characterization of 60-GHz indoor radio channels[END_REF], [START_REF] Xu | Spatial and temporal characteristics of 60-GHz indoor channels[END_REF].

II. MEASUREMENT CAMPAIGN

For the measurements equipment, we considered a set-up composed of a 4-ports Vector Network Analyzer (VNA), 2 mmW frequency converters operating in the frequency range A bistatic configuration has been considered with the TAs spaced apart of 0.16 m in order to mitigate the antenna coupling and to separate the transmitting and the receiving channels. Specifically, as we can see in Fig. 1, where the plan of the office and the corridor are reported, measurements have been collected in 10 and 12 different positions spaced of 0.405 m For each measurement position, the X-Y-Azimuth positioner permits to rotate the radar in the semi-plane from -90 • to 90 • with a step of 5 • (in accordance to the HPBW of the TA used) in order to emulate the beamforming operation, as the considered mmW TA is non-reconfigurable. Fig. 2 shows two photos taken during the measurements campaign.

The 1-bit phase compensation TAs used for measurements are fully described in [START_REF] Guerra | Application of transmitarray antennas for indoor mapping at millimeter-waves[END_REF], [START_REF] Kaouach | Wideband low-loss linear and circular polarization transmit-arrays in V-band[END_REF]. Note that it is composed of a focal source (a 10.2 dBi linearly-polarized pyramidal horn antenna working in the E-band) illuminating a planar array with 20×20 unit-cells composed of two patch antennas rotated in order to create a precise phase value to steer the beam [START_REF] Kaouach | Wideband low-loss linear and circular polarization transmit-arrays in V-band[END_REF]. For each position and each TA steering direction, the Smatrix was measured in the frequency domain from 55 to 70 GHz (i.e. the frequencies range corresponding to the TA 3 dB-bandwidth) with a step of 5 MHz [START_REF] Kaouach | Wideband low-loss linear and circular polarization transmit-arrays in V-band[END_REF].

III. DELAY SPREAD RESULTS

For the sake of simplicity, we suppose that, once fixed the steering direction, the angle of departure (AOD)-angleof-arrival (AOA) is the same for of all the rays due to the TA high directivity. Consequently, the antennas-embedded channel impulse response (CIR) for each radar position can be written as

CIR i (τ, θ) = Nsteer b=1 K l k=1 h i (τ -τ k , θ -θ b ) (1)
with i = 1, . . . , N pos , where N pos is the number of radar positions, N steer is the number of steering directions, K l is the number of rays, τ k and θ b are the delay and AOD-AOA of the kth ray, respectively. The power delay profile (PDP) for the ith radar position and the bth steering direction is defined as

PDP i (τ, θ b ) = K l k=1 |h b i (τ -τ k )| 2 . ( 2
)
Fig. 3 shows the reconstructed environments from the radar considering a complete angular scan. In both the polar plots, we have considered the radar in position p 1 and we have plotted the sum of the PDP i (τ, θ b ) value over the number of steering directions. Results have been normalized with respect to the maximum PDP value in the corridor (in Fig. 3-(left)) and in the office room (in Fig. 3-(right)). From the two polar plots, thanks to the arrays scanning, it is possible to identify the contour of the corridor and of the room walls.

The root mean square (RMS) delay spread is computed as

τ rms i (θ b )= K l k=1 |h b i (τ -τ k )| 2 (τ k -τ ) 2 K l k=1 |h b i (τ -τ k )| 2 (3)
where τ is the mean excess delay defined as

τ (θ b ) = K l k=1 |h b i (τ -τ k )| 2 τ k K l k=1 |h b i (τ -τ k )| 2 . ( 4
)
where

1 (x) = 1 if x ≥ 0, 0 otherwise.
As it is possible to observe from Fig. 4, the delay spread does not present large values. We ascribe this effect to the adoption of high directive antennas, which operate as a spatial filter, and to the particular environment configuration especially for what the corridor is concerned. Notably, such results are in agreement with [START_REF] Smulders | Statistical characterization of 60-GHz indoor radio channels[END_REF], where the authors observed a root delay spread of few nanoseconds, for the one-way channel, when high gain antennas are adopted. Moreover a small difference between the office and the corridor environment can be noticed.

IV. CONCLUSION

In this paper, the delay spread of two channel measurements campaigns has been derived. To achieve this goal, we first presented the measurements conducted in two indoor environments, and successively we characterized the RMS delay spread. Due to the high gain antennas considered, we found average values comprised between 3 and 7 ns. Further measurements will be conducted in different scenarios, to properly describe all the channel parameters.

p 1 p 2 p 3 p 4 p 5 p 6 p 7 p 8 p 9 p 10 (p 1 p 2 p 3 p 4 p 5 p 6 p 7 p 8 Fig. 1 .Fig. 2 .

 510812 Fig. 1. Plan of the indoor office environment.

Fig. 3 .

 3 Fig. 3. Radar map results for normalized PDP from radar position p 1 and considering all the steering directions in the corridor (left) and office (right). The maximum radius is 8 m.

Fig. 4 .

 4 Fig. 4. Left: Obtained RMS delay spread CDFs for the corridor and the office room. Right: average delay spread over steering directions for each position.

Fig. 4

 4 Fig. 4 shows the RMS delay spread cumulative distribution functions (CDFs) evaluated as

ACKNOWLEDGMENTS

This research was supported in part by the IF-EF Marie-Curie project MAPS (Grant 659067).


Enhancement of the low frequency performance of thin, film-faced layers of foam by surface segmentation

Stuart Bolton 1 , Benoit Nennig 2 and Nicolas Dauchez bolton@purdue.edu, benoit.nennig@supmeca.fr, nicolas.dauchez@utc.fr Objectives: Understand the absorption of film-faced foam with circumferential air gap New peak, at low frequency with viscous dissipation due to Diaphragm-Helmholtz resonator effect [START_REF] Schultz | Effect of solid phase properties on the acoustical performance of poroelastic materials[END_REF] • Very low motion of the membrane around 400 Hz • Membrane increases the structural dissipation • Gap enhances also the absorption at higher frequencies when compared with airtight membrane • Independently tunable membrane and air gap effects An approximation of the mass-spring resonance frequency, can be obtained with Ritz method

f mb ∼ 1 2π K ef f /h (ρ 11 + ρ 22 )h/3 + λ m . ( 1
)
where the effective compressibility of the material K ef f = E + K e is the sum of the stiffness of the fluid phase K e and the stiffness of the solid phase E.

Here, we get f mb = 930 Hz.

Simplified Diaphragm model

Assumptions :

• Rigid frame porous material • Diaphragm (motionless membrane)

• Plane wave regime porous h

The inductance L of a slit of height b in a rectangular duct of height a and of cross section a × d is given by [2, p. 487]

Ld ρ = - 2 π ln sin πb 2a , if b a. (2)
The impedance of the porous layer is Z p = -iρ p c p cot k p h and the impedance of the diaphragm is Z d = -iωLS. Assuming the best absorption arises when the imaginary part of Z = Z p + Z d vanishes, we get

f d ∼ 1 2π K e V L , (3)
where V = Sh porous material layer volume. This stiffness is the same as for the Helmholtz resonator. Putting ρ ≈ ρ 22 yields f d ≈ 600 Hz with is closed to the frequency of the 1st experimental peak.

Numerical simulations

Assumptions :

• FEM simulations with Quadratic elements [START_REF] Atalla | [END_REF] • Dissipated energy computed with [4] •

Normal incidence

Other incidences

Prospects :

• Combination with double porosity material [5] • Combination with Cuboid [6]

  Melamine foam : E = 50(1-0.076i) kPa, ν = 0.4, σ = 9000 Nm-4s, φ = 0.995, α ∞ = 1.01, Λ = 64µm, Λ = 143µm. Sample : radius r = 50 mm, height h = 25 mm, air gap b = 1 mm. Membrane density λ m = 76 g/m 2 . Measuremant B&K impedance tube Simplified airtight membrane model Assumptions : • membrane of surface mass λ m with no stiffness • No air leak • 1D Biot model

Free

  Clamped boundary condition, DtN map for radiation Effect of the air gap, ranging from 0.5, 1, 2, 5, 10 mm and radius r = 50 mm Periodic patch and oblique incidence effect : L x = 50 mm, L z = 50 mm, b = 1 mm, h = 25 mm

  3 Université de Technologie de Compiègne, Laboratoire Roberval UMR 6253, BP 20529, 60205 Compiègne cedex, France.

1 Ray W. Herrick Laboratories, School of Mechanical Engineering, 177 S. Russell Street, Purdue University, West Lafayette IN 47907-2099, USA 2 LISMMA EA2336, SUPMECA, 3 Rue Fernand Hainaut, 93407 Saint-Ouen Cedex, France. 3