ARCHIVES BUDEIRI LIBRARY FAMILY LIBRARIES FRANCISCAN PRINTING PRESS KHALIDI LIBRARY

by Vincent Lemire, Leyla Dakhli, Anouk Cohen This post is a short abstract of the report written after a mission realized by Leyla Dakhli, Vincent Lemire and Anouk Cohen in Jerusalem for a week, from 8 to 14 June 2015. The goal of this mission was only to launch this fieldwork and to open future tracks of works. The focus was to investigate the circulation of printed documents in Jerusalem 1840-1940 and their position, while paying crucial attention to the different languages of their production and distribution. This approach aimed to have a better view of the possible and/or impossible connected practices of knowledge. To this end, the mission focused on two kinds of places of production and circulation of knowledge:

-Family libraries: the goal of this investigation was to understand what kind of books made up the libraries of scholars from the period in question with a particular interest in practical knowledge of languages (by paying attention to dictionaries for example).

-Printing presses: this investigation aims to study the type of books produced between 1840 and 1940 and the languages in which they were produced (from archival documents and letters cases left in the old printing workshops for example). This mission sought to identify significant lieux de savoir (i-e places of knowledge) and documentation centers (archival or otherwise) and attempt to capture gestures and operations, instruments and supports, expertise and practices, interaction modes and validation procedures, forms of registration and dynamic transmission that contribute, in their articulation, in order to

Family libraries

The Khalidi library

The Khalidi library is arguably the best-preserved privately owned manuscript library in the old city.

The Khalidi library houses 1200 manuscripts, most of them microfilmed and stored in boxes in a temperature-controlled room in the attic of the old madrassa. We consulted the catalog of the collections of the library published in 2006 by Khader Salameh and Nazim Al-Ju'aba and prefaced by Walid Khalidi in Arabic . Then, we consulted Ruhi al-Khalidi's documents stored in an acidresistant gray box . Theses documents mainly consist of vocabulary books (organized into lists) and notes taken from various books including those of Ibn Khaldun. These notebooks may belong to his son (John al-Khalidi). Finally, we visited the room where 1900 books printed in Latin characters are stored, including many dictionaries, almanacs and guides that occupy an entire section of the bookcase. Instruments of knowledge such as magazines, dictionaries, grammar books, encyclopedias, manuals, guides, almanacs, atlases, official publications (such as court judgments of the mandate that would allow us to understand how they developed their research), as well as printers and publishers' catalogs could be a very interesting material for future research.

From the Excel table of the Latin characters printed catalog (available on Khalidi library's website ), it would be possible to define several sets of data to grasp some publication policies as well as identify the major production centers. Moreover, this statistical study would allow to identify the number of languages and to create clouds of information. Finally, crosscheck this different sets of information would help to understand in which languages the different users of the library were reading and what were the main themes they were interested in. The type of crosstabs that would be interesting to develop are the followings: owners / dates, owners / themes / languages, owners / places of publication.

The Budeiri library

The Budeiri library In this place, located next to the Haram esh-Sherif, we may distinguish three blocks of documents. 1) The old manuscripts: 1,166 manuscripts make up the collection. Today, 90% of them are digitized. The manuscripts are safely kept in a small room inside iron storages. It is worth noting that this room is located near a zâwiya where the ancestor allowed students to study. 2) The printed books: there are 158 printed books dating from the end of Ottoman period and Mandate period, mainly written in Ottoman and Arabic. All of them have been inventoried, receiving a singular quotation. Compared to the Khalidi library, Budeiri library's collection has fewer books in foreign (specially European) languages.

3) The archives are well preserved under plastic cover and stored in about twenty files. This is an original documentation in regard to different fields, especially the history of the Jerusalem Jordanian municipality (1949)(1950)(1951)(1952)(1953)(1954)(1955)(1956)(1957)(1958)(1959)(1960)(1961)(1962)(1963)(1964)(1965)(1966)(1967). There are also various policy papers, poetry and personal documents such as letters. Other documents dated 1950's consist in names of streets and locations. There are also maps, stamped collected letters and business cards. a few bound copies were printed. Also, the same book could be bound and/or not bound. 9. December 1917 -April 1919: "Registro dei lavori pel governo". This register, which looks like an ordinary notebook, is of a great importance: it lists all the printed works produced by the new British power in 1918-1919. There are pell-mell passes, labels, administrative stationery, forms, posters, legal notices, a music program, blank passports, maps of prisoners, telegram forms, Post records, daily reports from hospitals (Hospital Report Daily), records of wanted persons, documents related to the police force, building permits, conversion tables ("money rate"), Passes to enter City Walls, contract forms, Medical Ordinance, Jerusalem School of Music. It also contains valuable information on the price of commodities. These data are particularly important because they provide information on the implementation modalities of the new British administration in 1918-1920. Indeed, the Tipografia custody, being the most important of the city, did functions at that time as a kind of official printing press; and thereby, for historians, as a main way of getting information). 10. 1857 June 15 -1879 December 31: "Giornale delle prestazioni della tipografia". By 1857, this register describes in great detail the work done by the workers during the months worked. Beginning ten years after the founding of the Tipografia, this register may be the first of its kind in a context of reorganization of the printing press. 11. 1847 -1873: "Opere stampate nella Tipografia". This register is composed of two parts: on the one hand, it shows all the documents printed from January 1847 to March 1873. 607 documents had been printed over 26 years. Besides books, there are also travel tickets (biglietti di viaggio) for "il Municipio". A mention "biglietto di Transito" appears in 1871. On the other hand, are listed the outdoor expeditions to convents. Each page is devoted to one book [mostly catechisms, mass books, or Italian grammar books, etc.]. The main recipient cities are: Jerusalem, Cairo, Damascus, Aleppo, Saida, Tripoli, Nicosia, Fajum [Fayoum] Tiberias, Caiffa [Haïffa]… Thus, the Tipografia of Jerusalem operated as the as an important production center throughout the Levant.

12. 1890 January 2 -1909 January 11: "Stampi della tipografia". This is the register of orders for commands or manufacturing of special characters. In one column, there appears the drawing of the new engraved character. This register is particularly interesting because it allows us to date the production or purchase of letter cases in different alphabets (Russian, Arabic, etc.). By leafing through the register, we learn which characters (and languages) were mainly produced at each period.

Conclusion

This first mission to the family libraries and printing presses in Jerusalem consisted in laying the groundwork for case studies that could be conducted in the coming years. Basically, the goal of this fieldwork would be to better understand how knowledge was shared and/or divided and how texts were circulated (or not) in networks through Jerusalem 1840-1940. Of course there are a lot of other places to include in the future dashboard of this fieldwork (Hebraic, Armenian, Greek, Russian, German, British Printing Press; old Hebraic libraries in West Jerusalem). It should be noted here that this mission not only helped to locate existing archives but also, through chance encounters with people, relationships and spontaneity of the search, gave birth to archives. With regards to future studies that could better seize on modalities for constituting public knowledge in Jerusalem during the period in question, two types of questions could be considered, focusing on family libraries or on the printing presses. For example, a case study of Khalidi or Budeiri library could focus on how knowledge materializes in a particular architectural feature or furniture, in a group of books or specific objects. Indeed, the architectural and physical environment of the library could be placed at the heart of the investigation in order to study its essential role in the representation or self-representation of scholarly activity or of a certain idea of knowledge itself. Libraries give to see a certain idea of knowledge, specialist or generalist, local or universal, and how they materialize a representation of culture within its limits and in its ambitions. An investigation on Printing Press and printers could inform the history of scholarly work that is actually determined by the development of techniques and instruments. Sites that attract considerable attention -for authors, readers, and technicians -printing presses are also diffusion locations, where knowledge is reticulated or radiated. At the crossroads of flow, they are devoted to the establishment of texts themselves and their reproduction. Thus, the study of printing would open the way to a dynamic approach to knowledge: in terms of spheres of influence, centrifugal and centripetal forces, but also in terms of circulation and translation. In this way, one would better understand the central nodes of a network in relation to the others, located in more distant peripheries. A huge challenge, to be continued!

1 FAMILY

 1 LIBRARIES AND PRINTING PRESSES IN JERUSALEM (1840-1940): PRODUCTION, CIRCULATION AND RECEPTION OF MULTILINGUAL DOCUMENTS 27/11/2015 | OPEN JERUSALEM understand the concrete practices of shared and/or divided knowledge in the Holy city in the period 1840-1940.

Picture 1 . 3 Picture 2 .

 132 Ruhi / or John al-Khalidi's vocabulary notebook2 An example of dictionary in the Khalidi Library

Picture 3 .

 3 Manuscripts kept inside iron storages in the Budeiri LibraryPicture 4. One of the manuscripts from the Budeiri Library

Picture 8 .

 8 Letter cases in Arabic (Beitphage)Picture 9. Special stamps engraved in zinc(Beitphage)

  Picture 11. 1847 -1873: "Opere stampate nella Tipografia" (Archivi della Custodia) Picture 12. Aprile 1918 -Luglio 1919: "Registro dei lavori pel governo" (Archivi della Custodia)

Picture 5. An example of map stored in the archives from the Budeiri Library

Printing presses

Franciscan printing press (FPP) The Franciscan printing press (FPP) was founded in 1847, the first book printed this year was a catechism in Arabic. The old printing machines and letter cases are well preserved. With the exception of some of them that were displaced recently in Beitphage (the new site for FPP Franciscan Printing Press), three machines remained within the walls of Saint Saviour's Convent. One of them stands in the great hall of the Custody. Thanks to Fr. Sergey Loktionov, archivist in chief at the Custody, the historical archives of the FPP are well preserved and perfectly described (under the category "Tipografia"). For sure, the Franciscan printing press was the largest printing press in Jerusalem during the period in question, especially (but not only) for books printed in Latin characters. Thereby, it's a sort of laboratory / observatory through which it's possible to see the way in which the city did functioned on a social and cultural level. Indeed, we found a lot of mentions of business cards, printed invitations to concerts, receptions, etc. In the archives there are also precise indications on the printing process, including of activities of production and its different stages. Here are some examples of the consulted files:


Nesrine Houhat

Sébastien Ménigot

Tarek Boutkedjirt

Redouane Drai

Jean Marc Girault

Jean Marc

Sébastien Ménigot Eseo

Optimal Stochastic Excitation for Linear Flaw Detection in a Solid Material

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

Optimal Stochastic Excitation for Linear Flaw Detection in a Solid Material

Nesrine Houhat (1,4) , Sébastien Ménigot (2,3) , Tarek Boutkedjirt (4) , Redouane Drai (1) and Jean-Marc Girault (2,3) n.houhat@crti.dz, sebastien.menigot@eseo.fr, tboutkedjirt@usthb,dz, r.drai@crti.dz, jean-marc.girault@eseo.fr

(1) Research Center on Industrial Technologies CRTI (Ex CSC), P. O. Box 64, Chéraga, 16014, Algiers, Algeria, (2) ESEO Group, Angers, France. (3) LUNAM Université , LAUM, CNRS UMR 6613, Université du Maine, Le Mans, France.

(4) Physics of Ultrasound Research Team, Faculty of Physics, USTHB, Agiers, Algeria

Context

The present study constitutes a first step in the validation of the stochastic optimization approach of the excitation waveform for linear systems since our ultimate goal remains its application on nonlinear systems. The optimal excitation wave enables us the best discrimination between two media: a reference one without flaw and a medium with a linear flaw, without a priori informations about the medium and the transducer.

• Conclusion and Perspectives

In the present study, the best excitation wave which enabled to maximize the linear flaw detection was investigated through two process types, parametric and shape optimization. An experimental closed loop system was set up to find automatically the suboptimal/ optimal command. The cost function considered was an Euclidean distance between two media: a reference one and a medium with a linear flaw. The main outcome was the performance similarity between the parametric and the stochastic approaches. Nevertheless, the stochastic approach didn't require any a priori knowledge nor initialization values. For a linear flaw type, the optimal stochastic command tends to the time reversal solution which is a sinusoidal pulse at a fixed frequency. The stochastic approach may be more efficient for nonlinear systems, even for cases where the cost function is no longer convex, for which the use of simple parametric optimization is compromised.

• Results

• Closed Loop Optimization System

• Experimental Setup

Algorithms

• Genetic Algorithm 𝒇 𝒌 is the transmitted frequency at the iteration k. The coefficient 𝝁 𝒌 governs the speed of convergence and 𝛁𝒅(𝒇 𝒌 ) is the Euclidean distance gradient at the iteration k for the transmitted frequency 𝒇 𝒌 and is given by the expression:

𝛻𝑑(𝑓 𝑘 ) = 𝜕𝑑 𝜕𝑓 ≈ 𝑑 𝑘-1 -𝑑 𝑘 𝑓 𝑘-1 -𝑓 𝑘
Where ∆𝒇 =1MHz, for k={1,2,3}, the first three frequencies must be given initially.

𝝁 𝒌 governs the speed of convergence as follows:

Generate initial population

  𝐑𝐞𝐜𝐜𝐮𝐫𝐞𝐧𝐜𝐞 𝐫𝐞𝐥𝐚𝐭𝐢𝐨𝐧: 𝑓 𝑘+1 = 𝑓 𝑘 + 𝜇 𝑘 (∇𝑑(𝑓 𝑘 ))

Find

  12 stochastic signals are randomly chosen from a continuous uniform distribution, each signal is composed from 40 samples with a duration of 1µs.


Jean-Pierre Delmas

Abdelkader Oukaci

Pascal Chevalier

ASYMPTOTIC DISTRIBUTION OF GLR FOR IMPROPRIETY OF COMPLEX SIGNALS

In this paper, we consider the problem of testing impropriety (i.e., second-order noncircularity) of a complex-valued random variable (RV) based on the generalized likelihood ratio test (GLRT) derived under Gaussian distributions. Asymptotic (w.r.t. the data length) distributions of the GLR are given under the hypothesis that data are proper or improper, and under the true, not necessarily Gaussian distribution of the data. This enables us to derive in particular the receiver operating characteristics (ROC) of this test, an issue previously overlooked. Finally illustrative examples are presented in order to strengthen the obtained theoretical results.

Index Terms-Generalized likelihood ratio (GLR), receiver operating characteristics (ROC), asymptotic distribution of noncircularity coefficients estimate, improper, secondorder noncircular complex random variables.

INTRODUCTION

Many papers (see, e.g., [START_REF] Picinbono | Widely Linear Estimation with Complex Data[END_REF][START_REF] Schreier | Detection and estimation of improper complex random signals[END_REF][START_REF] Delmas | Asymptotically minimum variance second-order estimation for non-circular signals with application to DOA estimation[END_REF]) show that significant performance gains can be achieved by second-order algorithms exploiting the statistical information contained in C x = E(xx T ) (where x is the observation), provided it is non-zero in addition to that contained in the standard covariance matrix R x = E(xx H ), but with an additional complexity. Moreover, some such algorithms (see e.g., [START_REF] Abeida | MUSIC-like estimation of direction of arrival for non-circular sources[END_REF]) adapted for improper or second-order noncircular signals, i.e., with non-zero matrices C x , fail or suffer of too slow convergence when they are used for proper or second-order circular signals.

Thus, the question that arises as to how we can classify a signal as proper or improper. This problem is a binary hypothesis test H 0 : C x = O versus H 1 : C x = O. In practice, the parameters R x and C x are often unknown, thus only the GLR detector can be employed. Based on independent and identically distributed Gaussian samples (x k ) k=1,...,K , this detector was introduced independently by Ollila and Koivunen [START_REF] Ollila | Generalized complex elliptical distributions[END_REF] and Schreier, Scharf and Hanssen [START_REF] Schreier | A generalized likelihood ratio test for impropriety of complex signals[END_REF], its performance was illustrated by a Monte Carlo simulation only. Formulating this testing problem in terms of real-valued Gaussian random vectors, Walden and Rubin-Delanchy [START_REF] Walden | On testing for impropriety of complex-valued Gaussian vectors[END_REF] derived recently this GLRT as well. In addition, they presented a theoretic analysis of the null asymptotic distribution of the GLR and several numerical analysis based on Monte Carlo simulations for the alternative distribution under the Gaussian distribution of the signals.

The aim of this paper is to complement the theoretical asymptotical analysis of [START_REF] Walden | On testing for impropriety of complex-valued Gaussian vectors[END_REF] by considering the null and alternative asymptotic distribution of the GLR derived under the Gaussian distribution, but used in practice under arbitrary not necessarily Gaussian distributions. This paper is organized as follows. In Section 2, the GLRT is recalled for the convenience of the reader. Using the asymptotic distributions of the noncircularity coefficients given in [START_REF] Delmas | Asymptotic distribution of circularity coefficients estimate of complex random variables[END_REF], the asymptotic distribution of the GLR under the hypothesis that data are proper or improper, and under the true, not necessarily Gaussian distribution of the data are given in the scalar case, and then extended to the multidimensional case in Section 3. Due to the simplicity of the asymptotic distribution of the GLR, an interpretable closed-form expression of the ROC is given in the scalar case. Finally some illustrative examples are given in Section 4.

GENERALIZED LIKELIHOOD RATIO DECISION RULE

Assume that (x k ) k=1,...K ∈ C N is a realization of K independent identically zero-mean complex Gaussian distributed RVs of covariance matrices R x = E(xx H ) and C x = E(xx T ). Consider the following binary composite hypothesis testing problem:

H 0 : C x = O, R x H 1 : C x = O, R x .
The GLR replaces the unknown parameters R x and C x by their maximum likelihood (ML) estimates in the likelihood ratio. It is straightforward to derive its expression which is given by [START_REF] Ollila | Generalized complex elliptical distributions[END_REF] [6]

L(x, K) def = p (x k ) k=1,...K ; R x , C x , H 1 p (x k ) k=1,...K ; R x , O, H 0 = det( R x ) K det( R x) K/2 (1) with R x def = 1 K K k=1 x k x H k and R x def = 1 K K k=1 xk xH k where, xk def = [x T k , x H k ] T . The GLRT decides H 1 if L(x, K) > λ (2)
and otherwise H 0 . Note that no uniformly most powerful (UMP) linearly 1 invariant test for impropriety exists for N > 1 [START_REF] Walden | On testing for impropriety of complex-valued Gaussian vectors[END_REF]. In the scalar case N = 1, the GLRT is the UMP linearly invariant test [START_REF] Walden | On testing for impropriety of complex-valued Gaussian vectors[END_REF]. It becomes especially simple

L(x, K) = (1 -γ 2 x ) -K/2 with γ x = | 1 K K k=1 x 2 k |/ 1 K K k=1 |x k | 2 is the ML estimates [8] of the noncircularity coefficient γ x def = |E(x 2 k )|/E|x k | 2 and the GLRT decides H 1 if γ x > λ , (3)
which is quite intuitive.

ASYMPTOTIC DISTRIBUTION OF GLR

Throughout in the sequel, we consider this GLRT is used for independent identically zero-mean non necessarily Gaussian distributed data (x k ) k=1,...K . For nonGaussian data, the decision rule ( 2) is no longer a GLRT. However, it is simple to implement and generally provides good performance in practice (see e.g., for the detection of a known signal corrupted by noncircular interference [START_REF] Chevalier | GLRT-Based array receivers to detect a known signal corrupted by noncircular interferences[END_REF]).

Scalar complex random variable

In the scalar-valued case, the following result is proved in the Appendix.

Result 1 Under the respective hypothesis H 0 and H 1 , the following convergences in distribution hold when

K → ∞ √ K γ x L → R(1 + κ x 2 ) (4) √ K( γ x -γ x ) L → N (0, σ 2 γ ) if γ x < 1, (5)
where R(1 + κ x 2 ) and N (0, σ 2 γ ) denote the Rayleigh distribution with parameter 1 + κ x 2 and the zero-mean Gaussian distribution with variance σ 2 γ respectively, with

σ 2 γ = 1-2γ 2 x +γ 4 x +γ 2 x κ x + κ x 2 + γ 2 x (κ x ) 2 -2γ 2 x (κ x ) (6) if σ 2 γ = 0,
where under H 0 and H 1 , κ x , κ x and κ x are the normalized-like cumulants

Cum(x k ,x k ,x * k ,x * k ) (E(|x k | 2 )) 2 , Cum(x k ,x k ,x k ,x k ) (E(x 2 k )) 2 and Cum(x k ,x k ,x k ,x * k ) E(|x k | 2 )E(x 2 k )
respectively which are invariant to any rotation of the distribution of x k .

Remark 1: This theoretical result means in practice that for K 1, the estimate γ x is approximately Rayleigh (of parameter

1+ κx 2 K ) or Gaussian N (γ x , σ 2 γ K
) distributed under H 0 and H 1 , respectively. We note that the domain of validity of 1 Linear w.r.t. C, which thus includes rotation and scaling, but not widely linear operation.

this approximation depends on γ x and σ 2 γ through the rough relation γ x -2σγ √ K > 0. Concerning the practical use of this result, i.e., for P D = 1 and P FA = 0, we note that the distribution of γ x under H 0 and H 1 must overlap. This is roughly

achieved for γ x - 2σγ √ K < 4 √ 1+κx/2 √ K . Remark 2:
In the particular case of rectilinear RVs for which γ x = 1, we have x k = r k e iφ with r k is a real-valued RV and with φ fixed. Consequently, the noncircularity coefficient γ x is perfectly estimated, i.e., γ x = 1. In this case, the detection problem is singular and for a threshold λ close to 1, the probability of detection P D and false alarm P FA are equal to 1 and 0 respectively. Remark 3: It is possible that σ 2 γ = 0 with γ x < 1 (such a situation is given in [START_REF] Delmas | Asymptotic distribution of circularity coefficients estimate of complex random variables[END_REF]). In this case, the sequence K( γ x -γ x ) converges in distribution [10, Th.B, p.124] to an Hermitian form z H Ωz, where z is a two dimensional zero-mean complex Gaussian RV, whose distribution is defined by the right hand side of [START_REF] Serfling | Approximation Theorems of Mathematical Statistics[END_REF], and where our first order analysis does not allow one to specify the matrix Ω. Remark 4: We note that for γ x close to zero and K 1,

2 ln L(x, K) = -K ln(1 -γ 2 x ) ≈ K γ 2
x . Furthermore for x k Gaussian distributed, κ x = 0. In these cases (4) gives

2 ln L(x, K) L → χ 2 2 , under H 0 .
This convergence in distribution is consistent with the constant false alarm rate (CFAR) detector where the number 2 of degree of freedom of the chi-square distribution is equal to the number of real-valued components of c x def = E(x 2 k ), given by the Wilk's theorem [13, p.132]. Remark 5: For Gaussian distributed data, κ x = κ x = κ x = 0 and the variance σ 2 γ of the asymptotic distribution of γ x under the hypothesis H 1 equal to 1-2γ 2

x +γ 4

x , is a decreasing function of γ x . Consequently for a fixed P FA , i.e., for fixed threshold λ , P D is an increasing function of γ x that does not depend on the power of x k , which is very intuitive.

For an arbitrary, not necessarily Gaussian distribution of x k , Result 1 allows us to derive

P FA = P ( γ x > λ /H 0 ) ≈ Q χ 2 2 Kλ 2 1 + κ x 2 P D = P ( γ x > λ /H 1 ) ≈ Q N √ K(λ -γ x ) σ γ ,
where Q χ 2 2 (.) and Q N (.) denote the complementary cumulative distribution functions of the chi-square distribution with 2 degrees of freedom and the zero-mean, unit-variance Gaussian distribution respectively, and where σ γ is given by ( 6). This gives the following closed form expression of the ROC of GLR detector [START_REF] Delmas | Asymptotically minimum variance second-order estimation for non-circular signals with application to DOA estimation[END_REF].

P D ≈ Q N   (1 + κ x 2 )Q -1 χ 2 2 (P FA ) - √ Kγ x σ γ   . ( 7
)
We clearly see from this expression that for fixed P FA , P D is an increasing function of the data length K and for Gaussian distributed data, an increasing function of the noncircularity coefficient γ x .

Multidimensional complex random variable

In the multidimensional case (N > 1), no simple interpretable expression of P FA and P D are available, but the following result is proved in the Appendix.

Result 2 Under hypothesis H 0 and H 1 , the following convergences in distribution hold when K → ∞ for the decision statistic (x, K)

def = [L(x, K)] -2/K √ K(l(x, K) -i ) L → N (0, σ 2 i ), under H i i = 0, 1, (8)
where 0 = 1 and the expression of 1 < 1 is given in the Appendix.

Remark 6: Note that Wilk's theorem [13, p.132] gives

2 ln L(x, K) L → χ 2 N (N +1) under H 0 ,
where the number of degrees of freedom of the chi-squared distributions is equal to N (N + 1) of real-valued independent parameters in C x . Furthermore under H 1 , in the case of C x is "close" to O (see a more formal definition in [START_REF] Stuart | Advanced Theory of Statistics[END_REF]Ch.23.7]), the analysis of [11, Sec.II] is valid and gives the following approximation of distribution when K 1 :

2 ln L(x, K) a ∼ χ 2 N (N +1) (µ) under H 1 ,
where χ 2 N (N +1) (µ) represents a noncentral chi-squared distribution with N (N + 1) degrees of freedom and noncentral parameter µ. This parameter is a measure of the discrimination between the two hypothesis, whose general expression is given by [11, exp.(4)].

ILLUSTRATIVE EXAMPLES

We consider the following MIMO channel (extension of the example given in [START_REF] Schreier | A generalized likelihood ratio test for impropriety of complex signals[END_REF]) that transmits Q independent equiprobable BPSK symbols a q,k ∈ {-1, +1} over an additive noise channel, that also rotates independently the phase of the transmitted symbols a q,k by φ q,k

x k = Q q=1 σ q a q,k e iφ q,k s q + n k , ( 9
)
where σ q and s q are Q unknown amplitudes and steering vectors with unit first component, and where the components of n k are independent zero-mean complex circular Gaussian RV of variance σ 2 n . Under H 0 and H 1 , we assume that the phase (φ q,k ) k=1,..,K,q=1,..,Q are independent and respectively uniformly distributed on [0, 2π] or Gaussian distributed with mean φ q 0 and variance σ 2 φ q . So we are interested in classifying this channel as either incoherent or partially coherent, which is a binary composite hypothesis testing problem. We easily deduce that

R x = Q q=1 σ 2 q s q s H q + σ 2 n I Q , C x = O, resp., Q q=1 σ 2 q e 2iφ q 0 e -2σ 2 φq s q s T q under H 0 [resp. H 1 ]. For Q = 1, κ x = - 1 (1+ρ -1 x ) 2 under H 0 and γ x = e -2σ 2 φ 1 1+ρ -1 x , κ x = -1+e -4σ 2 φ 1 (1+ρ -1 x ) 2 κ x = e -4σ 2 φ 1 -3 and κ x = -2 1+ρ -1 x under H 1 , with a signal to noise ratio (SNR) of ρ x def = σ 2 1 /σ 2 n .
This model is compared to the Gaussian model obtained when φ q,k does not depend on k and a q,k are independent zero-mean complex circular or real-valued Gaussian RVs under H 0 and H 1 respectively. Fig. 1 shows the northwest corner of the ROC curve for the GLRT detector for Q = 1, K = 100 and ρ x = 0.63(-2dB) for BPSK model with a coherent channel (i.e., σ φ 1 = 0) and Gaussian model, and thus associated with the same value of γ x = 0.387. We note that the ROC curve is sensitive to the distribution of the data x k and that the empirical ROC fits the asymptotic theoretical ROC for the relatively small data length K = 100.

Fig. 2 shows the detection performance P D for different fixed P FA for model [START_REF] Chevalier | GLRT-Based array receivers to detect a known signal corrupted by noncircular interferences[END_REF] We see that the P D for fixed P FA is very sensitive to the coherence of the channel. When σ φ1 increases for a fixed SNR, the noncircularity coefficient γ x decreases and detection worsens.

CONCLUSION

In this paper, we have given the asymptotic distribution of the GLR for impropriety of complex signals, derived under the Gaussian distribution, but used under independent and identically arbitrary non necessarily Gaussian distributions of the data. Extension of this work to independent non identically distributed data will enable us to consider in a future contribution, more practical situations such that BPSK signals with nonnull carrier residues.

APPENDIX

Proof of Result 1 Under H 1 , ( 5) is directly issued form [8, Result 3]. Under H 0 , [8, Result 3] is not valid, but the analysis of [START_REF] Delmas | Asymptotic distribution of circularity coefficients estimate of complex random variables[END_REF] applies. The standard central limit 2 applied to the independent identically distributed bidimensional complex RVs

r x c x with r x = 1 K K k=1 |x 2 k | and c x = 1 K K k=1 x 2 k yields √ K r x -r x c x -c x L → N C 0 0 , σ 4 x 1+κ x 0 0 2+κ x , σ 4 x 1+κ x 0 0 0 , (10)
where σ 2

x def = E|x 2 k |. Then, considering the mapping ( r x , c x ) -→ m x = c x r x -→ γ x = | m x |,
whose differential of the first step is dm = 1 c dr under H 0 , the standard theorem of continuity (see e.g., [10, Th.A, p.122]) on regular functions of asymptotically Gaussian statistics applies. Consequently, we obtain the following convergence in distribution to a complex circular zero-mean Gaussian distribution of variance 2(1

+ κ x 2 ) √ K ( m x -0) L → N C 0, 1 σ 4 x σ 4 x (2 + κ x ), 0 . With γ 2 x = | m x | 2 , convergence in distribution (4) is proved. Proof of Result 2 With (x, K) = det[I -( R -1 x C x ) * R -1 x C x ], where C x def = 1 K K k=1 x k x T
k , the proof of Result 2 follows the same steps that for Result 1. Deriving the asymptotic distribution of (x, K) under H 0 and H 1 is based on the following mapping: Using the asymptotic Gaussian distribution of ( R x , C x ) [START_REF] Delmas | Asymptotically minimum variance second-order estimation for non-circular signals with application to DOA estimation[END_REF] derived from the standard central limit theorem, the differential3 of the different sub mappings of [START_REF] Kay | Asymptotically optimal detection in incompletely characterized non-Gaussian noise[END_REF], the chain rule and standard properties of the vec operator [14, Ch.2.4], the standard theorem of continuity (see e.g., [10, p. 122]) on regular functions of asymptotically Gaussian statistics applies and Result 2 follows, with 0 = 1 and

( R x , C x ) -→ Ω x = R -1 x C x -→ Ξ x = Ω * x Ω x -→ (x, K) = det[I -Ξ x ].
1 = det[I -(R -1
x C x ) * R -1 x C x ] < 1 and where expressions of σ 2 0 and σ 2 1 are not given here due to space limitation.

Fig. 1

 1 Fig.1 Asymptotic theoretical and empirical (with 10000 Monte Carlo runs) ROC curve associated with BPSK and Gaussian model.

Fig. 2 P

 2 Fig.2 P D for different fixed P FA for model (9) as a function of SNR.

( 11 ) 2 N

 112 C (m, R, C) denotes the Gaussian complex distribution with mean m, and covariances R and C.

dΩ = -R -1 x dR x R -1 x C x + R -1 x dC x , dΞ = dΩ * x Ω x + Ω * x dΩ x with Ω x = R -1 x C x and d = -det[I-Ξ]Tr (I -Ξ) -1 dΞ (from [14, Th.1, p.149]) with Ξ = (R -1 x C x ) * R -1 x C x .


Soumaya Sallem

Jean-Pierre Delmas

Pascal Chevalier

OPTIMAL SIMO MLSE RECEIVERS FOR THE DETECTION OF LINEAR MODULATION CORRUPTED BY NONCIRCULAR INTERFERENCE

Keywords: Single input multiple output, Maximum likelihood sequence estimation, Noncircular interference

This paper derives the optimal single input multiple output (SIMO) maximum likelihood sequence estimation (MLSE) receiver for the detection of quadrature amplitude modulations corrupted by potentially noncircular, stationary white or colored zero-mean Gaussian noise. It is proved that this receiver is composed by a widely linear (WL) filter followed by a modified version of the Viterbi algorithm. This WL linear filter is interpreted for complex-valued signal of interest (SOI) symbols as two WL multidimensional matched filter (WL MMF) that reduce to a single WL MMF for real-valued SOI symbols. The performance improvements of this receiver with respect to the standard SIMO MLSE are proved and illustrated.

INTRODUCTION

The MLSE for a digital QAM sequence is an optimal equalization technique in the sense that is minimizes the probability of a sequence error. The original MLSE algorithm has been applied to a single antenna receiver under the assumption of Gaussian circular white noise in the baseband [START_REF] Forney | Maximum-likelihood sequence estimation of digital sequences in the presence of intersymbol interferences[END_REF]. Different extensions have been proposed that include Gaussian circular colored noise and multisensor receivers (see e.g., [START_REF] Ungerboeck | Adaptive maximum likelihood receiver for carrier-modulated data transmission systems[END_REF][START_REF] Vila | MLSE antenna diversity equalization of a jammed frequency selective fading channel[END_REF][START_REF] Lindskog | Multi-channel maximum likelihood sequence estimation[END_REF][START_REF] Bottomley | Unification of MLSE receivers and extension to time-varying channels[END_REF]).

In radio mobile communication systems, the noise is mainly due to co-channel interference that exhibits the same features as the SOI. The MLSE of the SOI symbols using the exact statistical properties of the interference would require joint detection of the SOI and the interference, a rather involved task. Here, we derive the SIMO MLSE receiver under the assumption that the complex amplitude of the interference is potentially noncircular, stationary white or colored and furthermore zero-mean Gaussian distributed. Despite that this latter assumption is not valid in the context of radio mobile communication systems, where the complex amplitudes of the SOI and interference are cyclostationnary and non-Gaussian distributed, we will prove that the derived MLSE receiver outperforms the MLSE receiver that does not take into account the potential noncircularity of the interference.

To the best of our knowledge, few works about SIMO MLSE receivers for noncircular interferences have been published in the literature. Among them, [START_REF] Chevalier | New Insights into optimal widely linear array receivers for the demodulation of BPSK, MSK and GMSK signals corrupted by noncircular interferences -Application to SAIC[END_REF] has derived the MLSE receiver in the context of real-valued SOI with no interference intersymbol and [START_REF] Forster | Multisensors receivers using a filtered reference: application to GSM[END_REF] has proposed a suboptimal MLSE receiver under the assumption of cyclostationarity and rectilinearity of the interference. Note that some other works have taken into account the potential noncircularity of the interference by proposing some suboptimal receivers (see, e.g., [START_REF] Olivier | Single Antenna interference cancellation for synchronised GSM networks using widely linear receiver[END_REF]).

The paper is organized as follows. In Section 2, the SIMO MLSE receiver is derived for the detection of linear modulation corrupted by noncircular interference, from a general detection approach. It is proved that this receiver is composed by a WL linear filter that catches the sufficient statistics followed by a modified version of the Viterbi algorithm. This WL linear filter is interpreted in Section 3 for complex-valued SOI symbols as two WL multidimensional matched filters (WL MMF) that reduce to a single WL MMF for real-valued SOI symbols. The SNR of the current symbol with respect to the noise power at its output is compared to those obtained for the MLSE receiver under the standard assumption of circular noise in Section 4. Finally, a numerical illustration is given in Section 5 to quantify the gain in performance of the MLSE receiver derived under the assumption of noncircular w.r.t. circular noise.

SIMO MLSE RECEIVERS

Detection problem

Consider the general framework of the detection of a deterministic multidimensional continuous-time signal s (m) (t) ∈ C N , m = 1, ..., M that is corrupted by an additive zero-mean Gaussian stationary noise n(t). The observed signal is

x(t) = s (m) (t) + n(t), m = 1, ..., M. (1)
To properly derive the ML detector from the continuous-time signal (1), we use an orthonormal representation of x(t) [START_REF] Van Trees | Detection, Estimation and Modulation Theory, part 1[END_REF]Chap. 4]. Once a convenient set S of orthonormal real-valued scalar signals {φ k (t)} k=1,...,K has been adopted, each signal (Re(s (m) (t)), Im(s (m) (t))) is completely determined by the 2Kth vector s of its real-valued coefficients in this basis. Projecting x(t) on this basis, the 2Kth vector x = s (m) + n ∈ R 2K is obtained.

When n(t) is now circular, spatially and temporally white with spectral power density σ 2 I N , it is straightforward to prove that the real-valued vector n is zero-mean Gaussian distributed with covariance matrix σ 2 2 I 2K . Note that x by itself does contain all data from x(t) that is relevant to the ML detection of m [START_REF] Wozencraft | Principles of Communications Engineering[END_REF]Chap. 4]. Consequently the ML estimator of m is given by m = Arg min m ||xs (m) || 2 . Then to return to continuous-time signals, we note that the complementary part of the projection of n(t) on S is orthonormal to S and we thus obtain equivalently

m = Arg min m +∞ -∞ ||x(t) -s (m) (t)|| 2 dt.
(

) 2
Then when n(t) is potentially noncircular, spatially and temporally colored, we can extend the whitening approach used for scalar-valued circular colored noise in [START_REF] Ungerboeck | Adaptive maximum likelihood receiver for carrier-modulated data transmission systems[END_REF] and [START_REF] Van Trees | Detection, Estimation and Modulation Theory, part 1[END_REF]Chap. 4]. Here the statistical properties of n(t) are characterized by the covariance matrix

R n (τ ) = E[ n(t) n H (t -τ )] = R n (τ ) C n (τ ) C * n (τ ) R * n (τ ) of the augmented vector n(t) def = [n T (t), n H (t)] T where R n (τ ) def = E[n(t)n H (t-τ )] and C n (τ ) def = E[n(t)n T (t-τ )]
are the covariance and complementary covariance of n(t). To derive the ML detector, we prove in the Appendix the following result:

If the Fourier transform 1 R n (f ) of R n (τ ) is invertible
for all values of f (this is the case when n(t) is comprised by a sum of independent interferers and circular spatially and temporally white background noise), there exists a WL N × 2N causal and causally invertible whitening filter (W 1 (t), W 2 (t)) such that:

n w (t) = W 1 (t) ⋆ n(t) + W 2 (t) ⋆ n * (t)
is both circular, and temporally and spatially white, i.e.,

E[n w (t)n H w (t -τ )] = δ(τ )I N and E[n w (t)n T w (t -τ )] = 0 N
where ⋆ is the convolution product and δ is the delta distribution.

Using the whitening model

x w (t) = s (m) w (t) + n w (t) where x w (t) def = W 1 (t)⋆x(t)+W 2 (t)⋆x * (t) and s (m) w (t) def = W 1 (t) ⋆ s (m) (t) + W 2 (t) ⋆ s * (m) (t) that is equivalent to (1), the ML of m is now given from (2) by m = Arg min m +∞ -∞ ||x w (t) -s (m) w (t)|| 2 dt, (3)
which is equivalent from the Parseval's theorem and the Appendix to

m=Arg min m +∞ -∞ [ x(f )-s (m) (f )] H R -1 n (f )[ x(f )-s (m) (f )]df (4) where x(f ) = [x T (f ), x H (-f )] T and s (m) (f ) = [s (m)T (f ) s (m)H (-f )] T , Fourier transforms of x(t) def = [x T (t), x H (t)] T and s (m) (t) def = [s (m)T (t), s (m)H (t)] T , respectively because R -1 n (f ) = W H n (f )W n (f )
where W n (f ) is defined by (15).

Derivation of the SIMO MLSE receiver

Let us apply the previous result to the detection of the sequence {a k } k=0,...,K-1 of the following MAQ's complex amplitude transmitted on a selective fading channel

s (m) (t) = √ π s K-1 k=0 a k g(t -kT ), with g(t) = v(t) ⋆ h(t) (5)
where v(t) and h(t) denote the transmitted and channel baseband impulse response, respectively and m = 1, ..., M with M = (cardA) K where A is the symbol set {a k }. Considering only terms that depend on the symbols a k in (4), the sequence {a k } k=0,...,K-1 which minimizes the MLSE criterion ( 4) is equivalently the sequence that minimizes the following metric

Λ(a 0 , .., a K-1 ) = √ π s K-1 k=0 K-1 k ′ =0 Re(a * k a k ′ r k-k ′ + a k a k ′ r ′ k-k ′ ) -2 K-1 k=0 Re(a * k y k ), (6)
where

y k = +∞ -∞ g H 1 (f )R -1 n (f ) x(f )e i2πkf T df, (7)
r k = +∞ -∞ g H 1 (f )R -1 n (f )g 1 (f )e i2πkf T df, r ′ k = +∞ -∞ g H 2 (f )R -1 n (f )g 1 (f )e i2πkf T df, with g 1 (f ) def = g(f ) 0 and g 2 (f ) def = 0 g * (-f )
, with

g(f ) = v(f )h(f ).
When the symbols a k are real-valued, 2Re(a * k y k ) = a k z k of [START_REF] Chevalier | New Insights into optimal widely linear array receivers for the demodulation of BPSK, MSK and GMSK signals corrupted by noncircular interferences -Application to SAIC[END_REF], where now

z k = 2Re(y k ) = +∞ -∞ g H (f )R -1 n (f ) x(f )e i2πkf T df, (8)
with with g(f ) = [g T (f ), g H (-f )] T . {y k } k=0,...,K-1 and {z k } k=0,...,K-1 are sufficient statistics of the detection problem that can be interpreted as the outputs at time kT of the WL multidimensional filters whose frequency responses are respectively

w H 1 (f ) = g H 1 (f )R -1 n (f ) and w H (f ) = g H (f )R -1 n (f ), (9)
where the input is x(t). Assuming that the output ISI of this filter is limited to L preceding and L following sampling instants, metric (6) satisfies Λ(a 0 , ..., a k ) = Λ(a 0 , ..., a k-1 ) + λ(y k , a k , ..., a k-L ) and consequently the Viterbi algorithm can be applied to efficiently minimize this metric.

INTERPRETATION OF THE SIMO MLSE RECEIVER

When n(t) is circular, the WL filters (9) reduce respectively to the linear filter g H (f )R -1 n (f ) and to the real part of this filtering, that are interpreted [START_REF] Vila | Filtre adapté multidimensionnel pour l'égalisation d'un canal sélectif fréquence et brouillé[END_REF] as the linear matched filter (called MMF in [START_REF] Vila | MLSE antenna diversity equalization of a jammed frequency selective fading channel[END_REF]) that maximizes the output power of the current symbol with respect to the noise power, i.e., the SNR at the symbol sampling instant.

On the other hand, when n(t) is noncircular, the interpretation of the WL filter ( 9) is more involved for complexvalued a k , because substituting ( 5) into [START_REF] Forster | Multisensors receivers using a filtered reference: application to GSM[END_REF], we obtain a contribution of both a k and a * k

y k = √ π s [r 0 a k + r ′ 0 a * k + l =k (r l a k-l + r ′ l a * k-l )] + n k , (10)
where

n k def = +∞ -∞ g H 1 (f )R -1 n (f ) n(f )e i2πkf T
df is proved to be zero-mean noncircular complex Gaussian distributed with

E(n l n * l-k ) = r k and E(n l n l-k ) = r ′ k . However with, x(t) = √ π s [ K-1 k=0 a k g 1 (t -kT ) + K-1 k=0 a * k g 2 (t -kT )] + n(t),
it is clear that the WL filter w H 1 (f ) (9) can be interpreted as a WL matched filter that maximizes the output power of the current symbol a k with respect to the noise power, among the WL filters.

In the particular case of real-valued symbols x(t) = √ π s K-1 k=0 a k g(t -kT ) + n(t) and thus w(f ) is clearly interpreted as a WL matched filter.

PERFORMANCE ANALYSIS FOR REAL-VALUED SYMBOLS

For real-valued symbols,

z k = √ π s [r r,0 a k + l =k r r,l a k-l ]+ n r,k (with r r,k def = 2Re(r k + r ′ k ) and n r,k def = 2Re(n k )) and the SNR = πsE(a 2 k )r 2 r,0 E(n 2 r,k )
at time kT which is maximized by the WL MMF w H (f ), takes the value

SNR nc = π s E(a 2 k ) +∞ -∞ g H (f )R -1 n (f ) g(f )df. (11)
As for the circular Gaussian case, the purpose of the Viterbi algorithm is to deal with the WL MMF output ISI and the performance of the MLSE algorithm is thus directly related to this SNR.

It is interesting to compare this SNR nc , to the SNR obtained at the output of the MMF derived under the assumption of real-valued symbols and circular noise for which

z k = 2Re[ +∞ -∞ g H (f )R -1 n (f )x(f )e i2πkf T df ],
but used with noncircular noise. This SNR is given by 2

SNR c = 2π s E(a 2 k ) +∞ -∞ g H (f )R -1 n (f )g(f )df 1 + Re( +∞ -∞ g H (f )R -1 n (f )Cn(f )R * -1 n (-f )g * (-f )df ) +∞ -∞ g H (f )R -1 n (f )g(f )df
where R n (f ) and C n (f ) are the Fourier transforms of respectively E[n(t)n H (t)] and E[n(t)n T (t)]. We have from the inclusion principle: SNR nc ≥ SNR c .

NUMERICAL ILLUSTRATION

To gain some insight into relation [START_REF] Anderson | Optimal Filtering[END_REF], we consider here the special case where v(t) is a raise cosine pulse shape filter (real valued 1/2 Nyquist filter with

+∞ -∞ v 2 (t)dt = 1
) and h(t) is a specular channel with a flat fading on each path, i.e.,

h(t) = M m=1 β m δ(t -τ m )s m , with β 1 = 1, (β m ) m =1 ∈ C,
Furthermore to obtain interpretable closed-form expressions, we assume that τ m -τ ′ m = k m,m ′ T with k m,m ′ ∈ Z and n(t) temporally white (i.e., R ñ(τ ) = δ(τ )R ñ). In this case, it is straightforward to prove that

SNR nc = 2π s E(a 2 k ) M m=1 |β m | 2 s H m A n s m +Re(β 2 m s T m D * n s m ) SNR c = 2π s E(a 2 k ) M m=1 |β m | 2 s H m R -1 n s m 2 M m=1 |β m | 2 s H m R -1 n s m +Re(β 2 m s T m R * -1 n C * n R -1 n s m ) , with R -1 ñ = A n D n D * n A * n
. Naturally, we note that SNR nc reduces to SNR c for circular noise (D n = C n = 0). We note that SNR nc is the sum of the SNR nc obtained for each path of the channels, in contrast to SNR c whose expression is more involved because the filter g H (f )R -1 n (f ) is no longer matched to non circular noise.

To further illustrate, we assume that the noise is composed of a single rectilinear interference (where v(t) has a zero roll-off) that is synchronized with the SOI and a circular spatially and temporally white background noise. More precisely [START_REF] Vila | Filtre adapté multidimensionnel pour l'égalisation d'un canal sélectif fréquence et brouillé[END_REF], under the assumption of both circular noise and complex valued symbols.

R n = π 1 j 1 j H 1 + η 2 I N and C n = π 1 e 2iφ1 j 1 j T 1 . 2 Note that this SNR is different from the SNR πsE(|a k | 2 ) +∞ -∞ g H (f )R -1 n (f )g(f )df obtained by
In this case Fig. 1, compares SNR nc to SNR c as a function of the phase φ 1 , in the scenario of BPSK SOI and interference symbols with N = 2, M = 2 with τ 1 = 0, the first components of the steering vectors s 1 , s 2 and j 1 are equal to 1, π s /η 2 = 10 and π 1 /η 2 = 100. This figure that exhibits the mean values of SNR nc and SNR c over 50000 random channels (uniformly and independently distributed SOI and interference DOAs and β 2 with |β 2 | ≤ 1) shows that in this scenario the MLSE receiver derived under the assumption of noncircular noise largely outperforms those derived under the assumption of circular noise.

CONCLUSION

We have introduced and analyzed the SIMO MLSE receiver under the assumption of stationary noncircular Gaussian noise. The extension of this SIMO MLSE receiver and its performance analysis under the more realistic assumption of cyclostationary noncircular Gaussian noise in under way.

APPENDIX

We extend here the celebrated spectral factorization theorem to wide-sense stationary (WSS) complex-valued continuoustime process n(t) ∈ C N (i.e., where real and imaginary parts are jointly WSS). For this, we consider the associated realvalued n(t)

If the spectral factorization theorem applies to the real-valued stationary continuous-time process3 n(t), there exists a causal and inversible causal transfer function G(f

) ∈ C 2N ×2N such that R n(f ) = G(f )G H (f ) with G * (f )= G(-f ). Conse- quently W n(f ) def = G -1 (f ) satisfies W n(f )R n(f )W H n (f ) = I 2N and W * n(f ) = W n(-f ).
(13) Then using relation [START_REF] Vila | Filtre adapté multidimensionnel pour l'égalisation d'un canal sélectif fréquence et brouillé[END_REF], R n(f ) = 1 4 T H R n (f )T, we obtain

W n (f )R n (f )W H n (f ) = I 2N , (14)
W n (f ) = W 1 (f ) W 2 (f ) W * 2 (-f ) W * 1 (-f ) (15

Fig. 1

 1 Fig.1 Mean values of SNRnc and SNRc.

2 T

 2 def = [Re(n T (t)), Im(n T (t))] T and augmented processes n(t) def = [n T (t), n H (t)] T related by n(t)=Tn(t) and n(t)= 1 H n(t) with T def = I iI I -iI .

  ) to (13). Its inverse Fourier transform W n (t) is struc-tured as W n (t) = W 1 (t) W 2 (t) W 1 (t) ⋆ n(t) + W 2 (t) ⋆ n * (t)both circular, and temporally and spatially white.

	W * 2 (t) W * 1 (t)	and consequently
	n w (t)

def =

All Fourier transforms of scalars x, vectors x and matrices X use the same notation where τ is simply replaced by f .

This is in particular the case for power spectral density R n (f ) rational function in e i2πf for which Rn(f ) is invertible for all values of f , see e.g.[START_REF] Anderson | Optimal Filtering[END_REF] Appendix A6.1].whereW n (f ) = 1 2 √ 2 TW n(f )T H is structured as


Jean Pierre Delmas

Yann Meurisse

ON THE CRAMER RAO BOUND AND MAXIMUM LIKELIHOOD IN PASSIVE TIME DELAY ESTIMATION FOR COMPLEX SIGNALS

Keywords: Time delay estimation, circular and noncircular complex signals, Whittle formula, cross correlation

This paper is devoted to time delay estimation for wide sense stationary complex circular or noncircular Gaussian signals.

Using a theorem by Whittle that we have extended to complex data, closed-form expressions of the Cramer Rao bound (CRB) are given for the time delay alone in presence of nuisance parameters. In particular, we prove that the CRB for the time delay is weakly reduced for noncircular signals w.r.t. circular signals, except for very low signal to noise ratios (SNR), for which the CRB for rectilinear signals is half of the CRB for circular signals. Then, the maximum likelihood (ML) estimate that extends the generalized cross correlation (GCC) estimate is derived.

INTRODUCTION

Time delay estimation plays an important role in radar, sonar, and seismology for localizing radiating sources. It has been extensively studied in the past decade (e.g., [START_REF] Hahn | Optimum processing for delayvector estimation in passive signal arrays[END_REF], [START_REF] Knapp | The generalized correlation method for estimation of time delay[END_REF] and the references therein), but to the best of our knowledge, only for real-valued signals.

Here we focus on the time delay estimation based on the continuous-time complex envelopes issued from two separate sensors in passive systems. To make the model mathematically tractable, these signals are assumed wide sense stationary complex circular or noncircular Gaussian. The main aim of this paper is to give closed-form expressions of the CRB for the time delay alone in presence of nuisance parameters and to derive the ML estimator for time delay given signal and noise spectra.

The paper is organized as follows. Section 2 introduces the data model and assumptions. Section 3 extends to complex data a theorem by Whittle [START_REF] Whittle | The analysis of multiple stationary time series[END_REF] for the derivation of the Fisher Information matrix (FIM). In Section 4, closed-form expressions of the CRB are given for the time delay alone in presence of nuisance parameters for circular and noncircular signals. In Section 5, the ML estimate for time delay given signal and noise spectra that extends the generalized cross correlation (GCC) introduced in [START_REF] Knapp | The generalized correlation method for estimation of time delay[END_REF] is derived. Finally Section 6 illustrates the comparison of the CRB for the time delay for noncircular signals w.r.t. circular signals for both white spectra and white complementary spectra.

DATA MODEL AND ASSUMPTIONS

A continuous-time signal r(t) whose spectrum is bandlimited around f 0 emanating from a remote source and monitored in the presence of noise at two separated sensors can be modeled as

r 1 (t) = r(t) + w 1 (t) and r 2 (t) = ar(t -τ 0 ) + w 2 (t), (1)
where a and τ 0 denote the relative attenuation and delay of r(t) w.r.t. the first sensor. After bandpass filtering and down conversion of these two signals to baseband, the in-phase and quadrature components are paired to obtain the complex envelopes:

x 1 (t) = s (t -τ g (f 0 )) + n 1 (t) x 2 (t) = κs (t -τ g (f 0 ) -τ 0 ) + n 2 (t) (2)
where r(t) = ℜ ( s(t)e i2πf0t ) , κ = ae -i2πf0(τ0+τ ϕ (f0)) and where τ ϕ (f 0 ) and τ g (f 0 ) are, respectively, the phase and group delays of the perfectly linear and identical passband filters at the output of the two sensors.

To make the derivation of the CRB on the delay τ 0 mathematically tractable, the complex baseband signals s(t), n 1 (t) and n 2 (t) are assumed mutually uncorrelated, zero-mean Gaussian and wide-sense stationary (WSS) (i.e., their real and imaginary parts are jointly WSS). The noise components n 1 (t) and n 2 (t) are assumed circular, in contrast to s(t) that is not necessarily circular 1 .

In this case, the statistical properties of the noise are characterized by the power spectral densities (PSD) N 1 (f ) and

N 2 (f ): Fourier transforms of R ni (τ ) = E (n i (t)n * i (t -τ )), i = 1, 2, whereas s(t) needs both the PSD S(f ) and com- plementary PSD C(f ): Fourier transforms of R s (τ ) = E (s(t)s * (t -τ )) and C s (τ ) = E (s(t)s(t -τ )), respec- tively [4, Sec: 8.1].
We assume that x 1 (t) and x 2 (t) are bandlimited in [-1 2Ts , + 1 2Ts ] and sampled at the Shannon rate 1/T s giving the discrete-time processes x i (n) = x i (t) |t=nTs , i = 1, 2. For simplicity T s = 1 in the following. With these assumptions, the PSD S x (f ) and complementary PSD C x (f ) matrices of the 2-variate process

x(n) = [x 1 (n), x 2 (n)] T are given from (2) by S x (f ) = [ S(f ) + N 1 (f ) κ * S(f )e i2πf τ0 κS(f )e -i2πf τ0 |κ| 2 S(f ) + N 2 (f ) ] (3)
and

C x (f ) = [ C(f ) κC(f )e i2πf τ0 κC(f )e -i2πf τ0 κ 2 C(f ) ] . (4)

COMPLEX WHITTLE FORMULA

To derive the FIM for our complex data model ( 2), we have to extend the following theorem due to Whittle [3, th. 9] to complex-valued data.

Theorem Let x(n) be a real-valued discrete time WSS zeromean Gaussian multivariate process with spectrum S x (f ) that depends on the real-valued parameter θ. Then the elements J k,l of the FIM associated with N sample values of x(n) are asymptotically (w.r.t. to N ), given by

J k,l = N 2 ∫ +1/2 -1/2 Tr ( ∂S x (f ) ∂θ k S -1 x (f ) ∂S x (f ) ∂θ l S -1 x (f ) ) df.
(5) Consider now a complex-valued WSS zero-mean Gaussian multivariate process x(n) with spectrum S x (f ) and complementary spectrum C x (f ). The Whittle formula (5) applies to 5) and using Tr(AB) = Tr(BA) with U -1 = 1 2 U H prove that (5) still applies2 by replacing S x (f ) with S x(f ), where

x(n) def = [ ℜ(x(n)) ℑ(x(n)) ] = 1 2 Ux(n) with U def = [ I I -iI iI ] and x(n) def = [ x(n) x * (n) ] , whose spectra S x(f ) and S x(f ) are related by S x(f ) = 1 4 US x(f )U H . Introducing the spectrum S x(f ) in (
S x(f ) = [ S x (f ) C x (f ) C * x (-f ) S * x (-f ) ] . ( 6
)
In the particular case of circular complex data, the complementary spectrum C x (f ) is zero and it is straightforward to prove that

J k,l = N ∫ +1/2 -1/2 Tr ( ∂S x (f ) ∂θ k S -1 x (f ) ∂S x (f ) ∂θ l S -1 x (f ) ) df. ( 7
)

CRAMER RAO BOUND

Generally the spectrum S(f ) and the complementary spectrum C(f ) of s(t) and the spectra N 1 (f ) and N 2 (f ) of the noise are not perfectly known. S(f ) and C(f ) [resp., N 1 (f ) and N 2 (f )] depend on unknown parameters α [resp., β], which, along with the complex attenuation κ, comprise the nuisance parameters. Thus the unknown parameter of S x(f ) is θ = (τ 0 , ℜ(κ), ℑ(κ), α T , β T ) T . In the sequel, we distinguish the circular case, where closed-form expressions are available and the noncircular case, where due to the inversion of the 4 × 4 matrix S x(f ), only numerical properties have been derived except in specific conditions.

Circular case

On differentiating S x (f ) and performing the required matrix inversion and multiplications of ( 5), we have proved in particular that

J τ0,τ0 = 8π 2 |κ| 2 N ∫ +1/2 -1/2 f 2 S 2 (f ) ∆ κ (f ) df J τ0,ℜ(κ) = -4πℑ(κ)N ∫ +1/2 -1/2 f S 2 (f ) ∆ κ (f ) df (8) J τ0,ℑ(κ) = -4πℜ(κ)N ∫ +1/2 -1/2 f S 2 (f ) ∆ κ (f ) df (9) J τ0,α = J τ0,β = 0, (10)
but J ℜ(κ),α , J ℑ(κ),α , J ℜ(κ),β and J ℑ(κ),β are nonzero, with

∆ κ (f ) def = det(S x (f )) = (S(f ) + N 1 (f ))(S(f ) + N 2 (f )) - |κ| 2 S 2 (f ).
After deriving the other terms of J, (which are not given here for want of space), we note that J does not depend on τ 0 . Consequently CRB(τ 0 ) = (J -1 ) 1,1 does not depend on τ 0 either.

The FIM of the joint estimation of (τ 0 , κ) and the parameters (α, β) of spectra S(f ), N 1 (f ) and N 2 (f ) is not block diagonal. Thus, the CRB for τ 0 is smaller or equal in the case of known spectra, despite (10). Generally J τ0,ℜ(κ) ̸ = 0 and J τ0,ℑ(κ) ̸ = 0. So, the CRB for τ 0 is smaller or equal when κ is known. However, we note from (8) and (9), that in the case where the spectra S(f ), N 1 (f ) and N 2 (f ) are all even3 , J τ0,ℜ(κ) = J τ0,ℑ(κ) = 0 and thus

CRB(τ 0 ) = J -1 τ0,τ0 = ( 8π 2 |κ| 2 N ∫ +1/2 -1/2 f 2 S 2 (f ) ∆ κ (f ) df ) -1 , ( 11
) which is naturally a decreasing function of the attenuation a = |κ| that vanishes for non overlapping signal S(f ) and noise (N 1 (f ), N 2 (f )) spectra. For a = |κ| = 1, (11) is half of the CRB for τ 0 obtained in [5, (10)] for the delay estimation associated with the real-valued model (1) without any nuisance parameter (i.e., with a = 1).

Finally, we note that for model [START_REF] Knapp | The generalized correlation method for estimation of time delay[END_REF] with no attenuation (a = 1), using now the unknown parameter θ = (τ 0 , Arg(κ), α T , β T ) T , we have proved that the CRB for τ 0 remains unchanged w.r.t. the previous parametrization for arbitrary spectra.

Noncircular case

To derive interpretable closed-form expressions of the CRB for τ 0 in the noncircular case, we have to consider the following specific four conditions:

(i) The complementary spectrum C(f ) of s(t) is related to its spectrum S(f ) by the relation C(f ) = ρS(f )e 2iϕ with ρ ∈ [0, 1] and where ρ and ϕ can be interpreted as, respectively, the second-order noncircularity modulus and phase of s(t) [START_REF] Abeida | MUSIC-like estimation of direction of arrival for non-circular sources[END_REF]. For example, for s(t) circular, ρ = 0 and for s(t) rectilinear (i.e., s(t) = |s(t)|e iϕ ), ρ = 1.

(ii) The spectra of n 1 (t) and n 2 (t) are identical:

N 1 (f ) = N 2 (f ) = N (f ).
(iii) The known spectra S(f ) and N (f ) are even.

(iv) There is no attenuation: a = |κ| = 1. In these conditions, θ = (τ 0 , Arg(κ), ρ, ϕ) T and we have proved from [START_REF] Friedlander | On the Cramer Rao bound for time delay and Doppler estimation[END_REF], where S x (f ) is replaced by S x(f ) given by ( 6)

J τ0,τ0 = 8π 2 N ∫ +1/2 -1/2 f 2 S 2 (f )(2S(f )(1-ρ 2 )+N (f )(1+ρ 2 )) N (f )(2S(f )(1+ρ)+N (f ))(2S(f )(1-ρ)+N (f )) df J τ0,Arg(κ) = J τ0,ρ = J τ0,ϕ = 0. (12)
Consequently CRB(τ 0 ) = J -1 τ0,τ0 , that gives respectively for ρ = 0 (consistent with (11)) and ρ = 1

CRB(τ 0 ) = ( 8π 2 N ∫ +1/2 -1/2 f 2 S 2 (f ) N (f )(2S(f ) + N (f )) df ) -1 , CRB(τ 0 ) = ( 16π 2 N ∫ +1/2 -1/2 f 2 S 2 (f ) N (f )(4S(f ) + N (f )) df ) -1
.

Furthermore, we have numerically noted (by extensive scenarios) that if the condition (iv) is withdrawn, all the nuisance parameters (including now |κ|) remain decoupled in the FIM.

ML ESTIMATOR OF THE TIME DELAY

To derive the ML estimator of the time delay based on the continuous-time observation (x 1 (t), x 2 (t)) t∈[-T /2,T /2] , we use its equivalence with the sequence of the Fourier coefficients of x j (t), j = 1, 2

X j (k) = 1 T ∫ +T /2 -T /2
x j (t)e -i2πkt/T dt.

Note that the sequence (X 1 (k), X 2 (k)) k∈Z , as linear transform of (x 1 (t), x 2 (t)), is complex zero-mean Gaussian distributed as (x 1 (t), x 2 (t)) t∈[-T /2,T /2] and circular if s(t) is circular. For real-valued x j (t), a well known theorem of statistics [START_REF] Davenport | An Introduction to the Theory of Random Signals and Noise[END_REF]Sec. 6.4] asserts that these Fourier coefficients associated with different frequencies k/T are asymptotically (w.r.t. to T ) uncorrelated. More precisely, if T is large compared to the delay τ 0 plus the correlation time of signal and noise we have

E ( X i (k)X * j (l) ) ≈ δ k,l T S xi,xj ( k T ), (13)
i, j = 1, 2, where δ k,l is the Kronecker delta function and S xi,xj (f ) are the components of the spectra S x (f ) (3). For complex-valued x j (t), we have proved that (13) remains valid and

E (X i (k)X j (l)) ≈ δ k,-l T C xj ,xi ( k T ), (14)
where C xi,xj (f ) are the components of the complementary spectra C x (f ) (4). We note that in contrast to the circular case, the Fourier coefficients are no longer independent in the noncircular case. As x(t) is bandlimited to [-1 2Ts , + 1 2Ts ], the sequence (X 1 (k), X 2 (k)) can be truncated to k ∈ {-K, ..., +K} (with K = ⌊T /2T s ⌋) and the probability density function of the complex Gaussian distributed vector x = [X 1 (-K), X 2 (-K), .., X 1 (K), X 2 (K)] T given the unknown parameter θ = (τ, κ) (we suppose in this section that S(f ), C(f ), N 1 (f ) and N 2 (f ) are perfectly known) is written as

p(x; θ) = 1 π 2(2K+1) det 1/2 ( 1 T S x) e -T 2 x H S -1 x x ,
where x def = [x T , x H ] T and S x is the 2(2K + 1) × 2(2K + 1) matrix

S x = [ diag(S x ( k T )) antidiag(C x (-k T )) antidiag(C * x ( k T )) diag(S * x (-k T ))
] .

Because we have proved that det(S x) does not depend on τ but only on κ, the ML estimate τ 0 is given by τ 0 = arg max τ [L κ (τ )] where L κ (τ ) = -T x H S -1 x x, κ is either the ML estimate of κ, or the known parameter κ and S x(f ) depends on τ and κ. Noting that S -1 x is structured as S x, i.e.,

S -1 x = [ diag(A x ( k T )) antidiag(B x (-k T )) antidiag(B * x ( k T )) diag(A * x (-k T ))
] where A x (f ) and B x (f )) are rational fractions of (S x (f ), C x (f )) (not given here for want of space), L κ (τ ) can be approximated for T ≫ T s by

L κ (τ ) ≈ - ∫ 1/2Ts -1/2Ts
x H (f )

[ A x (f ) B x (-f ) B * x (f ) A * x (-f ) ] x(f )df (15) where x(f ) = [x T (f ), x H (-f )] T with x(f ) = ∫ +T /2 -T /2 x(t)e -i2πf t dt. Using the periodogram esti- mates S x2,x1 (f ) = 1 T x 2 (f )x * 1 (f ) and C x2,x1 (f ) = 1 T x 2 (f )x 1 (f )
of respectively the cross spectrum and complementary spectrum of x 2 (t) and x 1 (t) (with x(f ) = [x 1 (f ), x 2 (f )] T ), (15) can be written in the following form

L κ (τ ) ≈ ∫ 1/2Ts -1/2Ts ℜ ( a κ (f ) S x2,x1 (f )e i2πf τ +b κ (f ) C x2,x1 (f )e i2πf τ
) df( 16)

where a κ (f ) and b κ (f ) are rational functions in S(f ), C(f ), N 1 (f ), N 2 (f ) and κ (which are not given here for want of space).

The ML estimate τ 0 given by the maximization of (16) extends the GCC estimate introduced by Knapp et al. [START_REF] Knapp | The generalized correlation method for estimation of time delay[END_REF] in the real case. It consists in widely filtering x 1 (t) and x 2 (t), cross correlating these outputs and then maximizing the real part of this cross correlation. We note that for white spectra and complementary spectra specified in Section 6), τ 0 is given by τ

0 = arg max τ ℜ ( a 1 ( κ) R x2,x1 (τ )+b 1 ( κ) C x2,x1 (τ ) ) , ( 17
) with R x2,x1 (τ ) = ∫ T /2 -T /2 x 2 (t)x * 1 (t -τ )dt, C x2,x1 (τ ) = ∫ T /2 -T /2 x 2 (t)x 1 (t -τ )dt
, where x 1 (t) and x 1 (t) are zero outside [-T /2, T /2]. a 1 ( κ) and b 1 ( κ) are functions of κ that are not given here for want of space. For s(t) circular the maximization of ( 16) and (17) reduce, respectively, to

τ 0 = arg max τ ∫ 1/2Ts -1/2Ts S(f ) ∆ κ (f ) ℜ ( κ * S x2,x1 (f )e i2πf τ ) df τ 0 = arg max τ ∫ 1/2Ts -1/2Ts ℜ ( κ * S x2,x1 (f )e i2πf τ ) df, ( 18
)
in contrast to the real case, both require the knowledge of κ.

In particular we note that from (18), despite |R x2,x1 (τ )| = a|R s (τ -τ 0 )| ≤ aR s (0), the ML estimate τ 0 is not given by arg max τ | R x2,x1 (τ )|.

NUMERICAL ILLUSTRATIONS

To gain some insight into relation (12), we consider here the special case of subsection 4.2 in which the signal and noise are ideal white processes, i.e.,

S(f

) = { σ 2 s , |f | ≤ 1/2 0, |f | > 1/2 , C(f ) = { ρe 2iϕ σ 2 s , |f | ≤ 1/2 0, |f | > 1/2 N (f ) = { σ 2 n , |f | ≤ 1/2 0, |f | > 1/2 .
In that case, the nuisance parameters of the spectra are α = (σ 2 s , ρ, ϕ) T and β = σ 2 n and we have proved that in addition to (12) J τ0,σ 2 s = J τ0,σ 2 n = 0 and the CRB for τ 0 that depends only on ρ and the SNR r = σ 2 s /σ 2 n , is monotonically decreasing with both ρ and r.

for ρ = 0 and ρ = 1, respectively. Fig. 1 plots the ratio CRB ρ (τ 0 )/CRB 0 (τ 0 ) w.r.t. to ρ for different values of the SNR and for a = 1 (deduced from (19)). We note that for arbitrary a, the behavior of this ratio (deduced from the numerical evaluation of J -1 τ0,τ0 ) w.r.t. ρ is very similar with now r defined by (1 + a 2 )σ 2 s /σ 2 n .

Fig. 1 CRB ρ (τ 0 )/CRB 0 (τ 0 ) w.r.t. ρ for different values of the SNR.

  CRB ρ (τ 0 ) = 3 2π 2 N r 2

(2r( 1 +

 1 ρ) + 1)(2r(1 -ρ) + 1) 2r(1 -ρ 2 ) + (1 + ρ 2 ),

Note that if r(t) were assumed WSS, s(t) would be WSS and circular, but if s(t) is WSS, r(t) is WSS or wide-sense cyclostationary with the cyclic frequency

2f 0 , depending on whether s(t) is circular or noncircular.

Note that the theorem also applies to our continuous-time process sampled at the Shannon rate.

This is the practical case where r(t) is a modulation with carrier f 0 and w 1 (t) and w 2 (t) are white noises in the bandwidth of r(t).

We see that the CRB for τ 0 is weakly reduced for noncircular signals s(t) w.r.t. to circular signals, except for very low SNR, for which the CRB for rectilinear signals is half of the CRB for circular signals, consistent with (20) and (21).


Jean-Pierre Delmas

Habti Abeida

ON THE DEGREE OF SECOND-ORDER NON-CIRCULARITY OF COMPLEX RANDOM VARIABLES

Keywords: circular/noncircular, proper/improper, rectilinear signal, coherence matrix, canonical correlations, circularity spectrum

This paper addresses the degree of second-order non-circularity or impropriety of complex random variables and its purpose is to complement previously available theoretical results. New properties of the non-circularity rate (also called circularity spectrum) are given for scalar and multidimensional complex random variables with a particular attention paid to rectilinear random variables, i.e., with maximum circularity spectrum. Finally, the maximum likelihood estimate of the circularity spectrum in the Gaussian case and asymptotic distribution of this estimate for arbitrary distributions are given.

INTRODUCTION

Recently, there has been an increased awareness that significant performance gains can be achieved by taking the information contained in the complementary covariance [START_REF] Schreier | Second-order Analysis of improper complex random vectors and processes[END_REF] matrix R z = E(zz T ) (termed as relation matrix in [START_REF] Picinbono | Second-order complex random vectors and normal distributions[END_REF], pseudo covariance matrix in [START_REF] Eriksson | Complex random vectors and ICA models: identifiability, uniqueness, and separability[END_REF] and second covariance in [START_REF] Abeida | MUSIC-like estimation of direction of arrival for non-circular sources[END_REF]) into account in second-order algorithms previously based on the standard covariance matrix R z = E(zz H ) only (see, e.g., [START_REF] Schreier | Detection and estimation of improper complex random signals[END_REF]). In the past, it was often assumed that R z = 0, a case that is referred to as either proper, second-order circular or circularly symmetric. However in digital communications, modulated signals may be improper or second-order non-circular but not necessarily with a maximum non-circularity rate, i.e., rectilinear as it has been often considered in the literature (e.g., in direction of arrival estimation, [START_REF] Chargé | A non-circular sources direction finding method using polynomial rooting[END_REF][START_REF] Haardt | Enhancements of unitary ESPRIT for non circular sources[END_REF]). For example, binary phase shift keying (BPSK) modulation is rectilinear in contrast to Gaussian minimum shift keying (GMSK) modulation which is improper but not rectilinear after derotation.

This paper addresses the measure of the degree of secondorder non-circularity or impropriety of complex random variables which can be used to come up with appropriate algorithms or to assess detection or estimation performances of algorithms adapted to improper signals. Its purpose is to complement previously available theoretical results [START_REF] Schreier | Second-order Analysis of improper complex random vectors and processes[END_REF][START_REF] Picinbono | Second-order complex random vectors and normal distributions[END_REF][START_REF] Eriksson | Complex random vectors and ICA models: identifiability, uniqueness, and separability[END_REF][START_REF] Scharf | Canonical cordinates and the geometry of inference rate and capacity[END_REF][START_REF] Cramer | Some symmetricinvariant measures of multivariate association[END_REF]. New properties of the canonical correlation between z and z * (also called non-circularity rate in [START_REF] Abeida | MUSIC-like estimation of direction of arrival for non-circular sources[END_REF] and circular-ity spectrum in [START_REF] Eriksson | Complex random vectors and ICA models: identifiability, uniqueness, and separability[END_REF]) and of the augmented covariance matrice R z = E(zz H ) with z def = (z T , z H ) T are given for scalar and multidimensional complex random variables with a particular attention paid to rectilinear random variables, i.e., with maximum circularity spectrum. Finally, maximum likelihood (ML) estimate of the circularity spectrum in the Gaussian case and asymptotic distribution of this estimate for arbitrary distributions are given.

The paper is organized as follows. Section 2 is dedicated to scalar complex random variables, while, Section 3 extends these results to multidimensional complex random variables.

SCALAR COMPLEX RANDOM VARIABLE

Let z = x + iy denote a zero-mean second-order scalar complex random variable with variance σ 2 z def = E|z 2 | and complementary variance E(z 2 ). The non-circularity rate ρ ∈ [0, 1] and the non-circularity phase φ ∈ [0, π[ of z are defined by

ρe 2iφ def = E(z 2 ) E|z 2 | . ( 1
)
If ρ = 0, z is called proper [START_REF] Schreier | Second-order Analysis of improper complex random vectors and processes[END_REF] or circular to the second-order [START_REF] Picinbono | Second-order complex random vectors and normal distributions[END_REF] and if ρ = 1, z is called rectilinear [START_REF] Chevalier | New insights into optimal widely linear array receivers for the demodulation of BPSK, MSK and GMSK signals corrupted by non circular interferences -Application to SAIC[END_REF] because in this case z = |z|e iφ and z lies in one line of C.

If ρ co def = E(xy) σ x σ y with σ x def = E(x 2
) and σ y def = E(y 2 ), denotes the correlation coefficient between the real x and imaginary y parts of z, we prove the following relations between ρ and ρ co Result 1 The non-circularity rate ρ of a scalar complex random variable z and the correlation coefficient ρ co between its real x and imaginary y parts are related by the following relations

• ρ = 1 ⇔ ρ co = 1, • ρ = 0 ⇒ ρ co = 0, the converse is false because ρ co = 0 does not imply σ x = σ y ,
• ρ ≤ ρ co and ρ = ρ co when σ x = σ y .

Proof These relations are straightforwardly deduced from the following expression of the non-circularity rate:

ρ = ( σ x σy - σ y σx ) 2 ( σ x σy + σ y σx ) 2 + 4ρ 2 co 1 ( σ x σy + σ y σx ) 2 .
To interpret the non-circularity phase φ of z, we prove the following result

Result 2 For a non-circular scalar complex random variable z, the orthogonal regression line of the couple (x, y) has a direction given by the non-circularity phase φ and the mean square orthogonal distance to this line is given

1 by E(d 2 ) = σ 2 z 2 (1 -ρ).
Proof The orthogonal regression line (see e.g., [START_REF] Jolliffe | Principal Component Analysis Series: Springer Series in Statistics[END_REF]) of the couple (x, y) is given by the line orthogonal to the eigenvector u associated with the minimum eigenvalue λ of the co-

variance matrix R w of w def = x y
and the mean square orthogonal distance E(d 2 ) to this line is given by λ.

To solve easily this problem, it is convenient to work with the augmented vector

z def = z z * whose covariance ma- trix R z is related to R w by R w = 1 2 T H R z T using z = √ 2Tw, where T is the unitary matrix 1 √ 2 1 i 1 -i
. Because the minimum eigenvalue and the associated unit eigen-

vector of R z = σ 2 z 1 ρe 2iφ ρe -2iφ 1 are λ = σ 2 z (1 - ρ) and u = i √ 2
e iφ -e -iφ , the minimum eigenvalue and the associated unit eigenvector of R w are 1 2 λ and

T H u = -sin φ cos φ ⊥ cos φ sin φ .
Consequently, the larger is ρ, the smaller is the mean square distance of (x, y) to the orthogonal regression line and this distance is zero if and only if z is rectilinear along this orthogonal regression line whose direction is given by the noncircularity phase φ. Now, let us consider the estimation of the non-circularity rate ρ from T independent identically distributed realizations (z t ) t=1,..,T for which the following result is proved in Appendix A.

Result 3 When z t is Gaussian distributed, the maximum likelihood (ML) estimate (ρ T , φ T ) of (ρ, φ) is given by

| P T t=1 z 2 t | P T t=1 |z 2 t | , 1 2 Arg( P T t=1 z 2 t P T t=1 |z 2 t | ) . Furthermore, when z t is arbitrarily dis- tributed, the sequence √ T (ρ T -ρ) converges in distribution
to the zero-mean Gaussian distribution of variance

c ρ = 1 -2ρ 2 + ρ 4 + ρ 2 κ + κ 2 + ρ 2 (κ ) 2 -2ρ 2 (κ )
where κ, κ and κ are the normalized-like cumulants

Cum(z,z,z * ,z * ) (E(|z| 2 )) 2 , Cum(z,z,z,z) (E(z 2 )) 2
and Cum(z,z,z,z * ) E(|z| 2 )E(z 2 ) respectively.

1 Note that the expression

(σ 2 x +σ 2 y )- q (σ 2 x +σ 2 y ) 2 -4σ 2 x σ 2 y (1-ρ 2 co ) 2
of this distance as a function of the correlation coefficient ρ co given by the minimum eigenvalue of R w is much involved.

Note that the covariance of the asymptotic distribution of ρ T is a decreasing function of ρ when z t is Gaussian distributed (κ = κ = κ = 0) and vanishes for rectilinear random variables. Furthermore, using a derotation made by the normalized-like cumulants, the covariance of this empirical estimate does not depend of the non-circularity phase φ for arbitrary distributions.

MULTIDIMENSIONAL COMPLEX RANDOM VARIABLE

Consider now a full K-dimensional zero-mean second-order complex random variable z. The canonical correlations between z and z * i.e., the circularity spectrum of z, are denoted by (ρ k ) k=1,...,K and are arranged in decreasing order

1 = ρ 1 = ... = ρ r > ρ r+1 ≥ ... ≥ ρ K ≥ 0. Let R z def = E(zz H ) = R z R z R * z R * z
denote the covariance matrix of the augmented vector

z def = z z * where R z is non- singular, and R w def = E(ww T ) = R x R x,y R y,x R y , those of w = x y
. Regarding the rank of these covariance matrices, the following result is proved Result 4 For a full K-dimensional random variable z, the rank of the covariance matrices R z and R w are equal to 2K -r with r ∈ {0, .., K}.

Proof As for the scalar case, R z and R w are related by R w = 1 2 T H R z T where T is the unitary matrix 1

√ 2 I iI I -iI , consequently rank(R z ) = rank(R w ). Now consider R z .
From the definition of the coherence matrix 2

M = R -1/2 z R z R -T /2
z associated with z and z, R z may be factored (see e.g., [START_REF] Scharf | Canonical cordinates and the geometry of inference rate and capacity[END_REF]) as

R z = R 1/2 z O O R * /2 z I M M * I R H/2 z O O R T /2 z .
Since M is complex symmetric, there exists a specular singular value decomposition (SVD), called Takagi's factorization, which is M = U∆U T , where ∆ = Diag(ρ 1 , ..., ρ K ) and U is a unitary matrix. Combining this decomposition of M in 2 Note that the coherence matrix M depends on the specific square root R 1/2 z of R z , unique only if it is imposed to be positive definite Hermitian, in contrast to the circularity spectrum (ρ 1 , ..., ρ K ) which is always unique [3, th.2].

the previous expression of R z , we obtain

R z = R 1/2 z O O R * /2 z U O O U * I ∆ ∆ I U H O O U T R H/2 z O O R T /2 z (2)
Consequently rank(R z ) = rank I ∆ ∆ I = rank(I) + rank(I -∆I -1 ∆) using [12, th.8.5.10] that gives the rank of a partitioned matrix. So rank(R

z ) = K + rank(I -∆ 2 ) = K + (K -r).
Regarding the maximum of the circularity spectrum, the following equivalence is proved Result 5 The circularity spectrum is maximum, i.e.,

ρ 1 = ρ 2 = ... = ρ K = 1 if and only if (i) rank(R z ) = K (i.e., z belongs to a K-dimensional subspace of C 2K ), (ii) rank(R w ) = K (i.e., w belongs to a K-dimensional sub- space of R 2K ), (iii) there exists a square root R 1/2 z of R z such that R z = R 1/2 z R * /2
z , (iv) there exists square roots R

1/2 x and R 1/2 y of R x and R y respectively, such that R x,y = R 1/2 x R 1/2 y . Proof The equivalences (i) and (ii) are a direct consequence of rank(R z ) = rank(R w ) = K + rank(I -∆ 2 ). If the circularity spectrum is maximum, ∆ = I and (iii) follows because (2) implies R z = R 1/2 z UU T R T /2 z where R 1/2 z U is a square root of R z . Conversely, (iii) implies that R z = R 1/2 z R * /2 z R 1/2 z R * /2 z
which involves that rank(R z ) = K and the circularity spectrum is maximum. Equivalence (iv) follows the same lines that equivalence (iii) by considering the canonical correlations associated with x and y and equivalence (ii). By analogy with the scalar case, we propose to call rectilinear such complex multidimensional random variables z whose circularity spectrum is maximum. Note that if the components (z 1 , ...., z K ) of z are all rectilinear, there are K linear relations y k = tan(φ z k )x k , (k = 1, .., K) between the components of w, consequently rank(R w ) = K and z is rectilinear 3 . But the converse is not true: if z is rectilinear, its components (z k ) k=1,..,K need not have maximum non-circular rates ρ z k . For example, let z = (z 1 , z 2 ) T where z 1 is circular and z 2 = x 2 + iy 2 with x 2 = ax 1 and y 2 = ay 1 . z is rectilinear because w belongs to a 2-dimensional subspace of R 4 but the non-circularity rates of z 1 and z 2 are ρ z 1 = 0 and

ρ z 2 = |a 2 -1| a 2 +1 with ρ z 2 = 0 for a = 1.
To extend to the multidimensional case, the non-circularity phase φ defined in the scalar case by [START_REF] Schreier | Second-order Analysis of improper complex random vectors and processes[END_REF], we propose a definition based on the K-dimensional orthogonal regression subspace of (x 1 , ..., x K , y 1 , ..., y K ) which is the support of w for a maximum circularity spectrum. The canonical angles (φ 1 , φ 2 , .., φ K 2 ) between this subspace and each of the K hyperspaces (y k = 0) k=1,...,K of R 2K satisfy this aim. However, two questions remain open. First, does one extend the expression of the mean square orthogonal distance to this Kdimensional orthogonal regression subspace given in Result 2? Second, does one prove that the parameter (ρ, φ, R z ) with φ def = (φ 1 , φ 2 , .., φ K 2 ) T makes up a one to one parametrization of (R z , R z )?

Now, let us consider the estimation of the circularity spectrum ρ = (ρ 1 , ρ 2 , ..., ρ K ) T from T independent identically distributed realizations (z t ) t=1,..,T for which the following result is proved in [START_REF] Delmas | On the degree of second-order non-circularity of complex random variables[END_REF] using the same steps that for Result 3.

Result 6 When z t is Gaussian distributed, the ML estimate ρ T of ρ is given by the vector containing the K singular values of the empirical coherence matrix

M T = R -1/2 z,T R z,T R -T /2 z,T where R z,T def = 1 T T t=1 z t z H t and R z,T def = 1 T T t=1 z t z T t .
Furthermore, when z t is arbitrarily distributed and when the circularity spectrum ρ has distinct elements, the sequence √ T (ρ T -ρ) converges in distribution to a zero-mean Gaussian distribution that extends Result 3, whose covariance is specified in [START_REF] Delmas | On the degree of second-order non-circularity of complex random variables[END_REF].

A. APPENDIX: PROOF OF RESULT 3

When z t is Gaussian distributed, the log-likelihood function associated with (z t ) t=1,..,T can be classically written after dropping the constants as

L(ρ, φ, σ 2 z ) = - T 2 ln[Det(R z )] + Tr(R -1 z R z,T ) (3)
with R z,T def = 1 T T t=1 zt zH t where the parameter (ρ, φ, σ 2 z ) is embedded in the covariance matrix R z . Due to the structure (×) ( ) ( ) * (×) * of R z the ML estimation of R z becomes a constrained optimization problem which is not standard. But maximizing the log-likelihood (3) without any constraint on the Hermitian matrix R z reduces to a standard maximization problem, whose solution is R z,T . Because

R z,T = 1 T T t=1 |z 2 t | 1 T T t=1 z 2 t 1 T T t=1 z * t 2 1 T T t=1 |z 2 t |
is also structured as (×) ( ) ( ) * (×) * , R z,T is the ML estimate of R z . Using the invariance property of the ML estimate implies that the ML estimate of (ρ, φ) is given by

| T t=1 z 2 t | T t=1 |z 2 t | , 1 2 Arg( T t=1 z 2 t T t=1 |z 2 t |
) .

Deriving the asymptotic distribution of the empirical estimate ρ T when z t is arbitrarily distributed, relies on the standard central limit theorem 4 applied to the independent identically distributed bidimensional complex random variables r z,T r z,T with r z,T = 1

T T t=1 |z 2 t | and r z,T = 1 T T t=1 z 2 t : √ T r z,T -r z r z,T -r z L → N C 0 0 , c r c r,r c r ,r c r , c r c r,r c r ,

4 z 1 + 4 z 1 + 1 2

 41411 r c r , where r z = E|z 2 t | = σ 2 z and r z = E(z 2 t ) = ρσ 2 z e i2φ . Using the identityE(z 1 z 2 z 3 z 4 ) = Cum(z 1 , z 2 , z 3 , z 4 ) + E(z 1 z 2 )E(z 3 z 4 ) + E(z 1 z 3 )E(z 2 z 4 ) + E(z 1 z 4 )E(z 2 z 3 ), we obtain c r c r,r c r ,r c r = σ ρ 2 + κ ρe -i2φ (2 + κ * ) ρe i2φ (2 + κ ) 2 + κ c r c r,r c r ,r c r = σ ρ 2 + κ ρe i2φ (2 + κ ) ρe i2φ (2 + κ ) ρ 2 e i4φ (2 + κ ) .Then, considering the following mappings(r z,T , r z,T ) -→ m T = r z,T r z,T -→ ρ T = m T m * T , dm + mdm * ) ,the standard theorem of continuity (see e.g.,[13, p. 122]) on regular functions of asymptotically Gaussian statistics applies. Consequently, we have with m =r z r z = ρe i2φ √ T (m T -m) L → N C (0, c m , c m ), (c m + (c m e -4iφ)). Result 3 follows thanks to simple algebraic manipulations of c ρ .

Note that the components (z k ) k=1,..,K of z do not need to be uncorrelated as it is usually assumed in DOA estimation of non-circular sources (see e.g.[START_REF] Chargé | A non-circular sources direction finding method using polynomial rooting[END_REF][START_REF] Haardt | Enhancements of unitary ESPRIT for non circular sources[END_REF]).

N R (m, C) and N C (m, C, C ) denote Gaussian real and complex distribution with mean, covariance and complementary covariance are m, C and C respectively.