Abdelmalek Mennad

Stefano Fortunati
email: stefano.fortunati@iet.unipi.it

Mohammed Nabil

El Korso

Arezki Younsi

Abdelhak M Zoubir

Alexandre Renaux

Slepian-Bangs-type formulas and the related Misspecified Cramér-Rao Bounds for Complex Elliptically Symmetric distributions

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

Introduction

The asymptotic performance analysis of an estimation algorithm mostly relies on two simplified assumptions: i ) the data are assumed to be Gaussian distributed and ii ) the data model used to derive the estimation algorithm is supposed to be correctly specified , that is the probability density function (pdf) assumed to derive an estimator of the parameters of interest and the true pdf that statistically characterizes the data are exactly the same.

Although these assumptions guarantee the possibility to perform "elegant" performance assessment, e.g. by evaluating the Cramér-Rao Bound (CRB) for the estimation problem at hand and/or by obtaining a closed form expression for the Mean Square Error (MSE) of a given estimator, the everyday engineering practice clearly calls the hypotheses i ) and ii ) into question. Regarding the Gaussian model assumption, large-scale measurement campaigns and the subsequent statistical analysis of the data gathered from a plethora of engineering applications, e.g. outdoor/indoor mobile communications, radar/sonar systems or magnetic resonance imag-ing (MRI), have highlighted the impulsive, heavy-tailed behaviour of the observations [START_REF] Zoubir | Robust estimation in signal processing: a tutorial-style treatment of fundamental concepts[END_REF] . These experimental evidences have motivated the need to go beyond the Gaussian model and develop new statistical models able to better characterize the data. One of the more flexible and general non-Gaussian model is represented by the set of the Complex Elliptically Symmetric (CES) distributions [START_REF] Ollila | Complex elliptically symmetric distributions: survey, new results and applications[END_REF] , also called Multivariate Elliptically Contoured distributions [START_REF] Richmond | Adaptive Array Signal Processing and Performance Analysis in Non-Gaussian Environments[END_REF] . CES distributions encompasses the complex Gaussian, the Generalized Gaussian and all the Compound Gaussian (CG) distributions, such as the complex t -distribution and the K -distribution, as special cases. The pdf of a CES distributed N -dimensional random vector x l ∈ C N is completely characterized by the mean value γ, the scatter (or shape) matrix and by a real valued function w (t) : R + → R , called the density generator , i.e. x l ∼ CES N ( γ, , w ) [START_REF] Ollila | Complex elliptically symmetric distributions: survey, new results and applications[END_REF][START_REF] Richmond | Adaptive Array Signal Processing and Performance Analysis in Non-Gaussian Environments[END_REF] . The CES distributions have been used in a variety of applications, in particular in the radar and array signal processing fields.

Other experimental evidences reveal recurring violations of the matched model assumption, that is the claim of a perfect match between the assumed and the true data model. The mathematical bases of a formal theory of the parameter estimation under model misspecification has been firstly developed by statisticians as Huber [START_REF] Huber | The behavior of maximum likelihood estimates under nonstandard conditions[END_REF] , White [START_REF] White | Maximum likelihood estimation of misspecified models[END_REF] and Vuong [START_REF] Vuong | Cramér-rao Bounds for Misspecified Models[END_REF] and recently rediscovered by the Signal Processing (SP) community [START_REF] Richmond | Parameter bounds on estimation accuracy under model misspecification[END_REF][START_REF] Fortunati | The Misspecified Cramér-Rao bound and its application to scatter matrix estimation in complex elliptically symmetric distributions[END_REF][START_REF] Fortunati | The constrained Misspecified Cramér-Rao bound[END_REF] and applied to a va-riety of well-known engineering problems: to Direction-of-Arrival (DoA) estimation in array and MIMO processing [START_REF] Richmond | Parameter bounds on estimation accuracy under model misspecification[END_REF][START_REF] Ren | Performance bounds under misspecification model for mimo radar application[END_REF] , to covariance/scatter matrix estimation in CES distributed data [START_REF] Fortunati | The Misspecified Cramér-Rao bound and its application to scatter matrix estimation in complex elliptically symmetric distributions[END_REF][START_REF] Greco | Maximum likelihood covariance matrix estimation for complex elliptically symmetric distributions under mismatched conditions[END_REF][START_REF] Fortunati | Matched, mismatched, and robust scatter matrix estimation and hypothesis testing in complex t-distributed data[END_REF] , to radar-communication systems coexistence [START_REF] Richmond | Bayesian framework and radar: on misspecified bounds and radar-communication cooperation[END_REF] and to waveform parameter estimation in the presence of uncertainty in the propagation model [START_REF] Parker | Methods and bounds for waveform parameter estimation with a misspecified model[END_REF] , just to name a few.

This brief discussion clearly highlights the need to overtake both the Gaussian and the matched model assumptions while assessing the (asymptotic) performance of an estimator. As extensively discussed in the SP literature, one of the main tool for the performance assessment is the CRB that provides a lower bound to the MSE achievable by any unbiased estimator for a given estimation problem (see e.g. [START_REF] Kay | Fundamentals of Statistical Signal Processing[END_REF] ). Under the matched model assumption, the CRB can be evaluated as the inverse of the Fisher Information Matrix (FIM), then having a convenient and easy way to evaluate the FIM would be of great practical utility. To this end, in array processing applications, the celebrated Slepian-Bangs (SB) formula has been introduced. Developed in the seminal works of Slepian [START_REF] Slepian | Estimation of signal parameters in the presence of noise[END_REF] and Bangs [START_REF] Bangs | Array Processing with Generalized Beamformers[END_REF] , the SB formula provides a useful and compact expression of the FIM for vector parameter estimation under both Gaussian and matched model assumptions [START_REF] Kay | Fundamentals of Statistical Signal Processing[END_REF]Chapter 3,Appendix 3C] . Specifically, let θ ∈ ⊂ R d be a d -dimensional, deterministic parameter vector and let x = { x l } L l=1 with x l ∈ C N , be a set of L independent (possibly) complex random vectors, usually called snapshots , representing the available observations. If we assume that each snapshot follows a (complex) Gaussian parametric model, such that x l ∼ CN (γ (θ) , (θ)) , then the FIM for the estimation of θ ∈ can be expressed by means of the SB formula.

The first generalization of the SB formula to a non-Gaussian, but still perfectly matched, data model has been proposed by Besson and Abramovich in [START_REF] Besson | On the fisher information matrix for multivariate elliptically contoured distributions[END_REF] . Specifically, Besson and Abramovich derived a compact expression for the FIM for the estimation of θ ∈ when each snapshot x l is characterized by a parametric CES distribution, i.e. x l ∼ CES N ( γ l ( θ), ( θ), w ). Note that the functional form of the parametrized mean value γ l ( θ) is allowed to vary from snapshot to snapshot, while the covariance matrix is assumed to be constant. Clearly, since the Gaussian model belongs to the CES class, this generalized SB formula collapses to the classical one if the data are Gaussian distributed.

The second important step ahead has been made by Richmond and Horowitz in [START_REF] Richmond | Parameter bounds on estimation accuracy under model misspecification[END_REF] and then by Parker and Richmond in [START_REF] Parker | Methods and bounds for waveform parameter estimation with a misspecified model[END_REF] where the classical, Gaussian-based, SB formula has been extended to estimation problems under model misspecification, i.e. when the assumed parametric Gaussian model, say CN (γ (θ) , (θ)) , could differ from the true (possibly non parametric) one, indicated as CN (μ, ) . In other words, we allow the assumed parametric mean value γ( θ) and the assumed parametric covariance matrix ( θ) to differ from the true μ and for every possible value of the parameter vector θ ∈ , i.e. CN (γ (θ) , (θ)) = CN (μ, ) , ∀ θ ∈ . It is worth to underline that in the estimation framework under model misspecification, the FIM loses its classical statistical sense and it has to be substituted by the matrices A ( θ) and B ( θ) defined in [START_REF] Fortunati | The Misspecified Cramér-Rao bound and its application to scatter matrix estimation in complex elliptically symmetric distributions[END_REF] , Eqs. [START_REF] Zoubir | Robust estimation in signal processing: a tutorial-style treatment of fundamental concepts[END_REF] and [START_REF] Richmond | Parameter bounds on estimation accuracy under model misspecification[END_REF], respectively (see also [START_REF] Vuong | Cramér-rao Bounds for Misspecified Models[END_REF][START_REF] Richmond | Parameter bounds on estimation accuracy under model misspecification[END_REF] ). Consequently, in this context, SB-type formulas could be exploited to obtain A ( θ) and B ( θ) needed to evaluate the counterpart of the CRB in the presence of model misspecification, i.e. the Misspecified CRB (MCRB) [START_REF] Huber | The behavior of maximum likelihood estimates under nonstandard conditions[END_REF][START_REF] Vuong | Cramér-rao Bounds for Misspecified Models[END_REF][START_REF] Richmond | Parameter bounds on estimation accuracy under model misspecification[END_REF][START_REF] Fortunati | The Misspecified Cramér-Rao bound and its application to scatter matrix estimation in complex elliptically symmetric distributions[END_REF][START_REF] Greco | Maximum likelihood covariance matrix estimation for complex elliptically symmetric distributions under mismatched conditions[END_REF] . In particular, in [START_REF] Richmond | Parameter bounds on estimation accuracy under model misspecification[END_REF] the authors derived SB-type formulas for the "decoupled" scenario in which the unknown parameter vector θ ∈ can be partitioned in two sub-vectors η and ν, named "deterministic" and "stochastic" parameter subvectors respectively, such that θ = [ η T , ν T ] T and CN (γ (θ) , (θ)) CN (γ (η) , (ν)) = CN (μ, ) , ∀ θ ∈ . The findings presented in [START_REF] Richmond | Parameter bounds on estimation accuracy under model misspecification[END_REF] have been extended in [START_REF] Parker | Methods and bounds for waveform parameter estimation with a misspecified model[END_REF] to include the coupling of deterministic and stochastic parameters. More formally, in [START_REF] Parker | Methods and bounds for waveform parameter estimation with a misspecified model[END_REF] , SB-type formulas have been derived for the following

misspecified scenario CN (γ (θ) , (θ)) CN (γ (η, ω) , (ν, ω)) = CN (μ, ) , ∀ θ ∈ where the unknown parameter vector θ ∈ is partitioned as θ = [ η T , ν T , ω T ] T .
The natural extension of the works of Besson and Abramovich [START_REF] Besson | On the fisher information matrix for multivariate elliptically contoured distributions[END_REF] , Richmond and Horowitz [START_REF] Richmond | Parameter bounds on estimation accuracy under model misspecification[END_REF] and Parker and Richmond [START_REF] Parker | Methods and bounds for waveform parameter estimation with a misspecified model[END_REF] would be to derive SB-type formulas for parametric estimation problems involving CES distributed data under model misspecification. This paper aims exactly at filling this gap and obtaining some general "misspecified" SB formulas for CES distributed data.

Remark : Throughout this paper, we consider only the case of real parameter vectors. This is not a limitation, since we can always maps a complex vector in a real one simply by stacking its real and the imaginary parts. Clearly, the proposed derivation of the SB-type formulas could also be developed directly in the complex field by means of the Wirtinger calculus as in [START_REF] Richmond | Parameter bounds on estimation accuracy under model misspecification[END_REF][START_REF] Fortunati | Misspecified Cramér-Rao bounds for complex unconstrained and constrained parameters[END_REF] .

Notation : Throughout this paper, italics indicates scalar quantities ( a, A ), lower case and upper case boldface indicate column vectors ( a ) and matrices ( A ) respectively. Each entry of a matrix A is indicated as a i, j á [ A ] i, j . * indicates the complex conjugation. The superscripts T and H indicates the transpose and the Hermitian operators, then A H = ( A * ) T . Let f ( t ) be a real scalar function, than f ( t ) á df ( t )/ dt . Let A ( θ) be a matrix (or possibly vector or even scalar) function of the vector θ, then A 0 á A ( θ 0 ) while

A 0 i ∂A (θ) ∂θ i | θ= θ 0 and A 0 i j ∂ 2 A (θ) ∂θ i ∂θ j
| θ= θ 0 , where the vector θ 0 will be always explicitly defined in the paper. For two matrices A and B, A ≥ B means that A -B is positive semi-definite. Finally, for random variables or vectors, the notation = d stands for "has the same distribution as".

Problem setup

Let x = { x l } L l=1 , with x l ∈ C N , be a set of L independent complex random vectors (or snapshots ) representing the available observations. We assume that each snapshot is sampled from a CES distribution [START_REF] Ollila | Complex elliptically symmetric distributions: survey, new results and applications[END_REF][START_REF] Richmond | Adaptive Array Signal Processing and Performance Analysis in Non-Gaussian Environments[END_REF] , i.e., x l ∼ CES N ( μ l , , g ), then its pdf can be expressed as:

p X (x l ) p X (x l ; μ l , ) = c N,g | | -1 g((x l -μ l ) H -1 (x l -μ l )) (1)
where c N, g is a normalizing constant, g(t ) : R + → R is the density generator , μ l = E p { x l } is the mean value and is a positive definite Hermitian matrix called scatter matrix . In the rest of this paper, we always assume that the scatter matrix is of full rank, i.e. rank ( ) = N. From the Stochastic Representation Theorem [2] , a CES distributed random vector can be expressed as:

x l = d μ l + R Tu l , (2)
where:

• u l ∼ U (C S N
) is a N -dimensional complex random vector uniformly distributed on the unit hyper-sphere with N -1 topological dimension. As reported in [START_REF] Ollila | Complex elliptically symmetric distributions: survey, new results and applications[END_REF] (Lemma 1), E p { u l } = 0 and Moreover, under the assumption that rank ( ) = N, we have that:

E p u l u H l = (1 /N) I
Q l (x l -μ l ) H -1 (x l -μ l ) = d Q , ∀ l ∈ N . ( 3
)
As shown in [START_REF] Ollila | Complex elliptically symmetric distributions: survey, new results and applications[END_REF] , the pdf of Q has a one-to-one relation with density generator:

p Q (t) = δ -1 N,g t N-1 g(t ) , (4)
where δ N,g

∞ 0 t N-1 g(t ) dt < ∞ .
As a consequence of ( 3) and ( 4) , the expectation of functions of the quadratic form Q l , say h ( Q l ), can be explicitly derived as:

E Q { h (Q l ) } ∞ 0 h (t ) p Q (t ) dt = δ -1 N,g ∞ 0 h (t ) t N-1 g(t ) dt , ∀ l ∈ N . ( 5
)
It is clear from (5) that such expectation does not depend on the index l , since the pdf in (4) of the quadratic form Q l is invariant with respect to (w.r.t.) l . For this reason, to avoid confusion, in the rest of the paper we always indicate

E Q { h (Q l ) } simply as E Q { h (Q ) } . • T is a complex N × N matrix with rank ( T ) = N, such that = TT H . • If E Q { Q} < ∞ and rank ( ) = N, then the covariance matrix M = E p { (x l -μ l )(x l -μ l ) H } of x l can be decomposed as M = σ 2 ,
where (see Theorem 4 in [START_REF] Ollila | Complex elliptically symmetric distributions: survey, new results and applications[END_REF] ):

σ 2 E Q { Q} N , ( 6
)
and σ 2 can be interpreted as the statistical power of the CESdistributed vector x l .

From the Stochastic Representation Theorem, it is clear that the representation of a CES distributed vector x l is not uniquely determined by [START_REF] Ollila | Complex elliptically symmetric distributions: survey, new results and applications[END_REF] . In fact,

x l = d μ l + R Tu l = d μ l + (c -1 R )(cT ) u l , ∀ c > 0 .
From a different, yet equivalent, standpoint, this identifiability problem can be understood as an implicit consequence of the functional expression of a CES distribution. It is immediate to verify from (1) that CES N ( μ l , , g ( t )) ≡ CES N ( μ l , c 2 , g ( t / c 2 )), ∀ c > 0. As amply discussed in [START_REF] Ollila | Complex elliptically symmetric distributions: survey, new results and applications[END_REF] , this identifiability issue can be solved by posing a constraint on the modular variate R (and consequently, through (4) , on the density generator g ( t )), or on the scatter matrix . For convenience, we choose to put a constraint on R 2 = Q . Specifically in the rest of this paper, we always assume that E Q { Q} = N and consequently, from [START_REF] Vuong | Cramér-rao Bounds for Misspecified Models[END_REF] , M = , i.e. the scatter matrix equates the covariance matrix.

Due to the independence assumption, the joint pdf of the set x is given by the product of the marginal pdfs of each snapshot x l given in (1) :

p X (x ) p X (x 1 , . . . , x L ; μ 1 , . . . , μ L , ) = (c N,g ) L | | -L L l=1 g((x l -μ l ) H -1 (x l -μ l )) (7)
As discussed in the Introduction, the pdf in ( 7) is a general and powerful model that it is able to statistically characterize the impulsive, heavy-tailed data behaviour in a variety of applications and allow us to overtake the Gaussianity assumption. Let us now focus on the clearing of the matched model assumption. Following the recent developments on this field, in this paper we consider the following mismatched situation:

• The acquired dataset x = { x l } L l=1 is characterized by the true but unknown joint pdf given in [START_REF] Richmond | Parameter bounds on estimation accuracy under model misspecification[END_REF] . In particular, each data snapshot x l follows a CES distribution with mean value μ l , scatter matrix and density generator g ( t ), i.e. x l ∼ CES N (μ l , , g) , l = 1 , . . . , L .

• In order to derive an inference algorithm, we assume that each snapshot of the dataset x is sampled from a CES distribution with a density generator w ( t ), possibly different from g ( t ) for all t ∈ R + , and a mean value γ l ( θ) and a scatter matrix ( θ) parametrized by a deterministic parameter vector θ ∈ ⊂ R d to be estimated. In particular, we allow the assumed marginal model CES N ( γ l ( θ), ( θ), w ) to differ from the true one, CES N ( μ l , , g ) for every θ ∈ . This is a recurring scenario in array processing applications, where the mean value and/or the scatter matrix of the acquired snapshot vectors are assumed to be parametrized by a deterministic parameter vector whose components represent the Doppler frequency, the Direction of Arrivals (DOAs) of potential sources and so on.

The assumed marginal pdf of each snapshot x l can then be expressed as:

f X (x l ; θ) f X (x l ; γ l (θ) , (θ)) = c N,w | (θ) | -1 w ((x l -γ l (θ)) H (θ) -1 (x l -γ l (θ))) (8)
and, by exploiting the independence assumption, the joint pdf of x = { x l } L l=1 can be obtained as:

f X x ; θ f X x 1 , . . . , x L ; γ l θ , . . . , γ L θ , θ = ( c N,w ) L θ -L L l=1 w x l -γ l θ H θ -1 x l -γ l θ (9)
This scenario clearly represents an estimation problem in non-Gaussian data and in the presence of model misspecification. Let ˆ θ f ˆ θ f (x ) be a, possibly mismatched, estimator of the parameter vector θ ∈ derived under the assumed model f X ( x ; θ) in ( 9) while the data are characterized by the true model p X ( x ) in [START_REF] Richmond | Parameter bounds on estimation accuracy under model misspecification[END_REF] . Then, as discussed in [START_REF] Vuong | Cramér-rao Bounds for Misspecified Models[END_REF][START_REF] Richmond | Parameter bounds on estimation accuracy under model misspecification[END_REF][START_REF] Fortunati | The Misspecified Cramér-Rao bound and its application to scatter matrix estimation in complex elliptically symmetric distributions[END_REF] , under suitable regularity conditions, a lower bound on the error covariance matrix of any misspecified (MS)unbiased (see [START_REF] Vuong | Cramér-rao Bounds for Misspecified Models[END_REF][START_REF] Fortunati | The Misspecified Cramér-Rao bound and its application to scatter matrix estimation in complex elliptically symmetric distributions[END_REF] ) mismatched estimator ˆ θ f exists and it is given by the MCRB defined as:

C p ˆ θ f , θ 0 E p ( ˆ θ f -θ 0 )( ˆ θ f -θ 0 ) T ≥ 1 L A -1 (θ 0 ) B (θ 0 ) A (θ 0 ) -1 (10)
where θ 0 is the so-called pseudo-true parameter vector defined in [START_REF] Huber | The behavior of maximum likelihood estimates under nonstandard conditions[END_REF][START_REF] White | Maximum likelihood estimation of misspecified models[END_REF][START_REF] Vuong | Cramér-rao Bounds for Misspecified Models[END_REF] as the parameter vector that minimize the Kullback-Leibler divergence (KLD) between the true and the assumed models:

θ 0 arg min θ∈ { D ( p X f θ ) } = arg min θ∈ -E p { ln f X (x l ; θ) } , (11)
where

D ( p X f θ ) á E p {ln ( p X ( x l )/ f X ( x l ; θ))}.
The matrices A ( θ 0 ) and B ( θ 0 ) are defined as:

A (θ 0 ) i, j E p ∇ T θ ∇ θ ln f X (x l ; θ 0 ) i, j = E p ∂ 2 ln f X (x l ; θ) ∂ θ i ∂ θ j θ= θ 0 (12)
and

B (θ 0 ) i, j E p ∇ θ ln f X (x l ; θ 0 ) ∇ T θ ln f X (x l ; θ 0 ) i, j = E p ∂ ln f X (x l ; θ) ∂θ i θ= θ 0 • ∂ ln f X (x l ; θ) ∂θ j θ= θ 0 . ( 13
)
For a deep theoretical analysis of the existence, the properties and the practical applicability of the MCRB, we refer the reader to [START_REF] Vuong | Cramér-rao Bounds for Misspecified Models[END_REF][START_REF] Richmond | Parameter bounds on estimation accuracy under model misspecification[END_REF][START_REF] Fortunati | The Misspecified Cramér-Rao bound and its application to scatter matrix estimation in complex elliptically symmetric distributions[END_REF][START_REF] Fortunati | The constrained Misspecified Cramér-Rao bound[END_REF] and to the references therein. The goal of this paper is to provide a general closed-form expressions of A ( θ 0 ) and B ( θ 0 ) in the aforementioned context. In other words, we aim at deriving two SB-type formulas for the evaluation of the matrices A ( θ 0 ) and B ( θ 0 ) that represent a generalization of the classical FIM for estimation problem under model misspecification. It is important to note that in the following derivations we always assume the existence and the uniqueness of the pseudo-true parameter vector θ 0 defined in [START_REF] Greco | Maximum likelihood covariance matrix estimation for complex elliptically symmetric distributions under mismatched conditions[END_REF] . As it is clear from [START_REF] Ren | Performance bounds under misspecification model for mimo radar application[END_REF] and as it is amply discussed in [START_REF] Vuong | Cramér-rao Bounds for Misspecified Models[END_REF][START_REF] Richmond | Parameter bounds on estimation accuracy under model misspecification[END_REF][START_REF] Fortunati | The Misspecified Cramér-Rao bound and its application to scatter matrix estimation in complex elliptically symmetric distributions[END_REF] , finding the θ 0 that minimizes the KLD between the true and the assumed model is a prerequisite for the evaluation of the MCRB since all the derivatives involved in the two matrices A ( θ 0 ) and B ( θ 0 ) have to be evaluated at θ 0 . Finally, it is worth noticing that, under suitable regularity conditions on the true and the assumed pdfs, the definition of the pseudo-true parameter vector θ 0 in [START_REF] Greco | Maximum likelihood covariance matrix estimation for complex elliptically symmetric distributions under mismatched conditions[END_REF] can be expressed in an equivalent form as:

∂D ( p X f θ ) ∂θ i θ= θ 0 = -E p ∂ ln f X (x ; θ) ∂θ i θ= θ 0 = 0 , i = 1 , . . . , d. ( 14
)
As we will see in the next section, this equality will be exploited to evaluate explicitly the matrix B ( θ 0 ). We note in passing the similarity between the definition on θ 0 given in ( 14) and the condition in Eq. ( 9) of [START_REF] Besson | On the fisher information matrix for multivariate elliptically contoured distributions[END_REF] . This fact highlights the strong theoretical parallelism between the classical matched theory and the misspecified framework as detailed in [START_REF] Vuong | Cramér-rao Bounds for Misspecified Models[END_REF][START_REF] Richmond | Parameter bounds on estimation accuracy under model misspecification[END_REF][START_REF] Fortunati | The Misspecified Cramér-Rao bound and its application to scatter matrix estimation in complex elliptically symmetric distributions[END_REF] .

Slepian-Bangs formulas under misspecified CES models

This Section focuses on the derivation of the SB formulas for A ( θ 0 ) and B ( θ 0 ) defined in ( 12) and ( 13) , provided that there exists a unique pseudo-true parameter vector θ 0 satisfying [START_REF] Greco | Maximum likelihood covariance matrix estimation for complex elliptically symmetric distributions under mismatched conditions[END_REF] . We start by evaluating explicit expressions for the following quantities:

V i j (θ 0 ) = ∂ ln f X (x ; θ) ∂θ i θ= θ 0 • ∂ ln f X (x ; θ) ∂θ j θ= θ 0 , i, j = 1 , . . . , d, (15)
and

H i j (θ 0 ) = ∂ 2 ln f X (x ; θ) ∂θ i ∂θ j θ= θ 0 , i, j = 1 , . . . , d. ( 16
)
The entries of the matrices A ( θ 0 ) and B ( θ 0 ) can then be obtained by taking the expectation operator, w.r.t. the true distri-

bution p X ( x ), of H ij ( θ 0 ) and V ij ( θ 0 ), i.e. [ A (θ 0 )] i j = E p { H i j (θ 0 ) } and [ B (θ 0 )] i j = E p { V i j (θ 0 ) } .
Unfortunately, this expectation can be evaluated in closed form only in two particular cases, as we will detail in Sections 3.2 and 3.3 . In all the other cases, numerical integration techniques, e.g. the Monte Carlo integration, could be exploited.

3.1.

Evaluation of V ij ( θ 0 ) and H ij ( θ 0 ) and of their expectation w.r.t.

the true data distribution

According to the general mismatched estimation problem discussed in the previous section, we consider here the general case in which, for each available snapshot x l , a parametric CES model f X (x l ; θ) = CES N (γ l (θ) , (θ) , w ) is assumed, while actually each observation vector is distributed according to a different, possibly non-parametric CES data model, i.e. x l ∼ p X (x l ) = CES N (μ l , , g) .

From [START_REF] Fortunati | The constrained Misspecified Cramér-Rao bound[END_REF] , it is immediate to verify that:

∂ ln f X x ; θ ∂θ i θ= θ 0 = -L ∂ ln | (θ) | ∂θ i θ= θ 0 + L l=1 φ(G l (θ 0 )) ∂G l (θ) ∂θ i θ= θ 0 = -L tr (P 0 i ) + L l=1 φ(G 0 l ) ∂G 0 l ∂θ i ( 17
)
where

∂ ln | | ∂θ i θ= θ 0 = tr -1 0 0 i = tr (P 0 i ) , (18)
with φ(t) = w (t ) /w (t ) and

P 0 i -1 / 2 0 0 i -1 / 2 0 , ( 19
)
G 0 l G l (θ 0 ) (x l -γ 0 l ) H -1 0 (x l -γ 0 l ) , ( 20
)
where, for notational simplicity, γ 0 l γ l (θ 0 ) and 0 á ( θ 0 ). Then, an explicit expression of V ij ( θ 0 ) in ( 15) is given by:

V i j (θ 0 ) = L 2 tr (P 0 i ) tr (P 0 j ) -L tr (P 0 i ) L l=1 φ(G 0 l ) ∂G 0 l ∂θ j -L tr (P 0 j ) L l=1 φ(G 0 l ) ∂G 0 l ∂θ i + L l=1 L m =1 φ(G 0 l ) φ(G 0 m ) ∂G 0 l ∂θ i ∂G 0 m ∂θ j , ( 21
)
where:

∂G 0 l ∂θ i = -2 Re (x l -γ 0 l ) H -1 0 ∂γ 0 l ∂θ i -(x l -γ 0 l ) H S 0 i (x l -γ 0 l ) , (22)
and

S 0 i = -1 0 0 i -1 0 . ( 23
)
The term H ij ( θ 0 ) in ( 16) can be obtained, through direct calculation, from [START_REF] Bangs | Array Processing with Generalized Beamformers[END_REF] as:

H i j (θ 0 ) = L tr (P 0 i P 0 j -P 0 i j ) + L l=1 φ (G 0 l ) ∂G 0 l ∂θ j ∂G 0 l ∂θ i + L l=1 φ(G 0 l ) ∂ 2 G 0 l ∂ θ i ∂ θ j ( 24
)
where:

P 0 i j -1 / 2 0 0 i j -1 / 2 0 ( 25
)
and φ (t) = w (t ) /w (t ) -(w (t)) 2 /w 2 (t) . After having obtained the terms V ij ( θ 0 ) and H ij ( θ 0 ), we have to evaluate their expectations w.r.t. the true distribution p X ( x ). Since all the derivatives in (21) and (24) have to be evaluated in the pseudo-true parameter vector θ 0 , we can exploit the equality established in [START_REF] Parker | Methods and bounds for waveform parameter estimation with a misspecified model[END_REF] . Specifically, from [START_REF] Bangs | Array Processing with Generalized Beamformers[END_REF] , we have that:

∂D ( p X f θ ) ∂θ i θ= θ 0 = L tr (P 0 i ) - L l=1 E p φ(G 0 l ) ∂G 0 l ∂θ i = 0 , i = 1 , . . . , d, (26)
and consequently,

L l=1 E p φ(G 0 l ) ∂G 0 l ∂θ i = L tr (P 0 i ) , i = 1 , . . . , d. ( 27
)
Now, taking the expectation operator w.r.t. p X ( x ) of the term 21) and by exploiting the equality in (27) , the matrix B ( θ 0 ) can be expressed as:

V ij ( θ 0 ) in (
[ B (θ 0 )] i j =-L 2 tr (P 0 i ) tr (P 0 j ) + L l=1 L m =1 E p φ(G 0 l ) φ(G 0 m ) ∂G 0 l ∂θ i ∂G 0 m ∂θ j . ( 28
)
Similarly, the matrix A ( θ 0 ) can be obtained by taking the expectation operator w.r.t. p X ( x ) of the term H ij ( θ 0 ) in (24) as:

[ A (θ 0 )] i j = L tr (P

0 i P 0 j -P 0 i j ) + L l=1 E p φ (G 0 l ) ∂G 0 l ∂θ j ∂G 0 l ∂θ i + L l=1 E p φ(G 0 l ) ∂ 2 G 0 l ∂ θ i ∂ θ j . ( 29
)
As we can see from ( 21) and ( 24) , the expressions of V ij ( θ 0 ) and H ij ( θ 0 ) are highly involved from an analytical standpoint, and consequently it is impossible to derive in closed form their expectations in the general case. As discussed in Appendix A , closed form expressions can be obtained when the random terms in (21) and (24) satisfy certain independence conditions. Fortunately, there are two scenarios of great practical interest in which such conditions are met and consequently a closed form expression for A ( θ 0 ) and B ( θ 0 ) can be obtained. These special cases are:

Scenario 1: The true marginal pdf is an unspecified CES model, i.e. p X (x l ) = CES N (μ l , , g) , while the assumed pdf is a parametric complex Gaussian model, i.e. f X (x l ; θ) = CN (γ l (θ) , (θ)) .

Scenario 2: The Scenario 2 is characterized by two assumptions:

A1: The true and the assumed pdfs share the same parametric mean value γ l ( θ) and the same parametric scatter matrix ( θ) while the misspecification is caused by a wrong assumption on the density generator w ( t ). More formally, posit that the true marginal pdf of each snapshot is given by a parametric CES distribution such that p X (x l ) p X (x l ; θ) = CES N (μ l , , g) , where μ l = γ l ( θ) and = ( θ) for a given θ ∈ . The assumed model is instead another parametric CES distribution that share the same parameterization of the true one but may have a different density generator, i.e. f X (x l ; θ) = CES N (γ l (θ) , (θ) , w ) , θ ∈ , and g(t ) = w (t) , ∀ t ∈ R + . Note that this scenario is a particular case of the more general class of misspecified problems discussed in [START_REF] Fortunati | The Misspecified Cramér-Rao bound and its application to scatter matrix estimation in complex elliptically symmetric distributions[END_REF] , Section II.D. A2: The true parameter vector θ and the pseudo-true parameter vector θ 0 are equals. In particular, θ is the solution of the optimization problem in [START_REF] Greco | Maximum likelihood covariance matrix estimation for complex elliptically symmetric distributions under mismatched conditions[END_REF] .

Note that these two assumption are verified for the scatter matrix estimation problem discussed in [START_REF] Fortunati | The Misspecified Cramér-Rao bound and its application to scatter matrix estimation in complex elliptically symmetric distributions[END_REF] .

The Scenario 1 describes a common practice in array processing applications. In fact, when the true data model is unknown, a prevalent choice is to assume a simple Gaussian model that guarantees an easy derivation and a consequent real-time implementation of the estimation algorithm.

The Scenario 2 is a bit different, since it imply the a-priori knowledge of the functional form of the true parametric mean value γ l ( θ) and of the parametric scatter matrix ( θ). There are, however, a variety of practical applications in which this a-priori information is indeed available to the user. As an example, one can think of array signal processing applications in which the apriori knowledge of the array geometry leads to a correct specification of the parametrized mean value and covariance matrix of the collected snapshots, while the uncertainty on the statistical disturbance model could cause a wrong choice of the density generator. Of course, there are cases in which also the knowledge of the array geometry could be incorrect or partial, and consequently the assumed mean value and scatter matrix differ from the true ones (see [START_REF] Richmond | Parameter bounds on estimation accuracy under model misspecification[END_REF] for more details).

Scenario 1

In this subsection we provide the SB formulas, i.e. the closed form expressions of the matrices A ( θ 0 ) in [START_REF] Fortunati | Matched, mismatched, and robust scatter matrix estimation and hypothesis testing in complex t-distributed data[END_REF] and B ( θ 0 ) in [START_REF] Richmond | Bayesian framework and radar: on misspecified bounds and radar-communication cooperation[END_REF] , for the Scenario 1. We start by noticing that the assumed complex Gaussian model belongs to the CES class, i.e. f X (x l ; θ) = CES N (γ l (θ) , (θ) , w ) , where w (t) = exp (-t) . Consequently, it is immediate to verify that φ(t) = -1 and φ (t) = 0 . The matrix B ( θ 0 ) can be evaluated from (28) by using the fact that φ(G

l ) = -1 : [ B (θ 0 )] i, j = -L 2 tr (P 0 i ) tr (P 0 j ) + L l=1 L m =1 E p ∂G 0 l ∂θ i ∂G 0 m ∂θ j . ( 30
)
Following the procedure discussed in Appendix B , the matrix B ( θ 0 ) can be expressed as:

[ B (θ 0 )] i, j = 2 L l=1 Re 0 i -1 0 r 0 l + ∂γ 0 l ∂θ i H -1 0 -1 0 × 0 j -1 0 r 0 l + ∂γ 0 l ∂θ j + L E Q { Q 2 } N(N + 1) -1 tr (S 0 i ) tr (S 0 j ) + L E Q { Q 2 } N(N + 1) tr (S 0 i S 0 j ) , (31)
where:

r 0 l μ l -γ 0 l . ( 32
)
Note that all the derivatives have to be evaluated in the pseudotrue parameter vector θ 0 defined in [START_REF] Greco | Maximum likelihood covariance matrix estimation for complex elliptically symmetric distributions under mismatched conditions[END_REF] .

Let us evaluate the matrix A ( θ 0 ). From (24) and φ(G l ) = -1 , and φ (G l ) = 0 , through direct calculation (see also Appendix B ) we obtain:

α i j l (θ 0 ) E p ∂ 2 G 0 l ∂ θ i ∂ θ j = 2 Re ∂γ 0 l ∂θ i H -1 0 ∂γ 0 l ∂θ j + 2 Re r 0 l H S 0 j ∂γ 0 l ∂θ i + S 0 i ∂γ 0 l ∂θ j --1 0 ∂ 2 γ 0 l ∂ θ i ∂ θ j + r 0 l H -1 / 2 0 (P 0 i P 0 j + P 0 j P 0 i -P 0 i j ) -1 / 2 0 r 0 l + tr (P 0 i P 0 j + P 0 j P 0 i -P 0 i j ) -1 / 2 0 -1 / 2 0 . ( 33
)
Moreover, as in (31) , we used constraint on Q l . Finally, by inserting (33) in (24) , we obtain:

[ A (θ 0 )] i, j = L tr (P

0 i P 0 j -P 0 i j ) - L l=1 α i j l (θ 0 ) . ( 34
)
The expressions (31) and (34) represent the SB formulas for the mismatched Scenario 1.

Scenario 2

This Subsection focuses on the Scenario 2, i.e. the case in which the true and the assumed pdfs are CES distributions that share the same parametrized mean value and scatter matrix but are characterized by different density generators. From the proof provided in Appendix C , the matrices B ( θ 0 ) and A ( θ 0 ) can be expressed respectively as:

[ B (θ 0 )] i, j = [ B ( θ) ] i, j = 2 N E Q { Qφ 2 (Q ) } L l=1 Re ∂ γ 0 l ∂θ i H -1 0 ∂γ 0 l ∂θ j + L E Q { Q 2 φ 2 (Q ) } N(N + 1) -1 tr (P i ) tr (P j ) + L E Q { Q 2 φ 2 (Q ) } N(N + 1)
tr

(P i P j ) , ( 35
) [ A (θ 0 )] i, j = [ A ( θ) ] i, j = 2 N E Q { Qφ (Q ) } + NE Q { φ(Q ) } L l=1 Re × ∂ γ 0 l ∂θ i H -1 0 ∂γ 0 l ∂θ j + L E Q { Q 2 φ (Q ) } N(N + 1)
-1 tr (P

0 i P 0 j ) + L E Q { Q 2 φ (Q ) } N(N + 1)
tr (P 0 i ) tr (P 0 j ) .

(

) 36
The expressions ( 35) and (36) represent the SB formulas for the mismatched Scenario 2. It is worth to recall that the previous two formulas can be applied only if, in the particular estimation problem an hand, the pseudo-true parameter vector equates the true parameter vector, i.e. when θ 0 = θ. The reason for this restriction will be clarified in Appendix C .

Relationship to previous results

The aim of this Section is to show that the SB formulas derived for the Scenarios 1 and 2 encompass all the previously derived SB formulas as special cases. In particular, both the SB formula for CES distributions under perfect model specification, proposed in [START_REF] Besson | On the fisher information matrix for multivariate elliptically contoured distributions[END_REF] , and the SB formulas for the scatter/covariance matrix estimation of CES distributed vectors under misspecification of the density generator, proposed in [START_REF] Fortunati | The Misspecified Cramér-Rao bound and its application to scatter matrix estimation in complex elliptically symmetric distributions[END_REF][START_REF] Greco | Maximum likelihood covariance matrix estimation for complex elliptically symmetric distributions under mismatched conditions[END_REF] , can be obtained as special cases of the SB formulas shown here for the Scenario 2. In addition, the SB formulas for misspecified Gaussian models, proposed in [START_REF] Richmond | Parameter bounds on estimation accuracy under model misspecification[END_REF] , are special cases of the SB formulas discussed here for the Scenario 1. We note, in passing, that the SB-type formulas derived in [START_REF] Parker | Methods and bounds for waveform parameter estimation with a misspecified model[END_REF] can also be obtained as a particular case of the ones proposed in this paper in (31) and ( 34) . This can be easily done by partitioning the unknown parameter vector as θ = [ η T , ν T , ω T ] T and by taking into account the particular parameterization of the assumed mean value and of the assumed scatter matrix supposed in [START_REF] Parker | Methods and bounds for waveform parameter estimation with a misspecified model[END_REF] , i.e. γ (θ) = γ (η, ω) and (θ) = (ν, ω) .

The SB formula under correctly specified CES models [18]

The SB formula for correctly specified CES model has been derived in [START_REF] Besson | On the fisher information matrix for multivariate elliptically contoured distributions[END_REF] . Using the formalism introduced in this paper, two parametric CES models are said to be correctly specified if there exists a vector θ ∈ , such that the assumed CES distribution in [START_REF] Fortunati | The constrained Misspecified Cramér-Rao bound[END_REF] equates the true CES distribution in [START_REF] Richmond | Parameter bounds on estimation accuracy under model misspecification[END_REF] . More formally, the CES model f X ( x ; θ) is said to be correctly specified if there exists θ ∈ , such that f (x ; θ) = p X (x ) and, in particular, γ l ( θ) = μ l , ( θ) = and g ( t ) ≡ w ( t ). As proved in [START_REF] Vuong | Cramér-rao Bounds for Misspecified Models[END_REF] , under correctly specified model, we have that θ = θ 0 and B ( θ) = -A ( θ) , where B ( θ) is the classical FIM evaluated at the true parameter vector θ.

In the following, we will show that the SB formula derived in [START_REF] Besson | On the fisher information matrix for multivariate elliptically contoured distributions[END_REF] can be considered as a special case of the one obtained in Section 3.3 . In fact, according to the matched model assumption, we can define the true model as p (θ) , w ) with θ ∈ . With this in mind, we can exploit the result in Eq. ( 11) of [START_REF] Besson | On the fisher information matrix for multivariate elliptically contoured distributions[END_REF] , that is

X (x ) = f X (x ; θ) = CES N (γ l ( θ) , ( θ) , w ) , while the assumed parametric model is f X (x ; θ) = CES N (γ l (θ) ,
E Q { Q l φ(Q l ) } = -N,
where φ(t ) g (t ) /g(t ) . Finally, as shown in [START_REF] Vuong | Cramér-rao Bounds for Misspecified Models[END_REF] , we have that

B (θ 0 ) = B ( θ) = -A ( θ) = FIM ( θ) .
Then by exploiting (35) , we obtain:

[ B ( θ)] i, j = [ FIM ( θ)] i, j = 2 N E Q { Q φ2 (Q ) } L l=1 Re ∂ γl ∂θ i H ¯ -1 ∂ γl ∂θ j + L E Q { Q 2 φ2 (Q ) } N(N + 1) -1 tr ( ¯ -1 ¯ i ) tr ( ¯ -1 ¯ j ) + L E Q { Q 2 φ2 (Q ) } N(N + 1) tr ( ¯ -1 ¯ i ¯ -1 ¯ j ) , ( 37
)
where ¯ ( θ) and ¯ i

∂ (θ) ∂θ i | θ= θ .
It is immediate to verify that the matrix defined in (37) is exactly the same as the FIM given in Eq. ( 20) of [START_REF] Besson | On the fisher information matrix for multivariate elliptically contoured distributions[END_REF] .

4.2.

The SB formulas for scatter matrix estimation under misspecification of the density generator [START_REF] Fortunati | The Misspecified Cramér-Rao bound and its application to scatter matrix estimation in complex elliptically symmetric distributions[END_REF][START_REF] Greco | Maximum likelihood covariance matrix estimation for complex elliptically symmetric distributions under mismatched conditions[END_REF] In [START_REF] Fortunati | The Misspecified Cramér-Rao bound and its application to scatter matrix estimation in complex elliptically symmetric distributions[END_REF][START_REF] Greco | Maximum likelihood covariance matrix estimation for complex elliptically symmetric distributions under mismatched conditions[END_REF] , SB formulas for the scatter matrix estimation in CES distributed vectors under misspecification of the density generator have been proposed. It is easy to verify that this scenario is a special case of the more general Scenario 2 discussed in Section 3.3 . In particular, according to [START_REF] Fortunati | The Misspecified Cramér-Rao bound and its application to scatter matrix estimation in complex elliptically symmetric distributions[END_REF][START_REF] Greco | Maximum likelihood covariance matrix estimation for complex elliptically symmetric distributions under mismatched conditions[END_REF] , the dataset x is considered to be composed of L independent, zero mean snapshots distributed as x l ∼ CES N ( μ, , g ) where μ ≡ 0 and = ( θ) , for a given θ ∈ that is the true parameter vector. For each snapshot, we assume a marginal pdf f X (x l ; θ) = CES N (0 , (θ) , w ) , i.e.

we misspecified the density generator. Since we aim at finding SB formulas for the estimation of the scatter matrix itself, we have that θá vecs( ( θ)) ≡ vecs( ), where vecs is the operator that maps a symmetric N × N matrix in a N(N + 1) / 2 -dimensional vector whose entries are the elements of the upper (or lower) submatrix of . A similar notation holds for the true parameter vector, and in particular θ = vecs ( ¯ ) . Note that, in the following, we assume that the entries of θ (and then of the scatter matrix) are real numbers. Finally, the results in [START_REF] Fortunati | The Misspecified Cramér-Rao bound and its application to scatter matrix estimation in complex elliptically symmetric distributions[END_REF][START_REF] Greco | Maximum likelihood covariance matrix estimation for complex elliptically symmetric distributions under mismatched conditions[END_REF] can be readily derived by ( 35) and (36) by posing

L = 1 , ∂ γ 0 l ∂θ i ≡ 0 and E Q { Q φ(Q ) } = -N (see Appendix B ): [ B (θ 0 )] i, j = E Q { Q 2 φ 2 (Q ) } N(N + 1)
tr (P 0 i ) tr (P 0 j ) + tr (P

0 i P 0 j ) -tr ( P 0 i ) tr ( P 0 j ) , (38)
[

A (θ 0 )] i, j = E Q { Q 2 φ (Q ) } N(N + 1)
tr (P

0 i P 0 j ) + tr (P 0 i ) tr (P 0 j ) -tr (P 0 i P 0 j ) , (39)
where, in this case, the matrix P 0 i (or P 0 j ) can be expressed as

P 0 i = -1 0 A i = ¯ -1
A i where 0 is the matrix that minimizes the KLD between the true and the assumed distributions, and then, due to the Assumption A2, it is equal to the true covariance matrix, i.e.

0 = ¯ = ( θ) = . A i is a matrix defined as A i ∂ (θ) ∂θ i ≡ ∂ ∂θ i . 4.3.
The SB formulas for misspecified Gaussian models [START_REF] Richmond | Parameter bounds on estimation accuracy under model misspecification[END_REF] In this subsection, we briefly show how to obtain the SB formulas provided in [START_REF] Richmond | Parameter bounds on estimation accuracy under model misspecification[END_REF] by using the general results discussed here for the Scenario 1. The SB formulas for misspecified Gaussian models can, in fact, be derived as a special case of the ones given in (31) and (34) by positing as true distribution the parametric complex Gaussian model, i.e. p X (x l ) = CES N (μ l , , exp (-t)) = CN (μ l , ) , while the assumed marginal distribution is still given by f X (x l ; θ) = CES N (γ l (θ) , (θ) , exp (-t)) = CN (γ l (θ) , (θ)) . Note that the true and the assumed density generators are equal and it is simply given by g(t ) = w (t) = exp (-t) . This fact can be used to evaluate the term

E Q { Q 2 } as: E Q { Q 2 } = ∞ 0 exp (-t ) t N+1 δ -1 N,g = -[ exp (-t ) t N+1 δ -1 N,g ] ∞ 0 + (N + 1) ∞ 0 t N exp (-t ) δ -1 N,g dt . ( 40
) Since, [ exp (-t ) t N+1 δ -1 N,g ] ∞ 0 = 0 , we have that E Q { Q 2 } = (N + 1) ∞ 0 t N exp (-t) δ -1 N,g dt = -[ exp (-t ) t N δ -1 N,g ] ∞ 0 + N(N + 1) ∞ 0 t N-1 exp (-t) δ -1 N,g dt = N(N + 1) , ( 41
)
in which we used the fact that

∞ 0 t N-1 exp (-t) δ -1
N,g dt = 1 . Consequently, the entries of the matrix B ( θ 0 ) in (31) can be readily expressed as:

[ B (θ 0 )] i, j = 2 L l=1 Re 0 i -1 0 r 0 l + ∂γ 0 l ∂θ i H -1 0 -1 0 × 0 j -1 0 r 0 l + ∂γ 0 l ∂θ j + L tr (S 0 i S 0 j ) , (42)
where r 0 l μ l -γ l (θ 0 ) .

4.3.1.

The so-called Generalized Slepian formulas [START_REF] Richmond | Parameter bounds on estimation accuracy under model misspecification[END_REF] :

is constant w.r.t. θ
In this particular case, we have that all the derivatives of w.r.t. θ are zero. In particular, S i = S j = P i = P j = P i j = 0 . Then, by posing L = 1 as in [START_REF] Richmond | Parameter bounds on estimation accuracy under model misspecification[END_REF] , the entries of the matrices B ( θ 0 ) and A ( θ 0 ) can be obtained from ( 42) and (34) respectively, as:

[ B (θ 0 )] i, j = 2 Re ∂γ 0 ∂θ i H -1 0 -1 0 ∂γ 0 ∂θ j (43) and [ A (θ 0 )] i, j =-2 Re ∂γ 0 ∂θ i H -1 0 ∂γ 0 ∂θ j -2 Re ( r 0 ) H -1 0 ∂ 2 γ 0 ∂ θ i ∂ θ j , ( 44
)
where r 0 μγ (θ 0 ) . It is immediate to verify that these two expression are exactly the same of Eqs. ( 38) and (40) derived in [START_REF] Richmond | Parameter bounds on estimation accuracy under model misspecification[END_REF] .

4.3.2.

The so-called Generalized Bangs formulas [START_REF] Richmond | Parameter bounds on estimation accuracy under model misspecification[END_REF] : γ is constant w.r.t. θ Here we suppose that the mean value γ is independent of the parameter vector θ, consequently we have that ∂γ ∂θ i = 0 and

∂ 2 γ ∂ θ i ∂ θ j = 0 .
Then, if L = 1 , the entries of the matrix B ( θ 0 ) in (42) can be readily expressed as:

[ B (θ 0 )] i, j = 2 Re 0 i -1 0 r 0 H -1 0 -1 0 0 j -1 0 r 0 l + tr (S 0 i S 0 j ) , ( 45
)
from which one deduces straightforwardly Eq. ( 44) of [START_REF] Richmond | Parameter bounds on estimation accuracy under model misspecification[END_REF] with r 0 μγ (θ 0 ) . Finally, let us now derive the matrix A ( θ 0 ) for the particular scenario at hand. From (34) , we get:

[ A (θ 0 )] i, j = tr (P

0 i P 0 j -P 0 i j ) -r 0 l H -1 / 2 0 (P 0 i P 0 j + P 0 j P 0 i -P 0 i j ) -1 / 2 0 r 0 l + tr (P 0 i P 0 j + P 0 j P 0 i -P 0 i j ) -1 / 2 0 -1 / 2 0 . ( 46
)
which is the same as that given in Eq. ( 46) of [START_REF] Richmond | Parameter bounds on estimation accuracy under model misspecification[END_REF] .

Conclusion

The aim of this paper was to provide SB formulas for CES tributed data under model misspecification and thus to fill a theoretical and practical gap in the Signal Processing literature. Furthermore, we have shown that the proposed SB formulas encompass all the previously derived expressions as special cases. Moreover, these new SB formulas involved relatively slight modifications with respects to the classical counterpart obtained for Gaussian and correctly specified data models with only an expectation of some scalar functions to derive or to evaluate numerically. The practical importance of the proposed expressions is in the fact that allow us to easily evaluate Misspecified Cramér-Rao Bounds for a lot of applications that are characterized by a non-Gaussian and heavy-tailed data behaviour along with a model misspecification.

Future works will focus exactly on the application of the derived SB formulas to a plethora of engineering problems such as the Direction of Arrivals (DOA) estimation in array processing and the structured (i.e. Toeplitz) covariance/scatter matrix estimation for adaptive detection algorithm in non-Gaussian scenarios. and then it trivially satisfy the independence condition discussed in Appendix A . In order to obtain a closed form expression of the matrix B ( θ 0 ), the second term in (30) can be rewritten as:

L l=1 L m =1 E p ∂G 0 l ∂θ i ∂G 0 m ∂θ j = L l=1 E p ∂G 0 l ∂θ i ∂G 0 l ∂θ j + L l=1 L m =1 , m =l E p ∂G 0 l ∂θ i ∂G 0 m ∂θ j . (B.1)
By using (A.4) and the Stochastic Representation Theorem, when l = m we have that:

E p ∂G 0 l ∂θ i ∂G 0 m ∂θ j = E 2 Q { Q} E p z H l S 0 i z l E p z H m S 0 j z m + E Q { Q} A jm E p z H l S 0 i z l + A il E p z H m S 0 j z m + A il A jm , (B.2)
where A il is defined in (A.3) and where we used the facts that z l and z m are independent and

E p { z l } = E p { z m } = 0 . When l = m, the term in (B.
2) has to be recast as:

E p ∂G 0 l ∂θ i ∂G 0 l ∂θ j = E Q { Q 2 } E p z H l S 0 i z l z H l S 0 j z l + E Q { Q} A jl E p z H l S 0 i z l + A il E p z H l S 0 j z l + E p z H l h il h H jl z l + E p h H il z l z l h jl + A il A jl , (B.3)
where h il is defined in (A.2) and where we used the fact that the third-order moments of u l (and then of z l ) vanish (see Lemma 1 in [START_REF] Ollila | Complex elliptically symmetric distributions: survey, new results and applications[END_REF] ). Moreover, as a consequence of the circularity property of z l [START_REF] Ollila | Complex elliptically symmetric distributions: survey, new results and applications[END_REF] , the following equalities hold:

E p { z l z T l } = E p { z * l z H l } = 0 , (B.4) E p { z H l h il z H l h jl } = E p { h T il z * l z H l h jl } = 0 , (B.5) E p { h H il z l h H jl z l } = E p { h H il z l z T l h * jl } = 0 . (B.6)
From the properties of the trace operator, we have that:

E p z H l S 0 i z l = N -1 tr (S 0 i ) , (B.7) E p z H l h il h H jl z l = N -1 tr (h il h H jl ) , (B.8) E p h H il z l z l h jl = N -1 tr (h jl h H il ) , (B.9)
while, by exploiting exactly the same procedure used in [START_REF] Greco | Maximum likelihood covariance matrix estimation for complex elliptically symmetric distributions under mismatched conditions[END_REF][START_REF] Greco | Cramér-Rao lower bounds on covariance matrix estimation for complex elliptically symmetric distributions[END_REF] , it can be shown that:

E p z H l S 0 i z l z H l S 0 j z l = tr (S 0 i ) tr (S 0 j ) + tr (S 0 i S 0 j ) N(N + 1) . (B. 10
)
Another useful relation can be obtained by exploiting the equality in (27) . In fact, by combining (A.5) with (27) , we have that:

E Q { Q} E p { z H l S 0 i z l } = -tr (P 0 i ) -A il , i = 1 , . . . ,
E p ∂G 0 l ∂θ i ∂G 0 l ∂θ j = tr (P 0 i ) tr (P 0 j ) + tr (h il h H jl ) + tr (h jl h H il ) + E Q { Q 2 } N(N + 1) -1 tr (S 0 i ) tr (S 0 j ) + tr (S 0 i S 0 j ) (B.14)
Finally, by combining all the previous results, we have that:

L l=1 L m =1 E p ∂G 0 l ∂θ i ∂G 0 m ∂θ j = L 2 tr (P 0 i ) tr (P 0 j ) + L l=1 tr (h il h H jl ) + tr (h jl h H il ) + L E Q { Q 2 } N(N + 1) -1 tr (S 0 i ) tr (S 0 j ) + L E Q { Q 2 } N(N + 1) tr (S 0 i S 0 j ) , (B .15)
from which (31) follows immediately. Regarding the calculation of the matrix A ( θ 0 ) in (34) , no particular simplification can be made.

Specifically, the term E p

∂ 2 G 0 l ∂ θ i ∂ θ j
in (33) has to be obtained through direct calculation from (A.1) . Since the derivation is long, tedious and does not add any insightful considerations about the problem at hand, we decided to not report it here.

Appendix C. Proof of the SB formulas for the Scenario 2

As previously discussed in the paper, in the Scenario 2 we suppose that the true marginal pdf is given by a parametric CES distribution such that p X (x l ) p X (x l ; θ) = CES N (μ l , , g) , where μ l = γ l ( θ) and = ( θ) for a given θ ∈ . The assumed pdf is itself a parametric CES distribution that share the same parametrization of the true one but may have a different density generator, i.e. f X (x l ; θ) = CES N (γ l (θ) , (θ) , w ) , θ ∈ , and possibly g(t ) = w (t) , ∀ t ∈ R + . In order to guarantee the correct identifiability of the true and the assumed CES distributions, as before, we may impose a constraint of both the modular variate, that is:

E p { (x l -μ l ) H -1 (x l -μ l ) } E p { (x l -γ l ( θ) ) H ( θ) -1 ×(x l -γ l ( θ)) } = E Q { Q l } = N, (C.1) E f θ { (x l -γ l (θ)) H (θ) -1 (x l -γ l (θ)) } = N, ∀ θ ∈ . (C.2)
Let us suppose now that, for a given true pdf p X (x l ) p X (x l ; θ) and for a given assumed pdf f X ( x l ; θ), the pseudo-true parameter vector θ 0 equates the true parameter vector θ, and consequently, from [START_REF] Parker | Methods and bounds for waveform parameter estimation with a misspecified model[END_REF] :

∂D ( p X f θ ) ∂θ i θ= θ = -E p ∂ ln f X (x ; θ) ∂θ i θ= θ = 0 , i = 1 , . . . , d. (C.3)
Under this assumption, we have that:

G 0 l (x l -γ 0 l ) H -1 0 (x l -γ 0 l ) = (x l -γ l ( θ)) H ( θ) -1 (x l -γ l ( θ)) = (x l -μ l ) H -1 (x l -μ l ) Q l = d Q = R 2 , (C.4)
and consequently, the Stochastic Representation Theorem allows us to write the following equality:

x l -γ 0 l = x l -μ l = d R Tu l = d R 1 / 2 0 u l . (C.5)
where γ 0 l γ l (θ 0 ) and 0 á ( θ 0 ). It is worth to highlight that the equality chains in (C.4) and (C.5) hold true if and only if θ 0 = θ . We are quite confident that, in the context of Scenario 2 (i.e. when the misspecification is only due to a wrong assumption on the density generator), the equality θ 0 = θ always holds true. Since we have not a proof of this fact yet, we considered it as an assumption (Assumption A2 in Section 3.1 ).

As a consequence of (C.4) , the expectation of functions of G 0 l , say h (G 0 l ) , can be explicitly derived as:

E Q { h (G 0 l ) } ∞ 0 h (t) p Q (t) dt = δ -1 N,g ∞ 0 h (t) t N-1 g(t ) dt = E Q { h (Q ) } , ∀ l ∈ N . (C.6)
Since such expectation does not depend on the index l and on the unknown parameter vector θ, in the rest of this Appendix we always indicate

E Q { h (G 0 l ) } simply as E Q { h (Q ) } .
As in Appendix B for the Scenario 1, in order to obtain a closed form expression of the matrix B ( θ 0 ), the second term in (28) can be rewritten as: Let us now recall from [START_REF] Besson | On the fisher information matrix for multivariate elliptically contoured distributions[END_REF] the following three equalities:

L l=1 L m =1 E p φ(G 0 l ) φ(G 0 m ) ∂G 0 l ∂θ i ∂G 0 m ∂θ j = 2 E Q { Q φ 2 (Q ) } N L l=1 Re ∂γ 0 l ∂θ i H -1 0 ∂γ 0 l ∂θ j + L E Q { Q 2 φ 2 (Q ) } N(N + 1)
tr (P i ) tr (P j ) + tr (P i P j )

+ L (L -1)

E 2 Q { Q φ(Q ) } N 2
tr (P i ) tr (P j ) .

(C.12)

Let us now impose the equality in (27) . In particular, we have that:

L l=1 E p φ(G 0 l ) ∂G 0 l ∂θ i = -E Q { Q φ(Q ) } L l=1 E p × ∂γ 0 l ∂θ i H -1 / 2 0 u l + u H l -1 / 2 0 ∂γ 0 l ∂θ i -E Q { Q φ(Q ) } L l=1 E p u H l P 0 i u l = -LN -1 E Q { Q φ(Q ) } tr (P i ) , (C.13)
where we used the fact that the fact that E p { u l } = 0 while the last equality follows from (C.10) . Then, by exploiting the equality (27) , we obtain the following relation:

E Q { Q φ(Q ) } = -N, (C.14)
where φ(t) = w (t ) /w (t ) and w ( t ) is the density generator of the assumed CES distribution f X ( x l ; θ). In particular, all the derivatives have to be evaluated from (22) , while, to evaluate the expectation operator w.r.t. the true distribution p X ( x ), one has to use the Stochastic Representation in (C.5) by keeping in mind that θ = θ 0 . Since this calculation is tedious and not informative, we decided to not report it here.

  obtained by firstly taking the derivative of the terms in (22) and then by substituting in the obtained expression the Stochastic Representation of the difference vector x l -γ 0 l in (C.5) , we have that:

E p { u Hl

  P i u l u H l P j u l } = tr (P i ) tr (P j ) + tr (P i P j ) u H l P i u l } = E p { tr (P i u l u H l ) } = N -1 tr (P i ) (C.10) and E p { (u H l a )(u H l Du l ) } = 0 (C.11) for some vector a and hermitian matrix D . By substituting (C.8) in (C.7) and by exploiting the equalities (C.9) -(C.11) , and remembering that E p u l u H l = (1 /N) I (see Section 2 ), we get:

  Finally, by substituting (C.14) in (C.12) , and then replacing the obtained term in (28) , we get the closed form expression of the matrix B ( θ 0 ) given in (35) . The evaluation of the matrix A ( θ 0 ) in (36) follows directly from the direct calculation of the terms E p φ (G 0 l )

  then, from (B.7) and from the identifiablity condition for CES distribution assumed in this paper, i.e. E Q { Q} = N, we obtain: Using (B.11) and the condition E Q { Q} = N (and consequentlyE 2 Q { Q} = N 2 ), the term in (B.2) can be easily evaluated as:

	A il = -tr (P 0 i ) -tr (S 0 i	) , i = 1 , . . . , d.	(B.12)
	E p	∂G 0 l ∂θ i	∂G 0 m ∂θ j	= tr (P 0 i ) tr (P 0 j ) .	(B.13)
	Similarly, using (B.12), (B.8), (B.9), (B.10) and the condition E Q { Q} =
	N, the term in (B.3) can be expressed as:

d, (B.11) and

Acknowledgement

The work of Stefano Fortunati has been supported by the Air Force Office of Scientific Research under award number FA9550-17-1-0065 .

The work of M. N. El Korso and A. Renaux has been supported by ANR-Astrid MARGARITA.

Appendix A. Some considerations on the expectation of the terms V ij ( θ 0 ) and H ij ( θ 0 )

The aim of this appendix is to show under which conditions the expectations E p { V ij ( θ 0 )} and E p { H ij ( θ 0 )} could be evaluated in closed form.

Keeping in mind the Stochastic Representation of a CES distributed random vector x l given in (2) , let us define the vector z l = Tu l . Then, by recalling that r 0 l μ l -γ 0 l (see (32) ), the term 22) can be rewritten as:

where S 0 i is defined in (23) . Let us define the following vector and scalar quantities:

and

Note that, according to the Stochastic Representation Theorem [START_REF] Ollila | Complex elliptically symmetric distributions: survey, new results and applications[END_REF] , we have that

By exploiting the expression in (A.4) , we have that:

It must be noted now that, since G 0 l has been defined in [START_REF] Greco | Cramér-Rao lower bounds on covariance matrix estimation for complex elliptically symmetric distributions[END_REF] as,

the random variable φ(G 0 l ) and the other random quantities Q 1 / 2 l and z l are mutually dependent and consequently the closed form evaluation of the expectation operator in (A.5) is not feasible. Similar considerations hold true for all the other terms involving the expectation operator w.r.t. the true pdf p X ( x ), that are (24) . There are, however, two important cases in which φ(G 0 l ) is statistically independent of Q 1 / 2 l (and consequently of Q l ) and z l : these are the Scenario 1 and the Scenario 2 which are discussed in the following Appendices.

Appendix B. Proof of the SB formulas for the Scenario 1

As discussed in Section 3.2 , in the Scenario 1, the true marginal pdf is an unspecified CES pdf p X (x l ) = CES N (μ l , , g) , while the assumed one is a parametric complex Gaussian pdf f X (x l ; θ) = CN (γ l (θ) , (θ)) . As a consequence, φ(G l (θ)) ≡ -1 , ∀ θ ∈


Laurent Capocchi
email: capocchi@univ-corse.fr

Fabrice Bernardi
email: bernardi@univ-corse.fr

Dominique Federici
email: federici@univ-corse.fr

Paul Bisgambiglia

Transformation of VHDL Descriptions into DEVS Models for Fault Modeling *

Keywords: VHDL descriptions, DEVS formalism, modeling, simulation, fault

We propose in this article an approach for the transformation of VHDL descriptions into DEVS models for an easy and fast fault simulation. VHDL allows description of the structure of a design, that is how it is decomposed into sub-designs, and how those designs are interconnected. The specification of the function of designs are performed using familiar programming language forms. One of the main problems is that today tools are unable to quickly and easily create and simulate fault models directly from the VHDL descriptions. A way to solve this problem is to encapsulate these descriptions in easily simulable and evolutive models. We propose to use the DEVS formalism to achieve this encapsulation.

Introduction

Tests are a very important part in the digital systems design process. Fault modeling is one part of the tests that can be performed. We call "fault" the detection of an incorrect step, process or data definition in the design process [START_REF] Sieh | Comparing different fault models using verify[END_REF][START_REF] Celeiro | Vhdl fault simulation for defectoriented test and diagnosis of digital ics[END_REF]. A fault model identifies targets for testing and makes analysis possible.

VHDL is an hardware description language widely used in the digital systems industry [START_REF] Ashenden | The Designer Guide to VHDL[END_REF]. It allows description of the structure of a design, that is how it is decomposed into sub-designs, and how those designs are interconnected. The specification of the function of designs are performed using familiar programming language forms. One of the main problems is that today tools are unable to quickly and easily create and simulate fault models directly from the VHDL descriptions. A way to solve this problem is to encapsulate these descriptions in easily simulable and evolutive models. We propose to use the DEVS formalism to achieve this encapsulation.

Introduced by B.P. Zeigler in the early 70's, DEVS is a set-theoretic formalism that provides a mean of modeling discrete event systems in a hierarchical and * 0-7803-7952-7/03/$17.00 c 2003 IEEE.

modular way [START_REF] Zeigler | Theory of the Modeling and Simulation, 2nde Edition[END_REF]. Using this formalism, we can perform more easy modeling in decomposing a large system into smaller component models with coupling specifications between them. DEVS defines two kinds of models: atomic models and coupled models. An atomic model is a basic model with specifications for the dynamics of the model. It describes the behavior of a component, which is indivisible, in a timed state transition level. Coupled models tell how to couple several component models together to form a new model. This kind of model can be employed as a component in a larger coupled model, thus giving rise to the construction of complex models in a hierarchical fashion.

Our choice for DEVS has been motivated by three main points. First, digital systems can be represented by discrete event systems, and DEVS is the leading formalism in discrete event modeling and simulation. Second, a great advantage of DEVS is that it separates the modeling and simulation parts in a very efficient way, since the simulator is built directly from the model. Last, DEVS uses a modular approach that allows an easy introduction of the fault models [START_REF] Federici | Behavioral fault simulation: Implementation and experiment results[END_REF].

In a first section, we describe the VHDL language, and the DEVS formalism in a second one. The main part of this article is the third section. We propose a modeling approach for the VHDL descriptions based on the definition of four basic DEVS atomic models. Finally, we propose an example of modeling and simulation and we conclude this article.

VHDL Fault Modeling

VHDL allows to describe a digital structure following two types of descriptions: the structural description and the behavioural description. In the structural description, the design is described as a module with inputs and/or outputs. The electrical values on the outputs are some function of the values on the input. One way to represent the function of the module is to describe how it is composed of sub-models. However, in many cases, it is not appropriate to describe a module structurally. In the behavioural description, the function is described without reference to the actual internal structure of the module. We can find two forms for the behavioural description: the data flow form or the algorithmic form. In this paper, we study only this last form, since it has been demonstrated that any combinatory structure can be described using sequential statements [START_REF] Ashenden | The vhdl-cookbook[END_REF].

A digital system in VHDL consists of a design entity containing other entities that are then considered components of the top-level entity. Each entity is modeled by an entity declaration and an architecture body. One can consider the entity declaration as the interface to the outside world that defines the input and output signals, while the architecture body contains the description of the entity and is composed of interconnected entities, processes and components, all operating concurrently. entity NAME_OF_ENTITY is generic (generic_declarations); port (signal_names: mode type; signal_names: mode type; signal_names: mode type); : end [NAME_OF_ENTITY];

The architecture body specifies how the circuit operates and how it is implemented. As discussed earlier, an entity or circuit can be specified in a variety of ways, such as behavioral, structural (interconnected components), or a combination of the above.

architecture arch_name of NAME_OF_ENTITY is --components declarations --signal declarations --constant declarations --function declarations --procedure declarations --type declarations begin --Statements end architecture_name;

The statements in the body of the architecture make use of logic operators. Logic operators that are allowed are: and, or, nand, nor, xor, xnor and not. In addition, other types of operators including relational, shift, arithmetic are allowed as well.

The basis for sequential modeling is the process construct. A process statement is the main construct in behavioral modeling that allows you to use sequential statements to describe the behavior of a system over time. The syntax for a process statement is: A process is declared within an architecture and is a concurrent statement. However, the statements inside a process are executed sequentially. Like other concurrent statements, a process reads and writes signals and values of the interface (input and output) ports to communicate with the rest of the architecture.

[process_label:] process [(sensitivity_list)] [

DEVS Formalism

DEVS formalism introduces two kind of models, atomic models from which larger ones are built, and coupled models (also called network of models) that connects those models in a hierarchical fashion [START_REF] Zeigler | Object-Oriented Simulation with Hierarchical, Modular Models[END_REF]. Like in general systems theory, a DEVS model contains a set of states and transition functions that are triggered by the simulator.

A DEVS atomic model AM is a structure :

AM =< X, S, Y, δ int , δ ext , λ, t a >
where:

• X: {(p, v)|(p∈ input ports, v∈ Xp)} is the set of input ports and values for the reception of external events,

• Y: {(p, v)|(p∈ output ports, v∈ Yp)} is the set of output ports and values for the emission of events,

• S is the set of internal sequential states,

• δ int : S → S is the internal transition function that will move the system to the next state after the time returned by the time advance function,

• t a : S → + is the time advance function, that will give the life time of the current state (returns the time to the next internal transition),

• δ ext : Q×X → S is the external transition function that will schedule the states changes in reaction to an input event,

• λ : Q×X → S is the output function that will generate external events just before the internal transition takes places.

The dynamic interpretation is the following:

• Q = {(s, e)|(s ∈ S, 0 < e < ta(s)} is the total state set.
• e is the elapsed time since last transition, and s the partial set of states for the duration of ta(s) if no external event occur.

• δ int : the model being in a state s at ti, it will go into s', s' =δ int (s), if no external events occurs before ti + ta(s).

• δ ext : when an external event occurs, the model being in the state s since the elapsed time e goes in s', s' =δ ext (s,e,x).

-The next state depends on the elapsed time in the present state.

-At every state change, e is reset to 0.

• λ : the output function is executed before an internal transition, before emitting an output event the model remains in a transient state.

• A state with an infinite life time is a passive state (steady state), else, it is an active state (transient state). If the state s is passive, the model can evolve only with an input event occurrence.

The DEVS coupled model CM is a structure :

CM =< X, Y, D, {M d ∈ D}, EIC, EOC, IC >
where:

• X is he set of input ports for the reception of external events,

• Y is the set of output ports for the emission of external events,

• D is the set of components (coupled or basic models),

• M d is the DEVS model for each d ∈ D,
• EIC is the set of input links, that connects the inputs of the coupled model to one or more of the inputs of the components that it contains,

• EOC is the set of output links, that connects the outputs of one or more of the contained components to the output of the coupled model,

• IC is the set of internal links, that connects the output ports of the components to the input ports of the components in the coupled models.

In a coupled model, an output port from a model M d ∈ D can be connected to the input of another M d ∈ D but cannot be connected directly to itself.

VHDL Descriptions Transformations

Our basic approach for the transformation is to associate each VHDL instruction with a DEVS atomic model, and each process with a DEVS coupled model. We defined four basic atomic models kinds. The first one is the "allocation model" kind that represents the VHDL allocation instructions. We can note that this model is able to handle time delayed allocations. The second is the "junction model" kind that represents how two atomic models are linked together. The third is the "conditional model" kind that represents the classical control instructions (if, case, for,...). Finally, the last one is the "parallel model" kind that manages the parallel execution of processes. All these basic models can be easily personalized and combined in order to create complex models.

The Allocation Model

The allocation model (Figure 1) is used to describe any type of allocation that does not take into account the notion of "delay".

In Out

ALLOCATION

AM

Figure 1: The Allocation Model

The associate atomic model is the following: AM (alloc) =< X, S, Y, δ ext , δ int , λ, ta > where:

X = {("In", v)|v ∈ V } Y = {("Out", v)|v ∈ V } S = {"active", "passive"} × + 0 × {allocation} ∅ δ int (phase, σ, task) = ("passive", ∞, ∅)
δ ext (phase, σ, task, e, (In, message)) = ("active", 0, allocation), if phase = "passive" λ(phase, σ, task) = (Out, message) ta(phase, σ, task) = σ When an event occurs through the input port, the component becomes "active" using the δ ext function. Then, the main task to achieve is to evaluate the assignment statement and to poke the output message using the λ function.

The Junction Model

The junction model (Figure 3) is used to describe sequential instructions like "end if, end elsif, end case". The associate atomic model is the following: AM (junction) =< X, S, Y, δ ext , δ int , λ, ta > where:

X = {("In 0 ", v), • • • , ("In N ", v)|v ∈ V } Y = {("Out", v)|v ∈ V } S = {"active", "passive"} × + 0 × {"In 0 ", • • • , "In N "} × V δ int (phase, σ, input port) = ("passive", ∞, input port) δ ext (phase, σ, input port, e, (p, v)) if (phase = "passive") and (p ∈ {In 0 , • • • , In N }) = ("active", 0, p, v) else = ("passive", ∞, input port)
λ(phase, σ, input port) if (phase = "active") and (In 0 , message) = (Out, In 0 ) . . . if (phase = "active") and (In N , message)

= (Out, In N ) ta(phase, σ, input port) = σ
If the component is in the "passive" phase and if an event occurs on an input port, the phase becomes "ac-tive" using the δ ext function and the input port is selected. Then the λ function is actived and returns the selected input message.

The Conditional Model

The conditional model (Figure 4) occur in control structures like "if..then..else, when..else, with..select..when, case".

In

AM (conditional) =< X, S, Y, δ ext , δ int , λ, ta > where X = {("In", v)|v ∈ V } Y = {("Out 0 ", v), • • • , ("Out N ", v)|v ∈ V } S = {"active", "passive"} × + 0 × {"Sw 0 ", • • • , "Sw N "} δ int (phase, σ, Sw) = ("passive", ∞, Sw)
δ ext (phase, σ, Sw, e, (In, message)) if (phase = "passive") and (Sw = "Sw 0 ") = ("active", 0, "Sw 0 ") . . . if (phase = "passive") and (Sw = "Sw N ") = ("active", 0, "Sw N ") = ("passive", ∞, Sw) other λ(phase, σ, Sw) if (phase = "active") and (Sw = "Sw 0 ") = (Out 0 , message) . . . if (phase = "active") and (Sw = "Sw 0 ") = (Out N , message) ta(phase, σ, Sw) = σ

When an event occurs through the input port, the component becomes "active" using the δ ext function. Then, the main task to achieve is to evaluate the conditional statement and to poke the output message into the selected port using the λ function. We can note that it exists as many output ports than values into the conditional statement.

The Parallel Model

The parallel model (Figure 6) is used in order to synchronize the parallel execution of the processes. It is also used in order to manage the symbolic time.

In1 InN

Out1

In2 In3

OutFeedback_N OutM OutFeedback_1

PARALLEL AM Figure 6: The Parallel Model This model is described by the following atomic model:

AM (parallel) =< X, S, Y, δ ext , δ int , λ, ta > where X = {("In 0 ", v), • • • , ("In N ", v)|v ∈ V } Y = {("OutF eedback 0 ", v), • • • , ("OutF eedback N ", v)| v ∈ V } ∪ {("Out 0 ", v), • • • , ("Out M ", v)|v ∈ V } S = {"active", "passive"} × + 0 × + 0
×{"physique", "symbolique"} × {"P 0", "P 1", ...} δ int (phase, σ, c, status, process) if phase = "active" = ("passive", ∞, activeprocesses, "physical", null) else = (phase, ∞, c, "physical", null) δ ext (phase, σ, c, status, process, e, (p, v)) if (c = 0) and (process! = [])

= ("active", 0, length(process), status, process) if (c = 0) and (process = []) = ("passive", 0, n output ports -1, status, process) if (c! = 0) = ("passive", ∞, c, status, process) λ(phase, σ, c, status, process) if phase = "active" = {(OutF eedback 0 , msg),

• • • , (OutF eedback N , msg)} else = {(Out 0 , mesg), • • • , (Out M , mesg)} ta(phase, σ, c, status, process) = σ
In this model, we add a new state variable called "c" (for "counter") that permits to count the number of active processes during a VHDL delta period. This variable is reduced each time a new message occurs during an active phase. The state variable "process" is a list defining the active processes. It is used in the λ function in order to activate the feedback output ports toward the processes. This list is built in the external transition function by testing the sensitive signal stationarity. Finally, the "status" variable allows to distinguish between the physical (with a VHDL physical simulation time) or symbolical (with a VHDL symbolical delta time) modes.

Experiments and Results

As an example, we present the modeling of a 8-bit register. This register is composed by three processes, each of them defining a "Wait" instruction. We chose to present this example since it shows the parallelism notion.

The VHDL description is the following code: We can see in this code the three processes ("strobe", "enable" and "output").

entity
The first process is modeled using three atomic models: a conditional model for the "if" statement (CM0), an allocation model for the assignment statement (AM0) and a junction model for the "endif" statement (JM0).

The second process is modeled using only one allocation model (AM1) since it defines only one instruction.

Finally, the third process is modeled using four atomic models: a conditional model (CM1) for the "if" statement, two allocation models (AM2 and AM3) and a junction model (JM1) for the "endif" statement.

At the highest level, the coupled model associated with the register can be described in the following way: The simulation has been performed using the Python-DEVS simulator package [START_REF] Bolduc | A modeling and simulation package for classic hierarchical devs[END_REF] developped at the McGill University and the results are showned in Table 1. This table presents the signal evolutions of the register, for each physical time step of the simulation. We show also the internal cycles of the simulation denoted with a "+".

M C (register) =< X, Y, D, M d |d ∈ D, EIC, EOC, IC > where X = ∅ Y = {(OU T, v)|v ∈ V } D = {M C strobe , M C enable , M C output , M A} EIC = ∅ EOC = {(M A, Out1), (M C register , OU T )} IC = {((M A, OutF 1 ), (M C strobe , IN )), ((M A, OutF 2 ), (M C enable , IN )), ((M A, OutF 3 ), (M C output , IN )) ((M C strobe , Out), (M A, In 1 )), ( (
We can also point out that the simulation time step is not regular. For example, we pass directly from time "4" to time "6", and from time "6" to time "8". This is because we can predict the simulation time for the process activations since we know the input events.

Conclusion and Perspectives

This paper was devoted to the presentation of our approach to transform VHDL descriptions into DEVS models. We defined four basic kinds of atomic models and associated a process to a coupled model. We only presented in this paper the modeling and the simulation of a 8-bit register, but we performed also a successful modeling and simulation of the ITC'99 benchmarks. We have one main perspective for this work: we want to add a fault model to our descriptions in order to be able to perform an easy and fast fault simulation [START_REF] Kofman | Fdevs: A general devs-based formalism for fault modeling and simulation[END_REF]. We would also like to develop an integrated modeling and simulation environment dedicated to this task.

Figure 2

 2 Figure 2 presents the DEVS trajectories of an allocation model.When an event occurs through the input port, the component becomes "active" using the δ ext function. Then, the main task to achieve is to evaluate the assignment statement and to poke the output message using the λ function.

Figure 3 :

 3 Figure 2: Allocation model trajectories

Figure 4 :

 4 Figure 4: The Conditional Model It is described by the following atomic model:

Figure 5 :

 5 Figure 5: Conditional model trajectories

Table 1 :

 1 M C enable , Out), (M A, In 2 )), ((M C output , Out), (M A, In 3 ))} Figure 7 presents a representation of the modeled register. A circle corresponds to an allocation model, a diamond corresponds to a conditional model and a rectangle to a junction model. We can see a new atomic model called "PM" which is a parallel model. The main coupled model representing the register presents one output Simulation Resultsport, but no input ports, since, for the needs of the simulation, we insert a new atomic model acting as an event generator.

	CLK	D0	DI	STRB enbld	reg	NDS2 DS1
	0	0	00000000	0	1	1	0	0
	0+	11111111 00000000	0	0	1	0	0
	1	11111111 11100111	1	0	1	0	1
	1+	11111111 11100111	1	1	11100111	0	1
	2	11100111 00001111	1	1	11100111	1	1
	2+	11100111 00001111	1	0	11100111	1	1
	3	11111111 11110000	0	0	11100111	1	1
	4	11111111 10101010	0	0	11100111	1	0
	6	11111111 11111111	1	0	11100111	0	0
	6+	11111111 11111111	1	0	11111111	0	0
	8	11111111 00000000	0	0	11111111	1	0


Laurent Capocchi
email: capocchi@univ-corse.fr

Dominique Federici
email: federici@univ-corse.fr

Fabrice Bernardi
email: bernardi@univ-corse.fr

Paul Bisgambiglia

Behavioral Fault Simulation for VHDL Description using DEVS Formalism

INTRODUCTION

Due to the ever-increasing complexity of VLSI circuits, the use of VHDL behavioral descriptions (see [START_REF] Ashenden | The vhdl-cookbook[END_REF] for more explanations) in the fields of test generation and fault simulation becomes advised. Test generation tools research has been oriented for the last fifteen years towards generation tools for circuits modeled at a high abstraction level according to a behavioral view as shown by [START_REF] Ferrandi | Implicit test generation for behavioral VHDL models[END_REF][START_REF] Corno | Testability analysis and ATPG on behavioral RT-level VHDL[END_REF][START_REF] Ferrandi | Testing Core-Based Systems: A Symbolic Methodology[END_REF][START_REF] Oneill | Btg : a behavioral test generator[END_REF][START_REF] Norrod | An automatic test generation algorithm for hardware description language[END_REF][START_REF] Hummer | Functional tests for hardware derived from vhdl description[END_REF][START_REF] Courbis | Automatic test pattern generation for digital circuits[END_REF].

The aim of the test pattern generation process is to define patterns to apply on primary inputs. This is used to test eventual physical defects, eg. the defects can be detected only if they induce an irregular behavior called a fault. The fault effect or error is measured by a difference between the state of the fault-free model (reference model) and the state of the faulty model (model in which a fault hypothesis is injected). The fault simulation process consists of simulating a circuit in the presence of faults, and comparing the results of fault simulation with the fault free simulation of the same circuit with the same input test pattern. J. Lee and al. defined in [START_REF] Lee | Architectural level fault simulation using symbolic data[END_REF] a fault simulator for behavioral descriptions; the inconvenient is that the circuit description is not purely behavioral. In their approach described in [START_REF] Ward | Behavioral fault simulation in vhdl[END_REF], P.C. Ward and J.R. Armstrong proposed a Behavioral Fault Simulation (BFS) method based on simulation of behavioral VHDL descriptions. However this fault simulation process involves as many VHDL simulations as number of fault hypothesis.

Thus, one of the main problems is that today the tools for test generation are unable to quickly and easily create and simulate behavioral fault models directly from the VHDL descriptions. A way to solve this problem is to encapsulate these descriptions in easily simulable and evolutive models using DEVS formalism [START_REF] Zeigler | Theory of the Modeling and Simulation, 2nde Edition[END_REF] and to define a Behavioral Fault Simulator based on fast fault list propagation technique allowing the reduction of the number of VHDL simulation.

TRANSFORMATION OF VHDL DESCRIP-TIONS INTO DEVS MODELS

Introduced by B.P. Zeigler in the early 70's, DEVS is a settheoretic formalism that provides a mean of modeling discrete event systems in a hierarchical and modular way [START_REF] Zeigler | Theory of the Modeling and Simulation, 2nde Edition[END_REF].

Using this formalism, we can perform more easy modeling in decomposing a large system into smaller component models with coupling specifications between them. DEVS defines two kinds of models: atomic models and coupled models. An atomic model is a basic model with specifications for the dynamics of the model. It describes the behavior of a component, which is indivisible, in a timed state transition level. Coupled models tell how to couple several component models together to form a new model. This kind of model can be employed as a component in a larger coupled model, thus giving rise to the construction of complex models in a hierarchical fashion.

Our choice for DEVS has been motivated by three main points. First, digital systems can be represented by discrete event systems, and DEVS is the leading formalism in discrete event modeling and simulation. Second, a great advantage of DEVS is that it separates the modeling and simulation parts in a very efficient way, since the simulator is built directly from the model. Last, An important feature of DEVS is the possibility to define, in addition to the model scheme, the way the various components interact in the time evolution. We can easily add a desired faulty behavior to our models by changing the transition and output functions and by adding a new fault transition function.

Our basic approach for the transformation is to associate each VHDL sequential statements with a DEVS atomic model, and each process with a DEVS coupled model. We defined a library composed by four basic atomic models. The first one is the "allocation model" kind that represents the VHDL allocation instructions. We can note that this model is able to handle time delayed allocations. The second is the "junction model" kind that represents how two atomic models are linked together or classical control instructions (end if, end case). The third is the "conditional model" kind that represents the classical control instructions (if, case, for,...). Finally, the last one is the "parallel model" kind that manages the parallel execution of processes. All these basic models can be easily personalized and combined in order to create complex behavioral models.

BEHAVIORAL FAULT SIMULATION US-ING FAULT LIST PROPAGATION TECH-NIQUE

A fault modeling scheme allows to derive a set of fault hypothesis on the previously described model. Fault hypothesis are defined according to the elements involved in the model of the circuit under test. In order to have a behavioral fault model for which some measures of confidence are provided, we select fault hypothesis according to the fault model proposed in [START_REF] Ghosh | On behavior fault modeling for digital designs[END_REF]. The selected fault hypothesis are classified into the following three groups:

• F1 : Stuck-at of a signal or variable object.

• F2 : Faults on conditional model. F2t (resp. F2f) : the true branch (resp. the false branch) of the conditional model is always selected whatever the resulting value implies on the end reporting value.

• F3 : Jump of statement instruction.

The first step for BFS is performed by the parser and consists in determining the global fault list. In the second step, we have to choose the test sequence which will be applied at the input of the circuit under test. It is obtained using the pseudo-random generator. The third step is the fault simulation. The free simulation is performed concurrently with the fault simulation thanks to DEVS transformation of the circuit under test.

The fault propagation is achieved using fault lists that are propagated through the DEVS model associated with the VHDL description under test. The propagation is performed according to the rules detailled below:

• Principle 1 -Generalities : The fault lists propagation is driven by the architecture of DEVS model.

• Principle 2 -Assignment : During the simulation with fault list propagation, when an assignment model (D1 ≤ E(Di) with E is a function with VHDL operators) is found, the fault list associated to D1 is determined using the list associated to Di representing the locally observable faults on Di, the F3 type fault on D1 and the D1 list of faults calculated at the last cycle.

• Principle 3 -Selection: In the case of a selection model, the fault free path and the faulty path are determined. We put in no-safe ports all the faults involving a faulty path. The fault lists are propagated along the fault free path according to the principles 1, 2 and 3.

EXPERIMENTS AND RESULTS

The proposed BFS approach have been implemented in python and evaluated on behavioral VHDL descriptions designs of the ITC'99 benchmarks [START_REF] Davidson | Itc?99 benchmark circuits, preliminary results[END_REF].

Table 1 shows the following information : the number of behavioral faults, the length of the test sequence obtained by the pseudo-random generator, the detected faults and the achieved behavioral fault coverage.