Tao Bao

Mohammed Nabil

E L Korso

Arnaud Breloy

Conditional and Unconditional Deterministic Bounds on the MSE of the Non-Uniform Linear Co-centered Orthogonal Loop and Dipole Array

The co-centered orthogonal loop and dipole (COLD) array exhibits some interesting properties, which makes it ubiquitous in the context of polarized source localization. In the literature, one can find a plethora of estimation schemes adapted to the COLD array. Nevertheless, their ultimate performance in terms the so-called threshold region of mean square error (MSE), have not been fully investigated. In order to fill this lack, we focus, in this fast communication, on conditional and unconditional bounds that are tighter than the well known Cramér-Rao Bound (CRB). More precisely, we give some closed form expressions of the McAulay-Hofstetter, the Hammersley-Chapman-Robbins, the McAulay-Seidman bounds and the recent Todros-Tabrikian bound, for both the conditional and unconditional observation model. Finally, numerical examples are provided to corroborate the theoretical analysis and to reveal a number of insightful properties. II.

For the rest of this paper, the following notation will be used. A lowercase bold letter denotes a vector, and an uppercase bold letter denotes a matrix. Vectors are by default in column orientation unless specified. Upper scripts T , C and H are, respectively, the transpose, the conjugate and the trans-conjugate of a matrix. The operators tr {.} , |.|, . and R represent, respectively, the trace, the determinant of a matrix, the Euclidean norm and the real part. and ⊗ are the Hadamard and the Kronecker product, respectively. I L denotes the L × L identity matrix. 1 L is the L × L matrix filled by ones. [.] i and [.] i,j denote the i-th element of the vector and the i-th row and the j-th column element of the matrix. We define also the n-th norm as η n = ||η|| n . Finally, ξ is any an unbiased estimate of ξ.

IV. SYSTEM MODELS

Consider a non-uniform linear array composed of N COLD pairs with inter-element spacing d 1 .d, d 2 .d, . . . , d N .d that receives a signal emitted by a single far-field and narrow-band source. Let d n .d denotes the location of the n-th COLD sensor, in which d is unit. The array is collinear with the y-axis of an (O, xyz) coordinate system with its origin, O, in the center of the first pair of sensors (i.e., d 1 = 0). For each COLD sensor, the dipole is parallel to the z-axis and the loop is parallel to the x -y plane.

Assume a narrowband far-field source impinges on the array from directions described by the elevation angle φ and the azimuth angle θ. In this paper, we suppose that the source is contained in the x -y azimuthal plane, i.e. φ = π 2 , as illustrated in Fig. 1. For a given polarized signal, the vertical and the horizontal components of the incoming signal electric field can be specified by polarization state parameters ρ ∈ 0, π 2 and ψ ∈ [-π, π]. Consequently, the output of the n-th COLD sensor can be expressed by [START_REF] Li | Angle and polarization estimation using esprit with a polarizationsensitive array[END_REF] x

n (t) = [x loop , x dipole ] T = s(t)u(ψ, ρ) e jωdn.d + n n (t), n = 1, . . . , N, t = 1, . . . , T (1)
in which x loop and x dipole are the signals recorded on the small loop and the short dipole, respectively. The electrical angle ω = 2πd λ sin θ, where λ is the signal wavelength. T is the number of snapshots and s(t) is the source signal. The random process n n (t) = [n loop , n dipole ] T denotes a complex Gaussian circular noise with zero mean and a known covariance matrix Σ noise . The output vector received for the t-th snapshot can be written as

x(t) = [x T 1 (t), . . . , x T N (t)] T = s(t)e(ω) ⊗ u(ψ, ρ) + [n T 1 (t), . . . , n T N (t)]
T , where the steering vector is defined by e(ω) = [1, e jd2ω , . . . , e jd N ω ] T and the 2 × 1 polarization state vector is given by u(ψ, ρ) = [j2π A sl cos(ρ) λ , -L sd sin(ρ)e jψ ] T , in which L sd and A sl represent the length of the short dipole and the area of the small loop. From a modeling point of view, we can assume L sd = 2π A sl λ = 1. In the following, the unknown parameter vector is given by ξ = [ω, ψ, ρ] T , whereas, ξ 0 , ω 0 , ψ 0 and ρ 0 denote the real value of the candidate parameters ξ, ω, ψ and ρ, respectively. The joint probability distribution function (pdf) of the full observations χ = [x T (1), . . . , x T (T )] ∼ CN (µ(ξ 0 ), Σ(ξ 0 )) for a given ξ 0 , is expressed as follow

p(χ|ξ 0 ) = 1 |Σ(ξ 0 )|π 2N T e -(χ-µ(ξ 0 )) H Σ(ξ 0 ) -1 (χ-µ(ξ 0 )) (2)
Let E ( ξ0 -ξ 0 )( ξ0 -ξ 0 ) T be the covariance matrix of an estimate of ξ 0 . Let us assume that ξ is an asymptotically unbiased estimate of the true parameter vector ξ 0 , and define the CRB for the considered model [START_REF] Smith | Statistical resolution limits and the complexified Cramér Rao bound[END_REF]- [START_REF] El Korso | Statistical resolution limit for the multidimensional harmonic retrieval model: Hypothesis test and CramRao bound approaches[END_REF]. The covariance inequality principle states that, under quite general/weak conditions [START_REF] Cramér | Mathematical Methods of Statistics[END_REF],

MSE = E ([ ξ0 ] i -[ξ 0 ] i ) 2 ≥ CRB([ξ 0 ] i ),
where the CRB is given as the inverse of the Fisher information matrix (FIM) as CRB([ξ 0 ] i ) = [FIM -1 (ξ 0 )] i,i . Since we are working with a complex circular Gaussian observation model and using the Splepian-Bang formula [START_REF] Kay | Fundamentals of Statistical Signal Processing : Estimation Theory[END_REF], [START_REF] Mennad | Slepian-bangs-type formulas and the related misspecified cramrao bounds for complex elliptically symmetric distributions[END_REF], the i-th, k-th element of the FIM for the unknown real parameter vector ξ 0 can be written as

[FIM(ξ 0 )] i,k = tr Σ(ξ 0 ) -1 ∂Σ(ξ) ∂[ξ] i ξ=ξ 0 Σ(ξ 0 ) -1 ∂Σ(ξ) ∂[ξ] k ξ=ξ 0 + 2R ∂µ H (ξ) ∂[ξ] i ξ=ξ 0 Σ(ξ 0 ) -1 ∂µ(ξ) ∂[ξ] k ξ=ξ 0 (3)
There exist two different models depending on the assumptions about the signal sources, the conditional (deterministic) and unconditional (stochastic) cases.

A. The Conditional Case

The time-varying signal is modelled by s(t) = a(t)e j(2πf0t+γ(t)) , where a(t) denotes the real amplitude, γ(t) is the timevarying modulating shift phase and f 0 is the carrier frequency of the incident wave. Under this assumption, one has the following parameterized mean model given by χ ∼ CN (µ(ξ 0 ), Σ noise ), and in which

µ(ξ 0 ) = s ⊗ (e(ω 0 ) ⊗ u(ψ 0 , ρ 0 )) (4)
where s = [s(1), . . . , s(T )] T . Consequently, the FIM in (3) reduces to

[FIM(ξ 0 )] i,k = 2R ∂µ H (ξ) ∂[ξ] i ξ=ξ 0 Σ(ξ 0 ) -1 ∂µ(ξ) ∂[ξ] k ξ=ξ 0 (i, k) ∈ 1, 2, 3 2 (5)

B. The Unconditional Case

In the unconditional model, the signal is assumed to be complex circular Gaussian (with zero mean and variance σ 2 s I) independent from the noise. Consequently, the covariance parameterized observation model given by χ ∼ CN (0, Σ(ξ 0 )), in which

Σ(ξ 0 ) = σ 2 s (e(ω 0 ) ⊗ u(ψ 0 , ρ 0 ))(e(ω 0 ) ⊗ u(ψ 0 , ρ 0 )) H + Σ noise (6)
Then, by applying (3), one obtains

[FIM(ξ 0 )] i,k = T tr Σ(ξ 0 ) -1 ∂Σ(ξ) ∂[ξ] i ξ=ξ 0 Σ(ξ 0 ) -1 ∂Σ(ξ) ∂[ξ] k ξ=ξ 0 , i, k = 1, 2, 3. (7)
V. DETERMINISTIC LOWER BOUNDS BACKGROUND AND DERIVATION

The unification presented in [START_REF] Todros | General classes of performance lower bounds for parameter estimation Part I: Non-bayesian bounds for unbiased estimators[END_REF] gives analytical expressions of the McAulay-Seidman bound (MSB), the Hammersley-Chapman-Robbins bound (HCRB), the McAulay-Hofstetter bound (MHB) and the Todros-Tabrikian Bound (TTB) as follows:

C (L) MSB = ΦΨ -1 Φ T (8)
where

Φ = [ξ 1 -ξ 0 , . . . , ξ L -ξ 0 ] in which ξ l denotes the l-th test point for l = 1, . . . , L. Whereas, [Ψ] m,n = E χ|ξ 0 {ν(χ, ξ m )ν(χ, ξ n )} for m = 1, . . . , L, n = 1, .
. . , L and ν(χ, ξ l ) denotes the ratio-likelihood function, given by ν(χ, ξ l ) = p(χ|ξ l ) p(χ|ξ 0 ) . Furthermore, the HCRB and the MHB are given by

C (L) HCRB = Φ(Ψ -1 L 1 T L ) -1 Φ T (9)
and

C (L) MHB = C CRB + QR -1 Q T ( 10
)
where

Q = C CRB D -Φ, R = Ψ -D T C CRB D (11)
in which D = [d(ξ 1 ), . . . , d(ξ L )],
and each element of D is given by

d(ξ l ) = ∂KLD (p(χ|ξ l ) p(χ|ξ)) ∂ξ | ξ=ξ 0 T (12)
The term KLD(p(χ|ξ l ) p(χ|ξ)) denotes the Kullback-Leibler divergence [START_REF] Kullback | On information and sufficiency[END_REF] of p(χ|ξ) from p(χ|ξ l ). Finally, the TTB is given by

C (L,J) TTB = C CRB + QW H W RW H -1 W Q T (13)
in which, the three-dimensional-discrete-Fourier-transform (DFT) matrix is given by [START_REF] Todros | General classes of performance lower bounds for parameter estimation Part I: Non-bayesian bounds for unbiased estimators[END_REF] [W ] i,l = e -jΩ T i ξ l [START_REF] Todros | General classes of performance lower bounds for parameter estimation Part I: Non-bayesian bounds for unbiased estimators[END_REF] in which Ω i is expressed for the i-th frequency test bin by

Ω T i = 2π iω ∆ωLω , i ψ ∆ ψ L ψ , iρ
∆ρLρ , in which L ω , L ψ and L ρ represent the number of test points w.r.t. the unknown parameters ω, ψ and ρ, and L = L ω L ψ L ρ . The uniform inter-test points w.r.t. the unknown parameters are represented by ω, ψ and ρ ∆ ω , ∆ ψ and ∆ ρ , respectively. Similarly, i ω , i ψ and i ρ denote the index of test-bin in the three-dimensional of the frequency domain,

i ω ∈ 1, . . . , L ω , i ψ ∈ 1, . . . , L ψ and i ρ ∈ 1, . . . , L ρ .
In the remaining of this section, we derive analytical expressions of the MSB, the HCRB, the MHB and the TTB for conditional and unconditional observation model.

A. The Conditional Case

Let us partition the FIM w.r.t. the signal parameter ξ 0 as follow

FIM(ξ 0 ) =   F ω,ω F ω,ψ F ω,ρ F ψ,ω F ψ,ψ F ψ,ρ F ρ,ω F ρ,ψ F ρ,ρ   ( 15
)
in which, we used the notation F u,v , where the lower script u, v denotes the considered part of the FIM which corresponds to the derivation according to parameters u and v as shown in [START_REF] Krim | Two decades of array signal processing research: the parametric approach[END_REF]. Using [START_REF] Li | Angle and polarization estimation using esprit with a polarizationsensitive array[END_REF], the entries of the FIM are given by

F ω,ω = 2 (s ⊗ (je (ω 0 ) ⊗ u(ψ 0 , ρ 0 ))) H Σ -1 noise (s ⊗ (je (ω 0 ) ⊗ u(ψ 0 , ρ 0 ))) (16)
F ψ,ψ = 2 s H ⊗ e(ω 0 ) ⊗ ∂u(ψ, ρ 0 ) ∂ψ H | ψ=ψ0 Σ -1 noise s H ⊗ e(ω 0 ) ⊗ ∂u(ψ, ρ 0 ) ∂ψ | ψ=ψ0 (17)
and

F ρ,ρ = 2 s H ⊗ e(ω 0 ) ⊗ ∂u(ψ 0 , ρ) ∂ρ H | ρ=ρ0 Σ -1 noise s ⊗ e(ω 0 ) ⊗ ∂u(ψ 0 , ρ) ∂ρ | ρ=ρ0 (18)
in which e (ω 0 ) = [d 1 e jd1ω0 , . . . , d N e jd N ω0 ] T . The cross terms are given by

F ψ,ρ = F ρ,ψ = 2 s ⊗ e(ω 0 ) ⊗ ∂u(ψ, ρ 0 ) ∂ψ H | ψ=ψ0 Σ -1 noise s ⊗ e(ω 0 ) ⊗ ∂u(ψ 0 , ρ) ∂ρ | ρ=ρ0 (19)
F ω,ρ = F ρ,ω = 2 (s ⊗ (je (ω 0 ) ⊗ u(ψ 0 , ρ 0 ))) H Σ -1 noise s ⊗ e(ω 0 ) ⊗ ∂u(ψ 0 , ρ) ∂ρ | ρ=ρ0 (20)
and

F ω,ψ = F ψ,ω = 2 (s ⊗ (je (ω 0 ) ⊗ u(ψ 0 , ρ 0 ))) H Σ -1 noise s ⊗ e(ω 0 ) ⊗ ∂u(ψ 0 , ρ) ∂ρ | ρ=ρ0 (21)
In particular, if n n (t) is a complex circular white Gaussian noise with zero-mean and unknown variance σ 2 n , assumed to be uncorrelated both temporally and spatially (i.e., Σ noise = σ 2 n I 2N T ). The above expressions simplifies to

F ω,ω = 2 σ 2 n ||d n || 2 2 ||a|| 2 2 , F ψ,ψ = 2N σ 2 n sin(ρ 0 ) 2 ||a|| 2 2 and F ρ,ρ = 2N σ 2 n ||a|| 2 2 .
Whereas, the cross terms are given by

F ω,ψ = F ψ,ω = 2 σ 2 n sin(ρ 0 ) 2 ||d n || 2 1 ||a|| 2 2 and F ω,ρ = F ρ,ω = F ρ,ψ = F ψ,ρ = 0.
Considering an identifiable situation in which |FIM| = 0, one obtains after some calculs

CRB(ξ 0 ) = σ 2 n 2||a|| 2 2    N N ||dn|| 2 2 -sin(ρ0) 2 ||dn|| 4 1 - ||dn|| 2 1 N ||dn|| 2 2 -sin(ρ0) 2 ||dn|| 4 1 0 - ||dn|| 2 1 N ||dn|| 2 2 -sin(ρ0) 2 ||dn|| 4 1 ||dn|| 2 2 N sin 2 (ρ0)||dn|| 2 2 -sin(ρ0) 2 ||dn|| 4 1 0 0 0 1 N    (22)
On the other hand, in [START_REF] Chaumette | A class of weiss-weinstein bounds and its relationship with the bobrovsky-mayer-wolf-zakai bounds[END_REF] the KLD for conditional case is given by

KLD(p(χ|ξ l )||p(χ|ξ)) = p(χ|ξ l ) ln( p(χ|ξ l ) p(χ|ξ) )dχ = (µ(ξ) -µ(ξ l )) H Σ -1 noise (µ(ξ l ) -µ(ξ)) (23)
thus,

d(ξ l ) = - ∂KLD(p(χ|ξ l )||p(χ|ξ)) ∂ξ T | ξ=ξ 0 (24) = -2 (µ(ξ) -µ(ξ l )) H Σ -1 noise ∂µ(ξ) H ∂ξ T | ξ=ξ 0 (25)
Using, [START_REF] Chaumette | A new Barankin bound approximation for the prediction of the threshold region performance of maximum likelihood estimators[END_REF], each element of Ψ is of the form

[Ψ] m,n = E χ|ξ 0 {ν(χ, ξ m )ν(χ, ξ n )} = 1 π 2N T |Σ noise | e -(χ-µ(ξ m )) H Σ -1 noise (χ-µ(ξ m ))-(χ-µ(ξ n )) H Σ -1 noise (χ-µ(ξ n ))+(χ-µ(ξ 0 )) H Σ -1 noise (χ-µ(ξ 0 ))dχ = α(ξ m , ξ n ) 1 π 2N T |Σ noise | e -(χ-µ(ξ m )-µ(ξ n )+µ(ξ 0 )) H Σ -1 noise (χ-µ(ξ m )-µ(ξ n )+µ(ξ 0 )) dχ = α(ξ m , ξ n ) (26) in which α(ξ m , ξ n ) = e 2 {(µ(ξ m )-µ(ξ 0 )) H Σ -1 noise (µ(ξ m )-µ(ξ 0 ))} (27)
At last, plugging (4), ( 22)-( 27) into ( 8)-( 10) and ( 13) one obtains C

B. The Unconditional Case

Let us consider the unconditional model. For simplicity we define Γ(ξ 0 ) = e(ω 0 ) ⊗ u(ψ 0 , ρ 0 ) and let us recall the following matrix properties T r(XY ) = vec(X H ) H vec(Y ), vec(XY Z) = (Z T ⊗ X)vec(Y ), which hold for any matrices X, Y and Z [START_REF] Graham | Kronecker products and matrix calculus with applications[END_REF]. Using these properties along with [START_REF] Boyer | Study of the asymptotic cramer-rao bound for the cold uniform linear array[END_REF], we obtain

1 T FIM(ξ 0 ) = ∂r ∂ξ T H | ξ=ξ 0 Σ(ξ 0 ) -T ⊗ Σ(ξ 0 ) -1 ∂r ∂ξ T | ξ=ξ 0 =   g H ω g ω g H ω g ψ g H ω g ρ g H ψ g ω g H ψ g ψ g H ψ g ρ g H ρ g ω g H ρ g ψ g H ρ g ρ   (28)
where

r = vec(Σ(ξ 0 )) = σ 2 s (Γ c (ξ 0 ) ⊗ Γ(ξ 0 )) + vec(Σ noise ) and g ω = vec Σ(ξ 0 ) -1 2 ∂Σ(ξ) ∂ω | ξ=ξ 0 Σ(ξ 0 ) -1 2
If the sensor noise is both spatially and temporally white, as Σ noise = σ 2 n I 2N , using the matrix inversion lemma [START_REF] Golub | Matrix Computations[END_REF] into (6), one can obtain

Σ(ξ 0 ) -1 = 1 σ 2 n I 2N - σ 2 s σ 4 n + N σ 2 s σ 2 n Γ(ξ 0 )Γ H (ξ 0 )
On the other hand, in [START_REF] Chaumette | A class of weiss-weinstein bounds and its relationship with the bobrovsky-mayer-wolf-zakai bounds[END_REF] the KLD is obtained KLD(p(χ|ξ l )||p(χ|ξ

)) = E χ|ξ l χ H (I T ⊗ Σ(ξ)) -1 χ - E χ|ξ l χ H (I T ⊗ Σ(ξ l )) -1 χ + T ln |Σ(ξ)| -T ln |Σ(ξ l )| in which E χ|ξ l χ H (I T ⊗ Σ(ξ)) -1 χ = 2N T i=1 2N T j=1 E χ|ξ l [χ] c i [χ] c j [(I T ⊗ Σ(ξ)) -1 ] i,j = T tr Σ(ξ l )Σ(ξ) -1 and similarly, one obtains E χ|ξ l χ H Σ(ξ l ) -1 ξ = 2N T. Consequently, KLD(p(χ|ξ l )||p(χ|ξ)) = T tr(Σ(ξ l )Σ(ξ) -1 ) -2N T + T ln( |Σ(ξ)| |Σ(ξ l )| ) (29)
In addition, the l-th member of D in ( 12) is given by

∂KLD(p(χ|ξ l )||p(χ|ξ)) ∂ξ = T tr -Σ(ξ l )Σ(ξ) -2 ∂Σ(ξ) ∂ξ + T tr Σ(ξ) -1 ∂Σ(ξ) ∂ξ ( 30
)
and from [START_REF] Chaumette | A new Barankin bound approximation for the prediction of the threshold region performance of maximum likelihood estimators[END_REF], the element of Ψ is

[Ψ] m,n = E χ|ξ 0 {ν(χ, ξ m )ν(χ, ξ n )} = |I T ⊗ Σ(ξ 0 )| 2 |I T ⊗ Σ(ξ m )||I T ⊗ Σ(ξ n )| e -χ H ((IT ⊗Σξ m )) -1 +(I T ⊗Σ(ξ n )) -1 -2(I T ⊗Σ(ξ 0 )) -1 )χ p(χ|ξ 0 )dχ = |Σ(ξ 0 )||Σ(ξ m ) -1 + Σ(ξ n ) -1 -Σ(ξ 0 ) -1 | |Σ(ξ m )||Σ(ξ n )| (31) Finally, C (L) U-MSB , C (L) U-HCRB , C (L)
U-MHB , and C

(L,J)

U-TTB are given by replacing ( 6), ( 29) -( 31) into ( 8), ( 9), ( 10) and ( 13).

VI. NUMERICAL INVESTIGATION

Numerical results are presented in this section for a non-uniform linear COLD array with N = 8 sensors. One narrowband far-field source is located according to θ 0 = 60 • and the polarization state parameters are given by ρ 0 = 30 • and ψ 0 = 45 • . Simulations are performed for T = 15 snapshots.

A. Analytical and numerical analysis of the derived lower bounds

The aim of this part is to examine the usefulness of C TTB to predict the SNR threshold. Using, [START_REF] El Korso | Statistical resolution limit of the uniform linear cocentered orthogonal loop and dipole array[END_REF], [START_REF] Chaumette | A new Barankin bound approximation for the prediction of the threshold region performance of maximum likelihood estimators[END_REF], ( 13) and ( 23), we plot the derived lower bounds for the parameter ω 0 for both conditional and unconditional models in Fig. 2 and Fig. 3, respectively. These figures show that the derived bounds exhibit a threshold effect around -9dB. We note also that the new proposed TTB provides a better prediction of the SNR threshold as expected. Furthermore, for J < L, the computational cost of the TTB is lower in comparison to the MSB, HCRB and MHB. This is mainly due to the inversion of a J × J matrix due to presence of the discrete transform matrix W instead of the initial L × L matrix inversion.

In Fig. 4 and 5, we plot the TTB w.r.t. ω vs. SNR for different polarization parameters ψ 0 and ρ 0 in the conditonal case (the same conclusion are noticed for the unconditional case). From Fig. 4, we notice that, no matter how ψ 0 changes, the breakdown point is approximately fixed for a given ω 0 , meaning that the effect of the polarisation state parameter ψ can be neglected in designing the COLD array. In Fig. 5 we focus on the effect of ρ, in which, this figure shows that, for a different value of polarization parameter ρ and for a fixed SNR, the higher the ρ 0 is, the better are the performances (w.r.t. the MSE in the asymptotic region but also breakdown point).

B. Frequency test-bins and their effect on the TTB

One notes that increasing the number of sensors or test-points may improve the tightness of the bounds. This is also same for frequency-bins. However, this improvement comes at the expense of computational complexity. Considering there is an intuitive link between the performance of threshold prediction and the sequence of frequency test-bins. We give a sub-optimal method to design an optimal index of frequency test-bins in order to overcome an exhaustive search by computer over all possibilities. First, place only one frequency test-bin by minimizing the threshold SNR with respect to J positions. Second, iterate the first step by placing the n-th frequency test-bin at once sequentially until n = L with respect to J -n + 1 remaining positions. To illustrate and compare the accuracy and usefulness of the aforementioned approach, two numerical examples for TTB are obtained: case 1-With the same number of test-points L = 32, one notice that our sub-optimal method yields a very good agreement when compared with the searching solution performed by computer, as shown in Fig. 6. Furthermore, this figure also shows us that increasing the number of frequency test-bins gives a considerable improvement in the SNR threshold prediction; case 2-In Fig. 7, with the same number of test-points and frequency test-bins L = J = 32, the threshold prediction of optimal index is nearly 1dB better than the sequential one, which means that the index of frequency test-bins has an important effect on minimizing the ambiguity region threshold. As the number of snapshots T increases, this advantage will be more apparent. Consequently, we notice that the proposed method with low complexity is useful for a threshold prediction problem, especially for the sensor arrays with large test-points.

C. Designing a COLD array : Resolution factor vs threshold SNR

To better comprehend the system performance w.r.t the resolution factor (RF) and the threshold SNR as a function of array geometry we consider Fig. 8 in which we compare all possible configuration for N = 8 and a given aperture A = 23 (more precisely, such context gives 74613 possible sensor array configurations). The term resolution factor is the minus curvature of the magnitude squared of the beampattern at the peak of its mainlobe and is determined by the array geometry [START_REF] Bell | Ziv Zakai lower bounds in bearing estimation[END_REF] as:

RF = 8d 2 π 2 N 2 λ 2 κ T N (N I N -1 N )κ N in which κ N = [d 1 , . . . , d N ]
T denotes the N × 1 vector of array element index locations.

In general, we would like the threshold SNR to be as small as possible and the RF to be as large as possible [START_REF] Bell | Ziv Zakai lower bounds in bearing estimation[END_REF]. We see that the best performance is achieved at the upper-left corner of the distribution given in Fig. 8, where there is a tradeoff between resolution and threshold prediction.

As an example, eight types of array configuration are considered, as shown in Table 1. From comparison of simulation results, one can further notice that: 1) For the same array aperture and same number of sensors, the SNR threshold prediction and resolution factor are greatly affected by the array geometric configuration. 2) The configuration that puts two sensors at the extremity and the rest in the middle has the worst performance for resolution factor; Contrary, places two average parts of sensors on both sides seems to be the best geometry configuration (at least, it is the case for N = 8 and A = 23 sensors). From example, Type 1 to Type 4 arrays, which have the good performance for the threshold SNR, are nearly 3dB better than the relatively poor performance of type 7 and type 8, where this is due to the sensor geometry configuration.

VII. CONCLUSION

In this paper, we derive explicit closed-form expressions of different deterministic lower bounds on the mean square error for the so-called non-uniform linear co-centered orthogonal loop and dipole arrays in a passive polarization source localization context. Taking advantage of these expressions, we analyse and characterize the performances in the asymptotic region and non-asymptotic region in terms of breakdown point prediction for the conditional and unconditional observation models. Finally, numerical simulations show the effect of each polarization parameters and the array geometry on the optimal designing strategy.

Fig. 1 .

 1 Fig. 1. A non-uniform linear COLD array representation in the presence of one far-field source

Fig. 3 .

 3 Fig. 3. Lower bounds on the mean square error (unconditional case) w.r.t. ω for NULA-COLD array (L = J = 8).

Fig. 4 .Fig. 6 .

 46 Fig. 4. TTB on the mean square error (conditional case) w.r.t. ψ for NULA-COLD array. (L = J = 8)

Fig. 7 .

 7 Fig. 7. Comparisons of different index of frequency test-bins for TTB threshold prediction with test-points L = 32, frequency test-bins J = 32.

Fig. 8 .

 8 Fig. 8. Comparisons of different array geometric configurations for resolution factor w.r.t. SNR threshold predicted by the TTB (conditional case: N = 8, A = 23).

Array Type

Geometric Configuration Resolution Factor Threshold SNR Type 1

Table1. Different array geometric configurations among the 74613 possibilities for N = 8 and A = 23(• and • represent the position of sensor and missing sensors, respectively.)