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Robust joint relaxed maximum likelihood calibration and direction of arrivals estimation with sparse arrays
Abdellah Bouiba, Mohammed Nabil El Korso, Toufik Boukaba
To cite this version:
Abdellah Bouiba, Mohammed Nabil El Korso, Toufik Boukaba. Robust joint relaxed maximum like- lihood calibration and direction of arrivals estimation with sparse arrays. The 1st Conference on Electrical Engineering, Apr 2019, Alger, Algeria. �hal-02389095�
Robust joint relaxed maximum likelihood calibration and direction of arrivals estimation with sparse arrays
1Abdellah BOUIBA,2Mohammed Nabil EL KORSO, 3Toufik BOUKABA
1Aeronautical Science Laboratory, Institute of Aeronautics and Spatial Studies, University of Blida 1, Algeria,
2Paris Nanterre University, LEME EA4416, France
3Ecole Sup´´ erieure des Techniques de l’A´eronautique, Alger
abdellahbouiba@gmail.com,m.elkorso@parisnanter.fr,boukabatoufik@yahoo.fr
Abstract. In this paper, we focus on the joint calibration and estimation of direction of ar- rivals in a data aided scenario, i.e., presence of one or more calibrators. The considered context has the following specifications: i) the antenna sensors are characterized by an unknown com- plex gain, ii) the array is composed of small sub-arrays largely spaced leading to a sparse noise covariance matrix and iii) observations may contain outliers or non homogeneities. Nu- merical simulations assess that the proposed algorithm is robust to such disturbances while performing an accurate joint calibration and direction of arrival estimation and out preforms the state of the art.
Keywords: Maximum likelihood, compound Gaussian, calibration, direction of arrivals.
1 Introduction
Direction of arrivals (DOA) estimation is an important topic with a plethora of applications, e.g., wireless communication, radio-interferometer, radio-astronomy, RADAR/SAR/STAP applications, EEG localization, to cite a few [1–3]. In order to achieve high resolution, large systems are commonly considered. A typical example is the LOFAR (low frequency array) which consists of several thousands of elementary sensors forming a very large radio- interferometers [4]. Specifically, such instrument is decomposed of several sub-arrays (50 stations for the LOFAR) each containing several antennas. In this context, the whole array is characterized by i) its geometry, leading to a sparse block diagonal noise covariance matrix. This is naturally due to the decorrelation between the noise components between sub-arrays or stations (temporally and spatially decorrelation due to the large spacing between two distinct sub-arrays) [5–7]. ii) the presence of complex gain which is due to the individual distortions of sensor outputs [8, 9]. iii) finally, the possible presence of data aided sources or calibrators is taken into account (this is a typical scenario in wireless communication [10, 11] as well in radio imaging [12])
Apart from the specifications listed above, it is worth-mentioning that in a high res- olution scenario the Gaussian assumption of the noise is generally violated [13]. Typical examples are high resolution radar systems, the recent large radio-interferometers and high
resolution SAR imaging systems to cite a few [14]. Recently, an interesting alternative to Gaussian noise modeling appears: the so-called compound Gaussian noise modeling [15–17].
A compound Gaussian, known also as spherically invariant random process, is composed of two component random variables: thetexture parameter, a positive random variable which represents a local variation of the non homogeneities, and the speckle parameter which follows a zero mean complex Gaussian process. The multiplication of these two random parameters leads to a compound Gaussian random variable. The compound Gaussian pro- cess is known to be heavy tailed [13] (meaning that it takes into account the presence of outliers and non homogeneities) and it encompass several well known distributions, e.g., the Gaussian distribution, the K-distribution, the student’t distribution, the Weibell dis- tributions ect. The aforementioned specification of the compound Gaussian process makes it a perfect candidate for noise modeling in a high resolution context [18].
Consequently, in this paper we aim at designing scalable robust joint calibration and DOA estimation. The robustness is ensured by considering the compound Gaussian noise modeling. In addition, a block coordinate descent approach is considered in order to alle- viate the computational complexity cost.
Finally, it is worth-mentioning that the proposed algorithm unifies the existing work re- lated to the calibration and/or DOA estimation in the context of sparse arrays. Specifically, i) the proposed algorithm can be considered as a robust version of the so-called modified iterative maximum likelihood algorithm designed under Gaussian assumption [6] and ii) it generalizes the maximum likelihood DOA estimation in unknown noise fields using sparse sensor arrays [7] to the case of un-calibrated sparse arrays.
The paper is organized as follows: Section 2 focuses on the data model and the noise modeling, in section 3 we expose the design of the proposed algorithm. Section 4 presents the numerical simulation results and finally, the conclusion is drawn in Section 5. Re- garding the notations, scalars, vectors and matrices are represented by italic lower-case, boldface lower-case and boldface upper-case symbols, respectively. The transpose, the com- plex conjugate, the hermitian, the pseudo-inverse, the trace and determinant operator are represented, respectively, by (.)T, (.)∗, (.)H, (.)†, tr{.} and|.|. The operators bdiag{.}and diag{.} represent a block-diagonal and a diagonal matrix, respectively.
2 Data Model
2.1 Model description
In the following we consider an antenna with M sensors decomposed into L small and largely spaced sub-arrays (possibly non linear). This scenario is common in radio astron- omy and interferometer systems. Each sub-array is composed of Ml sensors, such that PL
l=1Ml=M (cf. Fig. 1). The whole antenna is exposed toDfar field signal sources. Part of these sources are known as calibrator sources, meaning that their positions and emitted signal sources are known (this is also known as data aided sources in the context of wireless
Robust calibration with sparse arrays 3
communication [11]). Let us denoteDK as the number of calibrator sources andDU as the number of the unknown sources, with D =DK+DU. Finally, we assume that the array (and consequently, the L sub-arrays) is not calibrated. This is mainly due to the instru- mentation perturbations such as phase shifts which introduce difference between sensors gains leading to a dramatic performances loss. In this context, an accurate calibration need to be done in order to avoid the aforementioned loss.
In the next subsection, we represent mathematically the adequate parametric model associated with the above description.
2.2 Model representation
The observation vector,y(t)∈CM, for the t-th snapshot is then given by
y(t) =GA(θ)s(t) +n(t) (1) where n(t) denotes the noise component which is explicated below, θ represents the DOA parameters, the steering matrix A(θ) = [A(θU),A(θK)], the signal sources s(t) = [sTU(t),sTK(t)]T, the matrix gain G= diag(g) in which the subscripts K and U represent the known (calibrator) and unknown components, respectively. The explicit steering matrix model depends on the considered case : azimuth only estimation or elevation and azimuth estimation. This can be found in classical textbooks as [1, 19].
2.3 Noise statistic
In an array processing context, it is commonly assumed to model the noise as a Gaussian process. This assumption is motivated by the central limit theorem. Nevertheless, with the crowing use of high resolution systems, the Gaussian assumption shows it limits due to the non applicability of the central limit theorem. Furthermore, the presence of outliers and the non homogeneity of observations clearly breaks the use of the Gaussian assumption.
An interesting alternative is the so-called compound Gaussian modeling (known also as spherically invariant random process modeling). The latter generalizes a large number of distributions (possibly heavy tailed), e.g., Student’s t-distribution, k-distribution, Gaus- sian distribution, Weibull distribution, to cite just a few. This means that considering a compound Gaussian modeling leads to a robust estimation w.r.t. to outliers (due to the consideration of heavy tailed distributions) as well being robust to all distributions repre- sented by compound Gaussian.
A compound Gaussian process,nl(t), for the l-th sub-array, can be represent by nl(t) =p
τl(t)xl(t) (2)
where the positif random variable τl(t) ∼ pτl(al, bl), in which a and b denote hyper- parameters characterizing the pdf pτ and xl(t)∼ CN(0,Σl). Specific considerations need to be take into account regarding the above noise modeling
– The sparse array structure, i.e., the presence ofLsub-arrays, leads to specific structure of the noise covariance matrixΣ. Specifically, due to the large inter sub-array distances w.r.t. the signal wavelength, the noise is typically considered as independent from a sub- array to a sub-array. This leads to a block-diagonal structure of theΣ as
Σ = bdiag(Σ1, . . . ,ΣL) (3)
in which Σl∈CMl×Ml withPL
l=1Ml=M.
– the random variable τl(t) follows a pdf pτl(al, bl). Taking into account the true dis- tribution of τl leads to a complex algorithm (due to the need to estimate the hyper parametersal andbl) and limits the robustness efficiency (since, we have to specify the pdf, which may be unknown in some applications are in radio-astronomy and radar.) An interesting alternative is to consider deterministic but unknown realizations ofτl(t) to be estimated.
– From (3) we can notice that an ambiguity exists. Consequently, we impose that Tr(Σ) = 1. This assumption does not affect the estimation accuracy of the parameter of interest which are contained in the parameterized mean [20].
Fig. 1.Array configuration
2.4 Vector of unknown parameters
Now, we are ready to specify the vector of unknown parameters,η, which is considered for the algorithm design. Specifically,
η=h
θUT,sTU(1), . . . ,sTU(T),gT,τT(1), . . . ,τT(T),{[Σ1]p1,q1}p
1≥q1, . . . ,{[ΣL]pL,qL}p
L≥qL
iT
(4) with τ(t) = [τ1(t), . . . , τL(t)]T.
Robust calibration with sparse arrays 5
3 Proposed algorithm
From (1) and (2), the likelihood function reads p(y(1), . . . ,y(T)|η) =
T
Y
t=1
−exp(β(t)HΣ(t)−1β(t))
|πΣ(t)| (5)
in which we defineΣ(t) = bdiag(τ1(t)Σ1, . . . , τL(t)ΣL) andβ(t) =y(t)−GA(θU)sU(t)− GA(θK)sK(t). Consequently, the optimization problem to be considered is
arg min
η Λ(η) (6)
s.t. Λ(η),
T
X
t=1
β(t)HΣ(t)−1β(t)−log|Σ(t)|
τ(t)>0,[Σ]p,p>0
Maximizing (6) w.r.t. ηis infeasible due to the non convexity of the objective function. In the following we propose to use a block coordinate descent algorithm in which we minimize the objective function (log likelihood) w.r.t. to a subset of η while fixing the other pa- rameters/subsets at their previously estimated values. Then, we iterate until convergence.
Theoretical study of the convergence of such algorithm is beyond the scope of this paper and is considered as a future work.
3.1 Update of τ(t) andΣ Equating ∂τ∂Λ
l(t) to zero and after some calculus, we obtain ˆ
τl(t) = 1
MlβlH(t)Σl−1βl(t) (7)
with
βl(t) = [y(t)−GA(θU)sU(t)−GA(θK)sK(t)]Pl−1
i=1Mi+1:Pl i=1Mi
In the same manner, equating ∂Σ∂Λ
l to zero and using (7), we obtain after some calculus Σˆl= Mi
T
T
X
t=1
βl(t)βl(t)H
βl(t)HΣˆl−1βl(t) (8) The fixed point equation above (8), and due to its G-convexity, it is known to have a unique solution whatever is the starting point.
3.2 Update of g
We can notice that
∂Λ
∂[g]i =
T
X
t=1
yH(t)Σ−1(t)eieTi A(θ)s(t) +s(t)AH(θ)GHΣ−1(t)eieTi A(θ)s(t)
(9)
in whichei is a zero element vector except thei-th element which is equal to one. Equating the above equation to zero leads to
zi =
M
X
m=1
gm∗[X(i)]m,m (10)
in which
zi =
T
X
t=1
y(t)HΣ(t)−1eieTi A(θ)s(t) and
X(i) =
T
X
t=1
Σ−1(t)eieTiA(θ)s(t)sH(t)A(θ)H
Considering the diversity of (10) w.r.t. sensor elements, we obtain after some calculus g= (B∗)†z∗ (11) with z= [z1, . . . , zM]T and [B]i,j = [X(i)]j,j,∀i, j= 1, . . . , M
3.3 Update of sU(t) Again, equating ∂s∂Λ
U(t) to zero and after some calculus, we obtain sU(t) = A(θU)HGHΣ(t)−1GA(θU)−1
(GA(θU))HΣ(t)−1(y(t)−GA(θK)sK(t))
= ( ¯AHU(t) ¯AU(t))−1A¯HU(t) ˜y(t) (12) in which the whitened steering matrix of the unknown and known sources are given re- spectively, by ¯AU(t) = Σ(t)−12 GAU and ¯AK(t) = Σ(t)−12 GAK, whereas, the whitened residuals reads ˜y(t) = ¯y(t)−A¯HK(t)sK(t) in which ¯y(t) =Σ(t)−12 y(t).
Robust calibration with sparse arrays 7
3.4 Update of θU
Minimizing (6) w.r.t. θU while fixing the other parameters, leads to θˆU = arg max
θU
T
X
t=1
˜
yH(t) ¯AU(t)sU(t) +sHU(t) ¯AHU(t) ˜y(t)−sHU(t) ¯AHU(t) ¯AU(t)sU(t)
= arg max
θU
˜
yH(t) ¯AU(t) A¯HU(t) ¯AU(t)−1A¯HU(t) ˜y(t)
= arg min
θU
T
X
t=1
||PA⊥¯
U(t)y(t)||˜ 22 (13)
in which we used the expression given in (12) and where PA⊥¯
U(t)=I−A¯U(t) A¯HU(t) ¯AU(t)−1A¯HU(t).
Summary 1:Proposed Algorithm
input :θK,sK, DK, DU
initialize:Σˆ =I,τˆ=1
while stop criterion unreacheddo Update of ˆθU by (13)
Update of ˆsU(t) by (12),∀t Update of ˆgby (11) Update of ˆτ(t) by (7),∀l,∀t
Update of ˆΣby (8), followed by a normalization step ˆΣ:= Σˆ
Tr( ˆΣ)
output :ηˆ
4 Numerical simulations
In the following simulations, we consider two equally powered far field sources,θU = 30◦and θK = 10◦, impinging on a full array composed of L= 3 linear sub-arrays. Each sub-array is uniformly linear array composed of, respectively, 5, 4 and 5 sensors with half-wavelength spacing. The inter-sub-array space is about five to six wavelengths spacing, respectively.
The texture parameter is generated as Gamma distribution with al = 0.5 and bl = 1, ∀l and we considered 20 snapshots. The amplitude and phase gain are generated uniformly on [0.5,1.5] and [0,2π] and the noise covariance matrix of each sub-array follows the model given in [21], i.e., [Σl]p,q= 0.9|p−q|ej π2(p−q).
In Fig. 1, we consider the accuracy of the direction of arrival using algorithms based on the context of sparse arrays. Specifically, [7] considers the implementation of the maximum
likelihood DOA estimation in unknown noise fields using sparse sensor arrays assuming perfectly calibrated arrays. This algorithm shows its limits due to the effect of the mis- calibration represented by the calibration matrixG. Furthermore, we consider the modified iterative maximum likelihood algorithm designed under Gaussian assumption for non cali- brated arrays [6]. The aforementioned algorithm behaves well under Gaussian assumption but shows it limits in the case of non Gaussian environment or in the presence of outliers as shown in Fig. 1. Finally, the proposed algorithm shows it usefulness in the case of un- calibrated or mis-calibrated sparse arrays in the context of non Gaussian environment. This is also validated by a closer look at the mean square error of the calibration parameters represented in Fig. 2 (we omit the algorithm proposed in [7] since it does not take into account the estimation of G).
Fig. 2.SNR vs. MSE DOA
5 Conclusion
In this paper, we proposed a novel robust algorithm that, jointly, calibrate the sensors and estimate the direction of arrival in a context of sparse arrays. The proposed algorithm is based on a relaxed version of the maximum likelihood in a non Gaussian environment. To reduce the computational cost, a block coordinate descent approach has been used in which we managed to obtain closed form updates for most of the parameters. Finally, numerical simulations assess the usefulness of the proposed algorithm regarding an accurate joint calibration and direction of arrival estimation by out preforming the state of the art.
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