Use of a sub-array statistical approach for the detection of a buried object

INTRODUCTION

Non-invasive identification of buried objects in the near-field of a receiver array is a subject of great interest, due to its application to the remote sensing of the earth’s subsurface (Daniels 2004; Armand and Polyakov 2005), environment reclamation (such as the detection of landmines; Geneva International Centre for Humanitarian Demining 2006), civil engineering (like the localization of pipes and conduits; Xiaojian, Haizhong and Huiliang 1997), archaeological investigation and more (Pirri, Lombardo and Bucciarelli 2000).

In general, the reconstruction of a scenario starting from the scattered electromagnetic field is a heavy problem and its solution has involved many authors with different approaches. Because of the intrinsic non-linearity of the inversion, some approximations (Born, Rytov, Kirchhoff, etc.) are considered in many works (e.g., Devaney 1981; Zapalowski, Fiddy and Leeman 1984; Mayer 2002; Soldovieri and Solimene 2007); moreover, numerical or iterative methods to solve the non-linear problem (e.g., Bojarski 1982; Chiu and Kiang 1992) have been proposed.

Some techniques, very popular in a smart antennas context, like the direction of arrival ones, are still adopted in radar applications to localize signal sources and targets (Shan, Wax and Kailath 1985; Chandran 2005; Balanis and Ioannides 2007; Dmochowski, Benesty and Affes 2007; Grice et al. 2007). The synergy between signal processing and electromagnetic areas was introduced several years ago in many works, as in Bruckstein and Kailath (1987) and Şahin and Miller (2001).

In this paper, we present a statistical approach to process the data calculated by a forward scattering solver, in order to find an estimation for the position of an object using a sub-array processing technique (Bouvet and Bienvenu 1991; Krim and Viberg 1996) similar to the one presented in Şahin and Miller (2001). Moreover, sub-array processing techniques have been considered in order to achieve localization also for objects in the proximity of the array (Schmidt 1986).

We apply the approach to many directions of arrival estimation methods and compare their performances in several scenarios. The forward scattering problem is solved with the cylindrical-wave approach (Di Vico et al. 2005).

The direction of the arrival approach assumes that the sources are in the far-field region of a receiving array, so that the received wavefront could be considered as planar. By dividing the whole array in a certain number of sub-arrays and finding the dominant direction of arrival for each one, it is possible to localize objects in the far-field of the sub-array even if they are in the near-field of the array.

In the first step of the detection procedure, we apply several directions of arrival techniques to obtain a set of angles for each sub-array. By triangulating these directions we obtain a collection of crossings crowding in close proximity of the object.

In a second stage, the crossings are modelled as Poisson distributed points, with a large rate parameter around the target and a low rate parameter in the ground region. In this way we can separate two types of regions: one, corresponding to the large rate distribution, is represented by a large number of crossings very close to each other, while the second region is a cloud of less dense crossings that are considered as the background. A testing procedure is employed to determine a set of window regions corresponding to the target area and finally the cylinder coordinates are extracted. The procedure is schematized in Fig. 1.

We apply the procedure in two different situations. First, we simulate several settings in a vacuum, in order to verify the intrinsic capabilities of the direction of arrival methods and the detection procedure in terms of object dimension and its distance from the array. We then simulate some cases in half-space geometry, with the array sensors adhering to the medium in which the cylinder is buried. Our purpose is to verify the performances of the direction of arrival algorithms in a dielectric homogeneous half-space different from air.

The paper is organized as follows. In the section ‘Signals and noise model’, a model for the received signals and noise is presented and the steering vector representation is introduced. In the section ‘Pseudospectrum and direction of arrival estimation’, a brief description of several directions of arrival methods is provided. The section ‘Sub-array localization procedure’ is devoted to the sub-array processing statistical procedure, while in the section ‘Numerical results’ the numerical results are presented and discussed. Finally, conclusions are drawn.

SIGNALS AND NOISE MODEL

We consider an array of identical sensors uniformly spaced along a line, with period , as depicted in Fig. 2. It is commonly referred to as a uniform linear array.

Let indicate the direction of arrival of the signal coming toward the array at the -th receiver . Then, under the plane-wave hypothesis, we find that the output of the generic sensor is

(1)

where and are the angular response of the -th sensor (supposed to be the same for all the array elements) and the incoming signal, respectively; denotes the sensor position. Since and considering isotropic sensors, the steering vector is defined as:

(2)

In equation (2), is the wavenumber associated to the plane wave propagating through the medium in which the object is buried. If this medium is different from the one in which the array is placed, the data obtained by solving the forward-scattering problem with the cylindrical wave approach take into account the refraction by the planar interface, while the inversion procedure neglects it. However, when the array is very close to the interface (as happens in several ground-penetrating radar (GPR) applications), the numerical error that affects the direction of arrival estimation assuming a homogenous medium is very small.

The is a Vandermonde-like vector; thus a matrix of steering vectors can be built, with independent columns (Van Trees 2002). In the uniform linear array case, is uniquely defined if and only if the condition is fulfilled, which is always satisfied if the period is less or equal to half a wavelength. Also to avoid aliasing in space it must be (Şahin and Miller 2001).

In Fig. 3, a receiving array with narrowband signals coming from different directions is sketched, with . Each received signal includes additive, zero mean, Gaussian noise. Time is represented by the -th time sample ( where is the total number of time samples); thus the array output can be given as (Gross 2005):

(3)

where

(4)

being the array weights; and are the incident signals vector and the noise vectors, respectively, at the time instant .

The received signals are time-varying and our calculations are based upon time snapshots of the incoming signal. Suppressing the time dependence and supposing an uncorrelated noise, the Hermitian array correlation matrix can be defined as:

(5)

where is the source correlation matrix, while is the noise correlation matrix; symbolizes the hermitian transposition.

The exact statistics for the noise and signals are not a priori known but assuming the process ergodic the spatial correlation can be approximated by using a time-averaged correlation (the superscript symbol ‘^’ denotes an estimation of the true value):

(6)

PSEUDOSPECTRUM AND DIRECTION OF ARRIVAL ESTIMATION

The pseudospectrum (Stoica and Moses 2005) is a function that gives an indication of the signal angles of arrival based upon maxima versus angle. The mean output power of the array, at the time instant , is given by the square magnitude of the array output, which is

(7)

Assuming that the components of can be modelled as zero-mean stationary processes, for a given weight vector , the mean output power of the array system is obtained by calculating the conditional expectation over

(8)

It is clear that the pseudospectrum function is strongly depending (through ) on and the particular choice of the vector differentiates the direction of arrival estimation algorithms. We analysed and implemented several methods: Bartlett and Capon beamforming, maximum entropy, linear prediction, Pisarenko harmonic decomposition, minimum norm and multiple signals classification.

The non-parametric methods of spectral estimation (Bartlett and Capon algorithms) do not make any assumption on the covariance structure of the data. They suppose that the functional form of the steering vector is known, characterizing the array as a spatial sampling device. The parametric, or modelbased, methods (linear prediction, maximum entropy, minimum norm, Pisarenko harmonic decomposition, multiple signals classification), instead, assume that the signal satisfies a generating model with known functional form and then proceed by estimating the parameters in the chosen model.

Some parametric algorithms considered (linear prediction, Pisarenko harmonic decomposition, maximum entropy) are based on the autoregressive moving average models of the signals (Stoica and Moses 2005). The other algorithms are eigenvalue decomposition techniques (minimum norm and multiple signals classification) (Godara 1997).

The Bartlett estimate is obtained by the maximization of the signal-to-noise ratio. If the array is uniformly weighted, we can define the Bartlett direction of the arrival estimate (Bartlett 1961) as

(9)

Moreover, if represents uncorrelated monochromatic signals and there is no system noise, the Bartlett estimation is equivalent to the square magnitude of a spatial finite Fourier transform of all the arriving signals.

(10)

The Capon direction of the arrival estimate is also known as a minimum variance distortionless response. It is essentially a maximum likelihood estimate of the power arriving from one direction, while all other sources are assumed as interference; the goal is to maximize the signal-to-interference ratio. The Capon pseudospectrum is given by (Capon 1969):

(11)

The Capon method has a better resolution than the Bartlett direction of the arrival estimate. But, in the cases when the sources are correlated, the Capon resolution may become worse. The main advantage of these two estimation methods is that they are non-parametric solutions; therefore a priori information of the statistical properties of the signals is not needed.

The aim of the linear prediction method is to minimize the prediction error between the output of each sensor and the actual output (Makhoul 1975). This is equivalent to minimizing the mean-squared error. It can be shown that

(12)

where is the Cartesian basis vector that is the -th column of the identity matrix. The particular choice of has a big influence on the prediction (Johnson 1982). This method is an autoregressive technique and the pseudospectrum peaks are proportional to the square of the signals power.

The goal of the maximum entropy is to maximize the entropy function in :

(13)

Here is the pseudospectrum having the following expression (Burg 1975):

(14)

where is the -th column of the inverse array correlation matrix .

The Pisarenko harmonic decomposition method (Pisarenko 1973) is based on a minimum mean-squared error approach, under the constraint that the norm of the weight vector be equal to unity. In particular, the minimum eigenvalue of can be used to minimize the mean-squared error. The corresponding pseudospectrum is given by

(15)

where is the eigenvector associated with the smallest eigenvalue .

The minimum norm technique was explained by Ermolaev and Gershman (1994). This method is only relevant in the uniform linear array case and it optimizes the weight vector yielding the pseudospectrum

(16)

where is the subspace of the M–D noise eigenvectors. It can be noted that this method combines all noise eigenvectors whereas the Pisarenko harmonic decomposition only uses the first noise eigenvector (Gross 2005).

The multiple signals classification technique was first introduced by Schmidt (1986). It is a high resolution eigenstructure method able to estimate the number of signals, the angle of arrival and the strength of the waveform, under the assumption that the noise is uncorrelated. The multiple signals classification algorithm generates two subspaces: (noise) and (signal), from the eigenvectors associated with the eigenvalues of the array correlation matrix .

In particular, since the noise is uncorrelated, the smallest M–D eigenvalues of are all equal among them and to the noise variance such as . The remaining D eigenvalues are greater, so where . is the -th positive eigenvalue of the hermitian matrix .

The noise subspace eigenvectors are obtained by M–D eigenvectors associated with the smallest M–D eigenvalues of and are orthogonal to the array steering vectors at the angles of arrival . Because of this condition, the Euclidean distance is equal to zero for each direction of arrival. The multiple signals classification pseudospectrum is finally given as

(17)

SUB-ARRAY LOCALIZATION PROCEDURE

The direction of the arrival approach assumes that the sources are in the far-field region of a receiving array, so that the received wave-front could be considered as planar. By dividing the whole array in a certain number of sub-arrays and finding the dominant direction of arrival for each one, it is possible to localize objects in the far-field of the sub-array even if they are in the near-field of the array. Therefore if represents the size of the sub-array, the minimum distance attended for a successful object localization is .

After estimating the directions of arrival, we proceed by triangulating them, obtaining a crossing pattern, as the one reported in the upper side of Fig. 4. Two regions can be distinguished, the first one including a big number of crosses that most probably locates the target and a background region in which the amount of crosses is quite small and spread. The two classes of crossings are modelled using a pair of spatial Poisson distributions (Hoaglin 1980; Şahin and Miller 2001): the rate parameter of the background is considerably smaller than the one of the target.

For a given crossing pattern, we divide the region of interest into uniform nonoverlapping windows of fixed size and count the number of crossings . The windows must be nonoverlapping to guarantee the independence of the random variable , which is the number of crossings in each window. The window dimensions affect the performances of the procedure and they must be optimized taking into account the size of the object to detect.

We suppose that is Poisson distributed and check this assumption by using the graphical technique presented in Hoaglin (1980). It suggests to plot, in an iterative way, the quantity versus the count of crossings , where , is the number of occurrences of finding crossings in a window. If the plotted trend forms a straight line, with a slope approximating ln , it properly approximates a Poisson distribution of rate parameter .

A typical plot thus obtained is reported in Fig. 5. It is possible to interpolate the circles by tracing two different lines: one of them, with a positive slope, corresponds to a large value of (greater than ) while the second one, with a negative slope matches the circles relative to smaller values of the crossing count (between 0 and ).

The rate parameters for the background and target regions are given by their maximum likelihood estimates:

(18)

(19)

where the quantities having subscript and are relevant to background and target, respectively, moreover and represent the total amount of background and target windows, respectively.

Once the two distributions parameters (18) and (19) have been estimated, the probability for each window (containing crossings) to belong to the background is:

(20)

while to the target is:

(21)

To extract crossing clusters, we sweep the region of interest with a test window having the same area of the non-overlapping windows used before. At each location of the test window, we count the number of crossings , with , where , is the number of overlapping windows in the region of interest. The hypothesis test is formally a binary one:

• is Poisson distributed with a small rate parameter ;
• is Poisson distributed with a large rate parameter .

We decide if the window belongs to background or target, by applying the generalized likelihood ratio test

(22)

The decision threshold can be determined from the desired false-alarm rate expressed as

(23)

We finally obtain an estimated crossing pattern like the one shown in Fig. 6.

NUMERICAL RESULTS

A uniform linear array of 51 equally spaced elements is considered, with period . A three-element sub-array partition is applied (the total number of sub-arrays is therefore 17). A perfectly conducting circular cylinder characterized by a radius , which has its centre located at a distance from the array axis. For this configuration, the cylinder is in the far zone of the sub-array if its burial depth is at least . The incident wave is TM polarized (electric field parallel to the cylinder axis). The noise variance is the false-alarm probability is 10⁶.

We first apply the localization procedure in a vacuum, for different values of and and using in the sub-array processing procedure square windows with side-length , also for the test windows. Objects and depths, both small and comparable to the operating wavelength, are considered. Moreover, a central and a peripheral position of the target are treated, with respect to the array geometry. We define the localization error as

(24)

where and are the real and estimated cylinder axis coordinates, respectively. The quantity is the ratio between the Euclidean distance of the estimated centre from the real one and the actual cylinder radius. When , the estimated centre is collocated in a point inside the cylinder transverse section, otherwise, if , the estimation of the centre falls outside the object. As a reference threshold, we assume that a target can be considered correctly localized if the error is less than 1. In Tables 2–5, some results are presented, with and assuming the values here listed (the symbols a1–a5 and A–E will be used for reference in the following):

	a1	a2	a3	a4	a5
Radius
	A	B	C	D	E
burial depth

Table 2. Error estimation of different directions of arrival techniques. Target in a vacuum, central position

		A	B	C	D	E
Bartlett/Multiple signals classification	a1	6.236	6.060	5.824	4.930	5.597
	a2	2.224	2.120	1.512	1.525	1.537
	a3	0.569	0.571	0.451	0.207	0.206
	a4	0.271	0.079	0.078	0.015	0.145
	a5	0.028	0.120	0.207	0.159	0.145
Capon	a1	6.623	6.568	6.917	6.308	5.880
	a2	2.246	1.847	2.276	1.596	1.726
	a3	0.731	0.704	0.451	0.318	0.223
	a4	0.249	0.117	0.204	0.100	0.201
	a5	0.060	0.112	0.167	0.170	0.201
Linear prediction	a1	7.925	6.847	6.588	5.830	6.314
	a2	2.640	2.473	1.976	1.516	1.734
	a3	0.759	0.646	0.591	0.428	0.224
	a4	0.266	0.208	0.129	0.685	0.099
	a5	0.220	0.299	0.312	0.174	0.005
Maximum entropy	a1	9.552	9.029	7.629	6.391	5.894
	a2	2.249	1.521	1.697	1.150	2.271
	a3	0.577	0.578	0.550	0.608	0.365
	a4	0.294	0.040	0.182	0.080	0.075
	a5	0.211	0.277	0.288	0.301	0.397
Minimum norm	a1	60.034	5.945	50508	5.306	4.647
	a2	2.182	2.025	1.852	1.527	1.431
	a3	0.617	0.600	0.417	0.4069	0.270
	a4	0.216	0.132	0.115	0.098	0.149
	a5	0.125	0.115	0.199	0.222	0.232
Pisarenko harmonic decomposition	a1	6.965	5.985	5.967	4.478	5.128
	a2	2.037	2.009	1.771	1.545	1.285
	a3	1.068	0.654	0.514	0.276	0.206
	a4	0.314	0.332	0.094	0.156	0.074
	a5	0.177	0.057	0.186	0.131	0.113

Table 3. Error estimation of different directions of arrivals techniques. Target position in a medium with , central position. Same geometrical parameters as Table 2

		A	B	C	D	E
Bartlett	a1	4.482	3.349	4.819	3.189	3.725
	a2	1.517	1.584	2.063	0.980	1.801
	a3	0.732	0.670	0.331	0.249	0.191
	a4	0.114	0.153	0.167	0.196	0.114
	a5	0.590	0.447	0.063	0.057	0.273
Capon	a1	4.356	10.463	4.898	9.834	7.822
	a2	1.541	4.226	2.359	1.555	1.294
	a3	1.203	0.393	0.867	0.544	0.483
	a4	1.044	0.244	0.156	0.099	0.223
	a5	0.209	0.041	0.432	0.309	0.166
Linear prediction	a1	4.062	3.535	7.644	5.241	4.342
	a2	5.365	2.108	1.253	2.806	0.991
	a3	1.820	0.360	0.365	0.348	0.813
	a4	0.271	0.570	0.164	0.260	0.236
	a5	0.091	0.163	0.055	0.128	0.156
Maximum entropy	a1	13.275	2.880	3.767	8.505	14.780
	a2	1.557	3.570	1.503	1.235	0.878
	a3	0.962	0.557	0.547	0.307	0.604
	a4	0.060	0.303	0.277	0.039	0.201
	a5	1.787	1.101	0.105	0.066	0.226
Minimum norm	a1	6.514	4.482	7.715	4.861	3.577
	a2	1.206	3.856	1.471	1.031	1.762
	a3	0.701	0.372	0.358	0.400	0.277
	a4	0.579	0.154	0.322	0.207	0.364
	a5	0.050	0.198	0.132	0.042	0.121
Pisarenko harmonic decomposition	a1	14.059	6.375	6.029	3.723	4.843
	a2	3.722	1.593	0.877	1.691	1.130
	a3	0.7163	0.738	0.766	0.304	0.406
	a4	0.6743	0.276	0.920	0.204	0.080
	a5	0.238	0.179	0.250	0.465	0.138
Multiple signals classification	a1	4.024	3.349	4.819	3.189	3.725
	a2	1.210	1.584	2.063	0.980	0.967
	a3	0.670	0.828	0.350	0.248	0.191
	a4	0.114	0.153	0.167	0.196	0.114
	a5	0.651	0.371	0.063	0.013	0.273

Table 4. Error estimation of different directions of arrival techniques. Target in a medium with , central position

		A	B	C	D	E
Bartlett/Multiple signals classification	a1	4.386	4.605	4.296	3.618	3.143
	a2	1.746	1.456	1.110	1.338	0.984
	a3	0.640	0.461	0.557	0.365	0.408
	a4	0.343	0.159	0.148	0.190	0.245
	a5	0.105	0.106	0.113	0.073	0.094
Capon	a1	5.719	3.913	3.022	4.473	4.076
	a2	1.602	1.304	0.765	0.903	1.014
	a3	0.486	0.563	0.522	0.556	0.445
	a4	0.576	0.197	0.115	0.228	0.214
	a5	0.156	0.116	0.124	0.068	0.125
Linear prediction	a1	3.851	3.196	3.724	3.714	3.058
	a2	1.523	2.837	0.942	1.305	1.197
	a3	0.580	0.407	0.679	0.625	0.554
	a4	0.358	0.224	0.217	0.181	0.282
	a5	0.083	0.153	0.087	0.112	0.223
Maximum entropy	a1	5.309	30613	3.484	4.156	3.633
	a2	1.024	1.004	1.020	1.395	1.126
	a3	0.595	0.438	0.849	0.464	0.318
	a4	0.329	0.138	0.139	0.213	0.234
	a5	0.133	0.148	0.132	0.162	0.183
Minimum norm	a1	4.054	4.517	4.200	3.652	3.089
	a2	1.702	1.444	1.073	1.395	1.069
	a3	0.678	0.432	0.551	0.361	0.416
	a4	0.326	0.191	0.154	0.201	0.218
	a5	0.116	0.116	0.121	0.087	0.120
Pisarenko harmonic decomposition	a1	4.851	5.872	4.379	3.930	3.395
	a2	1.420	1.433	1.224	1.295	1.488
	a3	0.376	0.518	0.626	0.297	0.550
	a4	0.178	0.243	0.124	0.232	0.305
	a5	0.554	0.087	0.130	0.124	0.253

Table 5. Error estimation of different directions of arrival techniques. Target in a peripheral position both in a vacuum and in a medium with

		A	E	A	E
Bartlett	a1	12.305	12.932	4.451	5.596
	a2	4.880	4.169	1.466	2.240
	a3	1.931	1.211	0.519	0.902
	a4	1.213	0.643	0.379	0.416
	a5	0.785	0.297	0.139	0.202
Capon	a1	15.384	13.420	3.365	3.143
	a2	4.899	4.526	1.780	1.883
	a3	2.014	1.439	0.329	0.758
	a4	1.124	0.690	0315	0.298
	a5	0.707	0.198	0.168	0.223
Linear prediction	a1	14.382	15.102	5.116	4.324
	a2	4.532	3.529	2.546	2.009
	a3	2.088	1.649	1.271	0.967
	a4	1.384	1.059	0.391	0.434
	a5	1.960	0.354	0.163	0.263
Maximum entropy	a1	14.342	10.810	3.753	9.097
	a2	6.102	4.296	0.957	1.962
	a3	2.067	1.081	1.568	0.683
	a4	1.796	0.835	0.697	1.039
	a5	0.590	0.599	0.394	0.123
Minimum norm	a1	11.630	12.624	3.707	4.760
	a2	4.741	4.422	1.629	2.294
	a3	1.840	1.250	0.649	0.924
	a4	1.270	0.661	0.295	0.482
	a5	0.754	0.321	0.164	0.240
Pisarenko harmonic decomposition	a1	21.490	14.080	4.425	3.951
	a2	5.134	3.946	1.344	2.618
	a3	2.511	1.289	0.421	0.749
	a4	1.008	0.614	0.428	0.397
	a5	0.758	0.163	0.641	0.236
Multiple signals classification	a1	12.305	12.932	3.244	5.037
	a2	4.880	4.169	1.215	2.240
	a3	1.931	1.211	0.426	0.902
	a4	1.213	0.643	0.207	0.416
	a5	0.785	0.297	0.139	0.202

In order to make the tables more readable, the cases showing an estimation error greater than 1 are depicted in bold. In Table 2, results concerning a cylinder in a vacuum, in a central position compared with the array axis, are summarized. The covered area is a square of side equal to ; the window side-length here considered is and the total number of windows is 625. It can be noted that the described procedure shows a good capability of localization when the object radius is equal to or longer. Anyway the error of all the implemented methods does not decrease monotonously with distance .

In Table 3 the object is placed inside a homogeneous medium having a relative permittivity , with the same values of geometrical parameters as in Table 2.

Thus, some configurations of Table 3 are now in the near-field of the sub-arrays: these cases are indicated with a grey background. The increased number of cases showing an error greater than 1 depends on a partial satisfaction of the far-field condition. Moreover, it is important that the procedure still shows a good behaviour even in many of the near-field cases, especially when radius is greater than .

In Table 4, results concerning a cylinder buried in a medium with are summarized. Now it has been assumed that . Therefore, a dimensional scaling of the scenarios considered in Table 2 has been performed. On the whole, behaviour analogous to Table 2 can be appreciated: most of the cases having are crowded in a1) and a2) configurations, that is when radius .

In Figs 7–10 the estimation error is plotted as a function of radius normalized to the wavelength, for a fixed value of the distance . The object is in a vacuum, in a central position. In particular, four window sizes are taken into account: and . The normalized radius varies uniformly from 0.02 and 1, in 50 steps. In the figure parts b), a zoom of the region where is given (that is when the object is properly localized).

In Fig. 7 it can be noticed that the estimation error is generally decreasing as radius grows but it tends to increase in the proximity of the value . In this case, in fact, the distance between the receiving antenna and the centre of the object approaches , which is the far-zone limit of the sub-array previously specified. In Figs 8–10 it can be argued that the trend of versus becomes more regular as the windows size increases and simultaneously the radius of the smallest object that can be properly localized increases toward .

Furthermore, the Bartlett and multiple signals classification methods give the same values of in most of the cases. As a general observation, we point out that they are, together with the maximum entropy, the methods that better localize the object.

In Table 5 some results concerning an object placed in a peripheral position compared with the array axis centre (), are presented. In particular we considered scenarios A and E of both the configurations studied in Tables 2 and 4.

It is clear that peripheral localization is affected by a higher estimation error when the object is placed in a vacuum, if compared with the results of Table 2. Instead, when the cylinder is buried in the medium with , the procedure seems to be more robust to a position offset. It has to be considered that, in this second case, all the dimensions (distance between antenna sensors, distance and radius ) are reduced by a factor so that the object is closer to the array axis than in the vacuum case. Moreover, the host medium is supposed to be homogeneous and lossless: if losses were considered, the localization would be more sensitive to the position offset.

CONCLUSION

An algorithm has been presented that can detect and localize an object in the proximity of a linear array of receiving antennas, this distance could be referred to as the far-field limit of the sub-arrays in which the array is partitioned. The adopted sub-array processing technique has low computational complexity, is easy and affordable to use. Several directions of arrival estimation algorithms have been implemented and employed.

The described procedure has been applied to numerous test scenarios involving a perfectly-conducting circular cylinder having radius not larger than a wavelength. It shows good capability in estimating the scatterer position, either when the cylinder is set in a vacuum or inside a dielectric half-space. In particular we checked the procedure performances varying the geometrical parameters and the size of the windows used to discretize and probe the region of interest. The numerical results show that the Pisarenko harmonic decomposition method gives an average error greater than the other directions of arrival techniques independently from the windows size. The Linear Prediction algorithm presents worst performances when the windows are small, showing a behaviour similar to the other methods as the windows size is increased.

Furthermore, several scenarios have been considered, with the object in a peripheral position with respect to the array axis, both when the cylinder is settled in a vacuum and in a dielectric medium. In particular, in this second case, the procedure shows good robustness to a positional offset of the target.

We intend to consider a more realistic scenario, with the object buried in a lossy medium under a rough surface. We also plan to study the effect on the localization error of different subdivisions of the array in sub-arrays.

Moreover, we look forward to estimate also the scatterer dimension and to apply our sub-array processing strategy in multiple objects localization.