HF radio direction finding operating on a heterogeneous array: Principles and experimental validation

Transcription

1 RADIO SCIENCE, VOL. 39,, doi: /2002rs002860, 2004 HF radio direction finding operating on a heterogeneous array: Principles and experimental validation Yvon Erhel Centre de Recherches des Ecoles de Coëtquidan Saint-Cyr, Guer, France Dominique Lemur, Louis Bertel, and François Marie Institut d Electronique et Télécommunications de Rennes, UMR 6164, Université de Rennes 1, Rennes, France Received 16 December 2002; revised 2 October 2003; accepted 23 October 2003; published 15 January [1] Array processing usually operates on identical sensors. In this paper an investigation is made to apply the direction finding algorithm MUSIC to an antenna array with nonidentical elements (heterogeneous array). This original solution is polarizationsensitive though it resorts to only one sensor at each point of the spatial sampling. As a major consequence of its structure, the computation of the MUSIC algorithm is based on an expression of the steering vector that integrates the spatial response of each sensor. This concept is then validated in the special context of HF radio direction finding, for which a deterministic model of the polarization at the exit of the ionospheric channel is derived. A multichannel radio receiving system has been developed and experimented using transmitters located more than 1000 km from the receiving site. INDEX TERMS: 0604 Electromagnetics: Antenna arrays; 0674 Electromagnetics: Signal processing and adaptive antennas; 6934 Radio Science: Ionospheric propagation (2487); 6994 Radio Science: Instruments and techniques; KEYWORDS: high resolution direction finding, heterogeneous array, wave polarization, ionosphere Citation: Erhel, Y., D. Lemur, L. Bertel, and F. Marie (2004), HF radio direction finding operating on a heterogeneous array: Principles and experimental validation, Radio Sci., 39,, doi: /2002rs Introduction [2] Radio direction finding generally resorts to homogeneous arrays made up with identical antennas. The signal processing implemented in the receiving system is based on a model of the inter-element geometrical phases as function of the angles of arrival. [3] A first set of so-called classical methods for the estimation of the directions of arrival (D.O.A.) considers the outputs of the sensors as valid data to be fitted with the model. For example, the conventional beamformer realizes a weighted combination of the observations searching to maximize the power of the resulting signal, the corresponding tap vector being defined in accordance with the expressions of the geometrical phases. On the other hand, interferometry consists in a direct estimation of the inter-channel phases (resorting to a Fourier analysis) as a first step, and then a computation of the angles of arrival based on the same expressions. These methods are well known and present several advantages such as a Copyright 2004 by the American Geophysical Union /04/2002RS low complexity and a significant robustness. However, they suffer from a lack of angular resolution so that they may fail in the separation of incident waves with close angles of arrival. [4] Therefore, a second class of methods have been developed which are said to present a high resolution in the sense that the minimum resolvable difference between two angles of arrival is much less than the width of the main directional lobe of the conventional beamformer. A large number of methods (Capon, Esprit, Maximum likelihood, Weighted Subspace Fitting,...) are based on the data covariance matrix, the elements of which are statistics of the second order of the sensors outputs. An overview of array processing can be found in [Krim and Viberg, 1996]. A very popular method is the MUSIC algorithm (MUltiple SIgnal Classification) that presents several advantages: (1) It asymptotically reaches the Cramer-Rao bound (guarantees theoretically the best angular accuracy that any estimator can propose) in presence of a single incident wave. (2) It does not demand a particular geometry; only the condition of geometrical nonambiguity must be verified. (3) Its computational cost is moderate since it appears as a local method: in presence of several incident waves, the 1of14

2 directions of arrival are successively searched in a single angular loop. On the contrary, global algorithms (such as the maximum of likelihood) jointly search the total set of angles of arrival in a multidimensional loop with a significant increase of the computational charge. [5] This paper describes a system of radio direction finding in the particular context of HF transmission through the ionosphere: it has to cope with specific problems such as the existence of highly correlated multipaths due to a small angular separation between the incident waves. Several authors [Ferrara and Parks, 1983; Demeure et al., 1993] have investigated to decrease this correlation by taking the possible differences of the incident polarizations into account. The corresponding arrays are homogeneous and build up with couples of identical antennas at each point of the spatial sampling. [6] The heterogeneous array is an alternative to realize a polarization-sensitive system of direction finding. Its main originality stands in the spatial distribution of nonidentical antennas. [7] The spatial response (or field-to-signal transfer function) of each antenna depends on the direction of arrival (D.O.A.) and may be computed resorting to a reliable electromagnetic model of the wave polarization at the exit of the ionosphere. In section 2, the principle of the calculation of the spatial response is derived for receiving antennas with small sizes as compared to the wavelength. [8] Section 3 underlines the originality of the signal processing implemented on an heterogeneous array. The formulation for the received signals is presented, underlining that the steering vector of the heterogeneous array contains the set of the different spatial responses in addition to the classical interelement geometrical phases. Consequently, taking into account the two orthogonal polarizations (ordinary O and extraordinary X) generated by the ionospheric channel, the derivation of the MUSIC algorithm in this context leads to the computation of two pseudospectra, the angular estimations being separated for each expected polarization O or X. [9] Section 4 presents an eight coherent channel system for reception that has been designed and build up in order to achieve direction finding in the HF band. Several experimental validations have been carried out involving broadcast transmitters located more than 1000 km from the receiving site. In this operational system, the computation of the pseudospectra is completed by a sorting of the extracted angular values. An original sorting criterion, based on the coherence of the angular estimations for the two expected polarizations, is derived in this paper and the resulting estimated D.O.A. are compared with the geometrical azimuths of the transmitters considered as references. [10] The experimental results underline the effective high resolution capability of the system, considering that several incident signals, incoming with a small difference of the elevations and a high temporal correlation, are separately detected. This improvement in the angular estimation is based on the polarization sensitivity of the system which is effective though a single sensor is set up at each position within the array, reducing by the way the number of receiving channels if compared with other polarization sensitive techniques. 2. Spatial Response of an HF Receiving Antenna 2.1. Polarization at the Exit of the Ionosphere [11] The ionosphere is an inhomogeneous medium as the electron density, which is the main electrical parameter, is a function of the position. It is also an anisotropic medium because the direction of the Earth s geomagnetic field has a particular role in the solution of the Maxwell equations. The solution proposed by Appleton and Hartree in the context of the magneto-ionic theory underlines the presence of two propagation modes [Ratcliffe, 1962; Kelso, 1964; Davies, 1990] named ordinary (denoted O) and extraordinary (denoted X). The different refractive indices of the ionosphere associated with the two polarizations distinguish them. [12] The magneto-ionic theory gives the ratios between the different components of the electric field in the plasma for the two modes. The corresponding expressions being time independent, the radio waves are considered to have a deterministic polarization. [13] The solution for the polarization has an elliptic shape but, generally, in the plasma the electric field is not transverse and two ellipticity ratios must be defined, one in the plane of the wave front (perpendicular to the wave vector) and the second in a plane which is orthogonal to the first one and contains the direction of propagation and the Earth s geomagnetic field. [14] For the following applications, only the polarization at the exit point of the ionosphere is required for a given HF radio link. It is calculated considering the limit conditions of Budden [1952] which express that the electron density tends to zero (and jointly the longitudinal component of the electric field) at the lower points of the ionosphere. In these conditions, the incident wave is T.E.M. polarized (transverse electromagnetic) from the exit of the ionosphere to the receiving station, considering that this part of the path is a free space propagation without any change in the polarization. Its elliptical shape is fully described with two parameters. For that purpose, let us consider the Davies system of coordinates which is defined as follows (Figure 1): axis 1, direction of the propagation vector k; axis 2, direction of the projection of the Earth s geomagnetic field Bo in the plane orthogonal 2of14

3 Figure 1. Davies system of coordinates. to axis 1; axis 3, complement of axis 1 and 2 in a direct system of coordinates. [15] The first parameter is the polarization ratio (real) h: its absolute value quantifies the respective lengths of the two axes of the ellipse along which the electrical field rotates: E 3M = jhje 2M ; its sign indicates the clockwise (+) or counterclockwise ( ) rotation. Figure 2 represents the corresponding ellipse in the plane of the wave front. [16] The phasor vector, defined as w = ð01jh Þ T, gives an expression of the electric field in the plane of the wave front, including the transmitted scalar signal s(t, r): Eðt; rþ ¼ B 1 A st; ð rþ ¼ w st; ð r Þ; ð1þ jh where t is time and r is the position vector. [17] For a given D.O.A., the two possible modes are characterized by two values h O and h X for the polarization ratio satisfying the relation: h O :h X ¼ 1: ð2þ The main axes of the corresponding ellipses are orthogonal. The second parameter is the inclination angle a evaluated in the plane of the wave front between the main axis and the local horizontal as shown in Figure 3. [18] For a given receiving site and a D.O.A. characterized by the azimuth and elevation angles q = (Az, El), the coordinates of the point at the exit of the ionosphere are easily estimated. Applying the results of the magneto-ionic theory with models of the electron density and of the Earth s geomagnetic field, it is then possible to calculate both parameters h and a as functions of the D.O.A. q. The expressions of h O and h X have been first derived by Appleton [Kelso, 1964; Davies, 1990]. and " sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi# h O ¼ 1 Y 2 T 2Y L 1 X jz Y 4 T ð1 X jzþ 2 þ 4Y2 L " sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi# h X ¼ 1 Y 2 T 2Y L 1 X jz þ Y 4 T ð1 X jzþ 2 þ 4Y2 L with ð3þ ð4þ 2 X ¼ f p =f ð5þ f p is the plasma frequency of the medium at point M: p f p ðmþ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi NM ð Þ:e 2 =4p 2 e o m ð6þ Figure 2. Definition of the polarization ratio h. Case jhj < 1. 3of14

4 Figure 3. Definition of the inclination angle a. where N(M) is the electron density at point M, m and e are the electron mass and charge, and f is the wave frequency. Y L ¼ jyjcosðþ d jyj ¼ f H =f f H is the plasma gyrofrequency for point M: f H ¼ e:b o =2pm ð7þ ð8þ ð9þ where Bo is the modulus of the local terrestrial magnetic field and d is the angle between the propagation vector and the terrestrial magnetic field. Y T ¼ jyjsinðþ d Z ¼ n=2pf ð10þ ð11þ where n is the frequency of collisions between electrons and neutrals. This term is generally supposed to be equal to zero for frequencies f in the interval 3 30 MHz (HF band). [19] To achieve the determination of the incoming polarization, it can be noticed that (1) at the exit of the ionosphere, X = 0 as the electron density N(M) tends to zero; (2) the angle d depends on the variable q; (3)a database on the Earth s geomagnetic field Bo(M) is supposed to be available Spatial Response of a Receiving Antenna [20] The knowledge of the incident polarization yields an expression of the signal at the output of a receiving 4of14 antenna with a small size compared to the wavelength. In the case of a wire antenna with a simple geometry (dipole antenna for example), the expression results in the following: (1) a rotation matrix Q(a) characterizing, in the plane of the wave front, the change of the system of coordinates from the main axis of the ellipse to a second system regarding the local horizontal direction as a reference B C axis QðaÞ ¼ 0 cosðaþ 0 sinðaþ sinðaþ C A cosðaþ ð12þ (2) a second rotation matrix B(q = (Az, El)) characterizing the transition from the previous system to a topocentric system attached to the receiving antenna; its axis are for example the west-east, south-north and vertical directions 0 1 cos Az sin El: sin Az cos El: sin Az BðÞ¼ q sin Az sin El: cos Az cos El: cos Az B A 0 cos El sin El ð13þ (3) an antenna specific vector V combining the three components of the electric field in the received signal. For example, the expression of V for a vertical monopole with the length L is simply V ¼ ð0 0 LÞ ð14þ

5 With this description, the output signal s r (t) can be written as: s r ðþ¼v:b t ðþq q ðaþ B 1 A st ðþ ð15þ jh where s(t) is the temporal expression of the transmitted signal evaluated at the phase center of the antenna. Assuming that the parameters a and h are D.O.A. dependant in a deterministic way, the previous formula can be shortened as: s r ðþ¼f t ðþst q ð16þ where F(q) is named spatial response of the antenna [Rojas-Varela, 1987; Bertel et al., 1989]. F(q) is generally complex due to the structure of the phasor w. It is real valued only for linear polarizations (h =0). [21] For a magnetic loop antenna, the output signal s r (t) is expressed with the components of the magnetic vector H, deduced from the electric field E by the following transformation: with Hðt; rþ ¼ M em :Eðt; rþ ð17þ M em ¼ m o c B A ð18þ where m o is the magnetic constant and c is the speed of light. s r (t) is then written as s r ðþ¼ jwm t o V m :BðÞQ q ðaþm em B 1 A st ðþ ð19þ jh where V m is a tap vector specific of the magnetic antenna combining the components the magnetic vector in the expression of the magnetic flux and w is the pulsation of the incident wave. As an example, for an east-west oriented vertical loop with a surface S, V m is expressed as V m ¼ ð0 S 0Þ ð20þ For a given D.O.A. q, at least two magneto-ionic modes are expected: two spatial responses F O (q) and F X (q) are 5of14 then computed after the determination of the two polarization ratios h O and h X. 3. Signal Processing Operating on a Heterogeneous Array 3.1. Expressions for Array Processing [22] The current assumptions for array processing are as follows: (1) reception of narrowband signals (with a small bandwidth compared to the central frequency of the spectrum) so that the concept of geometrical phase is valuable; (2) spatially white noise generated on the NC sensors, denoted by {n n (t)} n = 1,...,NC are mutually uncorrelated with identical power Isotropic Sensors [23] An array is made up with NC isotropic sensors; one of them is the reference for the geometrical phase. NS waves are supposed to be incident and the corresponding output signals generated on the reference sensor are denoted by {s k (t)} k = 1,..,NS. Each D.O.A. is identified by the angle q k (or couple of angles in a 3-D search). The geometrical phase existing for that direction between the reference and the nth sensor is denoted by j n (q k ). It is calculated as the scalar product of the incident wave vector k(q k ) and the position vector r n specifying the location of the nth sensor relatively to the reference: j n ðq k Þ ¼ kðq k Þ:r n ð21þ The output on the nth antenna is expressed as: x n ðþ¼ t XNS k¼1 e jj nðq k Þ s k ðþþn t n ðþ t ð22þ The NC output signals on the array are written in the output column vector X(t) as: XðÞ¼ t XNS k¼1 aðq k Þs k ðþþn t ðþ t ð23þ where a(q k ) is the steering-vector for the D.O.A. q k and N(t) is the noise vector. The components of a(q k ) contain the different inter element phases: T aðq k Þ ¼ 1; e jj 2ðq k Þ ;...::; e jj NCðq k Þ ð24þ Associating the NS steering vectors in the matrix A provides the classical linear model: XðÞ¼AS t ðþþn t ðþ t ð25þ where S(t) is the signal vector Heterogeneous Array: General Case [24] A heterogeneous array is made up of sensors which are different from one another. An a priori knowledge is supposed for their respective spatial

6 responses denoted by {F n (q)}, n = 1,...,NC. In this context, the linear model for the output signals of the heterogeneous array is expressed as: X h ðþ¼ t XNS k¼1 a h ðq k Þs k ðþþn t h ðþ: t ð26þ The components of the steering-vectors a h (q k ) combine the spatial responses and the exponentials which represent the phases j n (q k ) calculated with respect to the array geometry: a h ðq k Þ ¼ F 1 ðq k F NC ðq k Þe jj 1ðq k ð Þe jj NC q k Þ ; F 2 ðq k Þe jj 2ðq k Þ ; Þ T...:; ð27þ Denoting by F(q) =(F 1 (q),...,f NC (q)) T the vector of the antenna responses for the D.O.A. q, the whole steering vector is expressed as: a h ðþ¼f q ðþa q ðþ q ð28þ where represents the Schur-Hadamard product for two matrices. It can be noticed that a h (q) does not have a constant norm; this remark will be taken into account when applying the MUSIC algorithm on this particular type of array Heterogeneous Array for HF Applications [25] For applications in the HF band, it has been underlined that two possible types of polarization (O and X) are expected at the exit of an ionospheric radio link. Consequently, for a given D.O.A. q, two steeringvectors are defined and attached to the corresponding incident modes by a hp ðþ¼ q F 1P ðþe q jj 1 ðþ q ; F 2P ðþe q jj 2 ðþ q ;...:; F NCP ðþe q jj T NCðÞ q ð29þ with P representing the polarization type O or X. [26] In other words, the heterogeneous array is polarization-sensitive considering that the steering-vector is polarization-dependant by means of the spatial responses of the antennas {F n0 (q)} and {F nx (q)}. That argument turns to be an advantage in a context of spatially highly correlated multipaths: several incident signals with a small angular separation can be distinguished by a heterogeneous array assuming that their polarizations are different. This point of view is quantified in the following paragraph Reduction of the Spatial Correlation [27] This section underlines the advantages of the heterogeneous array considered as a means to reduce the spatial correlation of two incident sources in certain conditions particularly as the angular separation is small. 6of Expressions of the Spatial Correlation [28] For an array of isotropic sensors the spatial correlation of two incident sources with D.O.A. q 1 and q 2 is expressed as: aðq 1 Þ H aðq 2 Þ r ¼ k aðq 1 Þ k : k aðq 2 Þ k ¼ X NC ej ½ j nðq 2 Þ j n ðq 1 ÞŠ NC ð30þ Using a heterogeneous array with the same geometry, the expression of the spatial correlation becomes: a h ðq 1 Þ H a h ðq 2 Þ r h ¼ k a h ðq 1 Þ k : k a h ðq 2 Þ k ½Fðq 1 Þaðq 1 ÞŠ H Fðq 2 Þaðq 2 Þ ¼ k Fðq 1 Þaðq 1 Þ k : k Fðq 2 Þaðq 2 Þ k ð31þ This quantity can be written under the following form: X NC F* n ðq 1 Þ:F n ðq 2 Þe j j n q 2 ð Þ j n ðq 1 Þ r h ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X NC X jf n ðq 1 Þj 2 NC : jf n ðq 2 Þj 2 ð ð32þ The modulus of the spatial correlation is modified for this type of array: jr h j 6¼ jrj Improvement Due to the Heterogeneous Array [29] The detection capability of a direction finding method is enhanced with a reduction on the spatial correlation of the incident waves. To compare the performances of homogeneous and heterogeneous arrays, the relative values of the two quantities jr h j and jrj considered as functions of the (two) angles of arrival can be computed. As illustrated with an example in section 3.2.3, the heterogeneity of the array does not systematically decrease the spatial correlation. However, it can be mentioned that current situations of reception are characterized by very close angles of arrival though the spatial responses are significantly different for the D.O.A. q 1 and q 2 : it is the case with diversely polarized electromagnetic waves impinging from the same direction. [30] The geometrical phases are nearly equal in these conditions (j n (q 1 ) ffi j n (q 2 )) and the modulus of the spatial correlation for the homogeneous array is close to 1: jrj 1. On the other hand, considering the correlation for the heterogeneous array, the modulus jr h j approaches the value: X NC F* n ðq 1 Þ:F n ðq 2 Þ jr h j ffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð33þ X NC X jf n ðq 1 Þj 2 NC : jf n ðq 2 Þj 2

7 Figure 4. Homogeneous circular array. This ratio is less or equal to one, according to the Schwartz inequality: the spatial correlation decreases in this case Numerical Simulation [31] In the following simulation, HF radio waves are supposed to be received after propagation through the ionosphere. It involves two uniform circular arrays made up with 8 loop antennas. This geometry has a practical advantage since the accuracy of the angular estimation does not depend on the actual azimuth of arrival in this case. The first array is made up with 8 identical sensors (homogeneous array, Figure 4) and its spatial correlation is r. In the second one, the antennas are subject to a rotation of 30 degrees around a vertical axis every two positions within the array (Figure 5), so that it becomes heterogeneous with a spatial correlation r h. This angular value realizes a compromise between a significant diversity of the spatial responses and the existence of redundancy if the rotation exceeds 180 degrees. The spatial responses of the antennas are computed according to the model described in section 2 (expressions (17) to (20)). The numerical parameters are: radius R = 0.5 l, l denoting the wavelength, and carrier frequency fo = 15 MHz. [32] Two incident waves are impinging with a common elevation and orthogonal polarizations. The first signal has a fixed azimuth of arrival (Az 1 = 180 ), and the second azimuth Az 2 varies in the range of 0 to 360. The simulation compares the spatial correlation of this couple of incident waves (considered as a function of the variable azimuth Az 2 ) for the homogeneous and the heterogeneous array. Let us remember that whatever the method of direction finding, the less is the modulus of the spatial correlation, the easier is the separation of the signals. The critical case corresponds to a modulus close or equal to one for which the separation fails. 7of14 [33] Figure 6 plots the spatial correlations for both arrays (modulus of r and r h ) as function of the variable azimuth Az 2. The benefit provided by the heterogeneous array appears in an interval of Az 2 containing the value of Az 1 = 180. In that area, the modulus jr h j is significantly less than jrj. [34] Particularly, in the critical case of two superimposed incident waves (Az 2 =Az 1 = 180 ), the modulus jr h j is equal to 0.82 and jrj to 1: the heterogeneous array can separate the incident waves (with orthogonal polarizations) while the homogeneous array cannot. jr h j is not less than jrj whatever the azimuth, but when it exceeds this reference value, the corresponding values are significantly less than 1, so that the D.O.A. estimation works correctly Original Computation of the MUSIC Algorithm [35] Several methods of direction finding use the eigen decomposition of the data covariance matrix R xx [Marcos et al., 1996]. A frequently implemented algorithm is MUSIC [Schmidt, 1981; Bienvenu and Kopp, 1979] which is based on the orthogonality between an incident steering vector and the noise subspace spanned by the eigenvectors of R xx associated with its smaller eigenvalues Implementation on a Heterogeneous Array (General Case) [36] The different steps of the MUSIC implementation on a heterogeneous array are (1) estimation of the covariance matrix ^R xxh ¼ 1 N ech X Nech X h ðnþx h ðnþ H ð34þ with Nech snapshots of data; (2) eigen decomposition of the covariance matrix and estimation of the number of Figure 5. Heterogeneous circular array.

8 Figure 6. Comparison of the spatial correlations. sources NSE based on the dispersion of the eigen values; (3) computation, for all the potential D.O.A. q, ofthe angular function evaluating the norm of the projection of the steering-vector in the noise subspace. The terms pseudospectrum (introduced by Schmidt) or directional spectrum can equivalently be used for that angular function. [37] The orthogonality is only approximated as the computation uses the estimated covariance matrix (nonasymptotic case). Consequently, it is necessary to project in the noise subspace variable vectors with a constant norm to provide a reliable estimation of the D.O.A. Keeping in mind that the steering-vector a h (q) ofthe heterogeneous array precisely has a variable norm as noticed in section 3.1.2, the MUSIC pseudospectrum is computed according to the following formula where b h (q) = a hðþ q j ðþj is the normalized steering-vector: a h q PSSP h ðþ¼ q X NC m¼nseþ1 1 v H m :b hðþ q 2 ð35þ The set of {v m }, m = NSE + 1,..,NC are the eigenvectors spanning the noise subspace and the expression of b h is [Erhel and Bertel, 1998; Erhel et al., 1999]: b h ðþ¼ q 1 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! B u X t jf n ðþ q j 2 0 F 1 ðþe q jj 1ðÞ q... F NC ðþe q jj NCðÞ q 1 C A ð36þ Implementation in the Context of HF Direction Finding [38] The implementation of the MUSIC algorithm on a heterogeneous array in order to achieve HF direction finding is now presented. As shown in section 2, two propagation modes are expected for a given D.O.A. at the end of a radio link through the ionosphere. Therefore, in the computation of the algorithm two normalized steering vectors are computed for each angle of arrival q under test. The corresponding expression is b hp ðþ¼ q 1 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! B u X t jf np ðþ q j 2 0 F 1P ðþe q jj 1ðÞ q :: F NCP ðþe q jj NCðÞ q 1 C A ð37þ where the index P denotes the polarization type (O or X). At this step, and for each value of q, the spatial responses for the O mode (respectively X mode) are computed with the expression of the polarization ratio h O (respectively h X ). [39] The directions of arrival being estimated for both polarizations, the two sets of steering-vectors are projected in the noise subspace. Consequently, in this original version of the algorithm, two pseudospectra are computed according to the following expression: PSSP P ðþ¼ q X NC m¼nseþ1 1 ; P ¼ OorX v T m :b hpðþ q 2 ð38þ 8of14

9 Figure 7. Pseudospectrum (homogeneous array). The D.O.A.s are estimated with two different array manifolds, which illustrates that the heterogeneous array is polarization-sensitive. Several authors took advantage of the polarization to separate the incident signals in receiving schemes for which the polarization was not linear [Ferrara and Parks, 1983; Demeure et al., 1993; Bertel et al., 1993]. Their solutions require at least two sensors at each point of the spatial sampling, the corresponding antennas being sensitive to different components of the electromagnetic field. On the contrary, the proposed technique needs only one sensor at each position of the array. The incident polarization is taken into account, though the total number of sensors is divided by two, as compared with the previous solutions. However, it must be noticed that a reliable electromagnetic model of the spatial responses is required. Additionally, this method gives implicitly an estimation of the incident polarizations (O or X); this information is required by the ray tracing software that runs with the results of the direction finding in a global project of single site localization Numerical Simulation [40] In order to illustrate the previous points, a simulation of HF received signals has been performed. A single transmitted signal (AM modulation with a bandwidth of 2 khz) is reflected by two different ionospheric layers (F1 and F2). The anisotropy of each layer generates two polarizations and the corresponding couple of signals are strongly correlated with very close elevations. Consequently, a total of 4 incident waves are impinging on the receiving station. [41] Assuming that no tilt exists (horizontal gradient of the electron density), the azimuths of arrival are supposed to be equal. The actual values of the elevations 9of14 are El(1) = 30, X mode; El(2) = 32, O mode; El(3) = 60, X mode; El(4) = 62, O mode. The four modes are assumed to be impinging on the two circular arrays described in section The radius is equal to the wavelength, the average signal to noise ratio is equal to 20 db and the BT product (bandwidth multiplied by the differential time delay) for two signals which are reflected by the same layer is equal to 0.2. This last value indicates that the corresponding temporal correlation is strong. [42] A first simulation refers to the homogeneous array depicted in Figure 4. Figure 7 plots the pseudospectrum (as a function of the elevation only) computed with an azimuth supposed to be exactly estimated. It can be seen that the pairs of signals reflected by a common layer are not separated. [43] The next simulation deals with the circular heterogeneous array (Figure 5) and the same receiving scheme. Two pseudospectra PSSP O and PSSP X (one for each expected polarization) are computed and jointly plotted on Figure 8. Two maxima are detected for each polarization, underlining that the separation of the four incident signals is efficient in this case and is due to the polarization-sensitivity of the heterogeneous array. 4. Experimental Validations 4.1. Receiving and Sampling System [44] The I.E.T.R. laboratory (Institut d Electronique et de Télécommunications de Rennes) has developed a receiving site in an area of low urban density situated 30 km west of Rennes (longitude W; latitude N). A receiving system has been set up,

10 Figure 8. Pseudospectra (heterogeneous array). including the following elements: 8 active loop antennas with a 1.3 meter diameter, with low noise preamplifiers and low intermodulation products; 8 low loss coaxial cables 100 meters long; 8 identical coherent receivers, the design of which uses two frequency mixers, two bandwidths being available (50 Hz or 3 khz); a synchronous 8 input analog to digital converting card with a 12 bit resolution and a sampling frequency being adjustable up to 120 khz. [45] For the experiments which are described in this section, the array has a circular uniform geometry and a radius of 25 m. Its setup has been realized in accordance with the description of the heterogeneous array depicted in section HF Direction Finding on the Heterogeneous Array [46] The main goal is to achieve the angular estimation of the incident waves for broadcast transmitters located more than 1000 km from the receiving site. The experiments intend to exhibit the possible separation of multipaths with a strong spatial correlation by means of the polarization. [47] In order to appreciate the reliability of the experimental results, the estimated azimuths are compared with the geographical azimuths of the transmitting sites and the estimated elevations are compared with the predictions provided by a ionospheric propagation software Data Processing [48] The acquisitions become complex valued with a Hilbert filtering and are then corrected according to the calibration files. This last operation aims to suppress the phase and modulus differences detected between the 8 receiving channels. This correction obviously needs a preliminary measurement of these differences at the different carrier frequencies for which the direction finding will be later processed. [49] The MUSIC algorithm then runs with the complex calibrated data. In order to provide a reliable estimation of the D.O.A., a large data file is processed (typically samples at a rate of 40 ksamples/s) and then is arranged to achieve an effective averaging of the pseudospectra and sorting of the angular estimations. Actually, if the acquisition file is globally processed, a single pseudospectrum will be computed (for each polarization) with samples. A single set of angles of arrival will be estimated with a low level of reliability: some secondary peaks, due to artifacts, will be considered as actual D.O.A. [50] To cope with this problem, the acquisition is divided into 20 consecutive parts containing 6000 samples; each part is divided in 5 series with a 50% overlapping and a length of 2000 samples. For each part, separated from the previous one with an interval = 0.15s, a pseudospectrum for each polarization is computed by averaging the angular functions provided by the 5 time series. [51] Considering that the total number of samples is constrained in an acquisition file (fixed by the capacity of the buffers in the synchronous A.D.C. card), a compromise has to be found: increasing the length of the time series (or the number of time series) gives pseudospectra with a better contrast but induces a reduction of the number of consecutive parts. Consequently, the angular statistics will be based on a reduced number of estimations and become less reliable. [52] The final adjustment of the parameters such as the length of each time series or the sampling frequency results from the observation of pseudospectra computed 10 of 14

11 Figure 9. Pseudospectra (a) (X mode) and (b) (O mode) June 20, 2002; 11h10 U.T. See color version of this figure at back of this issue. 11 of 14

12 in a large number of experiments in HF radio direction finding; it has essentially an empirical justification. In an ideal case, if each part of the data file gives a valid pseudospectrum for each polarization, statistics are available for the D.O.A. estimation (mean value and standard deviation for elevation and azimuth) upon a set of 20 data. However, the computed pseudospectra must comply with a list of criteria [Erhel et al., 2002] to be validated. The criteria for each time series consist of (1) application of a minimum threshold to the major eigenvalue of the covariance matrix to ensure that the level of signal is sufficient regarding the quantum of analog to digital conversion; (2) application of a minimum threshold to the ratio of the major to the minor eigenvalue to protect from a poor signal to noise ratio. The criteria for each part of the acquisition and for each expected polarization consist of (1) confirmation that the maximum peak of the averaged pseudospectrum is clearly higher than the mean value; the angles of arrival relative to this major peak are considered as references; (2) rejection of the local maxima with an azimuth separated from the reference azimuth by more than a fixed threshold. [53] Considering both polarizations, an additional original criterion is based on the computation of two pseudospectra (O and X): if several signals with different polarizations are detected, the corresponding estimated azimuths must be contained in a limited angular interval considering the existence of a single transmitter. If at least one averaged pseudospectrum (O or X or both) passes these tests, its maxima become valid angular estimations. The set of angular estimates are then sorted according to the following steps: (1) identification of the most occurring number of detected signals, (2) selection of the corresponding angular estimations and computation of the mean value, and (3) rejection of the estimations appearing too far from the mean value Experimental Results [54] Some experimental pseudospectra are presented in this section as well as the corresponding angular statistics. The transmitter is located in Nador (Morocco: longitude W; latitude N). The carrier frequency is MHz and the geographical azimuth (considered as a reference) is 178. Figure 9 displays 3 examples among the 20 pseudospectra computed for the two expected modes (O and X) with the same acquisition file. [55] This figure underlines the existence of one uppermost peak for each type of pseudospectrum: 2 incident signals are detected with complementary polarizations; Table 1. Statistics of the Angular Estimations Table 2. Statistics of the Angular Estimations (Azimuth) Az Mean Rough estimation Sorted estimation the corresponding estimations of the azimuth of arrival are coherent and close to the expected one (178 ). Some secondary peaks are visible, particularly near the azimuth zero for the O pseudospectrum: they are considered as artifacts and are rejected by the angular sorting according to the tests listed in section [56] The statistics of the angular estimations (mean value and standard deviation for azimuth and elevation) are presented in the Table 1. The estimated azimuths are close to the expected value. A slight bias, however, is present and can be justified by several reasons. At first, it may result from some defaults of the receiving system like a lack of resolution in the inter channel calibration. Secondly, it can be accounted for by an inclination of the reflecting ionospheric layer that may be not strictly horizontal or, equivalently, by the presence of horizontal gradient of the electron density. The greater value of this bias (4.4 ) is comparable to the deviations measured by other authors [Tedd et al., 1984] experimenting HF transmissions in the European midlatitudes. [57] The mean elevations are in a good agreement with the predicted ones. Their standard deviations are relatively high due to the lack of resolution in elevation for a horizontal array. [58] The improvement in the reliability of the angular estimations yielded by the sorting of the pseudospectra is illustrated on Figure 10. A long-term experiment has been running with a D.O.A. estimation every 10 mn during a period of 12 h. The involved transmitter is located in Hamburg (Germany; fo = MHz). The geographical azimuth, considered as a reference, equals 47. The figure compares, for the only O polarization, the results of the rough angular estimations (extracted from all the local maxima of the pseudospectra) plotted in blue and the sorted estimations plotted in red (according to the criterions listed in section 4.2.1) with a representation giving jointly the elevation and azimuth of the D.O.A.s. [59] The dispersion of the estimations is clearly reduced by the postprocessing. The corresponding mean values and standard deviations for the azimuth are given in the Table 2; similar data for the elevation are not relevant due to the nonstationarity of this parameter during the experiment. s Az Az (Mean Value) s Az El (Mean Value) s El Path Path of Conclusion [60] The implementation of the MUSIC algorithm has been described for a heterogeneous array. A reliable

13 Figure 10. Reduction of the dispersion of the angular estimations. See color version of this figure at back of this issue. model of the spatial responses of the antenna is required to ensure the robustness of the method. That condition can be verified by the choice of antennas with a small size compared to the wavelength and a simple geometry. Moreover, in the particular case of HF direction finding, a thorough knowledge of the electromagnetic propagation in the ionosphere is necessary. [61] The proposed technique takes advantage of the polarization-sensitivity that characterizes the heterogeneous array though only one sensor is placed at each point of the spatial sampling. The resulting decorrelation for the incident modes is significant, principally in reception schemes for which two types of polarizations are expected. The D.O.A.s are estimated from two different pseudospectra, which appears to be similar to polarization filtering. Thus the method can be transposed for example in a VHF receiving scheme with left or right circular polarizations. References Bertel, L., J. Rojas-Varela, D. Cole, and P. Gourvez (1989), Polarisation and ground effects on HF receiving antenna patterns, Ann. Télécommun., 44(7 8). Bertel, L., A. Edjeou, V. Massot, and Y. Erhel (1993), Méthode itérative de filtrage de polarisation: Application à la goniométrie HF, actes du colloque GRETSI, pp , Juanles-Pins, Sept. Bienvenu, G., and L. Kopp (1979), Principe de la goniométrie passive adaptative, actes du colloque Gretsi, pp. 106/1 10, Nice. Budden, K. G. (1952), The theory of the limiting polarisation of radio waves reflected from the ionosphere, Proc. R. Soc. London, Ser. A. Davies, K. (1990), Ionospheric Radio, Peter Peregrinus, London. Demeure, C., A. Ferreol, and J. L. Rogier (1993), Mesures d angles d élévation en HF par une méthode haute résolution utilisant la diversité de polarisation, paper presented at Fifty- Third Agard-EPP Symposium, Rotterdam, Netherlands, Oct. Erhel, Y., and L. Bertel (1998), A method of direction finding operating on a heterogeneous array, paper presented at EUSIPCO Conference, Rhodes, Sept. Erhel, Y., F. Marie, and L. Bertel (1999), Résultats expérimentaux d une méthode de goniométrie opérant sur un réseau hétérogène de capteurs, actes du colloque Gretsi, pp , Vannes, Sept. Erhel, Y., F. Marie, and L. Bertel (2002), Statistical analysis of HF direction finding estimations, paper presented at URSI - GA 2002, Maastricht, Aug. Ferrara, E. R., and T. M. Parks (1983), Direction finding with an array of antennas having diverse polarizations, IEEE Trans. Antennas Propag., 31, Kelso, J. (1964), Radio Ray Propagation in the Ionosphere, McGraw-Hill, New York. Krim, H., and M. Viberg (1996), Two decades of array signal processing research, IEEE Signal Process., of 14

14 Marcos, S., et al. (1996), Méthodes à Haute Résolution, Librairie Hermès, Annemasse, France. Ratcliffe, J. A. (1962), The Magneto-Ionic Theory and Its Application to the Ionosphere, Cambridge Univ. Press, New York. Rojas-Varela, J. (1987), Antennes filtre de polarisation dans la bande HF, Ph.D. dissertation, Univ. de Rennes 1, Rennes, Mai. Schmidt, R. O. (1981), A signal subspace approach to multiple emitter location and spectral estimation, Ph.D. dissertation, Stanford Univ., Stanford, Calif., Nov. Tedd, B. L., H. J. Strangeways, and T. B. Jones (1984), The influence of large scale TIDs on the bearings of geographically spaced HF transmissions, J. Atmos. Terr. Phys., 46(2), L. Bertel, D. Lemur, and F. Marie, Institut d Electronique et Télécommunications de Rennes, UMR 6164, Université de Rennes 1, Campus de Beaulieu, Rennes Cedex, France. Y. Erhel, Centre de Recherches des Ecoles de Coëtquidan Saint-Cyr, Guer Cedex, France. (yvon.erhel@st-cyr. terre.defense.gouv.fr) 14 of 14

15 Figure 9. Pseudospectra (a) (X mode) and (b) (O mode) June 20, 2002; 11h10 U.T. 11 of 14

16 Figure 10. Reduction of the dispersion of the angular estimations. 13 of 14