F Marie

M Oger

D Lemur

Y Erhel

G Le Bouter

L Bertel

C Brousseau

A Low Complexity Method for HF Direction Finding

This paper presents a new algorithm with a low complexity for HF (3-30 MHz) direction finding (DF). This method is applied to different types of signal and to different configurations of circular arrays (homogeneous or heterogeneous).

Introduction:

In order to check or calibrate HF receiving antenna arrays we proposed [START_REF] Oger | A method to calibrate HF receiving antenna arrays[END_REF] to compare the estimated differential responses of an array with the theoretical values for a given direction of arrival (DOA) of the incident wave. In this letter, this idea is generalized to the determination of directional vectors which minimize the differences between the phases of the estimated and theoretical values of the array responses. This criterion is the basis of a simple DF method. After a short description of the proposed method, some experimental results involving different types of signal and different configurations of array are presented.

A method to estimate the differential responses of an array: Let us consider a receiving array. In the case of one incoming path, the signal at the output of an antenna can be expressed as:

( )

) (t N Sr(t) a (t) X n n n + = θ (1)
X n (t) : acquisition at the output of the channel n, Sr(t) : signal corresponding to the received wave,

Sr(t (θ a (θ a (θ a (t) X (t) X (t) X

NC n NC n ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ = ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ (4) One channel is considered as a reference with a ref (θ) = 1 so that Sr(t) = X ref (t)
From these equations the estimated differential response n a ~of channel n is extracted by the following expression:

) ( . ) ( ). ( ) ( ~* * t X t X t X t X a n ref ref ref n = (
For the measurements presented in the next paragraph, channel 1 is the reference (REF= ref = 1). The choice of the channel of reference is not critical for a circular array; for a heterogeneous array made up of very different antennas, however the choice would be less evident. It is desirable to choose a well modelled antenna giving a good signal to noise ratio.

The DOA of the sources are estimated by searching the directional vectors which minimize the differences between the phases of the estimated and those of the theoretical array responses. These correspond to the angular values which maximize a directional function expressed as:

∑ = - = NC n nor n thnor n PSSP a a L 1 2 )) ãrg( )) ( (arg( 1 ) ( θ θ (8)
In the case of a heterogeneous array, the theoretical responses [2] for the ordinary (denoted O) mode and for the extraordinary (denoted X) mode are computed, providing two directional functions (one for each expected polarization) and thus two pseudo-spectra (Fig. 4).

  N n (t) : additive noise, θ : angle of arrival (azimuth, elevation).On the basis of equation (1) and neglecting the additive noise, the signals existing at the outputs of the channels can be expressed as:

  , the first ones must be normalized with respect to a channel of reference: the same channel of reference as in equation (5) (REF = ref),if not the estimated responses have to be also normalized

4

  www.vtplc.com/communications/product.asp?itemID=1021&s=&catid=486 Authors' affiliations: F. Marie, M. Oger, D. Lemur, L. Bertel, C. Brousseau (Institut d'Electronique et Télécommunications de Rennes, Université de Rennes 1, Campus de Beaulieu 35042 RENNES France) Y. Erhel (Centre de Recherches des Ecoles de St Cyr Coetquidan 56381 GUER France) G. Le Bouter (CRESAT Ecole Supérieure et d'Application des Transmissions Quartier Leschi 35998 Rennes Armées France) E-mail: francois.marie@univ-rennes1.fr

Figure captions : Fig. 1 :

 :1 Figure captions: Fig. 1: Homogeneous (a) and heterogeneous (b) circular array

Y. Erhel, D. Lemur, L. Bertel and F. Marie 'H.F. radio direction finding operating on a heterogeneous array: principles and experimental validation', Radio-Science, Jan-Feb 2004, vol

39, n°1, pp 1003-1; 1003-14.

The experimental array: It contains eight active loop antennas equally spaced on a horizontal circle with a 25 m radius. If all the antennas are oriented in the same direction, the array is homogeneous (Fig. 1