Multipath resolving correlation interferometer direction finding

• US 20070273576A1
• Filed: 05/27/2006
• Published: 11/29/2007
• Est. Priority Date: 05/27/2006
• Status: Active Grant

• Current Assignee: BAE Systems Information and Electronic Systems Integration Incorporated (BAE Systems Plc)
• Original Assignee: BAE Systems Information and Electronic Systems Integration Incorporated (BAE Systems Plc)
• Inventors: Struckman, Keith A., Martel, Robert T.
• Granted Patent: US 7,358,891 B2
• Time in Patent Office: Days
• US Class Current: 342/156
• CPC Class Codes: G01S 3/146 by comparing linear polarisation components
• CPC Class Codes: G01S 3/74 Multi-channel systems specially adapted for direction-finding, i.e. having a single antenna system capable of giving simultaneous indications of the directions of different signals systems in which the directions of different signals are determined sequentially and displayed simultaneously G01S3/04, G01S3/14

Abstract

Apparatus and a method utilizing correlation interferometer direction finding for determining the azimuth and elevation to an aircraft at long range and flying at low altitudes above water with a transmitting radar while resolving multipath signals. The signals from the radar are received both directly and reflected from the surface of the water using horizontally polarized and vertically polarized antenna arrays, are digitized and are stored in separate covariant matrices. Eigenvalues for the eigenvectors of the matrices processed on signal samples recorded on horizontally polarized X arrays are compared to the eigenvalues for the eigenvectors of the covariance matrices processed on signal samples recorded on vertically polarized X arrays. Incident field polarization is associated with the antenna array measurements that yield the strongest eigenvalue. The eigenvector and eigenvalues for the strongest signal are selected and used for subsequent signal processing. An initial global search assuming mirror sea-state reflection conditions using the signal eigenvector having the strongest eigenvalue is performed to yield an approximate elevation α and azimuth β to the aircraft. The approximate values are then used as the starting point for a subsequent conjugate gradient search to determine accurate elevation α and azimuth β to the aircraft.

Claims

1. A method for determining the azimuth and elevation to an aircraft at long range that is transmitting an electromagnetic signal while flying at a low altitude above water, the signal being received using a plurality of vertically polarized antennas and horizontally polarized antennas in an antenna array, the received signal is received directly from the aircraft subject to electromagnetic scattering perturbations and is received via multi-path reflections from the surface of the water, and there is a calibration manifold having stored array steering vectors that is used to correct for the perturbations, the method comprising the steps of:

(a) forming a first plurality of covariance matrices from a like plurality of sets of samples of the electromagnetic signal received by the plurality of vertically polarized antennas;

(b) forming a second plurality of covariance matrices from a like plurality of sets of samples of the electromagnetic signal received by the plurality of horizontally polarized antennas;

(c) calculating signal eigenvalues and associated eigenvectors for each of the plurality of covariance matrices formed in steps (a) and (b) for the signal received directly from the aircraft and received via multi-path reflections from the surface of the water;

(d) comparing the signal eigenvalues for the first plurality of covariance matrices with the signal eigenvalues for the second plurality of covariance matrices to determine if the received signal has stronger vertical or horizontal polarization, and selecting the signal eigenvector associated with the signal having the stronger polarization for subsequent processing in steps (e) and (f);

(e) performing a global correlation maximum search, assuming mirror sea-state reflection conditions using the signal eigenvector selected in step (d) with steering vectors retrieved from the calibration array manifold to determine a first azimuth and elevation; and

(f) performing a conjugate gradient based correlation search using the selected signal eigenvector and the array manifold in the region of the first azimuth and elevation to determine accurate azimuth and elevation information to the aircraft.

2. The method for determining the azimuth and elevation to an aircraft of claim 1 further comprising the step of:

(g) correlating the signal eigenvector selected in step (d) and used in step (e) with the calibration manifold to obtain array steering vectors stored in the calibration manifold to be used to determine accurate azimuth and elevation information to the aircraft.

3. The method for determining the azimuth and elevation to an aircraft of claim 2 further comprising the steps of:

(h) digitizing the samples of the electromagnetic signal received by the plurality of vertically polarized antennas as real and imaginary components before they are processed into the first covariance matrix in step (a); and

(i) digitizing the samples of the electromagnetic signal received by the plurality of horizontally polarized antennas as real and imaginary components before they are processed into the second plurality of covariance matrices in step (b).

4. The method for determining the azimuth and elevation to an aircraft of claim 3 wherein the electromagnetic signal received by the plurality of vertically polarized antennas is sampled at the Nyquist rate to form the first plurality of covariance matrices in step (a), and wherein the electromagnetic signal received by the plurality of horizontally polarized antennas is sampled at the Nyquist rate to form the first plurality of covariance matrices in step (b).

5. The method for determining the azimuth and elevation to an aircraft of claim 4 further comprising the step of:

(j) alternately sampling the electromagnetic signal received by the plurality of vertically polarized antennas and the plurality of horizontally polarized antennas of the antenna array to create the first plurality of covariance matrices in step (a) and the second plurality of covariance matrices in step (b).

6. The method for determining the azimuth and elevation to an aircraft of claim 5 further comprising the step of:

(k) constructing the calibration manifold having stored array steering vectors that is used to correct for the perturbations on an accurately calibrated antenna range.

7. The method for determining the azimuth and elevation to an aircraft of claim 6;

wherein the signals used to form the first plurality of covariance matrices in step (a) are digitized and converted to real and imaginary form before being stored in the matrices, and

wherein the signals used to form the second plurality of covariance matrices in step (b) are digitized and converted to real and imaginary form before being stored in the matrices.

8. A method comprising determining the azimuth and elevation to an aircraft at long range and transmitting an electromagnetic signal while flying at a low altitude above water, wherein the signal is received using a plurality of vertically polarized antennas and horizontally polarized antennas in an antenna array, wherein the received signal is received directly from the aircraft and is received via multi-path reflections from the surface of the water, wherein the azimuth and elevation to the aircraft are defined by a conjugate gradient based correlation search of covariance matrices containing the signals received by the vertically polarized antennas and the horizontally polarized antennas in digitized form representing the real and imaginary components of the received signal, and wherein the following equation is used to process the digitized information stored in the covariance matrices is searched to find the azimuth and elevation to the aircraft using the equation:

${<br><br>R<br><br>\left(\mathrm{\alpha <br><br>},\mathrm{\beta <br><br>}\right)<br><br>}^2=\frac{\underset{\mathrm{na}=1}{\overset{N}{\sum <br><br>}}<br><br>{<br><br>{\left\{\begin{array}{c}{\mathrm{\rho <br><br>}}_d\times <br><br>A_{\mathrm{pol}}<br><br>\left(\mathrm{\alpha <br><br>},\mathrm{\beta <br><br>},\mathrm{na}\right)+{\mathrm{\rho <br><br>}}_r\times <br><br>\\ A_{\mathrm{pol}}<br><br>\left(-\mathrm{\alpha <br><br>}+\mathrm{\Delta <br><br>\alpha <br><br>},\mathrm{\beta <br><br>}+\mathrm{\Delta <br><br>\beta <br><br>},\mathrm{na}\right)\end{array}\right\}}^{*}\times <br><br>U_{\mathrm{pol}}<br><br>\left(\mathrm{na}\right)<br><br>}^2}{\underset{\mathrm{na}=1}{\overset{N}{\sum <br><br>}}<br><br>{<br><br>\left\{\begin{array}{c}{\mathrm{\rho <br><br>}}_d\times <br><br>A_{\mathrm{pol}}<br><br>\left(\mathrm{\alpha <br><br>},\mathrm{\beta <br><br>},\mathrm{na}\right)+{\mathrm{\rho <br><br>}}_r\times <br><br>\\ A_{\mathrm{pol}}<br><br>\left(-\mathrm{\alpha <br><br>}+\mathrm{\Delta <br><br>\alpha <br><br>},\mathrm{\beta <br><br>}+\mathrm{\Delta <br><br>\beta <br><br>},\mathrm{na}\right)\end{array}\right\}<br><br>}^2\times <br><br>\underset{\mathrm{na}=1}{\overset{N}{\sum <br><br>}}<br><br>{<br><br>U_{\mathrm{pol}}<br><br>\left(\mathrm{na}\right)<br><br>}^2}$

where |R(α

,β

)|² is the correlation squared and the maximum value is searched for during the conjugate gradient based correlation search, α

is the elevation angle to the aircraft transmitting the signal, β

is the azimuth angle to the aircraft transmitting the signal, ρ

_d is the direct complex coefficient, ρ

_r is the reflected complex coefficient, * is a complex conjugate, na is the number of antennas in the beam forming array utilized to receive the signals transmitted by the aircraft, A_pol are calibration vectors as a function of α and

β

, and U_pol is the eigenvector resolved received signal vector and is composed of the directly received signal component and the reflected signal component and is equal the sum of the eigenvectors of the signals from the vertically polarized antennas and the horizontally polarized antennas.

9. The method for determining the azimuth and elevation to an aircraft of claim 8 wherein the signals received using the plurality of vertically polarized antennas are digitized and stored in a first covariant matrix, and the signals received using the plurality of horizontally polarized antennas are digitized and stored in a second covariant matrix;

wherein the first covariant matrix undergoes eigenspace decomposition to produce a first set of signal array eigenvectors U having first eigenvalues;

wherein the second covariant matrix undergoes eigenspace decomposition to produce a second set of signal array eigenvectors U having second eigenvalues;

wherein the first eigenvalues are compared to the second eigenvalues and the received electromagnetic signal is vertically polarized if the first eigenvalues are greater than the second eigenvalues, and is horizontally polarized if the first eigenvalues are less than the second eigenvalues; and

wherein the eigenvector with the associated highest eigenvalues are used to perform the conjugate gradient based correlation search.

10. The method for determining the azimuth and elevation to an aircraft of claim 9 wherein before a conjugate gradient based correlation search is performed using the equation in claim 8 to determine the azimuth and elevation to the aircraft, a global correlation search is performed using the equation in claim 8 but the terms Δ

α and

Δ

β

in the equation are set equal to zero and it is assumed that mirror reflection conditions for reflection of the electromagnetic signal from the surface of the water, the result obtained using the equation in claim 8 are approximate values for elevation α and

azimuth β

to the aircraft, the approximate values then being used as the starting point in the conjugate gradient based correlation search using the equation in claim 8.

11. The method for determining the azimuth and elevation to an aircraft of claim 10 wherein the maximization of the term |R(α

,β

)|² over ρ

_d, ρ

_r is achieved by a observing that |R(α

,β

)|² can be the ratio of quadratic forms in closed form when the numerator of the equation in claim 8 is defined as equal to

${<br><br>{\mathrm{\rho <br><br>}}_d<br><br>}^2\times <br><br>{<br><br>\underset{\mathrm{na}=1}{\overset{N}{\sum <br><br>}}<br><br>\left\{{A<br><br>\left(\mathrm{\alpha <br><br>},\mathrm{\beta <br><br>},\mathrm{na}\right)}^{*}\times <br><br>U(<br><br>\mathrm{na})\right\}<br><br>}^2+{\mathrm{\rho <br><br>}}_d^{*}\times <br><br>{\mathrm{\rho <br><br>}}_r\times <br><br>\underset{\mathrm{na}=1}{\overset{N}{\sum <br><br>}}<br><br>\left\{{A<br><br>\left(\mathrm{\alpha <br><br>},\mathrm{\beta <br><br>},\mathrm{na}\right)}^{*}\times <br><br>U<br><br>\left(\mathrm{na}\right)\right\}\times <br><br>\underset{\mathrm{na}=1}{\overset{N}{\sum <br><br>}}<br><br>{\left\{{A<br><br>\left(-\mathrm{\alpha <br><br>}+\mathrm{\Delta <br><br>\alpha <br><br>},\mathrm{\beta <br><br>}+\mathrm{\Delta <br><br>\beta <br><br>},\mathrm{na}\right)}^{*}\times <br><br>U<br><br>\left(\mathrm{na}\right)\right\}}^{*}+{\mathrm{\rho <br><br>}}_d\times <br><br>{\mathrm{\rho <br><br>}}_r^{*}\times <br><br>\underset{\mathrm{na}=1}{\overset{N}{\sum <br><br>}}<br><br>{\left\{{A<br><br>\left(\mathrm{\alpha <br><br>},\mathrm{\beta <br><br>},\mathrm{na}\right)}^{*}\times <br><br>U<br><br>\left(\mathrm{na}\right)\right\}}^{*}\times <br><br>\underset{\mathrm{na}=1}{\overset{N}{\sum <br><br>}}<br><br>\left\{{A<br><br>\left(-\mathrm{\alpha <br><br>}+\mathrm{\Delta <br><br>\alpha <br><br>},\mathrm{\beta <br><br>}+\mathrm{\Delta <br><br>\beta <br><br>},\mathrm{na}\right)}^{*}\times <br><br>U<br><br>\left(\mathrm{na}\right)\right\}+{<br><br>{\mathrm{\rho <br><br>}}_r<br><br>}^2\times <br><br>{<br><br>\underset{\mathrm{na}=1}{\overset{N}{\sum <br><br>}}<br><br>\left\{{A<br><br>\left(-\mathrm{\alpha <br><br>}+\mathrm{\Delta <br><br>\alpha <br><br>},\mathrm{\beta <br><br>}+\mathrm{\Delta <br><br>\beta <br><br>},\mathrm{na}\right)}^{*}\times <br><br>U<br><br>\left(\mathrm{na}\right)\right\}<br><br>}^2$

and the denominator of the equation in claim 8 is defined as equal to

$\underset{\mathrm{na}=1}{\overset{N}{\sum <br><br>}}<br><br>{<br><br>U<br><br>\left(\mathrm{na}\right)<br><br>}^2\times <br><br>\left\{{<br><br>{\mathrm{\rho <br><br>}}_d<br><br>}^2\times <br><br>\underset{\mathrm{na}=1}{\overset{N}{\sum <br><br>}}<br><br>{<br><br>A<br><br>\left(\mathrm{\alpha <br><br>},\mathrm{\beta <br><br>},\mathrm{na}\right)<br><br>}^2+{\mathrm{\rho <br><br>}}_d^{*}\times <br><br>{\mathrm{\rho <br><br>}}_r\times <br><br>\underset{\mathrm{na}=1}{\overset{N}{\sum <br><br>}}<br><br>\left({A<br><br>\left(\mathrm{\alpha <br><br>},\mathrm{\beta <br><br>},\mathrm{na}\right)}^{*}\times <br><br>A<br><br>\left(-\mathrm{\alpha <br><br>}+\mathrm{\Delta <br><br>\alpha <br><br>},\mathrm{\beta <br><br>}+\mathrm{\Delta <br><br>\beta <br><br>},\mathrm{na}\right)\right)+{\mathrm{\rho <br><br>}}_d\times <br><br>{\mathrm{\rho <br><br>}}_r^{*}\times <br><br>\underset{\mathrm{na}=1}{\overset{N}{\sum <br><br>}}<br><br>\left(A<br><br>\left(\mathrm{\alpha <br><br>},\mathrm{\beta <br><br>},\mathrm{na}\right)\times <br><br>{A<br><br>\left(-\mathrm{\alpha <br><br>}+\mathrm{\Delta <br><br>\alpha <br><br>},\mathrm{\beta <br><br>}+\mathrm{\Delta <br><br>\beta <br><br>},\mathrm{na}\right)}^{*}\right)+{<br><br>{\mathrm{\rho <br><br>}}_r<br><br>}^2\times <br><br>\underset{\mathrm{na}=1}{\overset{N}{\sum <br><br>}}<br><br>{<br><br>A<br><br>\left(-\mathrm{\alpha <br><br>}+\mathrm{\Delta <br><br>\alpha <br><br>},\mathrm{\beta <br><br>}+\mathrm{\Delta <br><br>\beta <br><br>},\mathrm{na}\right)<br><br>}^2\right\}$

and the closed form maximization process is further simplified by letting H and G terms be defined as

$H_{11}={<br><br>\underset{\mathrm{na}=1}{\overset{N}{\sum <br><br>}}<br><br>\left\{{A<br><br>\left(\mathrm{\alpha <br><br>},\mathrm{\beta <br><br>},\mathrm{na}\right)}^{*}\times <br><br>U<br><br>\left(\mathrm{na}\right)\right\}<br><br>}^2$ $H_{12}=\underset{\mathrm{na}=1}{\overset{N}{\sum <br><br>}}<br><br>\left\{{A<br><br>\left(\mathrm{\alpha <br><br>},\mathrm{\beta <br><br>},\mathrm{na}\right)}^{*}\times <br><br>U<br><br>\left(\mathrm{na}\right)\right\}\times <br><br>\underset{\mathrm{na}=1}{\overset{N}{\sum <br><br>}}<br><br>\left\{A<br><br>\left(-\mathrm{\alpha <br><br>}+\mathrm{\Delta <br><br>\alpha <br><br>},\mathrm{\beta <br><br>}+\mathrm{\Delta <br><br>\beta <br><br>},\mathrm{na}\right)\times <br><br>{U<br><br>\left(\mathrm{na}\right)}^{*}\right\}$ $H_{21}=\underset{\mathrm{na}=1}{\overset{N}{\sum <br><br>}}<br><br>{\left\{{A<br><br>\left(\mathrm{\alpha <br><br>},\mathrm{\beta <br><br>},\mathrm{na}\right)}^{*}\times <br><br>U<br><br>\left(\mathrm{na}\right)\right\}}^{*}\times <br><br>\underset{\mathrm{na}=1}{\overset{N}{\sum <br><br>}}<br><br>\left\{{A<br><br>\left(-\mathrm{\alpha <br><br>}+\mathrm{\Delta <br><br>\alpha <br><br>},\mathrm{\beta <br><br>}+\mathrm{\Delta <br><br>}<br><br><br><br>\mathrm{\beta <br><br>},\mathrm{na}\right)}^{*}\times <br><br>U<br><br>\left(\mathrm{na}\right)\right\}$ $H_{22}={<br><br>\underset{\mathrm{na}=1}{\overset{N}{\sum <br><br>}}<br><br>\left\{{A<br><br>\left(-\mathrm{\alpha <br><br>}+\mathrm{\Delta <br><br>\alpha <br><br>},\mathrm{\beta <br><br>}+\mathrm{\Delta <br><br>\beta <br><br>},\mathrm{na}\right)}^{*}\times <br><br>U<br><br>\left(\mathrm{na}\right)\right\}<br><br>}^2$ $G_{11}=\underset{\mathrm{na}=1}{\overset{N}{\sum <br><br>}}<br><br>{<br><br>U<br><br>\left(\mathrm{na}\right)<br><br>}^2\times <br><br>\underset{\mathrm{na}=1}{\overset{N}{\sum <br><br>}}<br><br>{<br><br>A<br><br>\left(\mathrm{\alpha <br><br>},\mathrm{\beta <br><br>},\mathrm{na}\right)<br><br>}^2$ $G_{12}=\underset{\mathrm{na}=1}{\overset{N}{\sum <br><br>}}<br><br>{<br><br>U<br><br>\left(\mathrm{na}\right)<br><br>}^2\times <br><br>\underset{\mathrm{na}=1}{\overset{N}{\sum <br><br>}}<br><br>\left({A<br><br>\left(\mathrm{\alpha <br><br>},\mathrm{\beta <br><br>},\mathrm{na}\right)}^{*}\times <br><br>A<br><br>\left(-\mathrm{\alpha <br><br>}+\mathrm{\Delta <br><br>\alpha <br><br>},\mathrm{\beta <br><br>}+\mathrm{\Delta <br><br>\beta <br><br>},\mathrm{na}\right)\right)$ $G_{21}=G_{12}^{*}$ $G_{22}=\underset{\mathrm{na}=1}{\overset{N}{\sum <br><br>}}<br><br>{<br><br>U<br><br>\left(\mathrm{na}\right)<br><br>}^2\times <br><br>\underset{\mathrm{na}=1}{\overset{N}{\sum <br><br>}}<br><br>{<br><br>A<br><br>\left(-\mathrm{\alpha <br><br>}+\mathrm{\Delta <br><br>\alpha <br><br>},\mathrm{\beta <br><br>}+\mathrm{\Delta <br><br>\beta <br><br>},\mathrm{na}\right)<br><br>}^2,$

then the G and H terms are inserted into the equation in claim 8 to get a revised equation

${<br><br>R<br><br>\left(\mathrm{\alpha <br><br>},\mathrm{\beta <br><br>}\right)<br><br>}^2=\frac{\begin{array}{c}{<br><br>{\mathrm{\rho <br><br>}}_d<br><br>}^2\times <br><br>H_{11}+{\mathrm{\rho <br><br>}}_d^{*}\times <br><br>{\mathrm{\rho <br><br>}}_r\times <br><br>H_{12}+\\ {\mathrm{\rho <br><br>}}_d\times <br><br>{\mathrm{\rho <br><br>}}_r^{*}\times <br><br>H_{21}+{<br><br>{\mathrm{\rho <br><br>}}_r<br><br>}^2\times <br><br>H_{22}\end{array}}{\begin{array}{c}{<br><br>{\mathrm{\rho <br><br>}}_d<br><br>}^2\times <br><br>G_{11}+{\mathrm{\rho <br><br>}}_d^{*}\times <br><br>{\mathrm{\rho <br><br>}}_r\times <br><br>G_{12}+\\ {\mathrm{\rho <br><br>}}_d\times <br><br>{\mathrm{\rho <br><br>}}_r^{*}\times <br><br>G_{21}+{<br><br>{\mathrm{\rho <br><br>}}_r<br><br>}^2\times <br><br>G_{22}\end{array}}$

where the characteristic equation of Hermitian forms [H−

γ

G]=0 and the largest root of this equation is the maximum of the ratio of Hermitian forms is

${\mathrm{\gamma <br><br>}}_{\max }=\mathrm{maximum}<br><br><br><br>\mathrm{of}<br><br><br><br>\left(\frac{H<br><br>\left(x,y\right)}{G<br><br>\left(x,y\right)}\right);<br><br>$

where |R(α

,β

)|² has this form and the maximum value of |R(α

,β

)|² at angles α

, β

, −

α

+Δ

α

, and, β

+Δ

β

is the solution to the determinant equation

$\hspace{1em}<br><br>\left[\begin{array}{cc}{H}_{11}-{\mathrm{\gamma <br><br>}}_{\max }\times <br><br>G_{11}& H_{12}-{\mathrm{\gamma <br><br>}}_{\max }\times <br><br>G_{12}\\ H_{21}-{\mathrm{\gamma <br><br>}}_{\max }\times <br><br>G_{21}& H_{22}-{\mathrm{\gamma <br><br>}}_{\max }\times <br><br>G_{22}\end{array}\right]$

where this determinant equation is solved by the quadratic equation

α

×

γ

_max²+b×

γ

_max+c=0

α

=G₁₁×

G₂₂+|G₁₂|²

b=−

(H₁₁×

G₂₂+H₂₂×

G₁₁)+(H₁₂×

G₂₁+H₂₁×

G₁₂)

c=H₁₁×

H₂₂−

H₁₂×

H₂₁=0

and the maximum value of |R(α

,β

)|² is therefore

$\max <br><br>\left({<br><br>R<br><br>\left(\mathrm{\alpha <br><br>},\mathrm{\beta <br><br>}\right)<br><br>}^2\right)={\mathrm{\gamma <br><br>}}_{\max }=-\frac{b}{a}.$

12. A method for determining the azimuth and elevation to an aircraft at long range that is transmitting an electromagnetic signal while flying at a low altitude above water, the signal being received using a plurality of vertically polarized antennas and horizontally polarized antennas in an antenna array, the received signal is received directly from the aircraft subject to electromagnetic scattering perturbations and is received via multi-path reflections from the surface of the water, and there is a calibration manifold having stored array steering vectors that is used to correct for the perturbations, the method comprising the steps of:

(a) forming a first plurality of covariance matrices from a like plurality of sets of samples of the electromagnetic signal received by the plurality of vertically polarized antennas;

(b) forming a second plurality of covariance matrices from a like plurality of sets of samples of the electromagnetic signal received by the plurality of horizontally polarized antennas;

(c) calculating signal eigenvalues and associated eigenvectors for each of the plurality of covariance matrices formed in steps (a) and (b) for the signal received directly from the aircraft and received via multi-path reflections from the surface of the water;

(d) comparing the signal eigenvalues for the first plurality of covariance matrices with the signal eigenvalues for the second plurality of covariance matrices to determine if the received signal has stronger vertical or horizontal polarization, and selecting the signal eigenvector associated with the signal having the stronger polarization for subsequent processing in steps (e) and (f); and

(e) performing a conjugate gradient based correlation search using the selected signal eigenvector and the array manifold in the region of the first azimuth and elevation to determine accurate azimuth and elevation information to the aircraft.

13. The method for determining the azimuth and elevation to an aircraft of claim 12 further comprising the step of:

(f) performing a global correlation maximum search, assuming mirror sea-state reflection conditions using the signal eigenvector selected in step (d) with steering vectors retrieved from the calibration array manifold to determine a first azimuth and elevation, and the first azimuth and elevation are used as the starting point for the conjugate gradient based correlation search performed in step (e).

14. The method for determining the azimuth and elevation to an aircraft of claim 13 further comprising the steps of:

(g) periodically updating the first plurality of covariance matrices in step (a);

(h) periodically updating the second plurality of covariance matrices in step (b);

(i) reperforming the conjugate gradient based correlation search of step (e) except using the azimuth and elevation from the previous conjugate gradient based correlation as the starting point for the conjugate gradient based correlation in this step (i).

15. The method for determining the azimuth and elevation to an aircraft of claim 14 further comprising the step of:

(j) correlating the signal eigenvector selected in step (d) and used in step (e) with the calibration manifold to obtain array steering vectors stored in the calibration manifold to be used to determine accurate azimuth and elevation information to the aircraft.

16. The method for determining the azimuth and elevation to an aircraft of claim 15 further comprising the steps of:

(k) digitizing the samples of the electromagnetic signal received by the plurality of vertically polarized antennas as real and imaginary components before they are processed into the first covariance matrix in step (a); and

(l) digitizing the samples of the electromagnetic signal received by the plurality of horizontally polarized antennas as real and imaginary components before they are processed into the second plurality of covariance matrices in step (b).

17. The method for determining the azimuth and elevation to an aircraft of claim 16;

wherein the signals used to form the first plurality of covariance matrices in step (a) are digitized and converted to real and imaginary form before being stored in the matrices, and

wherein the signals used to form the second plurality of covariance matrices in step (b) are digitized and converted to real and imaginary form before being stored in the matrices.

18. The method for determining the azimuth and elevation to an aircraft of claim 17 wherein the following equation is used to process the digitized information stored in the covariance matrices is searched to find the azimuth and elevation to the aircraft using the equation:

${<br><br>R<br><br>\left(\mathrm{\alpha <br><br>},\mathrm{\beta <br><br>}\right)<br><br>}^2=\frac{\underset{\mathrm{na}=1}{\overset{N}{\sum <br><br>}}<br><br>{<br><br>{\left\{\begin{array}{c}{\mathrm{\rho <br><br>}}_d\times <br><br>A_{\mathrm{pol}}<br><br>\left(\mathrm{\alpha <br><br>},\mathrm{\beta <br><br>},\mathrm{na}\right)+{\mathrm{\rho <br><br>}}_r\times <br><br>\\ A_{\mathrm{pol}}<br><br>\left(-\mathrm{\alpha <br><br>}+\mathrm{\Delta <br><br>\alpha <br><br>},\mathrm{\beta <br><br>}+\mathrm{\Delta <br><br>\beta <br><br>},\mathrm{na}\right)\end{array}\right\}}^{*}\times <br><br>U_{\mathrm{pol}}<br><br>\left(\mathrm{na}\right)<br><br>}^2}{\underset{\mathrm{na}=1}{\overset{N}{\sum <br><br>}}<br><br>{<br><br>\left\{\begin{array}{c}{\mathrm{\rho <br><br>}}_d\times <br><br>A_{\mathrm{pol}}<br><br>\left(\mathrm{\alpha <br><br>},\mathrm{\beta <br><br>},\mathrm{na}\right)+{\mathrm{\rho <br><br>}}_r\times <br><br>\\ A_{\mathrm{pol}}<br><br>\left(-\mathrm{\alpha <br><br>}+\mathrm{\Delta <br><br>\alpha <br><br>},\mathrm{\beta <br><br>}+\mathrm{\Delta <br><br>\beta <br><br>},\mathrm{na}\right)\end{array}\right\}<br><br>}^2\times <br><br>\underset{\mathrm{na}=1}{\overset{N}{\sum <br><br>}}<br><br>{<br><br>U_{\mathrm{pol}}<br><br>\left(\mathrm{na}\right)<br><br>}^2}$

where |R(α

,β

)|² is the correlation squared and the maximum value is searched for during the conjugate gradient based correlation search, α

is the elevation angle to the aircraft transmitting the signal, β

is the azimuth angle to the aircraft transmitting the signal, ρ

_d is the direct complex coefficient, ρ

_r is the reflected complex coefficient, * is a complex conjugate, na is the number of antennas in the beam forming array utilized to receive the signals transmitted by the aircraft, A_pol are calibration vectors as a function of α and

β

, and U_pol is the eigenvector resolved received signal vector and is composed of the directly received signal component and the reflected signal component and is equal the sum of the eigenvectors of the signals from the vertically polarized antennas and the horizontally polarized antennas.

19. The method for determining the azimuth and elevation to an aircraft of claim 17 wherein when performing the global correlation maximum search in step (f) the equation in claim 18 is utilized but the terms Δ

α and

Δ

β

in the equation are set equal to zero and it is assumed that mirror reflection conditions for reflection of the electromagnetic signal from the surface of the water to yield the approximate values for elevation α and

azimuth β

to the aircraft which are used as the starting point in the first conjugate gradient based correlation search using the equation in claim 18.

20. The method for determining the azimuth and elevation to an aircraft of claim 19;

wherein the signals used to form the first plurality of covariance matrices in step (a) are digitized and converted to real and imaginary form before being stored in the matrices, and

wherein the signals used to form the second plurality of covariance matrices in step (b) are digitized and converted to real and imaginary form before being stored in the matrices.