Determining the angle of arrival of a target signal received by an array of antenna elements

1. A method of determining the angle of arrival of a target signal received by an array of antenna elements, comprising the steps of:

(a) with a receiver, obtaining observations of a received target signal from an array of antenna elements; and

(b) with a computer, processing the obtained observations of the target signal with estimates of a target signal for different angles of arrival of a target signal received by the array of antenna elements to determine that the angle of arrival of the target signal is an angle of arrival at which dissimilarity between the observations of target signal and the estimates of the target signal for the different angles of arrival is minimized;

wherein step (b) comprises the step of:

(c) determining the angle of arrival φ by using the equation:

φ=argmin φ ∥y−δA′x∥ 2

wherein y is an observation of the target signal obtained from the individual elements of an array of antenna elements, and

δA′x is an estimate of a target signal received by a set of antenna elements for a particular angle of arrival.

2. A method according to claim 1 , wherein the target signal is estimated as δ times a linear combination of sin(t)/t translates, which in matrix form is written as δA′x,

wherein δ is a diagonal matrix incorporating equations for the array of antenna elements,

δ

=

(

ⅇ

ⅈψ

0

⋱

ⅇ

ⅈψ

0

ⅇ

ⅈψ

1

⋱

ⅇ

ⅈψ

1

⋱

)

wherein with each e iψ k block has a length equal to the number of samples that a receiver coupled to the array obtains from an individual antenna element.

3. A method according to claim 2 , wherein the sin(t)/t translates are windowed for different modulation bandwidths.

4. A method of determining the angle of arrival of a target signal received by an array of antenna elements, comprising the steps of:

(a) with a receiver, obtaining observations of a received target signal from an array of antenna elements; and

(b) with a computer, processing the obtained observations of the target signal with estimates of a target signal for different angles of arrival of a target signal received by the array of antenna elements to determine that the angle of arrival of the target signal is an angle of arrival at which dissimilarity between the observations of target signal and the estimates of the target signal for the different angles of arrival is minimized;

wherein step (b) comprises the step of:

(c) determining the angle of arrival φ by using the equation:

φ=argmax φ {y*A{tilde over (x)}}

wherein y is an observation of the target signal obtained from a set of antenna elements,

wherein {tilde over (x)}=(A*A) −1 A*y, with {tilde over (x)} denoting least-squares optimality;

wherein the target signal is estimated as δ times a linear combination of sin(t)/t translates, which in matrix form is written as δA′{tilde over (x)},

wherein δ is a diagonal matrix incorporating equations for the array of antenna elements,

δ

=

(

ⅇ

ⅈψ

0

⋱

ⅇ

ⅈψ

0

ⅇ

ⅈψ

1

⋱

ⅇ

ⅈψ

1

⋱

)

wherein with each e iψ k block has a length equal to the number of samples that a receiver coupled to the array obtains from an individual antenna element, and

wherein A*A=(δA′)*δA′=A′*δ*δA′.

5. A method according to claim 4 , wherein the sin(t)/t translates are windowed for different modulation bandwidths.

6. A method according to claim 4 , wherein a factorization of A*A uses that A*A is banded.

7. A method according to claim 4 , wherein for the computation of A*y, A is sparse.

8. A system for determining the angle of arrival of a target signal received by an array of antenna elements, comprising:

a receiver adapted for obtaining observations of a received target signal from an array of antenna elements; and

a computer that is programmed for processing the obtained observations of the target signal with estimates of a target signal for different angles of arrival of a target signal received by the array of antenna elements to determine that the angle of arrival of the target signal is an angle of arrival at which dissimilarity between the observations of target signal and the estimates of the target signal for the different angles of arrival is minimized;

wherein the angle of arrival φ is determined by using the equation:

φ=argmin φ ∥y−δA′x∥ 2

wherein y is an observation of the target signal obtained from the individual elements of an array of antenna elements, and

δA′x is an estimate of a target signal received by a set of antenna elements for a particular angle of arrival.

9. A computer system according to claim 8 , wherein the target signal is estimated as δ times a linear combination of sin(t)/t translates, which in matrix form is written as δA′x,

wherein δ is a diagonal matrix incorporating equations for the array of antenna elements,

δ

=

(

ⅇ

ⅈψ

0

⋱

ⅇ

ⅈψ

0

ⅇ

ⅈψ

1

⋱

ⅇ

ⅈψ

1

⋱

)

wherein with each e iψ k block has a length equal to the number of samples that a receiver coupled to the array obtains from an individual antenna element.

10. A system for determining the angle of arrival of a target signal received by an array of antenna elements, comprising:

a receiver adapted for obtaining observations of a received target signal from an array of antenna elements; and

a computer that is programmed for processing the obtained observations of the target signal with estimates of a target signal for different angles of arrival of a target signal received by the array of antenna elements to determine that the angle of arrival of the target signal is an angle of arrival at which dissimilarity between the observations of target signal and the estimates of the target signal for the different angles of arrival is minimized;

wherein the angle of arrival φ is determined by using the equation:

φ=argmax φ {y*A{tilde over (x)}}

wherein y is an observation of the target signal obtained from a set of antenna elements,

wherein {tilde over (x)}=(A*A) −1 A*y, with {tilde over (x)} denoting least-squares optimality;

wherein the target signal is estimated as δ times a linear combination of sin(t)/t translates, which in matrix form is written as δA′{tilde over (x)},

wherein δ is a diagonal matrix incorporating equations for the array of antenna elements,

δ

=

(

ⅇ

ⅈψ

0

⋱

ⅇ

ⅈψ

0

ⅇ

ⅈψ

1

⋱

ⅇ

ⅈψ

1

⋱

)

wherein with each e iψ k block has a length equal to the number of samples that a receiver coupled to the array obtains from an individual antenna element, and

wherein A*A=(δA′)*δA′=A′*δ*δA′.

11. A nontransitory computer readable storage medium for use with a computer in a system for determining the angle of arrival of a target signal received by an array of antenna elements, wherein the system comprises: a receiver adapted for obtaining observations of a received target signal from the array of antenna elements; and a computer,

wherein the computer readable storage medium contains computer executable program instructions for causing the computer to perform the step of:

(a) processing the obtained observations of the target with estimates of a target signal for different angles of arrival of a target signal received by the array of antenna elements to determine that the angle of arrival of the target signal is an angle of arrival at which dissimilarity between the observations of target signal and the estimates of the target signal for the different angles of arrival is minimized;

wherein step (a) comprises the step of:

(b) determining the angle of arrival φ by using the equation:

φ=argmin φ ∥y−δA′x∥ 2

wherein y is an observation of the target signal obtained from the individual elements of an array of antenna elements, and

δA′x is an estimate of a target signal received by a set of antenna elements for a particular angle of arrival.

12. A computer readable storage medium according to claim 11 , wherein the target signal is estimated as δ times a linear combination of sin(t)/t translates, which in matrix form is written as δA′x,

wherein δ is a diagonal matrix incorporating equations for the array of antenna elements,

δ

=

(

ⅇ

ⅈψ

0

⋱

ⅇ

ⅈψ

0

ⅇ

ⅈψ

1

⋱

ⅇ

ⅈψ

1

⋱

)

wherein with each e iψ k block has a length equal to the number of samples that a receiver coupled to the array obtains from an individual antenna element.

13. A computer readable storage medium according to claim 12 , wherein the sin(t)/t translates are windowed for different modulation bandwidths.

14. A nontransitory computer readable storage medium for use with a computer in a system for determining the angle of arrival of a target signal received by an array of antenna elements, wherein the system comprises: a receiver adapted for obtaining observations of a received target signal from the array of antenna elements; and a computer,

wherein the computer readable storage medium contains computer executable program instructions for causing the computer to perform the step of:

(a) processing the obtained observations of the target signal with estimates of a target signal for different angles of arrival of a target signal received by the array of antenna elements to determine that the angle of arrival of the target signal is an angle of arrival at which dissimilarity between the observations of target signal and the estimates of the target signal for the different angles of arrival is minimized;

wherein step (a) comprises the step of:

(b) determining the angle of arrival φ by using the equation:

φ=argmax φ {y*A{tilde over (x)}}

wherein y is an observation of the target signal obtained from a set of antenna elements,

wherein {tilde over (x)}=(A*A) −1 A*y, with {tilde over (x)} denoting least-squares optimality;

wherein the target signal is estimated as δ times a linear combination of sin(t)/t translates, which in matrix form is written as δA′{tilde over (x)},

wherein δ is a diagonal matrix incorporating equations for the array of antenna elements,

δ

=

(

ⅇ

ⅈψ

0

⋱

ⅇ

ⅈψ

0

ⅇ

ⅈψ

1

⋱

ⅇ

ⅈψ

1

⋱

)

wherein with each e iψ k k block has a length equal to the number of samples that a receiver coupled to the array obtains from an individual antenna element, and

wherein A*A=(δA′)*δA′=A′*δ*δA′.

15. A computer readable storage medium according to claim 14 , wherein the sin(t)/t translates are windowed for different modulation bandwidths.

16. A computer readable storage medium according to claim 14 , wherein a factorization of A*A uses that A*A is banded.

17. A computer readable storage medium according to claim 14 , wherein for the computation of A*y, A is sparse.