Determining the angle of arrival of a target signal received by an array of antenna elements
In a system for determining the angle of arrival of a target signal received by an array of antenna elements, a pair of receivers simultaneously obtain observations of a received target signal from multiple elements of an array of antenna elements; and a computer processes the simultaneously obtained samples of the target signal to determine a maximum likelihood estimation (MLE) of the angle of arrival φ of the target signal by using the following equation: φ MLE =argmax φ Re(α*β). The value of β is determined in accordance with whether the target signal is known or unknown. When the target signal is unknown, the computer also processes the simultaneously obtained samples of the target signal to estimate the bandwidth of the received target signal by using binary hypotheses and a generalized log likelihood ratio test (GLLRT) or by using multiple hypotheses and pair-wise generalized log likelihood ratio tests. The value of a bandwidth constraint M that is associated with the estimated bandwidth is used to derive the value of β that is used to determine the MLE of φ.
1. A method of estimating the angle of arrival of a target signal received by an array of antenna elements, comprising the steps of:
(a) with a pair of receivers, simultaneously obtaining samples of a received target signal from multiple elements of an array of antenna elements; and
(b) with a computer, processing the simultaneously obtained samples of the target signal to determine a maximum likelihood estimation (MLE) of the angle of arrival φ of the target signal by using the following equation:
φ MLE =argmax φ Re (α*β)
wherein α is a complex vector that represents a phase difference associated with the angle of arrival that should be observed upon receipt of the signal by two particular antenna elements from which the samples are obtained; and
β represents the phase difference that is observed upon receipt of the signal by the two particular antenna elements from which the samples are obtained;
wherein when the target signal is unknown,
β n =y* 2n−1 y 2n in the time domain and β n =Y* 2n−1 Y 2n in the frequency domain,
wherein y 2n−1 and y 2n are complex N-tuple vectors representing the samples obtained from the nth simultaneously sampled pair of antenna elements and Y is a Fourier transform of y.
2. A method according to claim 1 , wherein for a phased array of said antenna elements, α n =e i(μ 2n (Φ)−μ 2n−1 (φ)) , wherein μ is a function of the angle of arrival that depends upon the geometry of the array of antenna elements.
3. A method according to claim 1 , further comprising the steps of:
(c) estimating the bandwidth of the received target signal by using binary hypotheses and a generalized log likelihood ratio test (GLLRT):
E
(
M
1
)
2
-
E
(
M
2
)
2
≶
M
2
M
1
λ
(
M
1
,
M
2
)
wherein λ(M 1 , M 2 ) is an appropriately chosen constant threshold;
wherein M specifies a bandwidth constraint expressed by a set of tuples of X(φ) where X(φ) may be non-zero,
X(φ) is an estimate of the Fourier transform of the unknown target signal,
Y is a Fourier transform of the sample obtained from the sampled antenna element, and
E
(
M
)
2
=
<
Y
1
,
Y
1
>
+
<
Y
2
,
Y
2
>
-
c
(
<
Y
1
,
Y
1
>
M
+
2
<
Y
1
,
Y
2
>
M
+
<
Y
2
,
Y
2
>
M
)
;
and
(d) deriving the value of β n =<Y 2n−1 , Y 2n > M in the frequency domain or the value of β n =<y 2n−1 , y 2n > M in the time domain by using the respective value of M that is associated with the estimated bandwidth pursuant to said GLLRT.
4. A method according to claim 1 , further comprising the steps of:
(c) estimating the bandwidth of the received target signal by using multiple hypotheses and pair wise generalized log likelihood ratio tests in accordance with:
E
(
M
1
)
2
-
E
(
M
2
)
2
<
λ
(
M
1
,
M
2
)
E
(
M
1
)
2
-
E
(
M
3
)
2
<
λ
(
M
1
,
M
3
)
⇒
M
1
wherein M specifies a bandwidth constraint expressed by a set of tuples of X(φ) where X(φ) may be non-zero,
X(φ) is an estimate of the Fourier transform of the unknown target signal,
Y is a Fourier transform of the sample obtained from the sampled antenna element, and
E
(
M
)
2
=
<
Y
1
,
Y
1
>
+
<
Y
2
,
Y
2
>
-
c
(
<
Y
1
,
Y
1
>
M
+
2
<
Y
1
,
Y
2
>
M
+
<
Y
2
,
Y
2
>
M
)
;
(d) deriving the value of β n =<Y 2n−1 , Y 2n > M in the frequency domain or the value of β n =<y 2n−1 , y 2n > M in the time domain by using the respective value of M that is associated with the estimated bandwidth pursuant to said generalized log likelihood ratio tests.
5. A method of estimating the angle of arrival of a target signal received by an array of antenna elements, comprising the steps of:
(a) with a pair of receivers, simultaneously obtaining samples of a received target signal from multiple elements of an array of antenna elements; and
(b) with a computer, processing the simultaneously obtained samples of the target signal to determine a maximum likelihood estimation (MLE) of the angle of arrival φ of the target signal by using the following equation:
φ MLE =argmax φ Re (α*β)
wherein α is a complex vector that represents a phase difference associated with the angle of arrival that should be observed upon receipt of the signal by two particular antenna elements from which the samples are obtained; and
β represents the phase difference that is observed upon receipt of the signal by the two particular antenna elements from which the samples are obtained;
wherein when the target signal is known,
β
n
=
(
x
n
*
y
2
n
-
1
)
*
(
x
n
*
y
2
n
)
x
n
2
in the time domain and
β
n
=
(
X
n
*
Y
2
n
-
1
)
*
(
X
n
*
Y
2
n
)
X
n
2
in the frequency domain,
wherein X and Y are Fourier transforms of x and y respectively,
wherein x n and X n are complex N-tuple vectors representing the known target signal in the time domain and in the frequency domain respectively; and
wherein y 2n−1 and y 2n are complex N-tuple vectors representing the samples obtained from the nth simultaneously sampled pair of antenna elements and Y is a Fourier transform of y.
6. A method according to claim 5 , wherein for a phased array of said antenna elements, α n =e i(μ 2n (φ)−μ 2n−1 (φ)),
wherein μ is a function of the angle of arrival that depends upon the geometry of the array of antenna elements.
7. A method of estimating the bandwidth of a target signal received by an array of antenna elements, comprising the steps of:
(a) with a pair of receivers, simultaneously obtaining samples of a received target signal from multiple elements of an array of antenna elements; and
(b) with a computer, processing the simultaneously obtained samples of the target signal to estimate the bandwidth of the received target signal by using binary hypotheses and a generalized log likelihood ratio test (GLLRT).
8. A method according to claim 7 , wherein the GLLRT is:
E
(
M
1
)
2
-
E
(
M
2
)
2
≶
M
2
M
1
λ
(
M
1
,
M
2
)
wherein λ(M 1 , M 2 ) is an appropriately chosen constant threshold;
wherein M specifies a bandwidth constraint expressed by a set of tuples of X(φ) where X(φ) may be non-zero,
X(φ) is an estimate of the Fourier transform of the unknown target signal,
Y is a Fourier transform of the sample obtained from the sampled antenna element, and
E
(
M
)
2
=
<
Y
1
,
Y
1
>
+
<
Y
2
,
Y
2
>
-
c
(
<
Y
1
,
Y
1
>
M
+
2
<
Y
1
,
Y
2
>
M
+
<
Y
2
,
Y
2
>
M
)
.
9. A method according to claim 8 , wherein λ(M 1 , M 2 )=argmin V ar{φ MLE }.
10. A method of estimating the bandwidth of a target signal received by an array of antenna elements, comprising the steps of:
(a) with a pair of receivers, simultaneously obtaining samples of a received target signal from multiple elements of an array of antenna elements; and
(b) with a computer, processing the simultaneously obtained samples of the target signal to estimate the bandwidth of the received target signal by using multiple hypotheses and pair wise generalized log likelihood ratio tests in accordance with:
E
(
M
1
)
2
-
E
(
M
2
)
2
<
λ
(
M
1
,
M
2
)
E
(
M
1
)
2
-
E
(
M
3
)
2
<
λ
(
M
1
,
M
3
)
⇒
M
1
wherein M specifies a bandwidth constraint expressed by a set of tuples of X(φ) where X(φ) may be non-zero,
X(φ) is an estimate of the Fourier transform of the unknown target signal,
Y is a Fourier transform of the sample obtained from the sampled antenna element, and
E
(
M
)
2
=
<
Y
1
,
Y
1
>
+
<
Y
2
,
Y
2
>
-
c
(
<
Y
1
,
Y
1
>
M
+
2
<
Y
1
,
Y
2
>
M
+
<
Y
2
,
Y
2
>
M
)
.
11. A system for estimating the angle of arrival of a target signal received by an array of antenna elements, comprising:
a pair of receivers adapted for simultaneously obtaining samples of a received target signal from multiple elements of an array of antenna elements; and
a computer adapted for processing the simultaneously obtained samples of the target signal to determine a maximum likelihood estimation (MLE) of the angle of arrival φ of the target signal by using the following equation:
φ MLE =argmax φ Re (α*β)
wherein α is a complex vector that represents a phase difference associated with the angle of arrival that should be observed upon receipt of the signal by two particular antenna elements from which the samples are obtained; and
β represents the phase difference that is observed upon receipt of the signal by the two particular antenna elements from which the samples are obtained;
wherein when the target signal is unknown, β n =y* 2n−1 y 2n in the time domain and β n =Y* 2n−1 Y 2n in the frequency domain,
wherein y 2n−1 and y 2n are complex N-tuple vectors representing the samples obtained from the nth simultaneously sampled pair of antenna elements and Y is a Fourier transform of y.
12. A system according to claim 11 , further comprising:
means for estimating the bandwidth of the received target signal by using binary hypotheses and a generalized log likelihood ratio test (GLLRT); and
means for deriving the value of β n =<Y 2n−1 , Y 2n > M in the frequency domain or the value of β n =<y 2n−1 , y 2n > M in the time domain by using the respective value of M that is associated with the estimated bandwidth pursuant to said GLLRT.
13. A system according to claim 11 , further comprising the steps of:
means for estimating the bandwidth of the received target signal by using multiple hypotheses and pair wise generalized log likelihood ratio tests in accordance with:
E
(
M
1
)
2
-
E
(
M
2
)
2
<
λ
(
M
1
,
M
2
)
E
(
M
1
)
2
-
E
(
M
3
)
2
<
λ
(
M
1
,
M
3
)
⇒
M
1
wherein M specifies a bandwidth constraint expressed by a set of tuples of X(φ) where X(φ) may be non-zero,
X(φ) is an estimate of the Fourier transform of the unknown target signal,
Y is a Fourier transform of the sample obtained from the sampled antenna element, and
E
(
M
)
2
=
<
Y
1
,
Y
1
>
+
<
Y
2
,
Y
2
>
-
c
(
<
Y
1
,
Y
1
>
M
+
2
<
Y
1
,
Y
2
>
M
+
<
Y
2
,
Y
2
>
M
)
;
means for deriving the value of β n =<Y 2n−1 , Y 2n > M in the frequency domain or the value of β n =<y 2n−1 , y 2n > M in the time domain by using the respective value of M that is associated with the estimated bandwidth pursuant to said generalized log likelihood ratio tests.
14. A system for estimating the angle of arrival of a target signal received by an array of antenna elements, comprising the steps of:
a pair of receivers adapted for simultaneously obtaining samples of a received target signal from multiple elements of an array of antenna elements; and
a computer adapted for processing the simultaneously obtained samples of the target signal to determine a maximum likelihood estimation (MLE) of the angle of arrival φ of the target signal by using the following equation:
φ MLE =argmax φ Re (α*β)
wherein α is a complex vector that represents a phase difference associated with the angle of arrival that should be observed upon receipt of the signal by two particular antenna elements from which the samples are obtained; and
β represents the phase difference that is observed upon receipt of the signal by the two particular antenna elements from which the samples are obtained;
wherein when the target signal is known,
β
n
=
(
x
n
*
y
2
n
-
1
)
*
(
x
n
*
y
2
n
)
x
n
2
in the time domain and
β
n
=
(
X
n
*
Y
2
n
-
1
)
*
(
X
n
*
Y
2
n
)
X
n
2
in the frequency domain,
wherein X and Y are Fourier transforms of x and y respectively,
wherein x n and X n are complex N-tuple vectors representing the known target signal in the time domain and in the frequency domain respectively; and
wherein y 2n−1 and y 2n are complex N-tuple vectors representing the samples obtained from the nth simultaneously sampled pair of antenna elements and Y is a Fourier transform of y.
15. A system for estimating the bandwidth of a target signal received by an array of antenna elements, comprising the steps of:
a pair of receivers adapted for simultaneously obtaining samples of a received target signal from multiple elements of an array of antenna elements; and
a computer adapted for processing the simultaneously obtained samples of the target signal to estimate the bandwidth of the received target signal by using binary hypotheses and a generalized log likelihood ratio test (GLLRT).
16. A system according to claim 15 , wherein the GLLRT is:
E
(
M
1
)
2
-
E
(
M
2
)
2
≶
M
2
M
1
λ
(
M
1
,
M
2
)
wherein λ(M 1 , M 2 ) is an appropriately chosen constant threshold;
wherein M specifies a bandwidth constraint expressed by a set of tuples of X(φ) where X(φ) may be non-zero,
X(φ) is an estimate of the Fourier transform of the unknown target signal,
Y is a Fourier transform of the sample obtained from the sampled antenna element, and
E
(
M
)
2
=
〈
Y
1
,
Y
1
〉
+
〈
Y
2
,
Y
2
〉
-
c
(
〈
Y
1
,
Y
1
〉
M
+
2
〈
Y
1
,
Y
2
〉
M
+
〈
Y
2
,
Y
2
〉
M
)
.
17. A system according to claim 16 , wherein λ(M 1 , M 2 )=argmin V ar{φ MLE }.
18. A system for estimating the bandwidth of a target signal received by an array of antenna elements, comprising the steps of:
a pair of receivers adapted for simultaneously obtaining samples of a received target signal from multiple elements of an array of antenna elements; and
a computer adapted for processing the simultaneously obtained samples of the target signal to estimate the bandwidth of the received target signal by using multiple hypotheses and pair wise generalized log likelihood ratio tests in accordance with:
E
(
M
1
)
2
-
E
(
M
2
)
2
<
λ
(
M
1
,
M
2
)
E
(
M
1
)
2
-
E
(
M
3
)
2
<
λ
(
M
1
,
M
3
)
⇒
M
1
wherein M specifies a bandwidth constraint expressed by a set of tuples of X(φ) where X(φ) may be non-zero,
X(φ) is an estimate of the Fourier transform of the unknown target signal,
Y is a Fourier transform of the sample obtained from the sampled antenna element, and
E
(
M
)
2
=
〈
Y
1
,
Y
1
〉
+
〈
Y
2
,
Y
2
〉
-
c
(
〈
Y
1
,
Y
1
〉
M
+
2
〈
Y
1
,
Y
2
〉
M
+
〈
Y
2
,
Y
2
〉
M
)
.
19. A 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 pair of receivers adapted for simultaneously obtaining samples of a received target signal from multiple elements of an array of antenna elements; and a computer,
wherein the computer readable storage medium contains computer executable program instructions for causing the computer to process the simultaneously obtained samples of the target signal to determine a maximum likelihood estimation (MLE) of the angle of arrival φ of the target signal by using the following equation:
φ MLE =argmax φ Re (α*β)
wherein α is a complex vector that represents a phase difference associated with the angle of arrival that should be observed upon receipt of the signal by two particular antenna elements from which the samples are obtained; and
β represents the phase difference that is observed upon receipt of the signal by the two particular antenna elements from which the samples are obtained;
wherein when the target signal is unknown, β n =y* 2n−1 y 2n in the time domain and β n =Y* 2n−1 Y 2n in the frequency domain,
wherein y 2n−1 and y 2n are complex N-tuple vectors representing the samples obtained from the nth simultaneously sampled pair of antenna elements and Y is a Fourier transform of y.
20. A computer readable storage medium according to claim 19 , wherein for a phased array of said antenna elements, α n =e i(μ 2n (φ)−μ 2n−1 (φ)) , wherein μ is a function of the angle of arrival that depends upon the geometry of the array of antenna elements.
21. A computer readable storage medium according to claim 19 , further comprising the steps of:
(c) estimating the bandwidth of the received target signal by using binary hypotheses and a generalized log likelihood ratio test (GLLRT):
E
(
M
1
)
2
-
E
(
M
2
)
2
≶
M
2
M
1
λ
(
M
1
,
M
2
)
wherein λ(M 1 , M 2 ) is an appropriately chosen constant threshold;
wherein M specifies a bandwidth constraint expressed by a set of tuples of X(φ) where X(φ) may be non-zero,
X(φ) is an estimate of the Fourier transform of the unknown target signal,
Y is a Fourier transform of the sample obtained from the sampled antenna element, and
E
(
M
)
2
=
〈
Y
1
,
Y
1
〉
+
〈
Y
2
,
Y
2
〉
-
c
(
〈
Y
1
,
Y
1
〉
M
+
2
〈
Y
1
,
Y
2
〉
M
+
〈
Y
2
,
Y
2
〉
M
)
;
and
(d) deriving the value of β n =<Y 2n−1 , Y 2n > M in the frequency domain or the value of β n =<y 2n−1 , y 2n > M in the time domain by using the respective value of M that is associated with the estimated bandwidth pursuant to said GLLRT.
22. A computer readable storage medium according to claim 19 , further comprising the steps of:
(c) estimating the bandwidth of the received target signal by using multiple hypotheses and pair wise generalized log likelihood ratio tests in accordance with:
E
(
M
1
)
2
-
E
(
M
2
)
2
<
λ
(
M
1
,
M
2
)
E
(
M
1
)
2
-
E
(
M
3
)
2
<
λ
(
M
1
,
M
3
)
⇒
M
1
wherein M specifies a bandwidth constraint expressed by a set of tuples of X(φ) where X(φ) may be non-zero,
X(φ) is an estimate of the Fourier transform of the unknown target signal,
Y is a Fourier transform of the sample obtained from the sampled antenna element, and
E
(
M
)
2
=
〈
Y
1
,
Y
1
〉
+
〈
Y
2
,
Y
2
〉
-
c
(
〈
Y
1
,
Y
1
〉
M
+
2
〈
Y
1
,
Y
2
〉
M
+
〈
Y
2
,
Y
2
〉
M
)
;
(d) deriving the value of β n =<Y 2n−1 , Y 2n > M in the frequency domain or the value of β n =<y 2n−1 , y 2n > M in the time domain by using the respective value of M that is associated with the estimated bandwidth pursuant to said generalized log likelihood ratio tests.
23. A 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 pair of receivers adapted for simultaneously obtaining samples of a received target signal from multiple elements of an array of antenna elements; and a computer,
wherein the computer readable storage medium contains computer executable program instructions for causing the computer to process the simultaneously obtained samples of the target signal to determine a maximum likelihood estimation (MLE) of the angle of arrival φ of the target signal by using the following equation:
φ MLE =argmax φ Re (α*β)
wherein α is a complex vector that represents a phase difference associated with the angle of arrival that should be observed upon receipt of the signal by two particular antenna elements from which the samples are obtained; and
β represents the phase difference that is observed upon receipt of the signal by the two particular antenna elements from which the samples are obtained;
wherein when the target signal is known,
β
n
=
(
x
n
*
y
2
n
-
1
)
*
(
x
n
*
y
2
n
)
x
n
2
in the time domain and
β
n
=
(
X
n
*
Y
2
n
-
1
)
*
(
X
n
*
Y
2
n
)
X
n
2
in the frequency domain,
wherein X and Y are Fourier transforms of x and y respectively,
wherein x n and X n are complex N-tuple vectors representing the known target signal in the time domain and in the frequency domain respectively; and
wherein Y 2n−1 and y 2n are complex N-tuple vectors representing the samples obtained from the nth simultaneously sampled pair of antenna elements and Y is a Fourier transform of y.
24. A computer readable storage medium according to claim 23 , wherein for a phased array of said antenna elements, α n =e i(μ 2n (φ)−μ 2n−1 (φ)) , wherein μ is a function of the angle of arrival that depends upon the geometry of the array of antenna elements.
25. A computer readable storage medium for use with a computer in a system for estimating the bandwidth of a target signal received by an array of antenna elements, wherein the system comprises: a pair of receivers adapted for simultaneously obtaining samples of a received target signal from multiple elements of an array of antenna elements; and a computer,
wherein the computer readable storage medium contains computer executable program instructions for causing the computer to process the simultaneously obtained samples of the target signal to estimate the bandwidth of the received target signal by using binary hypotheses and a generalized log likelihood ratio test (GLLRT).
26. A computer readable storage medium according to claim 25 , wherein the GLLRT is:
E
(
M
1
)
2
-
E
(
M
2
)
2
≶
M
2
M
1
λ
(
M
1
,
M
2
)
wherein λ(M 1 , M 2 ) is an appropriately chosen constant threshold;
wherein M specifies a bandwidth constraint expressed by a set of tuples of X(φ) where X(φ) may be non-zero,
X(φ) is an estimate of the Fourier transform of the unknown target signal,
Y is a Fourier transform of the sample obtained from the sampled antenna element, and
E
(
M
)
2
=
〈
Y
1
,
Y
1
〉
+
〈
Y
2
,
Y
2
〉
-
c
(
〈
Y
1
,
Y
1
〉
M
+
2
〈
Y
1
,
Y
2
〉
M
+
〈
Y
2
,
Y
2
〉
M
)
.
27. A computer readable storage medium according to claim 26 , wherein λ(M 1 , M 2 )=argmin V ar{φ MLE }.
28. A computer readable storage medium for use with a computer in a system for estimating the bandwidth of a target signal received by an array of antenna elements, wherein the system comprises: a pair of receivers adapted for simultaneously obtaining samples of a received target signal from multiple elements of an array of antenna elements; and a computer,
wherein the computer readable storage medium contains computer executable program instructions for causing the computer to process the simultaneously obtained samples of the target signal to estimate the bandwidth of the received target signal by using multiple hypotheses and pair wise generalized log likelihood ratio tests in accordance with:
E
(
M
1
)
2
-
E
(
M
2
)
2
<
λ
(
M
1
,
M
2
)
E
(
M
1
)
2
-
E
(
M
3
)
2
<
λ
(
M
1
,
M
3
)
⇒
M
1
wherein M specifies a bandwidth constraint expressed by a set of tuples of X(φ) where X(φ) may be non-zero,
X(φ) is an estimate of the Fourier transform of the unknown target signal,
Y is a Fourier transform of the sample obtained from the sampled antenna element, and
E
(
M
)
2
=
〈
Y
1
,
Y
1
〉
+
〈
Y
2
,
Y
2
〉
-
c
(
〈
Y
1
,
Y
1
〉
M
+
2
〈
Y
1
,
Y
2
〉
M
+
〈
Y
2
,
Y
2
〉
M
)
.