System and method for hybrid minimum mean squared error matrix-pencil separation weights for blind source separation
View Patent ↗A technique for blind source separation (“BSS”) of statistically independent signals with low signal-to-noise plus interference ratios under a narrowband assumption utilizing cumulants in conjunction with spectral estimation of the signal subspace to perform the blind separation is disclosed. The BSS technique utilizes a higher-order statistical method, specifically fourth-order cumulants, with the generalized eigen analysis of a matrix-pencil to blindly separate a linear mixture of unknown, statistically independent, stationary narrowband signals at a low signal-to-noise plus interference ratio having the capability to separate signals in spatially and/or temporally correlated Gaussian noise. The disclosed BSS technique separates low-SNR co-channel sources for observations using an arbitrary un-calibrated sensor array. The disclosed BSS technique forms a separation matrix with hybrid matrix-pencil adaptive array weights that minimize the mean squared errors due to both interference emitters and Gaussian noise. The hybrid weights maximize the signal-to interference-plus noise ratio.
1. A method for separating M signals provided by M sources and received by an array comprising N elements, said method comprising:
generating a hybrid separation matrix as a function of:
time differences between receipt of said M signals by said N elements;
a spatial fourth order cumulant matrix pencil;
a spatial correlation matrix; and,
steering vectors of said M signals, and,
multiplying said hybrid separation matrix by a time series matrix representation of said M signals.
2. A method in accordance with claim 1 wherein the hybrid separation matrix is in accordance with the following equation:
ŵ j,hyb =|{circumflex over (v)} j H {circumflex over (K)} j −1 {circumflex over (v)} j | −1 {circumflex over (K)} j −1 {circumflex over (v)} j ;
wherein,
v j is the steering vector of the j th signal; and,
K j is the noise spatial covariance matrix of the j th signal.
3. A method in accordance with claim 1 wherein said spatial fourth order cumulant matrix pencil is a function of a spatial fourth order cumulant matrix.
4. A method in accordance claim 3 , wherein said spatial fourth order cumulant matrix is in accordance with the following equation:
C
x
4
(
τ
1
,
τ
2
,
τ
3
)
≡
∑
i
=
1
N
Cum
[
x
i
*
(
t
-
τ
1
)
x
i
(
t
-
τ
2
)
x
(
t
)
x
H
(
t
-
τ
3
)
]
,
wherein
C x 4 (τ 1 , τ 2 , τ 3 ) is said spatial fourth order cumulant matrix having a first time lag, τ 1 , a second time lag, τ 2 , and a third time lag, τ 3 , each time lag being indicative of a time delay from one of said M sources to one of said N elements;
N is indicative of a number of elements in said array;
Cum [x i *(t−τ 1 ) x i (t−τ 2 ) x(t) x H (t−τ 3 )] is a cumulant operator on arguments [x i *(t−τ 1 ) x i (t−τ 2 ) x(t) x H (t−τ 3 )];
t is a variable representing time;
x i *(t−τ 1 ) represents a complex conjugate of one of said M signals from an i th source at time t−τ 1 ;
x i (t−τ 2 ) represents one of said M signals from an i th source at time t−τ 1 ;
x(t) is a vector representation of said M signals; and
x H (t−τ 3 ) represents the Hermitian transpose of x(t−τ 3 ).
5. A method in accordance with claim 1 wherein said step of generating said hybrid separation matrix comprises performing a generalized eigenvalue analysis of said spatial fourth order cumulant matrix pencil.
6. A method in accordance with claim 1 wherein M=N.
7. A method in accordance with claim 1 wherein M<N.
8. A computer readable medium encoded with a computer program code for directing a processor to separate M signals provided by a Msources and received by an array comprising N elements, said program code comprising:
a first code segment for causing said processor to generate a hybrid separation matrix as a function of:
time differences between receipt of said M signals by said N elements;
a spatial fourth order cumulant matrix pencil;
a spatial correlation matrix; and,
steering vectors of said plurality of signals, and,
a second code segment for causing said processor to multiply said separation matrix by a time series matrix representation of said M signals.
9. A computer readable in accordance with claim 8 wherein the hybrid separation matrix is in accordance with the following equation:
ŵ j,hyb =|{circumflex over (v)} j H {circumflex over (K)} j −1 {circumflex over (v)} j | −1 {circumflex over (K)} j −1 {circumflex over (v)} j ;
wherein,
v j is the steering vector of the j th signal;
K j is the noise spatial covariance matrix of the j th signal.
10. A computer readable in accordance with claim 8 wherein
said spatial fourth order cumulant matrix pencil is a function of a spatial fourth order cumulant matrix being a summation of steering vector outer products scaled by an individual source signal's fourth order cumulant; and,
said steering vector is indicative of respective phase delays between ones of said N elements.
11. A computer readable medium in accordance claim 10 wherein said spatial fourth order cumulant matrix is in accordance with the following equation:
C
x
4
(
τ
1
,
τ
2
,
τ
3
)
≡
∑
i
=
1
N
Cum
[
x
i
*
(
t
-
τ
1
)
x
i
(
t
-
τ
2
)
x
(
t
)
x
H
(
t
-
τ
3
)
]
,
wherein:
C x 4 (τ 1 , τ 2 , τ 3 ) is said spatial fourth order cumulant matrix having a first time lag, τ 1 , a second time lag, τ 2 , and a third time lag, τ 3 , each time lag being indicative of a time delay from one of said M sources to one of said N elements;
N is indicative of a number of elements in said array;
Cum [x i *(t−τ 1 ) x i (t−τ 2 ) x(t) x H (t−τ 3 )] is a cumulant operator on arguments x i *(t−τ 1 ) x i (t−τ 2 ) x(t) x H (t−τ 3 );
t is a variable representing time;
x i *(t−τ 1 ) represents a complex conjugate of one of said M signals from an i th source at time t−τ 1 ;
x i (t−τ 2 ) represents one of said M signals from an i th source at time t−τ 1 ;
x(t) is a vector representation of said M signals; and
x H (t−τ 3 ) represents the Hermitian transpose of x(t−τ 3 ).
12. A computer readable medium in accordance with claim 8 , said program code further comprising:
a third code segment for causing said processor to perform a generalized eigenvalue analysis of said spatial fourth order cumulant matrix pencil.
13. A computer readable medium in accordance with claim 8 wherein M=N.
14. A computer readable medium in accordance with claim 8 wherein M<N.
15. A system for separating M signals provided by M sources, said system comprising:
a receiver for receiving said M signals and for providing received signals therefrom; and
a signal processor for receiving said received signals, generating a hybrid separation matrix, and multiplying said separation matrix by a time series matrix representation of said received signals, wherein:
said hybrid separation matrix is a function of time differences between receipt of
said M signals by said receiver, a spatial correlation matrix; steering vectors of
said M signals and a spatial fourth order cumulant matrix pencil.
16. A system in accordance with claim 15 , wherein the hybrid separation matrix is in accordance with the following equation:
ŵ j,hyb =|{circumflex over (v)} j H {circumflex over (K)} j −1 {circumflex over (v)} j | −1 {circumflex over (K)} j −1 {circumflex over (v)} j ;
wherein,
v j is the steering vector of the j th signal;
K j is the noise spatial covariance matrix of the j th signal.
17. A system in accordance with claim 15 wherein said receiver comprises N elements configured to form an array.
18. A system in accordance with claim 15 wherein
said spatial fourth order cumulant matrix pencil is a function of a spatial fourth order cumulant matrix being a summation of steering vector outer products scaled by an individual source signal's fourth order cumulant; and,
said steering vector is indicative of respective phase delays between ones of said N elements.
19. A system in accordance claim 18 wherein said spatial fourth order cumulant matrix is in accordance with the following equation:
C
x
4
(
τ
1
,
τ
2
,
τ
3
)
≡
∑
i
=
1
N
Cum
[
x
i
*
(
t
-
τ
1
)
x
i
(
t
-
τ
2
)
x
(
t
)
x
H
(
t
-
τ
3
)
]
,
wherein:
C x 4 (τ 1 , τ 2 , τ 3 ) is said spatial fourth order cumulant matrix having a first time lag, τ 1 , a second time lag, τ 2 , and a third time lag, τ 3 , each time lag being indicative of a time delay from one of said M sources to one of said N elements;
N is indicative of a number of a number of elements in said array;
Cum [x i *(t−τ 1 ) x i (t−τ 2 ) x(t) x H (t−τ 3 )] is a cumulant operator on arguments x i *(t−τ 1 ) x i (t−τ 2 ) x(t) x H (t−τ 3 );
t is a variable representing time;
x i *(t−τ 1 ) represents a complex conjugate of one of said M signals from an i th source at time t−τ 1 ;
x i (t−τ 2 ) represents one of said M signals from an i th source at time t−τ 1 ;
x(t) is a vector representation of said M signals; and
x H (t−τ 3 ) represents the Hermitian transpose of x(t−τ 3 ).
20. A system in accordance with claim 17 wherein M=N.
21. A system in accordance with claim 17 wherein M<N.
22. In a method for recovering low SNR signals in an multi-signal and noise environment with a multi-sensor array wherein a separation matrix is applied to the multi-sensor array data, the improvement of forming the separation matrix with hybrid minimum mean squared error weights, wherein said weights are generated as a function of a spatial correlation matrix; steering vectors of said multiple signals and a spatial fourth order cumulant matrix pencil.
23. A method in accordance with claim 22 wherein the number of said multiple signals is equal to the number of said multiple sensors in said array.
24. A method in accordance with claim 22 wherein the number of said multiple signals is less than the number of said multiple sensors in said array.
25. A method for recovering an unknown signal from a composite signal containing the unknown signal and at least one interferer signal and noise, said method comprising the step of generating a separation matrix to suppress the at least one interferer signal and the noise, wherein the separation matrix is a function of the spatial correlation matrix of the unknown signal, a steering vector, and a spatial fourth order cumulant matrix pencil of the unknown signal and the at least one interferer signal.
26. A method in accordance with claim 25 wherein said composite signal comprises M signals and is received on an N element array.
27. A method in accordance with claim 26 wherein M=N.
28. A method in accordance with claim 26 wherein M<N.