System and method for hybrid minimum mean squared error matrix-pencil separation weights for blind source separation

• US 6,931,362 B2
• Filed: 11/17/2003
• Issued: 08/16/2005
• Est. Priority Date: 03/28/2003
• Status: Expired due to Fees

• Current Assignee: North South Holdings Incorporated (Spherix Incorporated)
• Original Assignee: Harris Corporation (L3Harris Technologies, Inc.)
• Inventors: Beadle, Edward Ray, Anderson, Richard Hugh, Dishman, John Fitzgerald, Anderson, Paul David, Martin, Gayle Patrick
• Primary Examiner(s): Nghiem, Michael
• Assistant Examiner(s): Washburn, Douglas N.
• Application Number:
  US10/713,107
• Publication Number:
  US 20040204922A1
• Time in Patent Office: 638 Days
• Field of Search: 340/310.03, 367/901, 381/94.1, 455/63.1, 455/226.1, 455/501, 702/127, 702/179, 702/189, 702/190, 702/191, 702/194, 702/196
• US Class Current: 702/190
• CPC Class Codes: G06F 18/2134 based on separation criteria, e.g. independent component analysis

Abstract

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.

Claims

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<sup>−

1</sup>{circumflex over (v)}_j|<sup>−

1</sup>{circumflex over (K)}_j<sup>−

1</sup>{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:

$C_x^4<br><br>\left({\mathrm{\tau <br><br>}}_1,{\mathrm{\tau <br><br>}}_2,{\mathrm{\tau <br><br>}}_3\right)\equiv <br><br>\underset{i=1}{\overset{N}{\sum <br><br>}}<br><br>\mathrm{Cum}<br><br>\left[x_i^{*}<br><br>\left(t-{\mathrm{\tau <br><br>}}_1\right)<br><br>x_i<br><br>\left(t-{\mathrm{\tau <br><br>}}_2\right)<br><br>x<br><br>\left(t\right)<br><br>x^H<br><br>\left(t-{\mathrm{\tau <br><br>}}_3\right)\right],$

wherein

C_x⁴(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<sup>−

1</sup>{circumflex over (v)}_j|<sup>−

1</sup>{circumflex over (K)}_j<sup>−

1</sup>{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:

$C_x^4<br><br>\left({\mathrm{\tau <br><br>}}_1,{\mathrm{\tau <br><br>}}_2,{\mathrm{\tau <br><br>}}_3\right)\equiv <br><br>\underset{i=1}{\overset{N}{\sum <br><br>}}<br><br>\mathrm{Cum}<br><br>\left[x_i^{*}<br><br>\left(t-{\mathrm{\tau <br><br>}}_1\right)<br><br>x_i<br><br>\left(t-{\mathrm{\tau <br><br>}}_2\right)<br><br>x<br><br>\left(t\right)<br><br>x^H<br><br>\left(t-{\mathrm{\tau <br><br>}}_3\right)\right],$

wherein;

C_x⁴(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<sup>−

1</sup>{circumflex over (v)}_j|<sup>−

1</sup>{circumflex over (K)}_j<sup>−

1</sup>{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:

$C_x^4<br><br>\left({\mathrm{\tau <br><br>}}_1,{\mathrm{\tau <br><br>}}_2,{\mathrm{\tau <br><br>}}_3\right)\equiv <br><br>\underset{i=1}{\overset{N}{\sum <br><br>}}<br><br>\mathrm{Cum}<br><br>\left[x_i^{*}<br><br>\left(t-{\mathrm{\tau <br><br>}}_1\right)<br><br>x_i<br><br>\left(t-{\mathrm{\tau <br><br>}}_2\right)<br><br>x<br><br>\left(t\right)<br><br>x^H<br><br>\left(t-{\mathrm{\tau <br><br>}}_3\right)\right],$

wherein;

C_x⁴(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.