Method of broadband constant directivity beamforming for non linear and non axi-symmetric sensor arrays embedded in an obstacle
View Patent ↗A method is provided for designing a broad band constant directivity beamformer for a non-linear and non-axi-symmetric sensor array embedded in an obstacle having an odd shape, where the shape is imposed by industrial design constraints. In particular, the method of the present invention provides for collecting the beam pattern and keeping the main lobe reasonably constant by combined variation of the main lobe with the look direction angle and frequency. The invention is particularly useful for microphone arrays embedded in telephone sets but can be extended to other types of sensors.
1. A beamformer for correcting the beam pattern and beamwidth of a microphone array embedded in an obstacle whose shape is not axi-symmetric, comprising:
a multiplier for multiplying a signal d of a sound source from a directivity angle θ to each respective microphone of said array by a respective weighting vector w to generate a product that enhances the signal d while minimising noise n, where n is not correlated to the signal d, and where n and d are both dependant upon frequency ω; and
an adder for summing each respective product to generate an output signal such that w opt H d=1;
wherein optimised weighting vector w opt is a solution of
Min
w
1
2
w
H
R
nn
w
where R nn is a normalised noise correlation matrix, and wherein said solution is constrained by introducing symmetric vectors d 0+0 i and d 0−0 i on either side of d where θ i >0, with i={1, . . . , N θ } is a set of directions belonging to directivity angle θ for increasing beamwidth of said array, and at least one further vector to correct for beam pattern asymmetry resulting from said obstacle having a shape that is non-axisymmetric.
2. The beamformer of claim 1 , wherein said solution is constrained by a set of 2i (i={1,2, . . . , N const }) linear constraints w H d θ+θ 1 =α i and w H d 0−θ 1 =α −i such that
Min
w
1
2
w
H
R
nn
w
subject
to
w
H
d
=
1
under constraint becomes:
Min
w
1
2
w
H
R
nn
w
subject
to
C
H
w
=
g
where C is a rectangular matrix defined by:
C=[d|d θ+θ 1 |d θ−θ i | . . . ]
and g is a vector defined by:
g
=
[
1
α
i
α
-
i
⋮
]
resulting in said optimised weight vector w opt being given by:
w opt =R nn −1 C[C H R nn C] −1 g.
3. The beamformer of claim 1 , wherein said solution is constrained by a set of quadratic constraints whereby d 0+0 i and d θ−θ i are used to build a cross-correlation matrix:
D θ 1 =d θ+θ i d θ+θ 1 H +d θ− i d 0−θ i H
and the quadratic constraints are defined as:
w H D θ i w=β i
where β i is a set of values required for w H D θ 1 w, resulting in said optimised weight vector w opt being a minimisation of:
J
(
w
,
λ
,
λ
2
)
=
1
2
w
H
R
nn
w
+
λ
(
1
-
w
H
d
)
+
∑
i
λ
i
(
β
i
-
w
H
D
θ
i
w
)
+
σ
2
(
γ
-
w
H
w
)
where Lagrange coefficients λ,λ 1 , are dependant on frequency ω.
4. The beamformer of claim 2 , wherein said at least one further vector is a single vector d θ±θ i , and wherein angle θ j is chosen in the direction of the asymmetry.
5. The beamformer of claim 2 , wherein said at least one further vector is a pair of vectors d θ+θ i and d 0−0 i (with θ j ≠θ i ), such that a set of linear constraints w H (d θ+θ j −d θ−θ 1 )=0 with θ j ≠θ i is defined irrespective of w H d θ±θ i =α i .
6. The beamformer of claim 4 , wherein the cross-correlation matrix associated with said single vector is D θ j =d θ±θ j d θ±θ j H .
7. The beamformer of claim 5 , wherein the cross-correlation matrix associated with said pair of vectors is D θ 1 =d θ+θ 1 d θ+θ i H +d θ−θ 1 d θ−θ j H for a pair of symmetric (θ j =θ i ) vectors or asymmetric (θ j ≠θ i ) vectors.
8. A method for correcting the beam pattern and beamwidth of a microphone array embedded in an obstacle whose shape is not axi-symmetric, comprising:
positioning respective microphones of said array at selected locations on said obstacle such that the distance between microphones is less than one half of λ/2, where λ represents wavelength;
for each said microphone calculating a weighting vector w such that the Hermitian product w opt H d=1 enhances the signal d of a sound source for a given signal angle of arrival θ while minimising noise n due to the environment, where n is not correlated to the signal d, and where n and d are both dependant upon frequency ω;
wherein optimised weighting vector w opt is a solution of
Min
w
1
2
w
H
R
nn
w
subject
to
w
H
d
=
1
,
where R nn is a normalised noise correlation matrix, and wherein said solution is constrained by introducing symmetric vectors d θ+θ i and d θ−θ i on either side of d where θ i >0, with i={1, . . . . N θ } is a set of directions belonging to directivity angle θ for increasing beamwidth of said array, and at least one further vector to correct for beam pattern asymmetry resulting from said obstacle having a shape that is non-axisymmetric.
9. The method of claim 8 , wherein said solution is constrained by a set of 2i (i={1, 2, . . . N const }) linear constraints w H d θ+θ i =α i and w H d θ−θ 1 =α −i such that
Min
w
1
2
w
H
R
nn
w
subject
to
w
H
d
=
1
under constraint becomes:
Min
w
1
2
w
H
R
nn
w
subject
to
C
H
w
=
g
where C is a rectangular matrix defined by:
C=[d|d θ+θ i |d θ−θ i | . . . ]
and g is a vector defined by:
g
=
[
1
α
i
α
-
i
⋮
]
resulting in said optimised weight vector w opt being given by;
w opt =R nn −1 C[C H R nn C] −1 g.
10. The method of claim 9 , wherein said solution is constrained by a set of quadratic constraints whereby d θ+θ 1 and d θ−θ 1 are used to build a cross-correlation matrix:
D θ i =d θ+θ i d θ+θ i H +d θ−θ i d θ−θ i H
and the quadratic constraints are defined as:
w H D 0 i w =β i
where β i is a set of values required for w H D θ i w, resulting in said optimised weight vector w opt being a minimisation of:
J
(
w
,
λ
,
λ
2
)
=
1
2
w
H
R
nn
w
+
λ
(
1
-
w
H
d
)
+
∑
i
λ
i
(
β
i
-
w
H
D
θ
i
w
)
+
σ
2
(
γ
-
w
H
w
)
where Lagrange coefficients λ,λ i are dependant on frequency ω.
11. The method of claim 9 , wherein said at least one further vector is a single vector d θ±θ j , and wherein the angle θ j is chosen in the direction of the asymmetry.
12. The method of claim 9 , wherein said solution is further constrained by introducing at least a pair of vectors d θ+θ i and d θ−θ j with θ j ≠θ i ) to correct for beam pattern asymmetry resulting from said obstacle having a shape that is non-axisymmetric and re-orient the beam, such that a set of linear constraints w H (d θ+θ j −d θ−θ i )=0 with θ j ≠θ i of is defined irrespective of w H d θ±θ i =α i .
13. The method of claim 11 , wherein the cross-correlation matrix associated with said single vector is D θ j =d θ±θ j d θ±θj H .
14. The method of claim 12 , wherein the cross-correlation matrix associated with said pair of vectors is D θ i =d 0+θ 1 d θ+θ 1 H +d θ−θ j d θ−θ j H for a pair of symmetric (θ j =θ i ) vectors or asymmetric (θ j ≠θ 1 ) vectors.
15. A method of designing a broad band constant directivity beamformer for a non-linear and non-axi-symmetric sensor array embedded in an obstacle, comprising:
applying a numerical method to said obstacle to generate a boundary elements mesh;
positioning array sensors at selected nodes of the boundary element mesh for defining sectors all around the array,
modelling a set of potential sources to be detected by said sensors in said sectors and determining the acoustic pressure at each of said sensors for each of said sources;
defining a noise field characterised by a normalized noise correlation matrix (R nn ) at said array sensors;
for each sector, with a look direction θ, defining (i) a pair of vectors whose directions are symmetric relative to direction θ, and at least one of (ii) a pair of vectors whose directions are asymmetric relative to direction θ, and (iii) a single vector with a direction different from θ, and
applying a set of constraints to said vectors in each sector to obtain an optimal weighting vector w opt for correction of beamwidth and beampattern asymmetry.