Broadband sensor location selection using convex optimization in very large scale arrays
Systems and methods are provided to determine a subset of D microphones in a set of N microphones on a perimeter of a space to monitor a target location. The space is divided into L interference locations. An equation is solved to determine microphone weights for the N microphones by minimizing the maximum gain for signals related to the target location and interference locations, further optimized over an l 1 penalty by applying a Lagrange multiplier to an l 1 norm of the microphone weights in a manner that determines a set of D non-zero microphones weights and a set of (N-D) microphone weights that are zero or close to zero. Microphone weights are determined for at least 2 different frequencies.
1. A method for monitoring an acoustic target in a target location in a space monitored by N microphones, the method comprising:
providing a number D equal to or greater than 2 and smaller than N to a processor;
providing a number of L locations in the space, wherein each location in the L locations is treated by a processor as an interference source; and
determining by the processor a set of D microphones from the set of N microphones, the determining including:
minimizing a maximum interference gain relative to the target location and the L interference locations over at least 2 different frequencies expressed in microphone weights for all N microphones; and
performing a convex optimization of the microphone weights for all N microphones over an l 1 penalty to determine a set of D microphone weights not equal to zero for each of the at least 2 frequencies.
2. The method of claim 1 , wherein the set of D microphones is applied to each of the at least 2 frequencies.
3. The method of claim 1 , wherein the microphones are sampled selectively over the space-frequency domain and at least two different sets of D microphones are determined.
4. The method of claim 1 , wherein the l 1 penalty includes applying a Lagrange multiplier in an l 1 norm of the microphone weights of the N microphones.
5. The method of claim 4 , wherein an expression is evaluated that includes a transfer function for the at least 2 frequencies between each of the N sensors and each of the L interferences and the l 1 penalty with a value of the Lagrange multiplier to determine a first set of microphone weights.
6. The method of claim 5 , wherein the expression is evaluated for different values of the Lagrange multiplier until the first set of microphone weights has D non-zero sensor weights and (N-D) microphone weights in the first set of microphone weights are at least a factor 10 smaller than a D th microphone weight.
7. The method of claim 6 , wherein a final set of microphone weights is determined from the first set of microphone weights by debiasing the expression by setting the value of the Lagrange multiplier to zero.
8. The method of claim 5 , wherein the expression is provided by:
min
(
w
n
(
f
i
)
)
i
=
1
,
n
=
1
F
,
N
max
1
≤
l
≤
L
∑
i
=
1
F
∑
n
=
1
N
H
ln
(
f
i
)
w
n
(
f
i
)
+
λ
∑
n
=
1
N
max
1
≤
i
≤
F
w
n
(
f
i
)
subject to:
∑
n
=
1
N
H
0
n
(
f
i
)
w
n
(
f
i
)
=
1
for
i
=
1
,
2
,
…
,
F
,
wherein
i is an index of one of F frequencies,
n is an index of one of N microphones,
l is an index of one of L interferences,
f i indicates a frequency of index i,
H ln indicates a transfer function between a source with index l and a microphone with index n,
w n (f i ) indicates a weight of microphone n for frequency f i , and
λ is the Lagrange multiplier.
9. The method of claim 1 , wherein a perimeter of the space is surrounded by the set of N microphones.
10. The method of claim 1 , further comprising determining microphone weights for a plurality of target locations to determine a strength of a signal in each of the plurality of target locations.
11. A system to monitor an acoustic target in a target location in a space monitored by N microphones, the system comprising:
a memory enabled to store data including instructions;
a processor configured to retrieve and execute instructions from the memory to:
accept a number D equal to or greater than 2 and smaller than N;
accept a number of L locations in the space, wherein each location in the L locations is treated as an interference source;
determine a set of D microphones from the set of N microphones, including:
minimization of a maximum interference gain relative to the target location and the L interference locations over at least 2 different frequencies expressed in microphone weights for all N microphones; and
performance of a convex optimization of the microphone weights for all N microphones over an l 1 penalty to determine a set of D microphone weights not equal to zero for each of the at least 2 frequencies.
12. The system of claim 11 , wherein the set of D microphones is applied to each of the at least 2 frequencies.
13. The system of claim 11 , wherein the processor is configured to sample the microphones selectively over the space-frequency domain and at least two different sets of D microphones are determined.
14. The system of claim 11 , wherein the l 1 penalty includes application of a Lagrange multiplier in an l 1 norm of the microphone weights of the N microphones.
15. The system of claim 14 , wherein the processor is configured to evaluate an expression that includes a transfer function for the at least 2 frequencies between each of the N sensors and each of the L interferences and the l 1 penalty with a value of the Lagrange multiplier to determine a first set of microphone weights.
16. The system of claim 15 , wherein the processor is configured to evaluate the expression for different values of the Lagrange multiplier until the first set of microphone weights has D non-zero sensor weights and (N-D) microphone weights in the first set of microphone weights are at least a factor 10 smaller than a D th microphone weight.
17. The system of claim 16 , wherein the processor is configured to determine a final set of microphone weights from the first set of microphone weights by debiasing the expression by setting the value of the Lagrange multiplier to zero.
18. The system of claim 15 , wherein the expression is provided by:
min
(
w
n
(
f
i
)
)
i
=
1
,
n
=
1
F
,
N
max
1
≤
l
≤
L
∑
i
=
1
F
∑
n
=
1
N
H
ln
(
f
i
)
w
n
(
f
i
)
+
λ
∑
n
=
1
N
max
1
≤
i
≤
F
w
n
(
f
i
)
subject to:
∑
n
=
1
N
H
0
n
(
f
i
)
w
n
(
f
i
)
=
1
for
i
=
1
,
2
,
…
,
F
,
wherein
i is an index of one of F frequencies,
n is an index of one of N microphones,
l is an index of one of L interferences,
f i indicates a frequency of index i,
H ln indicates a transfer function between a source with index l and a microphone with index n,
w n (f i ) indicates a weight of microphone n for frequency f i , and
λ is the Lagrange multiplier.
19. The system of claim 11 , wherein a perimeter of the space is surrounded by the set of N microphones.
20. The system of claim 11 , further comprising the processor configured to determine microphone weights for a plurality of target locations to determine a strength of a signal in each of the plurality of target locations.