Digital Signal Processing Reference
In-Depth Information
where tr[
·
] is the trace of a square matrix. We observe how the gain is upper
bounded [as long as
Γ
v
(
ω
) is nonsingular] and depends on the number of
microphones as well as on the nature of the noise.
In our context, the distortionless constraint is desired, i.e.,
h
H
(
ω
)
d
(
ω,
cos0
◦
)=1
.
(2.37)
As a consequence, it is easy to see that the filter:
v
(
ω
)
d
(
ω,
cos0
◦
)
d
H
(
ω,
cos0
◦
)
Γ
−1
Γ
h
MAX
(
ω
)=
(2.38)
−
v
(
ω
)
d
(
ω,
cos0
◦
)
maximizes the gain, which is given by
MAX
(
ω
)=
d
H
(
ω,
cos0
◦
−1
v
◦
G
)
Γ
(
ω
)
d
(
ω,
cos0
)
.
(2.39)
We are interested in three types of noise.
•
The temporally and spatially white noise with the same variance at all
microphones
5
. In this case,
Γ
v
(
ω
)=
I
M
, where
I
M
is the
M ×M
identity
matrix. Therefore, the white noise gain is
2
h
H
(
ω
)
d
(
ω,
cos0
◦
)
G
WN
[
h
(
ω
)] =
h
H
(
ω
)
h
(
ω
)
1
h
H
(
ω
)
h
(
ω
)
,
=
(2.40)
where in the second line of (2.40), the distortionless constraint is assumed.
For
h
(
ω
)=
d
(
ω,
cos0
◦
)
M
,
(2.41)
we find the maximum possible gain, which is
G
WN,MAX
(
ω
)=
M.
(2.42)
In general, the white noise gain of an
N
th-order DMA is
1
h
H
(
ω
)
h
(
ω
)
G
WN,N
[
h
(
ω
)] =
≤ M.
(2.43)
We will see how the white noise may be amplified by DMAs, especially at
low frequencies.
•
The diffuse noise
6
, where
5
This noise models well the sensor noise.
6
This situation corresponds to the spherically isotropic noise field.
Search WWH ::
Custom Search