Digital Signal Processing Reference
In-Depth Information
4
ωτ
0
(1
− α
1,1
)
.
χ
(
V
2
)
≈
(3.50)
We see from (3.50) that in the range where
α
1,1
is defined, the condition
number is minimized for
α
1,1
=
−
1, i.e., the cardioid. When
α
1,1
gets closer
to 1,
χ
(
V
2
) gets larger, which means that
V
2
is not well conditioned and
white noise amplification should be expected. One easy way to improve this
issue is to regularize
V
2
but this may affect the shape of the beampattern.
This method is equivalent to optimizing the array gain under some constraint
on the white noise gain [5].
For
β
1,1
= 0, we find that the directivity factor is
1
−
cos[
ω
(
τ
0
− τ
2
)]
1
−
sinc(
ωτ
0
)cos(
ωτ
2
)
′
G
DN,1
[
h
(
ω
)] =
1
−
cos[
ωτ
0
(1
− α
1,1
)]
1
−
sinc(
ωτ
0
)cos(
ωτ
0
α
1,1
)
.
=
(3.51)
′
Figures 3.17 and 3.18 give plots of
G
(
ω
)] from (3.51), as a function of
frequency, for the hypercardioid and supercardioid, respectively, for different
values of
δ
.
For small values of
ωτ
0
(1
− α
1,1
), we get
DN,1
[
h
(
ω
)]
≈
(
ωτ
0
)
2
(1
− α
1,1
)
2
2
6
′
G
DN,1
[
h
·
(
ωτ
0
)
2
3
α
1,1
+1
≈
3
(1
− α
1,1
)
2
3
α
1,1
+1
.
(3.52)
This result corresponds to the theoretical value of the directivity factor for
diffuse noise. For
α
1,1
=
−
1, we get the directivity factor (equal to 3) of
the cardioid (see Section 3.3) and for
α
1,1
= 0, we get the directivity factor
(also equal to 3) of the dipole (see Section 3.2). It is well known that another
type of hypercardioid
2
is obtained by maximizing the gain with diffuse noise.
According to [1], this gain is exactly equal to 4 =
M
2
. The value of
α
1,1
corresponding to this hypercardioid is
−
1
/
3 and the beampattern is
(
ω
)
,θ
]
≈
4
3
C
1
4
+
3
′
B
[
h
4
cos
θ
.
(3.53)
Substituting
α
1,1
=
−
3
into (3.52), we find that indeed
′
G
DN,1
[
h
(
ω
)]
≈
4
.
(3.54)
2
We recall that the hypercardioid from Chapter 2 was derived by maximizing the gain in
the presence of cylindrically isotropic noise. Unless stated otherwise, this is our definition
of hypercardioid.
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