Digital Signal Processing Reference
In-Depth Information
10
10
0
0
−10
−10
−20
−20
−30
−30
−40
−40
−50
−50
−60
−60
0
0.5 1.0
1.5
2.0 2.5 3.0 3.5 4.0
0
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
F (KHZ)
F (KHZ)
(
a
)
(
b
)
10
10
0
0
−10
−10
−20
−20
−30
−30
−40
−40
−50
−50
−60
−60
0
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
0
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
F (KHZ)
F (KHZ)
(c)
(d)
FIG. 3.15
The white noise gain of the first-order hypercardioid, as a function of frequency,
for different values of
Δ
: (a)
Δ
= 1 cm, (b)
Δ
= 2 cm, (c)
Δ
= 3 cm, and (d)
Δ
= 5 cm.
We conclude that the white noise is amplified if
√
2
1
− α
1,1
.
ωτ
0
<
(3.45)
Interestingly, the farther away is the null from the main lobe, the smaller is
the range of frequencies that can be affected by white noise amplification.
This fact intuitively makes sense since by putting the null closer to the main
lobe, the two constraints that shape the directional pattern will be conflicting
more with each other. In other words, the linear system of two equations given
in (3.34) will be ill conditioned. As a result, the uncorrelated white noise at
the two microphones will be amplified. The best possible situation (where
the two constraints are the farthest away and, hence, the linear system is
best conditioned) is a null at
θ
= 180
◦
, which corresponds to the cardioid. To
show more rigourously this important point, let us denote by
V
2
the 2
×
2
matrix involved in (3.34). The condition number of
V
2
is defined as [4]
−1
2
χ
(
V
2
)=
V
2
∞
V
∞
,
(3.46)
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