Biomedical Engineering Reference
In-Depth Information
Displacement velocity [mm/s]
Acoustic
source
,
s(t)
Acoustic
input
u
d
θ
G
Sensor 2
Sensor 1
(b)
Mechanical
processor
Mechanical
response
-
Azimothal angle of incidence [deg]
(a)
(c)
FIGURE 2.8
Sensory mechanism in the parasitoid fly
Ormia ochracea
: (a) structure of the mechanically coupled eardrums
modeled as a vibrating cantilever, (b) a conceptual far-field model where by the acoustic wavefront impinges the sensors at an
angle
θ
, and (c) amplification of the ITD (
τ
p
1
+
τ
p
2
) at the acoustic, mechanical, and the neuronal levels. Adapted from Ref.
11
.
the inter-element distance. Therefore, the acoustic
signal wavefront is considered planar as it reaches
the sensor, as shown in
Figure 2.8
b. The signal
x
(
p
j
,
t
) recorded at the
j
th element (location
specified by a three-dimensional position vector
P
j
∈ R
3
with respect to the center
G
of the array)
can be expressed as a function of the bearing
θ
that is the angle between the position vector
p
j
and the source vector
u
. The signal
x
(
p
j,
t
) is
written as
where
a
(
p
j
) and
τ
(
p
j
) denote the attenuation and
delay for the source
s
(
t
), respectively, measured
relative to the center of the microphone array.
Equation
(2.8)
is expanded using the Taylor
theorem as
∞
(−τ(
P
j
))
k
k
!
s
(
k
)
(
t
),
x
(
P
j
,
t
) =
a
(
P
j
)
(2.9)
k
=0
where
s
(
k
)
(
t
) is the
k
th temporal derivative of
s
(
t
).
Under far-field conditions,
a
(
p
j
) is approximately
constant and we set
a
(
p
j
)
≈
1 without sufficient
loss of generality. Also, for far-field conditions
x
(
P
j
,
t
)
=
a
(
P
j
)
s
(
t
−
τ(
P
j
))
,
(2.8)
