Biomedical Engineering Reference
In-Depth Information
where we have used the relationship
d
[
n
]
=
sgn(
v
[
n
−
1]). Note that the update rate
ξ
does
not affect the stability of learning. The constraint
space
C
is restricted to transforms represented
by lower-triangular matrices with diagonal
elements set to unity that can be expressed as
W
ij
=
0;
∀
i
<
j
;
W
ii
=
1. Thus, to satisfy the con-
straint
W
∈ C
, only the lower-diagonal elements
are updated in Eq.
(2.21)
. It can be seen from
Eq.
(2.21)
that if ||
W
||
∞
≤
B
> 0 is true, then the
recursion (21) will asymptotically lead to
out of phase with respect to
Δ
x
1
(
t
), and the dif-
ferential signal
Δ
x
3
(
t
) is 180° out of phase with
respect to
Δ
x
1
(
t
).
To apply
ΣΔ
learning for bearing estimation,
three of the four differential signals in Eqs.
(2.23)
are chosen as inputs, and the synaptic matrix
W
is chosen to be of the form
W
11
10
W
21
W
22
.
W
=
(2.24)
1
When applied to the three differential signals of
the microphone array modeled by Eq.
(2.23)
, the
ΣΔ
recursions in Eq.
(2.16)
lead to
E
n
{
d
[
n
]sgn
(
X
[
n
]
)
T
}
n
→∞
−→ 0
(2.22)
v
1
[
n
]=
v
1
[
n
− 1]+
x
2
[
n
]+
W
11
[
n
]
x
1
[
n
]−
d
1
[
n
],
V
2
[
n
]=
V
2
[
n
− 1]+
x
3
[
n
]+
W
21
[
n
]
x
1
[
n
],
+
W
22
x
2
[
n
]−
d
2
[
n
],
for
W
∞
∈
C
. Equation
(2.22)
shows that the pro-
posed
ΣΔ
learning algorithm produces binary
(spike) sequences that are mutually uncorre-
lated to a nonlinear function of the input signals.
(2.25)
with
d
1
[
n
]
=
sgn(
v
1
[
n
]) and
d
2
[
n
]
=
sgn(
v
2
[
n
]). The
adaptation steps for the parameters
W
11
,
W
21
,
and
W
22
based on Eq.
(2.21)
can be expressed as
2.6.1
ΣΔ
Learning for Bearing
Estimation
The design of the bio-inspired acoustic source
localizer based on
ΣΔ
learning has been pre-
sented by Gore
et al
.
[39]
.
Figure 2.10
shows
the localizer constructed using an array of off-
the-shelf electret microphones interfacing with
a custom-made application-specific integrated
circuit (ASIC). Similar to the mechanical canti-
lever model in a parasitoid fly, the differential
electrical signals recorded by the microphones
can be approximated by
W
11
[
n
]=
W
11
[
n
− 1]−ξ
d
1
[
n
]sgn(
x
1
[
n
]),
W
21
[
n
]=
W
21
[
n
− 1]−ξ
d
2
[
n
]sgn(
x
1
[
n
]),
W
22
[
n
]=
W
22
[
n
− 1]−ξ
d
2
[
n
]sgn(
x
2
[
n
]).
(2.26)
One of the implications of
ΣΔ
adaptation
steps in Eq.
(2.26)
is that, for the bounded values
of
W
11
,
W
21
, and
W
22
, the following asymptotic
result holds:
N
,
1
N
x
1
(
t
)
=−
s
(1)
(
t
)
c
cosθ
x
2
(
t
)
=−
s
(1)
(
t
)
c
sinθ
x
3
(
t
)
=+
s
(1)
(
t
)
c
cosθ
x
4
(
t
)
=+
s
(1)
(
t
)
c
sinθ
lim
N
→∞
d
1,2
[
n
]→0.
(2.27)
n
=1
(2.23)
To demonstrate how Eqs.
(2.25)
and
(2.26)
can
be used for bearing estimation, consider two dif-
ferent cases based on the quality of common-
mode cancellation.
Case I: Perfect common-mode cancellation
. The
differential microphone is assumed to com-
pletely suppress the common-mode signal
x
cm
(
t
) in Eq.
(2.23)
. Also, the bearing of the
source is assumed to be located in the positive
which bear similarity to the differential signals
observed for the parasitoid fly.
Figure 2.10
also
shows a scope trace of the sample differential
output produced by the microphone array
when a 1 kHz tone is played from a standard
computer speaker. The scope trace clearly
shows that the differential signal
Δ
x
2
(
t
) is 90°
