Image Processing Reference
In-Depth Information
Fig. 11.1 Problem sketch
secondary source can be implemented with an array of number of discrete
loudspeakers.
As the result, Eq. (
11.1
) can be simplified into the discrete form of Rayleigh I
integral as shown in Eq. (
11.4
) as the following:
"
#
1
e
jk r r
m
j
j
j
ˁ
0
ω
2
~
V
m
~
P
ð
r
; ω
Þ¼
r
m
,
ω
x
:
ð
11
:
4
Þ
Δ
ˀ
~
r
~
r
m
m
¼
1
The driving function for the
m
-th driver in the discrete control source array is
obtained from (
11.4
) and is shown in Eq. (
11.5
).
e
jk r r
m
j
j
Q
m
~
A
m
~
;
ð
r
; ω
Þ¼
ð
r
; ω
Þ
W
ðÞ
ð
11
:
5
Þ
~
r
~
r
m
where
A
m
denotes a weighting factor of the
m
-th driver,
W
(
) is the frequency
spectrum of an audio signal. The driving function (
11.5
) is a modified interpretation
of the Rayleigh integral equation (
11.2
) describing the solution for reconstruction of
a wave field surrounded by only monopoles. However, (
11.5
) is valid only when a
virtual source is supposed to be situated outside
S
(see Fig.
11.1
).
The driving function (
11.5
) can be modified further so that the solution is valid
for reconstruction of a wave field both inside and outside
S
, in other words
everywhere except on the boundary
S
in Fig.
11.1
[
12
]:
ω
e
ð
ʸ ðÞþ
1
:
jk r
j
r
m
j
j
ʸ ðÞ
Q
m
~
ð
r
; ω
Þ¼
A
m
~
ð
r
; ω
Þ
W
ðÞ
:
ð
11
:
6
Þ
~
r
~
r
m
Here
A
m
denotes a weighting factor of the
m
-th driver,
W
(
ω
) is the frequency
ʸ
~
r
is an indicator function.
ʸ
~
r
has the value
spectrum of an audio signal and
1 for all
D
, i.e. a focused source; otherwise 0, where
D
is a problem domain and
S
is the boundary of
D
. It is important to emphasise that the driving function shown
~
r
2
