Digital Signal Processing Reference
In-Depth Information
exponential functions. Suppose also that there are altogether
m
signal sources.
Then the signal of the
m
th source can be written as
I
m
x
m
(
t
)
=
A
mi
sin(2
π
f
mi
t
+
ϕ
mi
)
,
(17.6)
i
=
1
where
A
mi
,
ϕ
mi
are the amplitudes, frequencies and phases of the signal
components respectively and
I
m
is the number of components in the signal emitted
by the
m
th source.
The first sensor will be chosen as a reference point in time and in distance.
Then, if the distance to the
k
th sensor is
d
k
and the arrival angle of the signal from
the
m
th source is denoted by
f
mi
and
m
, the component of the
k
th sensor output signal
coming from the
m
th signal source can be given as
Θ
A
mi
sin
2
π
f
mi
t
mi
I
m
2
π
f
mi
d
k
sin
Θ
m
x
mk
(
t
)
=
−
+
ϕ
.
(17.7)
c
i
=
1
The complete output signal of the
k
th sensor, containing components coming
from all signal sources, can then be defined as
M
x
k
(
t
)
=
x
mk
(
t
)
m
=
1
A
mi
sin
2
π
f
mi
t
+
ϕ
mi
I
m
M
Θ
2
π
f
mi
d
k
sin
m
=
−
c
m
=
1
i
=
1
A
mi
sin
2
π
f
mi
sin
mi
cos 2
π
f
mi
t
M
I
m
Θ
m
=
d
k
+
π
−
ϕ
c
m
=
1
i
=
1
A
mi
cos
2
π
f
mi
sin
d
k
−
ϕ
mi
sin 2
π
f
mi
t
Θ
m
+
.
(17.8)
c
The function
A
sin
(
2
π
ft
+
ϕ
) can be represented in the following form:
A
sin(2
π
ft
a
cos 2
π
ft
b
sin 2
π
ft
+
ϕ
)
=
+
,
where
a
.
The coefficients
a
and
b
are orthogonal projections of the sine functi
on on
cos 2
π
ft
and sin 2
π
ft
=
A
sin
ϕ
and
b
=
A
cos
ϕ
N
,
has been obtained and that these data have been decomposed by applying some
high-performance method and that parameters of the signal components as well
as the orthogonal projections have been estimated as a result of temporal spectral
.
Suppose that a snapshot multitude
{
x
k
(
t
n
)
}
,
n
=
1
,
