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clear that the learning algorithm on complex plane (
C
BP) that trains the CVNN must
not only obtain a convergence with respect to the magnitude, but also with respect
to the phase. This is equivalent to stating that the real as well as imaginary parts of
the complex numbers must be separately captured by the
C
BP.
2.3.1.1 Beauty of Complex Numbers
Complex domain (
C
) itself is gaining more attention because many real applications
involve signals that are inherently complex-valued. A complex number is directly
related to the two-dimension data. It comprises of two real numbers and comes with
phase information embedded into it. Addition and multiplication are much easier in
C
. Any complex number has a length and angle, hence forms a plane. Their oper-
ations are very related to two-dimensional geometry where one can use complex
arithmetic to do various geometric operations. Thus, it is more significant in prob-
lems where we wish to learn and analyze signal amplitude and phase precisely. Let
C
be the set of complex numbers and triplet
(
F
,
•
,
↗
)
be a Field equipped with
operations
•
,
↗
, satisfying the Closure, Commutative, Associative, Identity, Distrib-
ute (
), and Inverse properties for arbitrary elements belonging to
F
. A two-dimensional Field with basis
↗
distributes over
•
{
1
,
i
}
forms a two-dimensional vector space
of
C
over
R
.
Definition 2.1
The triplet
(
F
,
•
,
↗
)
is said to be a Field equipped with operations
•
,
↗
;
∀
c
1
,
c
2
,
c
3
∈
F
, if it satisfies the following axioms with respect to
•
:
1.
Closure
If
c
1
,
c
2
∈
F
=⃒
c
1
•
c
2
∈
F
.
2.
Commutative c
1
•
c
2
=
c
2
•
c
1
.
3.
Associative c
1
•
(
c
2
•
c
3
)
=
(
c
1
•
c
2
)
•
c
3
.
4.
Identity
∃
an element called '0' such that
c
1
•
0
=
0
•
c
1
=
c
1
.
5.
Inverse
∃
a unique
c
1
for every
c
2
such that
c
1
•
c
2
=
0.
and following axioms with respect to
↗
:
1.
Closure
If
c
1
,
c
2
∈
F
=⃒
c
1
↗
c
2
∈
F
.
2.
Commutative c
1
↗
c
2
=
c
2
↗
c
1
.
3.
Associative c
1
↗
(
c
2
↗
c
3
)
=
(
c
1
↗
c
2
)
↗
c
3
.
4.
Identity
∃
an element called '1' such that
c
1
↗
1
=
1
↗
c
1
=
c
1
.
5.
Inverse
∃
a unique
c
1
for every
c
2
(
=
0
)
such that
c
1
↗
c
2
=
1.
and also
Distributive Property
,
↗
distributes over
•
such that
c
1
↗
(
c
2
•
c
3
)
=
c
1
↗
c
2
•
c
1
↗
c
3
.
Definition 2.2
The Field of complex numbers is a degree two field extension over
the field of real numbers. The set of complex numbers is a Field equipped with
operations
+
,
×
such that for every two elements
a
+
jb
,
c
+
jd
:
•
(
a
+
jb
)
+
(
c
+
jd
)
=
(
a
+
c
)
+
j
(
b
+
d
)
.
•
(
a
+
jb
)
×
(
c
+
jd
)
=
(
ac
−
bd
)
+
j
(
bc
+
ad
)
.
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