Neurocomputing with High Dimensional Parameters - High Dimensional Neurocomputing: Growth, Appraisal and Applications - page 28

values. This indicates that the complex variable-based neural network may be useful
for such applications. CVNN is the extension of RVNN, in which all the parameters
and signals flowing through it are complex numbers (Two Dimension Parameters) in
contrast to real numbers in the RVNN. The different neurobiological studies revealed
that the action potential in human brain may have different pulse patterns and the
distance between pulses may be different. This justifies the introduction of complex
numbers representing phase and magnitude by a single entity into neural networks.
CVNN has been applied to various fields like adaptive signal processing, speech
processing, and communicating, but are not limited. These applications often use
signals, which have two types of information embedded in it, the magnitude and
the phase. In real domain it is not possible to represent both these quantities by
a single quantity; so the magnitude and phase is represented by two numbers and
then the neural network can be trained using these two quantities as separate inputs.
While the neural network trained on this topology may give satisfactory results, but
the relationship between phase and magnitude of a signal cannot be represented
by the model because of their separation. Chapter 5 demonstrate that the CVNN
can learn and generalize linear and bilinear transformation over typical geometric
structures on plane. These transformations cannot be learned using RVNN. The
CVNN shows excellent generalization capabilities for these transformations because
representation of magnitude and phase by single quantity i.e. complex number. It also
worth to mention here that CVNN has yielded far better result even in case of real-
valued problem (Chap. 4 ) , hence outperformed over equivalent real-valued neural
network.
2.3.1 Properties of Complex Plane
The complex plane is the geometric analog of complex numbers, which has a long and
mathematically rich history. It is the set of dual numbers over the real that possesses
one to one correspondence with the points of cartesian plane. The complex plane
is unlike real line, for it is two-dimensional with respect to real numbers and one-
dimensional with respect to the set of complex numbers (Halmos 1974). A point on
the plane can be viewed as a complex number with the x and y coordinates regarded
as the real and imaginary parts of the number. The set of complex numbers is a Field
equipped with both addition and multiplication operations, and hence makes a perfect
platform of operation. But the order that existed on the set of real numbers is absent
in the set of complex numbers, and as a result, no two complex numbers could be
compared as being big or small with respect to each other, but their magnitudes (which
are real numbers) could well be compared. The properties of the complex plane are
different from those of the real line. The set of real numbers was one-dimensional,
while as was pointed out, the set of complex numbers is one-dimensional if the field in
question is the set of complex numbers itself, while it is two-dimensional if the filed
is the set of real numbers. The complex numbers have a magnitude associated with
them and a phase that locates the complex number uniquely on the plane. It is hence