Neurocomputing in Complex Domain - High Dimensional Neurocomputing: Growth, Appraisal and Applications - page 56

wide spectrum of computer simulations. Their simulations for conformal mappings
(geometric transformations) were not possible without the ability to process sig-
nals maintaining magnitude and phase relationship properly in activation function,
Eq. 3.3 , explained in Chap. 5 . The accurate estimation of both magnitude and phase
of the complex-valued signals are important in many real-world applications. Thus,
learning in CVNN with activation function, Eq. 3.3 , ensures the flow of complex-
valued signals preserving the magnitude and phase relationship.
The real part of activation function defined with equation Eqs. 3.3 and 3.4 is
independent of y and its imaginary part is independent of x . The derivative of the
activation function with respect to the real part of the argument, x , is free from the
imaginary part. Similarly, the derivative of the activation function with respect to
the imaginary part of the argument, y , is pure imaginary. The surface plots of the
real and imaginary parts of the function and their derivatives are shown in Fig. 3.5 .
It can be easily seen that the function does not have any singular points and nor
Fig. 3.5 a Real part of split-type function (Nitta activation function), b Imaginary part of Nitta
activation function, c Real part of the derivative with respect to x, d Imaginary part of the derivative
with respect to y