Nature of complex number and complex-valued neural networks

We discuss the nature of complex number and its effect on complex-valued neural networks(CVNNs).After we review some examples of CVNN applications,we look back at the mathematical history to elucidate the features of complex number,in particular to confirm the importance of the phaseand-amplitude viewpoint for designing and constructing CVNNs to enhance the features.This viewpoint is essential in general to deal with waves such as electromagnetic wave and lightwave.Then,we point out that,although we represent a complex number as an ordered pair of real numbers for example,we can reduce ineffective degree of freedom in learning or self-organization in CVNNs to achieve better generalization characteristics.This merit is significantly useful not only for waverelated signal processing but also for general processing with frequency-domain treatment through Fourier transform.