The use of Mobius transformations in neural networks and signal processing

A framework for the use of Mobius transformations in general neural networks (NNs) and signal processing is provided. It is first shown that both a nonlinear activation function of a neuron and a first order all-pass filter section can be considered as Mobius transformations. Further the global input-output relationship in layered NNs is shown to belong to a modular group of compositions of Mobius transformations, whereas cascaded all-pass digital filters are shown to represent the Blaschke product of Mobius transformations. Finally, Routh stability in nonlinear field filters is briefly addressed in this context. For rigour, existence and uniqueness of such an approach is considered