Nonlinear Extension of Inverse Filters for Decomposition of Monotonic Multi-Component Signals

The article is devoted to improving quality of decomposition of monotonic multi-component timeand frequency-domain signals. Decomposition filters operating with data sampled at geometrically spaced times or frequencies (at equally spaced times or frequencies on a logarithmic scale) are combined with artificial neural networks. A nonlinear processing unit, which can be considered as a deconvolution network or a nonlinear decomposition filter, is proposed to be composed from several linear decomposition filters with common inputs, which outputs are nonlinearly transformed, multiplied by weights and summed. One of the fundamental findings of this study is a square activation function, which provides some useful features for the decomposition problem under consideration. First, contrary to conventional activation functions (sigmoid, radial basis functions) the square activation function allows to recover sharper peaks of distributions of time constants (DTC). Second, it ensures physically justified nonnegativity for the recovered DTC. Third, the square activation function transforms the Gaussian input noise into the nonnegative output noise with specific probability distribution having the standard deviation proportional to the variance of input noise, which, in most practical cases when noise level in the data is relatively low, increases radically the noise immunity of the proposed nonlinear algorithms. Practical implementation and application issues are described, such as network training, choice of initial guess, data normalization and smoothing. Some illustrative examples and simulations are presented performed by a developed deconvolution network, which demonstrate improvement of quality of decomposition for a frequency-domain multi-component signal. Key-Words: Decomposition, Monotonic Multi-Component Signals, Distribution of Time Constants, Decomposition Filters, Square Activation Function, Deconvolution Networks