Easy Control of the Transition Band Width in Hyperbolic Tangent Filters by a New Parameterization

A new parameterization of the hyperbolic tangent function is suggested for easy control of the width of the transition region between the limiting values of -1 and 1. The hyperbolic tangent function approaches the signum function as the suggested half-width parameter approaches zero. This permits definition of the rectangular function as the limiting case of a combination of two shifted hyperbolic tangent functions. Since all types of ideal frequency selective filters are derived from the combination of some rectangular functions, the hyperbolic tangent window can also be used for the same purpose. The suggested filters are continuities in the whole space and provide an opportunity for easy control of the width of the passband, transition band and stopband through adjustment of the half-width parameter. The formulation and examples are provided for low-, band-, high-pass, and band-stopping two-dimensional filters. However, the results can easily be generalized for any type of frequency selective filter.