Digital filter design using the hyperbolic tangent functions

Abstract: Low- and band-pass filters can be designed by a combination of hyperbolic tangent functions in the frequency domain using the scaling and shifting theorems of the Fourier Transforms. The corresponding filter function in the time domain can be derived analytically from the frequency domain expression.The smoothness parameters control the slopes at the cutoff regions and permit the construction of a relatively short filter while reducing the oscillations of the filter response in the time domain. Different smoothness parameters can be chosen for the low and high cutoff frequencies in the band-pass filter design. Following the proposed scheme can easily derive the other type of the filters (e.g. high-pass).