Digital Filter Design Using Fractional Bilinear Transform

The fractional bilinear transform used in analog-to-digital (A/D) conversion is proposed. This transformation is derived by means of fractional delay filter. Two forms are presented. One is approximated by Lagrange FIR fractional delay filter; the other is approximated by Thiran IIR fractional delay filter. Their magnitude responses are both linear in low frequency band, but differ in high frequency band for distinct applications.
According to this transformation, first-order digital differentiator and integrator can be easily designed. Additionally, some design examples are illustrated to show the linear frequency mapping property when performing A/D conversion. The designed digital differentiator and integrator are compared with existing differentiators and integrators. The designed low-order differentiator is suitable for real-time applications.
To improve highly selective 3-D recursive filters transformed from the bilinear transformation for tracking moving objects, first use 2-D examples to discuss the effects of diverse transformations. The experimental results show that the designed 3-D recursive plane filter can extract the desired moving object.
Index terms—Bilinear transformation, digital differentiator, digital integrator, fractional bilinear transformation, fractional delay filter, three-dimensional planar-resonant recursive filter, tracking moving objects.