Bilinear Transformation

The bilinear transformation may be defined by

$$s = c\frac{1-z^{-1}}{1+z^{-1}}\protect$$ (I.9)

$$z^{-1} = \frac{1-s/c}{1+s/c}\protect$$ (I.10)

where $c$ is an arbitrary positive constant that we may set to map one analog frequency precisely to one digital frequency. In the case of a lowpass or highpass filter, $c$ is typically used to set the cut-off frequency to be identical in the analog and digital cases. The bilinear tranform was introduced in 1947^I.2 for discrete-time filter analysis (a year after the first general-purpose computer--the ENIAC--was announced) by Arnold Tustin,^I.3 so it is also called ``Tustin's Method.''