where <math>T</math> is the sample time of the discrete system. The inverse transform is
An analog filter is stable if the poles of its transfer function fall in the negative real half of the complex plane. A digital filter is stable if the poles of its transfer function fall inside the unit circle in the complex plane. The bilinear transform maps the negative real half of the complex plane to the interior of the unit circle. This way, filters designed in the the analog domain can be easily converted to the digital domain while preserving their stability.
As an example take a simple RC-filter. This filter has a transfer function
If we wish to simulate this filter in a digital simulation, we can apply the bilinear transform by substituting for <math>s</math> the formula above, after some reworking, we get the following filter representation:
somebody who knows about this will need to write about warping and pre-warping