Confusion Regarding Bi Linear Transform I was reading my book where the z-transform of a signal is derived to be ${1-e^{-2bT}z^{-1}}$ . Then it goes on to say that by applying the bilinear transform we can get 

$$\frac{2(1+bT+(bT-1)z^{-1})}{T(1+z^{-1})}$$

I know a little bit about bilinear transform but this example is using it in a different context. Can someone please help me understand how the bilinear transform is being used here.
The bi linear transform is the transform from the Laplace Transform Domain to the Z Transform.

The Laplace Transform Domain is a regular plane.
This transform transforms vertical lines in the Laplace domain into circles in the Z Domain.
Hence the Fourier Vertical Line in Laplace Domain (The Y Vertical Lines) is transformed into the unit circle in the Z Transform.

I'm a little inaccurate here, since the whole left side of the plane is transformed into the inner part of the Z Plane and the right hand of the Laplace transform it transformed to the outer part of the Z Plane (Out of the unit circle).

This transform is used to derive Digital Filters from their Analog Laplace domain representation.