Relationship To Laplace Transform
The Bilinear transform is a useful approximation for converting continuous time filters (represented in Laplace space) into discrete time filters (represented in z space), and vice versa. To do this, you can use the following substitutions in H(s) or H(z):
from Laplace to z (Tustin transformation), or
from z to Laplace. Through the bilinear transformation, the complex s-plane (of the Laplace transform) is mapped to the complex z-plane (of the z-transform). While this mapping is (necessarily) nonlinear, it is useful in that it maps the entire axis of the s-plane onto the unit circle in the z-plane. As such, the Fourier transform (which is the Laplace transform evaluated on the axis) becomes the discrete-time Fourier transform. This assumes that the Fourier transform exists; i.e., that the axis is in the region of convergence of the Laplace transform.
Read more about this topic: Z-transform
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