Discrete time IIR filter designs based on transformation of continuous time designs are easy to carry out

Some problem with mathjax

Bilinear Transformation

Bilinear Transformation bewteen s-plane and z-plane

$$
s = \frac{ 2 }{ T } \frac{1-z^{-1}}{1+z^{-1}} \
z = \frac{1+(T/2)s}{1-(T/2)s}
$$

The imaginary axis $s = j\Omega$ is mapped onto the unit circle $e^{j\omega}$

$$
\Omega = \frac{2}{T} tan(\omega/2) \
\omega = 2 arctan(\Omega T/2)
$$

Here is the figure shows the s-plane and z-plane