I have the following transfer function in the S-Domain:
$$G(s)=\frac{a}{s+a}$$
I want to determine a transfer function in the Z-Domain that does a good job of modelling this system for a discrete time approximation.
Should I use the bilinear transformation:
$$s=\frac{2}{T}\frac{1-z^{-1}}{1+z^{-1}}$$
$$G(z)=G(\frac{2}{T}\frac{1-z^{-1}}{1+z^{-1}})$$
the impulse invariant transformation
$$G(z)=Z(L^{-1}(G(s))_{t=nT})$$
the step invariant transformation:
$$G(z)=(1-z^{-1})Z(L^{-1}(\frac{G(s)}{s})_{t=nT})$$
or something else? (In the above, $Z$ is for Z-Transform and $L$ is for Laplace transform, I couldn't figure out how to do the fancy symbols)
What are the instances when I should use each type of transformation?