Relationship Between Laplace and $z$-Transforms.

Question

I've recently come across the relation $s= \frac{2(z-1)}{T(z+1)}$ between the Laplace and $z$-Transforms with inverse $z= \frac{2+sT}{2-sT}$ in some lecture slides, however there was no elaboration and all I know is that T represents the sampling period and hence is a measure of time. My question is, has anyone come across this correspondence before and could point me in the right direction of what it means?

Thanks!

Answer 1

Yes, it's a bilinear transformation. It's used to convert between continuous-time and discrete-time systems (probably in your case, between the transfer functions of filters within the time-domain and discrete-time domain).

See

https://en.wikipedia.org/wiki/Bilinear_transform

for more details.