# Converting Analog Filter into Digital Filter, why Bilinear Transform?

What's the advantage of using the Bilinear Transform?

$$H_d(z) = H_c(s)\bigg|_{s=\frac{2}{T_s}\frac{z-1}{z+1}}$$

When you can just use this equation?

$$H_d(\omega) = H_c(\Omega)\bigg|_{\Omega=\omega/T_s}$$

In other words, why does bilinear transform exist? These two equations look almost the same to me...What are the tradeoffs between using one equation verses the other? Is there a case where you would use one verses the other?

$$\left . H_d(z) = H_c(s) \right |_{s = \frac{2}{T\,s}\frac{z-1}{z+1}}$$ describes a transfer function in the $$z$$ domain that you can easily translate into a difference equation and realize in software.
$$\left . H_d(\omega) = H_c(\Omega) \right |_{\Omega = \frac{\omega}{T\,s}}$$ describes an idealized frequency response that you would like $$H_d$$ to have when you are done realizing it physically in software.
Note that there are other ways of approximating $$H_c$$ with some $$H_d$$ -- the bilinear transform is just one way. It has a lot of currency because it's conceptually simple, and works pretty well. It also has a lot of currency because it's easy to do with pencil, paper, and a slide rule -- today, there's numerical optimization techniques that can get you closer to some desired filtering goal, for less work (but -- I can never remember the search terms :( )