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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.
They're different. Note particularly the "would like to have" -- any translation from the continuous-time domain to the discrete-time domain is an approximation; part of your job is to make sure it's both practically realizable and good enough.
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 :( )
