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 :( )

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    $\begingroup$ numerical optimization techniques to approximate a desired frequency response with a digital filter: The most famous one is probably the Remez exchange method. $\endgroup$ – Marcus Müller Jun 23 '20 at 19:39
  • $\begingroup$ And to add the least squares algorithm, which for most applications I deal with (optimizing overall SNR versus peak error) it usually ends up being the better choice. $\endgroup$ – Dan Boschen Jun 24 '20 at 14:49

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