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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?

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$$\left . H_d(z) = H_c(s) \right |_{s = \frac{2}{T\,s}\frac{z-1}{z+1}}$$ 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. $\endgroup$ Jun 23, 2020 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. $\endgroup$ Jun 24, 2020 at 14:49
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Why cannot we use:

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

which means infinitely rolling an infinite continuous frequency axis over the finite discrete frequency axis (the unit circle).

The answer lies in the answer to a different question:

What is the domain of $\omega$?

We could consider two cases:

  • $\omega \in (-\pi, \pi]$ (which is used for discrete transfer functions), means that we'll only get a part of $H_c(\Omega)$, namely, the one for $\Omega \in (-\pi/T_s, \pi / T_s]$,
  • $\omega \in (-\infty, \infty)$ (to cover the whole domain of $H_c(\Omega)$), means that $H_d(\omega)$ will get multiple values for each argument $\omega$ (because the continuous frequency axis is a line and the discrete frequency axis is a circle). Unless $H_d(\Omega)$ is periodic with a period corresponding to one circulation of the discrete frequency axis, which is a very rare and specific case.

So either we get a trimmed version of $H_c(\Omega)$ or an incorrectly defined (multi-valued) function.

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So here's the thing. Any linear time-invariant filter has three essential elemental building blocks.

  1. Adders
  2. Scalers (multiply by a constant)
  3. Something that discriminates w.r.t. frequency. That's the only way a filter can filter out some frequency vs some other frequency.

Both analog and digital filters do 1 and 2 in a straight-forward way.

The way that analog filters discriminate frequency is with devices that differentiate or integrate w.r.t. time. Differentiating a higher frequency sinusoid results in a larger amplitude than for a lower frequency.

The way that digital filters discriminate frequency is with devices that delay a signal w.r.t. time.

Both domains have their equivalent transforms, the Laplace Transform for analog in which the differentiator is an operator that is multiplication by $s$ (the integrator is $s^{-1}$) and the Z Transform for digital in which the the delay operator is multiplication by $z^{-1}$.

Now in both domains, the delay of the same unit of time (which is the sampling period, $T$) is

$$ z^{-1} = e^{-sT} $$

or

$$ z = e^{sT} $$

or

$$ s = \frac{1}{T} \ln(z) $$

Now if digital filters had an operator that could do the $\ln(z)$ operation, you could emulate an analog filter design exactly, whatever analog $s$-plane transfer function $H_\mathrm{a}(s)$ that is your model, you could make a $z$-plane digital filter $H(z)$ follow it exactly with

$$ H(z) = H_\mathrm{a}(s) \bigg|_{s=\frac{1}{T}\ln(z)} $$

but digital filters don't have that $\ln(z)$ operation. They have the $z^{-1}$ operation. But we know we can construct digital filters with adders, scalers, delay elements. We have already learned that any rational function

$$ H(z) = \frac{b_0 + b_1 z^{-1} + \cdots + b_M z^{-M}}{a_0 + a_1 z^{-1} + \cdots + a_N z^{-N}} $$

So we can't do $\ln(z)$ exactly. But we can approximate it.

$$ \ln(z) = 2\left[\left({z-1\over z+1}\right)+ {1\over3} \left({z-1\over z+1}\right)^3 + {1\over5} \left({z-1\over z+1}\right)^5+ \cdots\right] $$

The bilinear transform approximates that infinite series by keeping just the first-order term:

$$ \ln(z) \approx 2 \cdot \frac{z-1}{z+1} = 2 \cdot \frac{1-z^{-1}}{1+z^{-1}} $$

So then the substitution above becomes

$$ H(z) = H_\mathrm{a}(s) \bigg|_{ s \leftarrow \frac{2}{T}\frac{1-z^{-1}}{1+z^{-1}} } $$

If $H_\mathrm{a}(s)$ is a rational function with order $N$ (let's say that $M \le N$) then $H(z)$ must also be a rational function of the same order.

So that's the root motivation of the bilinear transform. It's about how to implement or approximate $\ln(z)$.

Now there are some neat mathematical features using that substitution because in analog filters, to do frequency response we make this $ s = j \Omega $ substitution which means we evaluate $H_\mathrm{a}(s)$ on the (imaginary) $j \Omega$ axis. With digital filters we say $z=e^{j\omega}$ for frequency response and evaluate $H(z)$ on the $|z|=1$ unit circle.

So the $z=e^{sT}$ function maps the $j \Omega$ axis in the $s$-plane to the unit circle $z=e^{j \omega}$ unit circle where

$$\omega = \Omega T$$

But so also does the bilinear transform. You can show that:

$$ z = \frac{1+\frac{sT}{2}}{1-\frac{sT}{2}} $$

which is the inverse bilinear transform, also maps the $j \Omega$ axis in the $s$-plane to the unit circle $z=e^{j \omega}$ unit circle but to a different spot on the unit circle.

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