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As I understand it, for a signal \$f(t)\$ in time, its Laplace transform \$\mathfrak{L}\left\{f(t)\right\}=F_1(s)\$ and Z transform \$\mathfrak{Z}\left\{f(t)\right\}=F_2(z)\$ are related by a transformation \$z=e^{sT}\leftrightarrow s=1/T\,\log(z)\$ where \$T\$ is the sampling period (since the Z transform is discrete in time).

In practice, this is approximated to the first degree as follows $$\begin{align*}z&=e^{sT}\\&=\frac{e^{sT/2}}{e^{-sT/2}}\\&\approx\frac{1+sT/2}{1-sT/2}\end{align*}$$and thus \$(1-sT/2)z\approx1+sT/2\$ so \$sT/2\approx(z-1)/(z+1)\$ and ultimately \$s\approx\frac2T\frac{z-1}{z+1}=\frac2T\frac{1-z^{-1}}{1+z^{-1}}\$.

Now, I understand up to here, but I fail to understand why we use this particular first-order approximation over, say, \$z=e^{sT}\approx1+sT\leftrightarrow s\approx(z-1)/T=\frac{1-z^{-1}}{Tz^{-1}}\$.

Does this approximation 'behave' in some significantly poorer way for most purposes?


Sorry about the tags -- I tried various things like 'bilinear-transform' but they did not exist and I lack the points to create them.

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The Forward Euler Transform $$z=e^{sT}\approx1+sT\leftrightarrow s\approx(z-1)/T=\frac{1-z^{-1}}{Tz^{-1}}$$ is easy to understand in that it is a direct translation and scaling from the \$s\$-domain to the z-domain. But the translation can transform stable \$s\$-domain poles into unstable \$z\$-domain poles.

To see this consider the diagram below.

The Left Half Plane in the \$s\$-domain (shaded) is scaled by \$T\$ and translated by \$1\$ to the \$z\$-domain. It should be clear to see that a pole X that is stable in \$s\$ can be unstable in \$z\$ by the Forward Euler transform.

enter image description here

In contrast, the bilinear transform $$s\approx\frac2T\frac{z-1}{z+1}=\frac2T\frac{1-z^{-1}}{1+z^{-1}}$$ translates the entire LHP of the s-domain in the unit circle of the \$z\$-domain by frequency warping via an intermediate w-plane.

Referring to the diagram below (ref: Ogata.K, Discrete Time Systems, 1995, Prentice-Hall), you can see that the entire LHP of the \$s\$-domain (a) is transformed to the unit-circle (b) via the w-plane scaling (c).

So that's why Bilinear is preferred in practice to the Forward Euler. However, there are other choices such as zero-pole matching (which I prefer), that may be employed over Bilinear due the frequency warping involved in Bilinear.

enter image description here

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    \$\begingroup\$ awesome! I understand now -- the bilinear transform as an approximation preserves system 'stability'. Both forward Euler and the bilinear result in frequency warping as they are simple approximations (as I understand) but pole-zero matching gives no such warping, correct? \$\endgroup\$ – oldrinb Dec 19 '14 at 21:37
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I can answer your question only in a context with the switched capacitor (S/C) technique which uses the mathematical tools of digital signal processing. Here four different approximations are in use:

(1) Euler forward (EF), (2) Euler backward (EB), (3) bilinear (BI) and (4) LDI (Lossless discrete integrator).

For S/C circuits, it is common practice to use S/C circuits based on integrators. Here are the important differences:

(1) EF-integrator: For rising frequencies the approximation causes POSITIVE phase errors

(2) EB-integrator: For rising frequencies the approximation causes NEGATIVE phase errors

(3) BI-integrator: No phase and amplitude errors, however, for rising frequencies there is a kind of "shrinking" of the frequency axis based on an arctan function. For all lowpass and bandpass functions this effect causes a real zero for a finite frequency w=0.5*wcl (wcl: clock frequency). This effect is appreciated because all periodic spectral repetitions do not overlap and, thus, do not disturb each other.

(4) LDI-integrator: Combination of two integrators with EF and EB approximation, respectively.

I hope this helps to answer a part of your question.

EDIT: The approximation (z-1)/T as mentioned by you is equivalent to the the EF transformation.

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In addition to the excellent point about stability made by akellyirl, another advantage of the bilinear transform (or Tustin method) compared to forward and backward differences is that it leads to a better approximation to the integral.

See the following image:

image of approximation for bilinear method compared to forward and backward differences from http://scilab.ninja/study-modules/scilab-control-engineering-basics/module-7-continuous-to-discrete-conversion-methods/

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