z-Transform Methods: Definition vs. Integration Rule

Question

The definition of the z-transform is defined as $z = e^{sT}$ where "s" is complex frequency for continuous-time systems and "T" is the sample period. Why are rules such as the forward rectangular rule, or Tustin's method used instead of the definition?

Forward rectangular rule: $s \leftarrow \frac{z-1}{T}$ Tustin's rule: $s \leftarrow \frac{2}{T} \frac{z - 1}{z + 1}$

I am asking about the characteristics of the transform methods. If I transform G(s) to G(z), then why use Tustin's rule or forward rectangular rule instead of the definition? I would think the integration rules are just an approximation to the definition. What are the characteristics of these maps and how do they compare with each other?

Answer 1

Just to avoid a misunderstanding: the $\mathcal{Z}$-transform is a transform defined for sequences, comparable to the Laplace transform for continuous functions. What you are talking about is not the $\mathcal{Z}$-transform, but methods for converting analog to digital (actually, discrete-time) systems. [And it doesn't help that one of those conversion methods is called the matched Z-transform, which is just an unfortunately chosen name.]

There are several methods to achieve such a conversion, and all of them have certain properties. There is no ideal or "correct" mapping, and some methods may be more well-suited for certain applications than others.

The methods mentioned in your question are direct mapping methods, where the variable $s$ in the continuous-time transfer function is simply replaced by a (rational) function of $z$. Apart from the two methods mentioned in your question (1. Tustin's method - also called bilinear transform, and 2. forward Euler), there is also the backward Euler method, which replaces $s$ by $(1-z^{-1})/T$. Note that Tustin's method (bilinear transform) is equivalent to the trapezoidal integration rule. The important advantage of the bilinear transform for the conversion of analog systems is that the frequency axis ($j\omega$-axis) of the $s$-plane is transformed to the unit circle of the $z$-plane, and, consequently, the frequency response of the resulting discrete-time system is just a compressed (warped) version of the analog frequency response. One consequence of this is that analog frequency selective filters with optimal magnitudes (Chebysev, Butterworth, elliptic) are mapped to discrete-time filters satisfying the same optimality criteria. Furthermore, since the left half-plane is mapped to the inside of the unit circle in the $z$-plane, stability (and the minimum-phase property) are preserved by the transformation.

This is different for the forward and backward Euler methods. With those methods, the $j\omega$-axis is not transformed to the unit circle of the $z$-plane, and the left half-plane is not mapped to the region $|z|<1$. In case of the forward Euler method, the $j\omega$-axis is mapped to the line $\textrm{Re}\{z\}=1$, and the left half-plane is mapped to the half-plane $\textrm{Re}\{z\}<1$. This means that a stable analog prototype filter might be transformed to an unstable discrete-time filter. The backward Euler method transforms the $j\omega$-axis to the circle $|z-\frac12|=\frac12$, and the left half-plane is mapped to the region inside that circle: $|z-\frac12|<\frac12$. Consequently, stability and the minimum phase property are preserved, but the analog frequency axis is not mapped to the unit circle of the complex plane, so the resulting frequency response is not just a warped version of the original analog frequency response.

Note that there are also other very common methods for converting analog to digital systems, which are not based on direct mappings from the $s$-plane to the $z$-plane. These methods are based on time domain criteria. The two most common of these methods are the impulse-invariant method and the step-invariant method. They preserve the shape of the impulse (or step) response of the analog system by defining the desired digital system as the one whose impulse (step) response is a sampled version of the corresponding analog impulse (step) response. These methods introduce aliasing in the frequency domain, but - unlike the bilinear transform - they do not introduce frequency warping.

Answer 2

First consider continuous-time (a.k.a "analog") LTI systems (sometimes we call these "filters").
Now we know from LTI system theory that the Laplace transform of the input, $X(s)$, and output, $Y(s)$, of an LTI system are related to each other with this multiplicative transfer function, $H(s)$:

$$ Y(s) = H(s) X(s) $$

It also turns out that, to get the frequency response, we evaluate the transfer function with $s=j\Omega$. So $H(j\Omega)$ is the frequency response containing both the magnitude $|H(j\Omega)|$ and phase $\phi = \arg \{H(j\Omega) \}$ information.

Now, for implementation, we have three basic classes of building blocks to create an LTI system:

1. signal adders or subtractors
2. signal scalers (multiplication by a constant)
3. integrators (which implement $s^{-1}$)

The third class is what is necessary to make a filter, an LTI system that can discriminate with respect to frequency. The first two classes behave identically with any frequency, so all you can make (in the audio world) with the first two are mixers and amplifiers. You need class number 3 to make a tone control or an equalizer.

Now, even though there are lots of different circuit topologies (the so-called "Sallen-Key" circuit is a common one), essentially, that's all you have for a continuous-time LTI: adders, scalers, and integrators. That's it.

There are canonical architectures that allow one to assemble from those three building blocks a general and finite-order transfer functions

$$ H(s) = \frac{b_0 + b_1 s^{-1} + b_2 s^{-2} ... + b_N s^{-N}}{a_0 + a_1 s^{-1} + a_2 s^{-2} ... + a_N s^{-N}} $$

This is called a rational transfer function

Usually, with no loss of generality, we divide both numerator and denominator with $a_0$, making that coefficient equal to 1 and changing the other coefficients to some other values. So this is still general:

$$ H(s) = \frac{b_0 + b_1 s^{-1} + b_2 s^{-2} ... + b_N s^{-N}}{1 + a_1 s^{-1} + a_2 s^{-2} ... + a_N s^{-N}} $$

There are canonical forms (two most common are Direct Form 1 and Direct Form 2) will allow us to construct an LTI system that implements the above $H(s)$. Using the Direct Form 2 does this with $N$ integrators.

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Now, consider discrete-time (a.k.a "digital") LTI systems (sometimes we call these "filters").
Now we know from LTI system theory that the Laplace transform of the input, $X(z)$, and output, $Y(z)$, of an LTI system are related to each other with this multiplicative transfer function, $H(z)$:

$$ Y(z) = H(z) X(z) $$

It also turns out that, to get the frequency response, we evaluate the transfer function with $z=e^{j\omega}$. So $H(e^{j\omega})$ is the frequency response containing both the magnitude $|H(e^{j\omega})|$ and phase $\phi = \arg \{H(e^{j\omega}) \}$ information.

Now, for implementation, we have three basic classes of building blocks to create an LTI system:

1. signal adders or subtractors
2. signal scalers (multiplication by a constant)
3. unit sample delays (which implement $z^{-1}$)

The third class is what is necessary to make a filter, an LTI system that can discriminate with respect to frequency. Only the delay behaves differently with different frequencies.

Again, with adders, scalers, and delays we can construct a general rational transfer function that looks like:

$$ H(z) = \frac{b_0 + b_1 z^{-1} + b_2 z^{-2} ... + b_N z^{-N}}{1 + a_1 z^{-1} + a_2 z^{-2} ... + a_N z^{-N}} $$

And the same canonical forms can be used.

Now, from above, it can be surmized that, to relate the two domains:

$$ z = e^{sT} $$

where $T=\frac{1}{f_\text{s}}$ is the sampling period and $f_\text{s}$ is the sample rate.

So then, it is also true that

$$ s = \tfrac{1}{T} \log(z) $$.

So if we had a design for a continuous-time system (or "filter") that worked for us satisfactorily, we could implement it exactly (at least for frequencies below Nyquist) if we could substitute $\tfrac{1}{T} \log(z)$ for every $s$. And if we had a building block that could implement $\log(z)$, we could do that.

But we don't have a building block that implements $\log(z)$. We have adders (no "$z$" in that), scalers (no "$z$" in that), and delays $z^{-1}$.

So the question then becomes "How do we approximate $\log(z)$ as a function of $z^{-1}$?"

When I get back to this I will try to answer that more specifically.