Matched Z-transform method

Builds on Pole Zero of a holomorphic function Z-transform

Definition

Let $T > 0$ be a sampling period. Let $H$ be a rational function of $s$ with nonzero leading constant $c$ , zeros $q_1, \ldots, q_M$ and poles $p_1, \ldots, p_N$ , listed with multiplicity: $H(s) = c \, \frac{\prod_{i=1}^{M} (s - q_i)}{\prod_{j=1}^{N} (s - p_j)}.$ The matched Z-transform of $H$ is the following rational function of $z$ . $H_d(z) = k \, \frac{\prod_{i=1}^{M} (1 - e^{q_i T} z^{-1})}{\prod_{j=1}^{N} (1 - e^{p_j T} z^{-1})}.$ The constant $k$ is chosen so that $H_d$ and $H$ agree at a chosen reference point. The usual choice is $H_d(1) = H(0)$ when $H(0)$ is finite and nonzero.

Remarks

Example: $H(s) = a/(s + a)$ with $a > 0$ has one pole $-a$ and $H(0) = 1$ . Its matched transform is $H_d(z) = k/(1 - e^{-aT} z^{-1})$ , and $H_d(1) = 1$ gives $k = 1 - e^{-aT}$ . Non-example: $H(s) = e^{-s}$ , a pure delay, is not a rational function, so the method does not apply to it as stated. A pole $p$ with $\operatorname{Re} p < 0$ goes to $e^{pT}$ with $\abs{e^{pT}} < 1$ , so a stable continuous filter gives a stable discrete one. The map $a \mapsto e^{aT}$ is the relation $z = e^{sT}$ between the Z-transform of samples and the Laplace transform. Some versions add zeros at $z = -1$ for the zeros of $H$ at infinity when $N > M$ .

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Tag	`FZ6KV2J5C9`
URI	`m2://FZ6KV2J5C9`
Kind	definition
Writing	W0 stub
MSC	93C62, 44A55, 94A12
Defines	Matched Z-transform method
Uses	Pole, Zero of a holomorphic function, Z-transform

Matched Z-transform method

Definition

Remarks

Reading path

Further reading

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