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Z-transform

Sequence	Transform
$\delta(k - n)$	$z^{-n}$
$1$	$\frac{z}{z - 1}$
$k$	$\frac{z}{(z - 1)^2}$
$k^2$	$\frac{z(z + 1)}{(z - 1)^3}$
$a^k$	$\frac{z}{z - a}$
$ka^k$	$\frac{az}{(z - a)^2}$
$\sin ak$	$\frac{z \sin a}{z^2 - 2z \cos a + 1}$
$\cos ak$	$\frac{z(z - \cos a)}{z^2 - 2z \cos a + 1}$
$a^k \sin bk$	$\frac{az \sin b}{z^2 - 2az \cos b + a^2}$
$a^k \cos bk$	$\frac{z^2 - az \cos b}{z^2 - 2az \cos b + a^2}$

properties

Linearity: if $x(n) = af_{1}(n) + bf_{2}(n)$ then $X(z) = aF_{1}(z) + bF_{2}(z)$

Time shifting:

\begin{aligned} Z[x(t)] &= X(z) \\ x(k-n) &= z^{-n}X(z) \\ x(k+n) &= z^{n}X(z) \end{aligned}

quantization error

sampling

The idea to convert analog to digital

$T$ is the sampling period, and $\frac{1}{T}$ is the sampling rate in cycles per second

\text{error} = \frac{M}{2^{n+1}}

where $n$ is number of bits used for digitalisation

resolution of A/D converter

minimum value of the output that can be represented as binary number, or $\frac{M}{2^n}$

sampled data system

reference input $r$ is the sequence of sample values $r(kT)$

A sampler is a switch that closes every $T$ seconds:

r^{*}(t) = \sum_{k=0}^{\infty} r(kT) \delta (t-kT) \quad (t>0)

Transfer function of sampled data:

R^{*}(s) = \mathcal{L} (r^{*}(t)) = \sum_{k=0}^{\infty} r(kT) e^{-ksT}

definition

Let $z = e^{sT}$ , we have the following definition:

z-transform

$Z \{r(t)\} = F(z) = Z(r^{*}(t)) = \sum_{k=0}^{\infty} r(kT)z^{-k}$

zero-order hold

Transfer function of Zero-Order hold

\mathcal{L}(u(t) - u(t-T)) = \frac{1}{s} - \frac{e^{sT}}{s}

finding the discrete transfer function

$G(s) = \frac{s^2 + 4s + 3}{s^3 + 6s^2 + 8s}$

\begin{aligned} G(s) &= \frac{s^2 + 4s + 3}{s^3 + 6s^2 + 8s} = \frac{0.375}{s} + \frac{0.25}{s+2} + \frac{0.375}{s+4} \\ G(t) &= \mathcal{L}^{-1}(G(s)) = 0.375 + 0.25 e^{-2t} + 0.375 e^{-4t} \\ G(z) &= Z(G(t)) = 0.375 \frac{z}{z-1} + 0.25 \frac{z}{z-e^{-2T}} + 0.375 \frac{z}{z-e^{-4T}} \end{aligned}

inverse z-transform

$G(z) \to x(k)$

power series

use: when G(z) is expressed as the ratio of two polynomials in z

G(z) = a_{0} + a_{1} z^{-1} + a_{2} z^{-2} + \ldots

partial fraction

For example: $G(z) = \frac{z}{(z-1)(z-2)} = \frac{-z}{z-1} + \frac{z}{z-2} = \sum_{k=0}^{\infty} (-1 + 2^k)z^{-k}$

thus, $g(kT) = 2^k-1$

stability

system	pole location criteria on z-plane
Stable	All poles inside unit circle
Unstable	Any poles outside unit circle
Marginally Stable	One or more poles on unit circle, remaining poles inside unit circle

poles on s-plane

mapping from s-plane to z-plane

$z = e^{\alpha T}(\cos \omega T + j \sin \omega T)$

we assume $s = \alpha + j \omega$

Location on s-plane	Value of $\alpha$	Value of $e^{\alpha T}$	Mapping on z-plane
Imaginary axis ( $j\omega$ )	$\alpha = 0$	$e^{\alpha T} = 1$	On unit circle
Right half-plane	$\alpha > 0$	$e^{\alpha T} > 1$	Outside unit circle
Left half-plane	$\alpha < 0$	$e^{\alpha T} < 1$	Inside unit circle

final value theorem

definition

If $\lim_{k \to \infty}x(k)$ exists, then the follow exists:
$\lim_{k \to \infty} x(k) = \lim_{z \to 1} (z-1) X(z)$

root locus on z-plane

derive open loop function $K \bar{GH}$
Factor numerator and denominator to get open loop zeros and poles
Plot roots of $1+K \bar{GH}=0$ in z-plane as k varies $\bar{GH(z)} = \frac{N(z)}{D(z)}$

reference: examples for z-transform

Sequence	Transform
$\delta(k - n)$	$z^{-n}$
$1$	$\frac{z}{z - 1}$
$k$	$\frac{z}{(z - 1)^2}$
$k^2$	$\frac{z(z + 1)}{(z - 1)^3}$
$a^k$	$\frac{z}{z - a}$
$ka^k$	$\frac{az}{(z - a)^2}$
$\sin ak$	$\frac{z \sin a}{z^2 - 2z \cos a + 1}$
$\cos ak$	$\frac{z(z - \cos a)}{z^2 - 2z \cos a + 1}$
$a^k \sin bk$	$\frac{az \sin b}{z^2 - 2az \cos b + a^2}$
$a^k \cos bk$	$\frac{z^2 - az \cos b}{z^2 - 2az \cos b + a^2}$

properties

Linearity: if $x(n) = af_{1}(n) + bf_{2}(n)$ then $X(z) = aF_{1}(z) + bF_{2}(z)$

Time shifting:

\begin{aligned} Z[x(t)] &= X(z) \\ x(k-n) &= z^{-n}X(z) \\ x(k+n) &= z^{n}X(z) \end{aligned}

quantization error

sampling

The idea to convert analog to digital

$T$ is the sampling period, and $\frac{1}{T}$ is the sampling rate in cycles per second

\text{error} = \frac{M}{2^{n+1}}

where $n$ is number of bits used for digitalisation

resolution of A/D converter

minimum value of the output that can be represented as binary number, or $\frac{M}{2^n}$

sampled data system

reference input $r$ is the sequence of sample values $r(kT)$

A sampler is a switch that closes every $T$ seconds:

r^{*}(t) = \sum_{k=0}^{\infty} r(kT) \delta (t-kT) \quad (t>0)

Transfer function of sampled data:

R^{*}(s) = \mathcal{L} (r^{*}(t)) = \sum_{k=0}^{\infty} r(kT) e^{-ksT}

definition

Let $z = e^{sT}$ , we have the following definition:

z-transform

$Z \{r(t)\} = F(z) = Z(r^{*}(t)) = \sum_{k=0}^{\infty} r(kT)z^{-k}$

zero-order hold

Transfer function of Zero-Order hold

\mathcal{L}(u(t) - u(t-T)) = \frac{1}{s} - \frac{e^{sT}}{s}

finding the discrete transfer function

$G(s) = \frac{s^2 + 4s + 3}{s^3 + 6s^2 + 8s}$

\begin{aligned} G(s) &= \frac{s^2 + 4s + 3}{s^3 + 6s^2 + 8s} = \frac{0.375}{s} + \frac{0.25}{s+2} + \frac{0.375}{s+4} \\ G(t) &= \mathcal{L}^{-1}(G(s)) = 0.375 + 0.25 e^{-2t} + 0.375 e^{-4t} \\ G(z) &= Z(G(t)) = 0.375 \frac{z}{z-1} + 0.25 \frac{z}{z-e^{-2T}} + 0.375 \frac{z}{z-e^{-4T}} \end{aligned}

inverse z-transform

$G(z) \to x(k)$

power series

use: when G(z) is expressed as the ratio of two polynomials in z

G(z) = a_{0} + a_{1} z^{-1} + a_{2} z^{-2} + \ldots

partial fraction

For example: $G(z) = \frac{z}{(z-1)(z-2)} = \frac{-z}{z-1} + \frac{z}{z-2} = \sum_{k=0}^{\infty} (-1 + 2^k)z^{-k}$

thus, $g(kT) = 2^k-1$

stability

system	pole location criteria on z-plane
Stable	All poles inside unit circle
Unstable	Any poles outside unit circle
Marginally Stable	One or more poles on unit circle, remaining poles inside unit circle

mapping from s-plane to z-plane

$z = e^{\alpha T}(\cos \omega T + j \sin \omega T)$

we assume $s = \alpha + j \omega$

Location on s-plane	Value of $\alpha$	Value of $e^{\alpha T}$	Mapping on z-plane
Imaginary axis ( $j\omega$ )	$\alpha = 0$	$e^{\alpha T} = 1$	On unit circle
Right half-plane	$\alpha > 0$	$e^{\alpha T} > 1$	Outside unit circle
Left half-plane	$\alpha < 0$	$e^{\alpha T} < 1$	Inside unit circle

final value theorem

definition

If $\lim_{k \to \infty}x(k)$ exists, then the follow exists:
$\lim_{k \to \infty} x(k) = \lim_{z \to 1} (z-1) X(z)$

root locus on z-plane

derive open loop function $K \bar{GH}$
Factor numerator and denominator to get open loop zeros and poles
Plot roots of $1+K \bar{GH}=0$ in z-plane as k varies $\bar{GH(z)} = \frac{N(z)}{D(z)}$

Z-transform

Étiquette

publié à

modifié à

durée

source

properties

quantization error

sampled data system

definition

zero-order hold

finding the discrete transfer function

inverse z-transform

power series

partial fraction

stability

we assume $s = \alpha + j \omega$

final value theorem

root locus on z-plane

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Liens retour

Z-transform

Étiquette

publié à

modifié à

durée

source

properties

quantization error

sampled data system

definition

zero-order hold

finding the discrete transfer function

inverse z-transform

power series

partial fraction

stability

we assume s=α+jωs = \alpha + j \omegas=α+jω

final value theorem

root locus on z-plane

Vous pourriez aimer ce qui suit

Liens retour

we assume $s = \alpha + j \omega$