Laplace Transformation
Se regneregler.
Overføringsfunktion
$$\text{impuls} \rightarrow \text{respons}$$
$$ H(s) = \mathcal{L}{h(t)} = \int_{-\infty}^{\infty} h(t) e^{-st} , dt, \s s = \sigma + j\omega$$
$h(t)$: Impulse response in the time domain. $s$: Input signal
$$H(s) = \frac{\text{output}(s)}{\text{input}(s)}$$
Poles
$$H(s) = \frac{\beta}{\alpha} \Rightarrow \begin{cases} \beta = 0 &\Rightarrow \text{nulpunkter} \ \alpha = 0 &\Rightarrow \text{poles} \end{cases} $$ $\alpha$: den karakteristikle ligning
Pol = Singularitet i frekvensdomæne
If poles are on the imaginary axis the impulse response will result in a constant oscillation at the frequency.
Table
Inverse Laplace Transformation
$$\text{respons} \rightarrow \text{impuls}$$
Try to use tables instead of this equation.
$$ f(t) = \frac{1}{2\pi j} \int_{\sigma_{c}-j\infty} TODO $$