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Differensligninger
Describe the output ($y$) as a function of an input ($x$) and previous values of $y$.
$$ y(n) = \sum_{i=0}^{N}a_{i}x(n-i) - \sum_{i=1}^{N}b_{i}y(n-i) $$
Første Orden
$N=1$
$$y(n) = a_{0}x(n) + a_{1}x(n-1) - b_{1}y(n-1)$$
Anden Orden
$N=2$
$$y(n) = a_{0}x(n) + a_{1}x(n-1) + a_{2}x(n-2) - b_{1}y(n-1) - b_{2}y(n-2)$$
Overføringsfuntion
Samme som i Laplace Transformation.
$$H(z) = \frac{Y(z)}{X(z)}$$
Differensligning til overføringsfunction
$$y(n) = 2y(n-1) + 3x(n) \Rightarrow Y(z) = 2Y(z) \cdot z^{-1} + 3X(n)$$
Example: transfer function from difference equation