Filters

Lektion 1 - Filterfunktioner.pdf

Filter Types

Ideal filters

We want three things from a filter:

1. Constant amplification on the pass band
2. Lots of dampening after cutoff frequency
3. Linear phase* (see >Delay through filter (gruppeløstid))

Actual filter types

These all only have poles no zero points.

• Butterworth Filter
• Chebyshev Filter
• Bessel (More linear phase)

Plots of the three filters

Poles of the filters. Note: Butterworth lies on a circle around $(0, 0)$, with a radius of $j\omega_{a}$.

Frekvensnormering

Normering: $s \rightarrow s \cdot \omega_{a}$. Denormering: $s \rightarrow \frac{s}{\omega_{a}}$.

Used for determining the order of the filter needed (form factor). $$F = \frac{\omega_{s}}{\omega_{a}}$$ $\omega_{s}$: Stop band frequency $\omega_{a}$: Cutoff frequency The result it the $x$-axis on a plot like this: You select the lowest order, that adheres to the specification, as higher order filter, are more complicated.

Filter Order ($N$)

Poles = $-20 \frac{\text{db}}{\text{dec}} \cdot N$ Zeros = $+20 \frac{\text{db}}{\text{dec}} \cdot N$

First order filter ($k$ and $a$ are constants): $$H_{1} = \frac{k}{s+a}$$ A **second order** filter could be written like this: $$H_{2} = \frac{k}{(s+a)(s+b)}$$ You can also multiply filters together to create higher order filters: $$H_{1} \cdot H_{1} = \frac{k^{2}}{(s+a)^2}$$ In practice this is the same at *putting the filters after each other*.

Delay through filter (gruppeløstid)

The delay of different frequencies is different. We can try to mitigate this in our filter.

The delay can be found as the derivative of phase as a function of frequency $\phi(\omega)$. $$T_{g} = -\frac{d\phi(\omega)}{d\omega} \s [\text{s}]$$ Because of this we want *linear phase*:

If not, the Step Response will ocilate: