Part 14 - Filters

Introduction

In this part we’ll understand the concept of filters.

Filters

We usually use filters for two different purposes:

• Frequency elimination
• Frequency detection

Example

Say we have the following system: $$ G(s) = \dfrac{a}{s + a} $$

What type of filter is this? Let’s find out. Plotting the bode plot will help us.

Note: This is for $a = 1$

We can see that this is a low-pass filter! We will call the part that lets through the frequency for passband. The range that gets eliminated/attenuated is the stopband.

For a high-pass filter, the general form is: $$ G(s) = \dfrac{s}{s + b} = \dfrac{\frac{s}{b}}{1 + \frac{s}{b}} $$

Note: This is for $b = 1$

Filter types

So far we’ve seen low-pass and high-pass filters. Let’s list all the filters that we’ll work with:

• Low-pass (LP - low pass, high gets filtered out)
• High-pass (HP - high pass, low gets filtered out)
• Band-stop filter (BS - frequencies within a specific range gets attenuated, rest are unfiltered)
• Band-pass filter (BP - frequencies within a specific range pass, rest are attenuated)
• Notch filter (Filters exactly one frequency)

Butterworth filters

When we need to perform filter realization, we usually will restort to Butterworth:

$$ \text{Order 1: } G(s) = \dfrac{\omega_1}{s + \omega_1} \\ \text{Order 2: } G(s) = \dfrac{\omega_1^2}{s^2 + 2 \zeta \omega_1 s + \omega_1^2} \ | \ \zeta = \frac{1}{\sqrt{2}} $$

Filter transformation

Say we have this filter: $$ G(s) = \dfrac{1}{s + 1} $$

This is a low-pass filter. Say we want another type of filter from this - what should we do?

1. Switch of break-frequency point to $\omega_c$ (cut-off)

Say we want to go from LP $\to$ HP, then we need to set: $$ s \to \frac{s}{\omega_c} $$

$$ G(s) = \dfrac{1}{\frac{s}{\omega_c} + 1} = \dfrac{\omega_c}{s + \omega_c} $$

In the case we had a second order filter: $$ G(s) = \dfrac{1}{\frac{s^2}{\omega_c^2} + \sqrt{2} \frac{s}{\omega_c} + 1} = \dfrac{\omega_c^2}{s^2 + \sqrt{2}\omega_c s + \omega_c^2} $$