Part 14 - Filters | rezarezvan.com

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 1

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.

Bode plot of the low-pass filter G(s) = \frac{a}{s + a} for a = 1. — Figure 1: Bode plot of the low-pass filter $G(s) = \frac{a}{s + a}$ for $a = 1$ .

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}}.

Bode plot of the high-pass filter G(s) = \frac{s}{s + b} for b = 1. — Figure 2: Bode plot of the high-pass filter $G(s) = \frac{s}{s + b}$ for $b = 1$ .

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 resort to Butterworth:

\begin{align*} \text{Order 1: } & G(s) = \dfrac{\omega_1}{s + \omega_1}, \newline \text{Order 2: } & G(s) = \dfrac{\omega_1^2}{s^2 + 2 \zeta \omega_1 s + \omega_1^2}, \quad \zeta = \frac{1}{\sqrt{2}}. \end{align*}

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?

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}.