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## Introduction

Hey it's a me again @drifter1!

Today we continue with my mathematics series about **Signals and Systems** to get into **Butterworth Filters**.

So, without further ado, let's dive straight into it!

## Introduction to Butterworth Filters

Butterworth filters apply topics that we discussed in the previous article, which was about discrete-time system design. This useful class of filters is specified by two main parameters:

- Cut-off Frequency (
*ω*)_{c} - Filter Order (
*N*)

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Applying techniques such as impulse invariance, it's possible to create discrete-time Butterworth filters. The filter specifications can be mapped to a corresponding set of specifications for the discrete-time filter, but there are of course some limitations. In order to avoid aliasing, the continuous-time filter needs to be band-limited. This can be done using a procedure known as bilinear transformation, where the entire imaginary axis in the s-plane is mapped around the unit circle. Since the imaginary axis is of infinite length, a non-linear distortion mapping between the two frequency axes occurs. Thus, even this technique can only be used when the filter is (at least) approximately a constant piecewise function (defined by cases).

## Butterworth Filter Equation and Pole-Zero Plot

Mathematically, Butterworth filters are defined as:

In the s-plane, the square of the magnitude of the frequency response gives us:

The poles of *B(s) B(-s)* are at:

which gives us pole-zero patterns of the following form:

## Mapping using Impulse Invariance

Let's first talk about mapping the filter using Impulse Invariance. The sampled signal is taken as follows:

As a Fourier series, the frequency response is defined as:

For the Z-Transform, we need to determine *B(s)* and using partial fraction expansion we come up with:

## Mapping using Bilinear Transformation

Using Bilinear Transformation, the following mapping occurs:

where *s* and *z* are given by:

For *z = e ^{jΩ}*,

*s*is given by:

Mapping from the continuous-time frequency axis (ω) to the discrete time frequency axis (Ω) is exactly what the Bilinear Transform is all about. So, in the end, it's basically the following mapping:

## RESOURCES:

### References

### Images

Mathematical equations used in this article were made using quicklatex.

Block diagrams and other visualizations were made using draw.io

## Previous articles of the series

### Basics

- Introduction → Signals, Systems
- Signal Basics → Signal Categorization, Basic Signal Types
- Signal Operations with Examples → Amplitude and Time Operations, Examples
- System Classification with Examples → System Classifications and Properties, Examples
- Sinusoidal and Complex Exponential Signals → Sinusoidal and Exponential Signals in Continuous and Discrete Time

### LTI Systems and Convolution

- LTI System Response and Convolution → Linear System Interconnection (Cascade, Parallel, Feedback), Delayed Impulses, Convolution Sum and Integral
- LTI Convolution Properties → Commutative, Associative and Distributive Properties of LTI Convolution
- System Representation in Discrete-Time using Difference Equations → Linear Constant-Coefficient Difference Equations, Block Diagram Representation (Direct Form I and II)
- System Representation in Continuous-Time using Differential Equations → Linear Constant-Coefficient Differential Equations, Block Diagram Representation (Direct Form I and II)
- Exercises on LTI System Properties → Superposition, Impulse Response and System Classification Examples
- Exercise on Convolution → Discrete-Time Convolution Example with the help of visualizations
- Exercises on System Representation using Difference Equations → Simple Block Diagram to LCCDE Example, Direct Form I, II and LCCDE Example
- Exercises on System Representation using Differential Equations → Equation to Block Diagram Example, Direct Form I to Equation Example

### Fourier Series and Transform

- Continuous-Time Periodic Signals & Fourier Series → Input Decomposition, Fourier Series, Analysis and Synthesis
- Continuous-Time Aperiodic Signals & Fourier Transform → Aperiodic Signals, Envelope Representation, Fourier and Inverse Fourier Transforms, Fourier Transform for Periodic Signals
- Continuous-Time Fourier Transform Properties → Linearity, Time-Shifting (Translation), Conjugate Symmetry, Time and Frequency Scaling, Duality, Differentiation and Integration, Parseval's Relation, Convolution and Multiplication Properties
- Discrete-Time Fourier Series & Transform → Getting into Discrete-Time, Fourier Series and Transform, Synthesis and Analysis Equations
- Discrete-Time Fourier Transform Properties → Differences with Continuous-Time, Periodicity, Linearity, Time and Frequency Shifting, Conjugate Summetry, Differencing and Accumulation, Time Reversal and Expansion, Differentation in Frequency, Convolution and Multiplication, Dualities
- Exercises on Continuous-Time Fourier Series → Fourier Series Coefficients Calculation from Signal Equation, Signal Graph
- Exercises on Continuous-Time Fourier Transform → Fourier Transform from Signal Graph and Equation, Output of LTI System
- Exercises on Discrete-Time Fourier Series and Transform → Fourier Series Coefficient, Fourier Transform Calculation and LTI System Output

### Filtering, Sampling, Modulation, Interpolation

- Filtering → Convolution Property, Ideal Filters, Series R-C Circuit and Moving Average Filter Approximations
- Continuous-Time Modulation → Getting into Modulation, AM and FM, Demodulation
- Discrete-Time Modulation → Applications, Carriers, Modulation/Demodulation, Time-Division Multiplexing
- Sampling → Sampling Theorem, Sampling, Reconstruction and Aliasing
- Interpolation → Reconstruction Procedure, Interpolation (Band-limited, Zero-order hold, First-order hold)
- Processing Continuous-Time Signals as Discrete-Time Signals → C/D and D/C Conversion, Discrete-Time Processing
- Discrete-Time Sampling → Discrete-Time (or Frequency Domain) Sampling, Downsampling / Decimation, Upsampling
- Exercises on Filtering → Filter Properties, Type and Output
- Exercises on Modulation → CT and DT Modulation Examples
- Exercises on Sampling and Interpolation → Graphical/Visual Sampling and Interpolation Examples

### Laplace and Z Transforms

- Laplace Transform → Laplace Transform, Region of Convergence (ROC)
- Laplace Transform Properties → Linearity, Time- and Frequency-Shifting, Time-Scaling, Complex Conjugation, Multiplication and Convolution, Differentation in Time- and Frequency-Domain, Integration in Time-Domain, Initial and Final Value Theorems
- LTI System Analysis using Laplace Transform → System Properties (Causality, Stability) and ROC, LCCDE Representation and Laplace Transform, First-Order and Second-Order System Analysis
- Exercises on the Laplace Transform → Laplace Transform and ROC Examples, LTI System Analysis Example
- Z Transform → Z Transform, Region of Convergence (ROC), Inverse Z Transform
- Z Transform Properties → Linearity, Time-Shifting, Time-Scaling, Time-Reversal, z-Domain Scaling, Conjugation, Convolution, Differentation in the z-Domain, Initial and Final value Theorems
- LTI System Analysis using Z Transform → System Properties (Causality, Stability), LCCDE Representation and Z Transform
- Exercises on the Z Transform → Z Transform and ROC Examples, ROC from Conditions, LTI System Analysis Example
- Continuous-Time to Discrete-Time Design Mapping → Discrete-Time System Design Techniques (Mapping from Derivatives to Differences, Mapping using Impulse Invariance), First- and Second-Order Systems and Z-Transform

## Final words | Next up

And this is actually it for today's post!

Next up is System Feedback...

See Ya!

Keep on drifting!