<commit_before>---
layout: post
mathjax: true
---

Filtering data and signals is a great application of mathematics in signal processing. But with all the Fourier transforms, Convolutions, and various properties like causality, time-invariance, linearity... It can get confusing. In this post, I want to clarify the relationship between a filter, it's impulse response and transfer function, and why applying a filter is the same as convoluting a transformed signal with a transfer function. Don't worry, it will sound less intimidating in the end.

## Prerequisites and Notation

I will use $\widehat{c}$ to refer to the Fourier Transform of $c$ (and $(c + d)\hat{ }$ for the transform of larger terms). Conversely, I will use the inverted hat for the inverse of the Fourier Transform: $f(t) = (\widehat{f}(\cdot))\check{ }(t)$.

To denote the Convolution of two signals $f$ and $c$, I will use $f \ast c$.
<commit_msg>fix: transfer function -> impulse response<commit_after>---
layout: post
mathjax: true
---

Filtering data and signals is a great application of mathematics in signal processing. But with all the Fourier transforms, Convolutions, and various properties like causality, time-invariance, linearity... It can get confusing. In this post, I want to clarify the relationship between a filter, it's impulse response and transfer function, and why applying a filter is the same as convoluting a transformed signal with an impulse response. Don't worry, it will sound less intimidating in the end.

## Prerequisites and Notation

I will use $\widehat{c}$ to refer to the Fourier Transform of $c$ (and $(c + d)\hat{ }$ for the transform of larger terms). Conversely, I will use the inverted hat for the inverse of the Fourier Transform: $f(t) = (\widehat{f}(\cdot))\check{ }(t)$.

To denote the Convolution of two signals $f$ and $c$, I will use $f \ast c$.

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