📄️ Prerequisite Theory
This section assumes familiarity with the below:
📄️ Keywords & Conventions
Keywords
📄️ Introduction
Fourier techniques let us express any signal as a sum of its constituent "pure" sinusoids (frequencies), meaning:
📄️ Fourier Series - the bridge (periodic signals)
The Fourier Series decomposes a periodic signal in the time domain into a sum of harmonics -
📄️ Continuous Fourier Transform
If you take the Fourier series and let the period $T \to \infty$, the discrete harmonic lines become continuous — resulting in the Continuous Fourier Transform (CFT).
📄️ Discrete-time signals and the DFT
When we sample a continuous-time signal at a sampling rate $f_s$ (in Hz), or when we work with inherently discrete data (such as digital audio or images),
📄️ How Fourier Transforms Work
To understand how Fourier transforms work “under the hood”, we must start by
📄️ Sign of the Exponent in the Fourier Kernel and Its Relationship to Rotational Direction
In the Fourier transform, the kernel contains a complex exponential of the form:
📄️ Relationship Between the Fourier Series and the Continuous and Discrete Fourier Transforms
- Fourier series: discrete frequencies for periodic continuous signals
📄️ Why Img2Num Uses the DFT
1. Inputs are digital — images and arrays are sampled and finite; the DFT exactly models the transform we can compute on them.
📄️ Fourier Cheat Sheet
Core identities