Discrete Fourier Transform: Introduction (2020) (www.chciken.com)

This post is an accessible, linear-algebra–first introduction to the Discrete Fourier Transform (DFT). It frames discrete signals as N-dimensional vectors and shows the DFT as a complex rotation (matrix multiplication) that reveals which sinusoidal frequencies make up a signal — a foundation for many ML/data tasks like audio feature extraction, denoising, and image filtering. The author builds from basic prerequisites (matrices and Euler’s formula) and motivates the transform with practical examples (guitar tuners, noise filtering), promising a hands‑on follow-up. Technically, the DFT is presented as X(n) = sum_{k=0}^{N-1} x(k) e^{-i2πkn/N}, which can be written as an N×N matrix whose rows are complex exponentials. The post proves those rows are mutually orthogonal; their squared length equals N, so a 1/√N normalization produces an orthonormal (unitary) matrix. That unitary view explains the inverse: the conjugate-transpose yields the inverse, and for the common non-normalized convention you recover x(k) = (1/N) sum_{n} X(n) e^{+i2πkn/N}. The write-up emphasizes the DFT’s role as a basis of complex exponentials — building blocks that can reconstruct any discrete signal — and touches on why normalization is omitted in DSP for computational convenience. This geometric perspective clarifies invertibility, energy preservation, and why fast algorithms (FFT) and spectral preprocessing matter in AI/ML workflows.