The Discrete Fourier Transform (DFT) is a fundamental tool in signal processing, but its mathematical complexity, involving summations with complex exponentials like e^(-iwt), can be daunting. This article demonstrates that, at its core, the DFT can be understood and calculated by hand as a series of matrix multiplications.

Understanding DFT as matrix operations demystifies its power, especially when compared to learned transforms like the U-Net. While U-Nets learn filters from data to process signals in the spatial domain, the DFT is a fixed, hand-designed transform that reveals a signal's frequency domain composition. This article visualizes signals A, B, and C in the frequency domain as weighted sums of cosine waves at different frequencies (1w, 2w, 3w, 4w).

By applying the Inverse DFT to convert these signals to their time-domain representations and then using the DFT to convert them back, the process is illustrated. For example, a time-domain signal X, sampled at 10 points, is shown as a weighted sum of four cosine waves. The DFT's task is to uncover these weights. The coefficients of signals A, B, and C are represented in a matrix F, with each row corresponding to a signal and each column to a frequency. Sampling cosine waves at discrete time points from 1t to 10t forms the basis for this calculation.