Hacker News 上发布的《手工计算AI》系列第19篇文章以实际示例演示了离散傅里叶变换（DFT）的计算原理[1]。DFT 的本质是一系列矩阵乘法运算，虽然涉及复数项 e^(-iwt) 的求和计算，但通过矩阵形式可以更直观地理解其机制[1]。
该文章通过具体案例展示了如何手工计算信号的频域系数[1]。以信号X为例，其在10个时间点的采样值分别为-2.5、-1.8、3、-0.7、-1.0、-0.7、3、-1.8、-2.5、5[1]。文章演示了将时域信号转换到频域的完整过程，用以分析其频率成分[1]。
DFT 作为一种经典的固定变换方法，与神经网络通过学习方式处理信号的方法形成对比[1]。神经网络通过训练来获得滤波器参数，而 DFT 则采用预先设定的数学变换来完成信号分析[1]。
A computational tutorial demonstrates how the discrete Fourier transform (DFT) operates fundamentally as a series of matrix multiplications [1]. The article, part of the "Computing AI by Hand" series, illustrates the mechanics of DFT by showing its manual calculation process, where the transform converts signals from the time domain to the frequency domain to analyze their frequency components [1].
The tutorial contrasts DFT's fixed mathematical structure with the adaptive approach used by neural networks such as U-Net, which learn their filtering parameters rather than applying predetermined transformations [1]. Through concrete examples, the piece walks through calculating frequency-domain coefficients for signals and determining time-domain samples by hand [1]. A sample signal X is presented with ten time-point values: [-2.5, -1.8, 3, -0.7, -1.0, -0.7, 3, -1.8, -2.5, 5] [1]. The computation involves summing terms with complex exponentials of the form e^(-iwt), yet the underlying operation remains matrix multiplication, making DFT an accessible classical technique for signal processing when approached systematically [1].