Implementing the Fourier Transform Numerically in Python: A Step-by-Step Guide

This guide explains how to numerically implement the continuous Fourier transform in Python, clarifying that standard libraries like NumPy and SciPy compute the Discrete Fourier Transform (DFT) instead. It presents a step-by-step implementation using a numerical quadrature method, specifically the left Riemann sum, to approximate the transform's integral. The mathematical derivation is provided, along with Python code functions to perform the calculation and compare it against SciPy's built-in integration tools. The discussion is motivated by a practical problem in stochastic process calibration, demonstrating the need for accurate Fourier transform computation in quantitative analysis and data science.