How the Fourier Transform Converts Sound Into Frequencies

2026-03-24 · Source: AI Advances - Medium · Field: Technology & Digital — Artificial Intelligence & Machine Learning, Data Science & Analytics · Depth: Intermediate, extended

Summary

This article provides an intuition-first guide to the Fourier Transform (FT), explaining how it converts time-domain sound signals into frequency-domain representations. It begins by reviewing digital sound storage concepts like sampling (e.g., 44.1 kHz for CD-quality, 16 kHz for speech) and quantization (e.g., 16-bit audio offers 65,536 levels). The core mechanism of the FT is described as a "winding machine" where the input signal g(t) is wrapped around a circle in the complex plane at a speed determined by a test frequency f. The output for each frequency f is a complex number, from which its magnitude (amplitude of contribution) and phase (starting offset) are extracted. A key concept is the Centre of Mass (COM) of this wound-up curve: a COM far from the origin indicates a strong presence of frequency f, while a COM near the origin means f is barely present. The article uses a concrete example of g(t) = sin(2π·300·t) + sin(2π·700·t) to illustrate how the FT correctly identifies constituent frequencies (300 Hz, 700 Hz) and dismisses non-constituent ones (500 Hz). It also clarifies how Euler's formula allows the FT to automatically capture phase information without explicit searching.

Key takeaway

For Machine Learning Engineers working with audio data, understanding the Fourier Transform's core "winding machine" and Centre of Mass concepts clarifies how `np.fft.rfft()` transforms raw audio into frequency features. This intuition is crucial for interpreting spectrograms and making informed decisions about feature extraction, especially when considering the importance of phase information in tasks like speech synthesis versus magnitude-only approaches.

Key insights

The Fourier Transform decomposes complex signals into constituent frequencies by winding the signal and analyzing the Centre of Mass.

Principles

• Matching frequencies create lopsided windings, yielding high magnitudes.
• Non-matching frequencies create balanced windings, yielding low magnitudes.
• Euler's formula enables simultaneous correlation with all phases.

Method

The FT winds a time-domain signal g(t) around the complex plane at a test frequency f, then calculates the Centre of Mass of the resulting curve to determine the frequency's amplitude and phase contribution.

In practice

• Use `np.fft.rfft()` to compute the Discrete Fourier Transform.
• Extract magnitude with `np.abs()` for frequency contribution.
• Extract phase with `np.angle()` for frequency starting offset.

Topics

• Fourier Transform
• Frequency Analysis
• Digital Signal Processing
• Audio Preprocessing
• Complex Numbers

Best for: Machine Learning Engineer, Data Scientist, AI Student

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Editorial summary, takeaway, and curation by AIssential. Original article published by AI Advances - Medium.