Fourier Transforms - Seeing Sounds

Have you ever wondered how your music app displays those wavy visual patterns while a song plays? Or how a smartphone can identify your voice and separate it from background noise? The answer lies in a powerful mathematical tool called the Fourier Transform.

What Is a Fourier Transform?

At its core, the Fourier Transform is a way of taking something that changes over time—like a sound wave—and breaking it down into its basic building blocks: sine and cosine waves. These are the purest, simplest types of waves, and amazingly, you can combine them in the right way to recreate any complex wave.

Think of it like a musical chord. A chord is made up of different notes played together. The Fourier Transform helps you figure out what those individual notes (frequencies) are in a complex sound.

Time Domain vs Frequency Domain

When you record sound, you usually see it as a waveform over time. This is called the time domain. But if you apply a Fourier Transform, you shift to the frequency domain, where you can see how much of each frequency is present in the sound.

This is what lets us “see” sounds!

Time Domain

Sound wave = Amplitude vs Time

Frequency Domain

Spectrum = Amplitude vs Frequency

Why Is This Useful?

Fourier Transforms have real-world applications in:

• Audio Compression (MP3, AAC): Identifies parts of the sound we can’t hear and removes them.
• Brainwave Analysis (EEG): Helps analyze patterns in brain signals.
• Voice Recognition: Helps identify unique frequency patterns in a person’s voice.
• Image Processing: JPEG uses a similar concept called the Discrete Cosine Transform.
• Wireless Communication: Helps break down signals into different channels.

A Simple Visual Example

Imagine clapping your hands. The sound isn’t a pure tone—it’s a mixture of many frequencies. If we record that sound and apply a Fourier Transform, we’ll get a spectrum showing which frequencies are involved.

Clap sound (Time domain) → Fourier Transform → Frequency Spectrum

This is what audio visualizers in music players actually show!

Fast Fourier Transform (FFT)

The math behind Fourier Transforms can be intensive, but there’s an efficient version called the Fast Fourier Transform (FFT) that computers use. It’s the reason real-time sound visualization is possible.

What Are Sine and Cosine Waves?

A sine wave is a smooth, periodic oscillation. If you were to watch a point move around a circle at constant speed and plot its vertical position over time, you’d get a sine wave. It has a perfectly repetitive and continuous form, oscillating between a maximum and minimum value.

The cosine wave is just the same wave, shifted in phase by 90 degrees (or (\frac{\pi}{2}) radians). While sine starts at zero, cosine starts at its maximum.

Mathematically:

sin(x) — starts at 0 and rises
cos(x) — starts at 1 and drops

They both repeat every (2\pi) units and are described as periodic functions. These functions are the foundation of trigonometry and appear in countless areas of physics and engineering.

Frequency, Amplitude, and Phase

To fully describe a sine or cosine wave, we need to know:

• Frequency – how fast it oscillates (measured in Hz)
• Amplitude – how tall the wave is (its energy or volume)
• Phase – where the wave starts in its cycle (measured in degrees or radians)

Different combinations of these parameters create the rich variety of tones we hear in music and nature.

Why Are They Important for Sound?

Every pure tone you hear (like a tuning fork or a flute note) is basically a sine wave of a specific frequency. These are the fundamental frequencies. But most real-world sounds—like speech, music, or noise—are combinations of many sine waves of different frequencies and amplitudes.

These waves interact, amplify, cancel, and shape each other, forming the full richness of sound. Understanding how to separate and analyze these building blocks is what makes sound engineering and digital signal processing possible.

Superposition of Waves

One of the most powerful properties of sine and cosine waves is superposition. This means that two or more waves can be added together to form a new, more complex wave.

Example:

If you add:

• \[\( \sin(x) \) — a low-frequency wave\]
• ( 0.5 \cdot \sin(3x) ) — a higher-frequency wave

you get a wave that still looks smooth, but it’s no longer just a simple tone—it’s a richer sound with multiple frequencies.

The ear and brain perceive this combination as a single sound, but math allows us to pull it apart.

Visualizing This with Manim

In the accompanying Manim animation, you’ll see:

1. A complex wave (sum of sine waves)
2. The individual components: one low-frequency sine wave and one higher-frequency sine wave
3. How these add up visually and mathematically to recreate the original wave

This process mirrors what the Fourier Transform does—it dissects a signal into a collection of sine and cosine waves.

Why This Matters

Understanding sine and cosine waves is like learning the letters of a language before writing poetry. Before we can decompose sound with Fourier analysis, we need to understand what it’s made of.

These waveforms are everywhere—in sound, in light, in vibrations, in radio waves, and even in the quantum world. Mastering their behavior is key to unlocking many of the secrets of nature and technology.

In the next post, we’ll explore how to transition from viewing a signal in time to analyzing it in frequency, giving us an entirely new way to understand sound.

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