MP3, JPEG, MRI, and more - Fourier in the real world
The Fourier transform is one of the most widely used mathematical tools in technology. Every time you listen to music, take a photo, or use WiFi, Fourier analysis is working behind the scenes.
Let's explore some of the most impactful applications.
MP3 uses a variant called the Modified Discrete Cosine Transform (MDCT) to convert audio into frequency components. It then uses psychoacoustic models to determine which frequencies can be removed without noticeable quality loss.
JPEG uses the 2D Discrete Cosine Transform (DCT) on 8x8 pixel blocks. High-frequency components (fine details) are quantized more heavily, exploiting the fact that human vision is less sensitive to high-frequency color variations.
The blocky artifacts in low-quality JPEGs are a direct result of the 8x8 block processing. Each block is transformed independently, leading to visible boundaries at high compression.
MRI machines don't directly measure pixels - they measure the Fourier transform of the body! Radio waves excite hydrogen atoms, and the signal received is naturally in the frequency domain.
The raw MRI data is called "k-space" - it's the 2D Fourier transform of the image. The inverse FFT reconstructs the anatomical image we're familiar with.
Modern wireless communication uses Orthogonal Frequency Division Multiplexing (OFDM), which is essentially parallel Fourier-based transmission on many sub-frequencies.
By splitting data across many frequencies, OFDM is robust against interference and multipath (signals bouncing off walls). Each frequency carries a small piece of data, and missing one doesn't crash the whole transmission.
Analyze incoming noise in frequency domain, generate opposite phase signal to cancel it.
Analyze earthquake waves to determine source location and Earth's internal structure.
Convert speech to spectrograms (time-varying frequency) for AI analysis.
The Quantum Fourier Transform is central to Shor's algorithm for factoring.
X-ray diffraction patterns are Fourier transforms of crystal structures.
Analyze reflected signals in frequency domain to detect objects and measure velocity.
The Fourier transform is ubiquitous because it captures something fundamental: many physical systems are governed by wave-like phenomena. Light, sound, radio waves, quantum particles - they all oscillate.
The FFT algorithm (1965) made the transform computationally practical, enabling the digital revolution. Some estimate that more FLOPs are spent on FFTs than any other algorithm in the world.