The Fourier Transform is a mathematical tool used to break down a function—typically a time-based signal—into its constituent frequencies. It transforms a signal from the time domain (where it’s expressed as amplitude over time) into the frequency domain (where it’s expressed as amplitude across different frequencies). This allows you to see the frequency components that make up the original signal, revealing patterns or behaviors that might not be obvious in the time domain.
Formally, for a continuous function ( f(t) ), the Fourier Transform is defined as:
In simpler terms, it’s like taking a musical chord and figuring out which individual notes (frequencies) are being played and how loudly. For example, a noisy audio signal could be analyzed to isolate a hum at 60 Hz or a high-pitched whine at 3000 Hz.
There’s also the Discrete Fourier Transform (DFT) for digital signals (like audio samples or stock price data), which is computed over a finite set of points. The Fast Fourier Transform (FFT) is an efficient algorithm to calculate the DFT, widely used in real-world applications.
Applications:
- Signal Processing: Filters out noise in audio or images.
- Engineering: Analyzes vibrations in mechanical systems.
- Finance: Detects cycles or periodicities in market data.
- Physics: Solves differential equations or studies wave behavior.