See how any periodic function can be broken down into sine and cosine waves. Draw your own shape and watch it transform into its frequency components.
The Fourier Transform is a mathematical technique that decomposes a function into its constituent frequencies. Named after the French mathematician Jean-Baptiste Joseph Fourier, this transform is fundamental to many fields including signal processing, quantum physics, and image compression.
For periodic functions, the Fourier Series represents the function as a sum of sine and cosine terms:
Where the coefficients aₙ and bₙ are calculated using:
The Fourier Transform converts a signal from the time domain (how the signal changes over time) to the frequency domain (what frequencies make up the signal). This allows us to:
Different functions have characteristic Fourier transforms:
Fourier transforms have numerous practical applications:
For digital applications, we use the Discrete Fourier Transform, which operates on discrete data points rather than continuous functions. The Fast Fourier Transform (FFT) is an efficient algorithm for computing the DFT, reducing computational complexity from O(n²) to O(n log n).
In this simulation, you can explore these concepts by drawing your own function or selecting a preset function, and observing how it decomposes into its frequency components through the Fourier transform.