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Fourier Transform Explorer

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.

Time Domain
Frequency Domain

Fourier Transform Controls

Draw Function
Preset Functions

Drawing Tools

Transform Settings

Fourier Coefficients

a₀ 0.000
a₁ 0.000
b₁ 0.000

Understanding the Fourier Transform

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.

The Fourier Series

For periodic functions, the Fourier Series represents the function as a sum of sine and cosine terms:

f(x) = a₀/2 + Σ[aₙcos(nx) + bₙsin(nx)]

Where the coefficients aₙ and bₙ are calculated using:

aₙ = (1/π) ∫f(x)cos(nx)dx
bₙ = (1/π) ∫f(x)sin(nx)dx

From Time Domain to Frequency Domain

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:

  • Identify the fundamental frequencies in a complex signal
  • Filter out unwanted frequencies (noise reduction)
  • Compress data by keeping only the most significant frequency components
  • Analyze the spectral characteristics of signals

Common Fourier Transform Pairs

Different functions have characteristic Fourier transforms:

  • Square Wave: Contains only odd harmonics with amplitudes that decrease as 1/n
  • Sawtooth Wave: Contains all harmonics with amplitudes that decrease as 1/n
  • Triangle Wave: Contains only odd harmonics with amplitudes that decrease as 1/n²
  • Impulse (Delta) Function: Contains all frequencies with equal amplitude

Applications of Fourier Transforms

Fourier transforms have numerous practical applications:

  • Audio Processing: Equalizers, noise reduction, and audio compression
  • Image Processing: JPEG compression, filtering, and feature extraction
  • Medical Imaging: MRI and CT scan reconstruction
  • Quantum Mechanics: Analyzing wave functions and energy states
  • Communications: Modulation, demodulation, and spectral analysis

The Discrete Fourier Transform (DFT)

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.