Fourier Transform Calculator

Understanding Fourier Transforms

What is a Fourier Transform?

The Fourier transform decomposes a signal into its constituent frequencies, revealing its frequency spectrum.

X(f) = ∫x(t)e^(-j2πft)dt

where:

  • X(f) is the frequency domain representation
  • x(t) is the time domain signal
  • f is frequency
  • t is time
  • j is the imaginary unit

Properties of Fourier Transforms

Linearity

  • F{ax(t) + by(t)} = aX(f) + bY(f)
  • Superposition principle
  • Scaling property

Time Shifting

  • F{x(t-t₀)} = X(f)e^(-j2πft₀)
  • Phase shift in frequency
  • Time delay property

Frequency Shifting

  • F{x(t)e^(j2πf₀t)} = X(f-f₀)
  • Modulation theorem
  • Spectrum shifting

Applications

Signal Processing

  • Filtering
  • Compression
  • Feature extraction
  • Pattern recognition

Communications

  • Modulation
  • Channel analysis
  • Bandwidth allocation
  • Error detection

Image Processing

  • Image enhancement
  • Edge detection
  • Frequency analysis
  • Compression

Advanced Topics

Discrete Fourier Transform

  • Sampling theory
  • Aliasing effects
  • FFT algorithms

Window Functions

  • Spectral leakage
  • Resolution vs. accuracy
  • Window types

Time-Frequency Analysis

  • Short-time Fourier transform
  • Wavelet transforms
  • Gabor analysis