Inverse Fourier Transform

Fourier Analysis · Axiom Academy

LESSON Inverse Fourier Transform Reconstructing the time-domain signal from its frequency components 1. The Inverse Fourier Transform Formula Given a frequency-domain representation F(ω), we can recover the original time-domain signal f(t) using the inverse transform: This integral sums up all frequency components e^(iωt), each weighted by F(ω), to reconstruct the original signal. The factor 1/(2π) normalizes the reconstruction. 2. Symmetry Between Forward and Inverse The forward and inverse transforms are remarkably symmetric. Compare the two formulas: Notice the only differences: the sign of the exponent (−iωt vs +iωt) and the normalization factor (1/2π). This symmetry reveals the deep duality between time and frequency domains. 3. Understanding the 1/(2π) Factor The normalization factor 1/(2π) ensures that transforming forward and then backward returns the original signal. Different conventions exist: Standard Convention (used here): Forward: no factor, Inverse: 1/(2π) Uses ω = 2πf, eliminating the factor entirely The key is consistency: whichever convention you choose, the product of forward and inverse factors must equal 1/(2π) for proper reconstruction. 4. The Transform Pair Relationship Functions f(t) and F(ω) form a transform pair, denoted by the bidirectional arrow: This relationship is fundamental to Fourier analysis. Many common functions have well-known transform pairs: Exponential decay ↔ Lorentzian

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