Fourier transform for dummies

Abstraction: Intuitive explanations of Fourier transform from physics and engineering perspectives

Key points:

• The Fourier transform decomposes any signal into a sum of circular motions (sinusoids); R(ω) in z(t) = ∫R(ω)e^(iωt)dω is the Fourier transform of z(t)
• Convolution in the time/space domain becomes pointwise multiplication in the frequency domain; this makes Fourier transforms essential for filtering, signal processing, and polynomial multiplication (FFT runs in O(n log n))
• The discrete Fourier transform diagonalizes all shift-invariant linear operators — its basis vectors are eigenvectors of the cyclic shift operator S, derived from N-th roots of unity
• Applications include signal/image processing, optics (diffraction patterns), quantum mechanics (position-momentum duality via Heisenberg uncertainty), spectroscopy, and guitar tuners (FFT to find fundamental frequency)
• Fourier series represent periodic functions as countably infinite sums of harmonics; the continuous transform generalizes this to non-periodic functions as T→∞
• Characteristic functions in probability theory are Fourier transforms of density functions; convolution of distributions becomes multiplication of characteristic functions

Connections: Fourier Transform · Signal Processing · Spectral Analysis · Linear Algebra

Source: https://math.stackexchange.com/questions/1002/fourier-transform-for-dummies