FourierTransformSpektroskopie

The Fourier transform is a mathematical technique that converts a time-domain function into a representation in the frequency domain. For an integrable function f(t) on the real line, the continuous Fourier transform F(ω) is defined as F(ω) = ∫_{-∞}^{∞} f(t) e^{-i ω t} dt, and the inverse transform recovers f from F via f(t) = (1/2π) ∫_{-∞}^{∞} F(ω) e^{i ω t} dω. Different normalization conventions exist, but the core idea is to express a signal as a sum of complex exponentials with frequencies ω.

In discrete settings, the discrete Fourier transform (DFT) converts a finite sequence f[n] into a sequence F[k] = ∑_{n=0}^{N-1} f[n] e^{-i 2π kn / N}, with the inverse f[n] = (1/N) ∑_{k=0}^{N-1} F[k] e^{i 2π kn / N}. The DFT is typically computed efficiently via the fast Fourier transform (FFT) algorithm, enabling practical analysis of sampled data.

Key properties include linearity, time shifting (shifting f by t0 multiplies F by e^{-i ω t0}), frequency shifting, and the convolution theorem (convolution in time corresponds to multiplication in frequency). Parseval’s identity links energy in time and frequency domains. The transform is invertible under appropriate conditions, allowing reconstruction of the original signal.

Applications span signal processing, audio and image analysis, communications, solving differential equations, and spectral analysis. Variants include unitary forms and multi-dimensional Fourier transforms used for images and volumes. The Fourier transform originated in the work of Jean-Baptiste Joseph Fourier and remains a foundational tool across science and engineering.