spektraleFourierMethoden

Spektrale Fourier-Transformation is a mathematical method used to express a time-domain signal as a function of frequency. It exposes the spectral content and amplitude-phase information of the constituent sinusoids, enabling analysis of how different frequencies contribute to the overall signal.

In continuous time, the Fourier transform F(ω) is defined as F(ω) = ∫ f(t) e^{-i ω t} dt, and its inverse is f(t) = (1/2π) ∫ F(ω) e^{i ω t} dω. In discrete time, the Discrete Fourier Transform computes X[k] = ∑_{n=0}^{N-1} x[n] e^{-i 2π kn/N}, with the inverse x[n] = (1/N) ∑_{k=0}^{N-1} X[k] e^{i 2π kn/N}. These expressions show how time-domain data relate to their frequency-domain representations.

The transform is linear and invertible under standard conventions. It relates to Fourier series: for periodic signals, Fourier series coefficients can be interpreted as samples of the spectrum at harmonic frequencies. The convolution theorem states that convolution in time corresponds to multiplication in frequency, providing a practical tool for filtering and system analysis.

Computationally, Fast Fourier Transform (FFT) algorithms reduce the complexity from O(N^2) to O(N log N), making spectral methods widely practical. Applications span audio and speech processing, image and video analysis, optics and radar, spectroscopy, seismology, and broader scientific data analysis, where identifying and manipulating frequency components is essential.