Fouriertransformation

The Fourier transformation, also known as the Fourier transform, is a mathematical operation that decomposes a function of time into its frequency components. For a continuous-time signal f(t), the Fourier transform F(ω) is defined by F(ω) = ∫_{-∞}^{∞} f(t) e^{-i ω t} dt, where ω is the angular frequency. The inverse transform reconstructs f from F via f(t) = (1/2π) ∫_{-∞}^{∞} F(ω) e^{i ω t} dω, under conditions ensuring existence. The transform converts time domain signals into the frequency domain, revealing spectral content and enabling analysis of frequency components. It linearizes convolution: F{f * g} = F(ω) G(ω). There are different conventions, such as the angular-frequency form above and the frequency in cycles per unit time: F(f) = ∫ f(t) e^{-2π i f t} dt, with corresponding inverse.

For discrete data, the discrete Fourier transform (DFT) and its efficient implementation, the fast Fourier transform (FFT), are used. The DFT samples the spectrum at N evenly spaced frequencies: F[k] = ∑_{n=0}^{N-1} f[n] e^{-i 2π kn / N}. Inverse DFT recovers f. Applications span spectral analysis, filtering, image processing, data compression, solving PDEs, and communication systems. The transform is closely related to Fourier series for periodic signals and to the Laplace transform for signals with growth. Extensions include multi-dimensional transforms, windowed/short-time Fourier transforms for nonstationary signals, and generalized functions. See also Fourier series, Laplace transform, DFT, FFT.