Fouriertranszformációval

Fourier transform is a mathematical tool that decomposes a function of time into its frequency components. For a function f(t) that is integrable on the real line, the continuous Fourier transform F(ω) is defined by F(ω) = ∫_{-∞}^{∞} f(t) e^{-i ω t} dt. The inverse transform recovers f from F via f(t) = (1/2π) ∫_{-∞}^{∞} F(ω) e^{i ω t} dω, under appropriate conditions.

In digital signal processing, the discrete Fourier transform (DFT) computes F_k = Σ_{n=0}^{N-1} f_n e^{-2π i k n / N}, with inverse f_n = (1/N) Σ F_k e^{2π i k n / N}. The fast Fourier transform (FFT) is an efficient algorithm to compute the DFT for large N.

Key properties include linearity and the time- and frequency-shift relations. The Fourier transform converts differentiation into multiplication: F{f'(t)} = i ω F(ω); higher derivatives scale accordingly. The convolution theorem states that the Fourier transform of a convolution is the product of transforms, and scaling in time corresponds to inverse scaling in frequency.

Applications span signal processing, audio, image and video analysis, physics, and the numerical solution of partial differential equations. Historically, Jean-Baptiste Joseph Fourier introduced the transform in the early 19th century to analyze heat flow, building on Fourier series to formalize spectral analysis. Limitations include a trade-off between time and frequency localization, meaning precise time localization comes at the expense of frequency precision; this has motivated methods such as the short-time Fourier transform and wavelet transforms.