Fouriertransformaatiossa

Fourier transform is a mathematical technique used to transform a function from its original domain (often time or space) into a representation in the frequency domain. This transformation is particularly useful in signal processing, image analysis, and various fields of physics and engineering. The Fourier transform of a function f(t) is defined as:

F(ω) = ∫[-∞, ∞] f(t) * e^(-iωt) dt

where F(ω) is the Fourier transform of f(t), ω is the angular frequency, t is the time variable, and i is the imaginary unit. The inverse Fourier transform is given by:

f(t) = (1/2π) * ∫[-∞, ∞] F(ω) * e^(iωt) dω

The Fourier transform provides a way to express a function as a sum of sinusoidal components, each with a specific frequency, amplitude, and phase. This decomposition allows for the analysis of the frequency content of a signal, making it a powerful tool for filtering, noise reduction, and other signal processing tasks. The Fourier transform has numerous applications, including the analysis of musical sounds, the design of electronic filters, and the study of wave phenomena in physics.