Fouriertransform

The Fourier transform is a mathematical transform that decomposes a function of time into its constituent frequencies. For a continuous-time signal f(t), the Fourier transform F(ω) is defined by F(ω) = ∫_{-∞}^{∞} f(t) e^{-i ω t} dt, with the inverse relation f(t) = (1/2π) ∫_{-∞}^{∞} F(ω) e^{i ω t} dω. Different conventions may use frequency f instead of angular frequency ω and place different normalization factors.

For discrete-time or sampled signals, several closely related transforms are used. The discrete-time Fourier transform (DTFT) treats frequency as a continuous variable, while the discrete Fourier transform (DFT) computes a finite set of samples X[k] = Σ_{n=0}^{N-1} x[n] e^{-i 2π kn / N} with inverse x[n] = (1/N) Σ X[k] e^{i 2π kn / N}. The fast Fourier transform (FFT) is an efficient algorithm to compute the DFT.

Key properties include linearity, time and frequency shifting, and the convolution theorem, which states that convolution in time corresponds to multiplication in frequency. Parseval's theorem relates the total energy in the time domain to that in the frequency domain. The transform is used to analyze and filter signals, solve differential equations, and reveal spectral content across physics and engineering.

Extensions include the Fourier series for periodic signals, multidimensional transforms for images and volumes, and generalized Fourier transforms with alternative kernels. Practice and theory often emphasize boundary conditions, sampling, and numerical considerations that affect accuracy and resolution.