DefinitionFourier coefficients are coefficients of Fourier series representing a periodic function. These coefficients are weighted averages of the values of f over one period with respect to the basis functions (sine or cosine waves). The Fourier coefficients help to determine the amplitude and phase of each harmonic component in the Fourier series; these in turn indicate the frequency content and periodic as well as aperiodic behavior of the function. In signal processing and data analysis and in many other fields of science and engineering such as image analysis, fourier coefficients are used to examine periodic signals and to dismantle complicated waveforms into simpler parts. Example: For a periodic function f(x) with period 2π, the nth Fourier coefficient an for the cosine terms is an=π1∫−ππf(x)cos(nx)dx, and the nth Fourier coefficient bn for the sine terms is bn=π1∫−ππf(x)sin(nx)dx.
