Fourier series represents periodic function as series expansion of infinite trigonometric function term. /kloc-was put forward by the French mathematician Fourier at the beginning of the 0/9th century, and has become an important tool in modern mathematics and signal processing.
For the function f(x) with period t, its Fourier series can be expressed as:
f(x)= A0+σ{ ancos(nωx)+bns in(nωx)}
Where a0, an and bn are coefficients, n is a positive integer, and ω is the angular frequency.
If the coefficients an and bn are all 0 for all parts where n is greater than 1, that is, for all high-order harmonic components, only a0 and a 1 remain, that is:
f(x)≈a0+a 1*cos(ωx)
Such a function can be regarded as a simple periodic function, which only contains DC component and fundamental frequency component, and has no component corresponding to higher harmonics.
Application of Fourier series
Fourier series can decompose a signal into sine and cosine signals with different frequencies, which can help people understand the frequency domain characteristics of the signal and be used for signal filtering, noise reduction and demodulation. Similarly, Fourier series can decompose an image into various frequency components, so as to compress, denoise and sharpen the image. Fast Fourier transform algorithm is widely used in digital image processing.
Many physical quantities can be expressed as periodic functions, such as sound waves and electromagnetic waves. These periodic functions can be expanded by Fourier series to get the corresponding spectrum distribution, which is helpful for people to understand and study physical phenomena. Fourier series is also widely used in mathematical analysis, such as proving Cauchy convergence criterion, studying function limit and continuity, etc.