Fourier Series

Fourier series are used to approximate functions with periodic inputs in terms of sums of sinusoids with different frequencies. They are named after Joseph Fourier, who showed that any periodic function could be represented as a sum of sinusoids with different frequencies. The frequencies of the sinusoids in the Fourier series are integer multiples of the fundamental frequency.

Fourier series are often used to approximate functions that are not periodic. In this case, the Fourier series is an infinite sum of sinusoids with different frequencies that converges to the function being approximated. The Fourier series coefficients quantify the amplitude and phase of the sinusoids making up the series.

Fourier series are used in a variety of applications, including signal processing, image processing, and communications. In signal processing, Fourier series are used to represent signals as sums of sinusoids with different frequencies. This representation can be used to analyze the properties of the signal, such as its bandwidth or its power. In image processing, Fourier series are used to represent images as sums of sinusoids with different frequencies. This representation can be used to analyze the properties of the image, such as its edges or its texture. In communications, Fourier series are used to represent signals as sums of sinusoids with different frequencies. This representation can be used to encode the signal in a bandwidth-efficient manner.