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Unit-II: FOURIER SERIES
Contents: Definition, Dirichlet’s conditions, Full range Fourier series, Half range Fourier series, Harmonic analysis, Parseval’s identity and Applications to problems in Engineering.
Introduction
A Fourier Series is an expansion of a periodic function f(x) in terms of an infinite sum of sines and cosines .The Fourier series is named in honor of Joseph Fourier who made important contributions to the study of trigonometric series.Fourier introduced the series for the purpose of solving the heat equation in a metal plate.
Trigonometric Series
Fourier Series
Example
Cannot be expressed as fourier series, since it has an infinite discontinuity at x = pi/2
Can be expressed as Fourier Series
Even & Odd Function Fourier Series
Half Range Sine Series and Half Range Cosine Series
Illustration
Applications
Fourier Series provide a way to efficiently store and regenerate the musical tones of various instruments. Pure tones have frequency and amplitude, which determine the pitch and strength of sound respectively.
Fourier Series and its variations are extensively used to clean up noisy signals.
Fourier Series are used to optimize the design of a telecommunication system using information about the spectral components of the data signal that the system will carry.
Fourier Series can be used by astronomers to deduce the chemical composition of a star by analyzing the frequency components, or spectrum, of the star's emitted light.
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