The document discusses Fourier series. A Fourier series expresses a periodic function as an infinite sum of sines and cosines. It is named after Jean-Baptiste Joseph Fourier who made important contributions to studying trigonometric series. A Fourier series breaks down a function into its constituent frequencies and determines the contribution of each frequency to the overall signal. The formula, properties, examples, advantages, and applications of Fourier series are provided.

1. FOURIER SERIES
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PRESENTED BY:
AKSHARA.V.B

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INTRODUCTION:
 the key to understanding the Fourier series is our ancient desire to
express everything in terms of circles or the most exceptionally simple and
elegant abstract human construct.
Most people prefer to say the same thing in a more historical manner: to
break a function into sines and cosines.
 The Fourier series is named in honour of Jean-Baptiste Joseph
Fourier (1768–1830)
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, publishing his initial results in his 1807

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DEFINITION:
 A Fourier series is an expansion of a periodic function in terms of an
infinite sum of sines and cosines.
 A Fourier series make use of the orthogonality relationships of
the sine and cosine functions.
 The computation and study of Fourier series is known as harmonic analysis
 Any set of functions that form a complete orthogonal system have a
corresponding generalized Fourier series analogous to the Fourier series
Using the method for a generalized Fourier series, the usual Fourier series
involving sines and cosines is obtained by taking f1(x)=cos x and f2(x)=sinx .
Since these functions form a complete orthogonal system over

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FORMULA:
WHERE,

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EXAMPLE:
 FIND THE FOURIER SERIES FOR THE JUNCTION f(x)dx = X2 IN THE
INTERVAL (0,2L)
1
TO FIND

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To find
2
1+ cos 0

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To find
3
=
=
From

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PROPERTIES:
Linearity
Time Shifting
Frequency Shifting
Conjugation
Time Reversal
Time scaling
Multiplication
Differentiation
Real and Even Signals
Real and Odd Signals

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ADVANTAGE:
Original time function can be uniquely recovered from it.
 Convolution integrals can be evaluated using fourier transform.
 The fourier transform is the most useful for analyzing signals involved
in communication systems because here the amplitude and phase
characteristics are readily known.

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APPLICATION:
 Communications
 Astronomy
 Geology
 Optics