4a Fourier Series 11

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MATEMATEKA REKAYASAMO 091204

Fourier Series 1

mahmud mustain


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Infinite Series - Numbers

A geometric progressionis a set of numberswith a common ratio.

Example: 1, 2, 4, 8, 16

A seriesis the sumof a sequence of numbers.

Example: 1 + 2 + 4 + 8 + 16

An infinite seriesthat converges to a particularvalue has a common ratio less than 1.

Example: 1 + 1/3 + 1/9 + 1/27 + ... = 3/2

When we expand functions in terms of someinfinite series, the series will convergeto thefunction as we take more and more terms.


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Infinite Series Expansions of Functions

We learned before in the Infinite Series

Expansionschapter how to re-express many

functions (like sinx, logx, ex, etc) as apolynomial with an infinite number of terms.

We saw how our polynomial was a good

approximation near some valuex = a(in the case

of Taylor Series) orx = 0 (in the case of

Maclaurin Series). To get a better approximation,

we needed to add more terms of the polynomial.
http://www.intmath.com/Series-expansion/Infinite-series-expansion.phphttp://www.intmath.com/Series-expansion/Infinite-series-expansion.phphttp://www.intmath.com/Series-expansion/Infinite-series-expansion.phphttp://www.intmath.com/Series-expansion/Infinite-series-expansion.phphttp://www.intmath.com/Series-expansion/Infinite-series-expansion.php


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Fourier Series - A Trigonometric Infinite Series

In this chapter we are also going to re-expressfunctions in terms of an infinite series. However,

instead of using a polynomial for our infiniteseries, we are going to use the sum of sineandcosinefunctions.

Fourier Series is used in the analysis of signals

in electronics. For example, later we will see theFast Fourier Transform, which talks about pulsecode modulation which is used when recordingdigital music.
http://www.intmath.com/Fourier-series/7_Fast-fourier-transform-FFT.phphttp://www.intmath.com/Fourier-series/7_Fast-fourier-transform-FFT.php


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Example

We will see functions like the following,which approximates a saw-tooth signal:

How does it work? As we add more terms to the series,

we find that it converges to a particular shape.


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Taking one extra term in the series each time

and drawing separate graphs, we have:

f(t) = 1 (first term of the series):


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Fourier Coefficients For Full Range Series Over

Any Range -L TOL


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Dirichlet Conditions

Any periodic waveform of periodp =2L, can beexpressed in a Fourier series provided that

(a) it has a finite number of discontinuities within the period2L;

(b) it has a finite average value in the period 2L;(c) it has a finite number of positive and negative maxima

and minima.

When these conditions, called the Dirichlet conditions,

are satisfied, the Fourier series for the function f(t) exists. Each of the examples in this chapter obey the Dirichlet

Conditions and so the Fourier Series exists.


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AL-HAMDULILLAH

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4a Fourier Series 11