BASICS OF FOURIER SERIES

INTRODUCTION:





                                               Fig.1 periodic square wave

Fourier series are used in the analysis of periodic functions.

Many of the phenomena studied in engineering and science are periodic in nature eg. The current and voltage in an alternating current circuit. These periodic functions can be analysed into their constituent components (fundamentals and harmonics) by a process called Fourier analysis.

We are aiming to find an approximation using trigonometric functions for various square, saw tooth, etc. waveforms that occur in electronics. We do this by adding more and more trigonometric functions together. The sum of these special trigonometric functions is called the Fourier series.

Periodic functions

De nition

A function f(x) is said to be periodic if there exists a number

T  > 0 such that f(x + T ) = f(x) for every x. The smallest such

T   is called the period of f(x).

Intiutively, periodic functions have repetitive behavior.

A periodic function can be de ned on a nite interval, then copied and pasted so that it repeats itself. 

Examples

I    sin x and cos x are periodic with period 2

I    sin(πx) and cos(πx) are periodic with period 2

I    If L is a x fised number, then sin(2πx/L) and cos(2πx/L ) have period L

                 Sine and cosine are the most \basic" periodic functions!

Fourier series

Let p > 0 be a fixed number and f(x) be a periodic function with period 2p, defined on (−p,p). The Fourier series of f(x) is a way of expanding the function f(x) into an infinite series involving sines and cosines:

f(x) = a_0/2+∑2_(n=1)^∞〖a_n cos(nπx/p) 〗+∑_(n=1)^∞sin(nπx/p)

where a0, an, and bn are called the Fourier coefficients of f(x), and are given by the formulas

a0=1/p ∫_(-p)^p〖f(x)dx=0〗

an=1/p ∫_(-p)^pf(x)cos(nπx/p)   dx=0

bn=1/p ∫_(-p)^pf(x)sin(nπx/p)   dx=0

Remarks:

1.      To find a Fourier series, it is sufficient to calculate the integrals that give the coefficients a0, an, and bn and plug them in to the big series formula.

2.      Typically, f(x) will be piecewise defined.

3.      Big advantage that Fourier series have over Taylor series: the function f(x) can have discontinuities!

                   Useful identities for Fourier series: if n is an integer, then

       4.                  sin(nπ) = 0

                   e.g. sin(π) = sin(2π) = sin(3π) = sin(20π) = 0

       5. cos(nπ) = (−1)ⁿ = 1 for even

                                      −1 for odd

                 e.g. cos(π) = cos(3π) = cos(5π) = −1, 

            but cos(0π) = cos(2π) = cos(4π) = 1.

Fourier coefficients of an even function

If f(x) is an even function, then the formulas for the coefficients simplify. Specifically, since f(x) is even, f(x)sin(nπx p ) is an odd function, and thus

 

bn=1/p ∫_(-p)^p▒f(x)sin(nπx/p)dx=0

Therefore, for even functions, you can automatically conclude (no computations necessary!) that the bn coefficients are all 0.

Fourier coefficients for an odd function:

If f(x) is odd, then we get two freebies:

 a0=1/p ∫_(-p)^p〖f(x)dx=0)

an=1/p ∫_(-p)^pf(x)cos(nπx/p)   dx=0

Note: In general, your function may be neither even nor odd. In those cases, you should use the original formulas for computing Fourier coefficients, given in equation

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