The Fourier Series

A Fourier series is an expansion of a periodic function f (t) in terms of an infinite sum of cosines and sines.

In other words, any periodic function can be resolved as a summation of constant value and cosine and sine functions:

The computation and study of Fourier series is known as harmonic analysis and is extremely useful as a way to break up an arbitrary periodic function into a set of simple terms that can be plugged in, solved individually, and then recombined to obtain the solution to the original problem or an approximation to it to whatever accuracy is desired or practical.

Symmetry Considerations:

· Symmetry functions:

(i) even symmetry

(ii) odd symmetry

Even symmetry: Any function f (t) is even if its plot is symmetrical about the vertical axis, i.e.

-->f(-t)=f(t)

The examples of even functions are:

The integral of an even function from −A to +A is twice the integral from 0 to +A

Odd symmetry: Any function f (t) is odd if its plot is antisymmetrical about the vertical axis, i.e.

-->f(-t)=-f(t)

The examples of odd functions are:

The integral of an odd function from −A to +A is zero

Even and odd functions:

The product properties of even and odd functions are:

· (even) × (even) = (even)

· (odd) × (odd) = (even)

· (even) × (odd) = (odd)

· (odd) × (even) = (odd)

Function defines over a finite interval

· Fourier series only support periodic functions

· In real application, many functions are non-periodic

· The non-periodic functions are often can be defined over finite intervals, e.g.

· Therefore, any non-periodic function must be extended to a periodic function first, before computing its Fourier series representation

· Normally, we prefer symmetry (even or odd) periodic extension instead of normal periodic extension, since symmetry function will provide zero coefficient of either an or bn

· This can provide a simpler Fourier series expansion

Half-range Fourier series expansion:

· The Fourier series of the even or odd periodic extension of a non-periodic function is called as the half-range Fourier series

· This is due to the non-periodic function is considered as the half-range before it is extended as an even or an odd function

· If the function is extended as an even function, then the coefficient bn= 0, hence

Which only contains the cosine harmonics.

· Therefore, this approach is called as the half-range Fourier cosine series

· If the function is extended as an odd function, then the coefficient an= 0, hence

Which only contains the sine harmonics.

· Therefore, this approach is called as the half-range Fourier sine series