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Fourier index

Fourier Series

Fourier Series

In various areas of engineering periodic wave forms are obtained which need to be analysed.    Simple examples include Sine waves, Cosine Waves, Square, Triangular, Sawtooth and various combinations of these forms. A Fourier series is an expansion of a periodic function in terms of an infinite sum of sines and cosines.    The derivation and study of Fourier series is known as harmonic analysis and is extremely useful as a way to break up an periodic function into a set of simple terms that can be selected, solved individually, and then recombined to obtain the solution to the original problem.   The accuracy of the solution is dependent on the level of application of the procedure.    The use of these series and their integrals provide a very powerful tool in connection with various problems involving ordinary and partial differential equations.

The theory of Fourier series is complicated , but the application of these series is simple.  Fourier series are to a degree more universal than Taylor series because many discontinous functions of interest can be developed in Fourier series which do not have Taylor expansions.

Consider a function f( x ) of a real variable x which has the property.

f(x + T) = f( x )

This function is a periodic function.  If T is the smallest value for which the equation is true then T is called the real period.

If n is an interger then it also follows that

f(x - nT) = f(x - T) = f( x ) = f(x+T) = f(x+nT)

Any multiple of T ( n 0) is also a period.    If f( x ) and g( x ) also have the period T then the function

h( x ) = af( x ) + bg( x )

Also has a period T

The most familiar examples of periodic functions are the sine and cosine functions.
It is possible to represent any function of the period T = 2π by use of the simple functions sin( x ), cos( x ) sin(2x) , cos(2x)....sin(nx), cos (nx).

It should be noted that the funtions sin( n.x + α ) and cos (n.x +α) are essential the same.    sin(nx + α ) = cos(nx + α - π/2. ).    The only difference being the location of the y axis along the horizontal x axis.    This position of the y axis is in most practical applications not important.     Both variants have the period 2.π /n. The general expansion which is used to enable the sine and cosine functions to be used to represent the periodic function under consideration is of the form

f(x) = a0 + a1 cos( x ) + b1 sin( x ) + a2 cos(2x)+ b2 sin(2x) + a3 cos(3x)+ b3 sin(3x).........

This is simplified to the trigonometric series The various coefficients can be calculated using the following equations. (Coefficient Equations) The resulting trigonometric series is called the fourier series for f( x ) and the relevant coefficients are called the fourier coefficients.

A very brief explanation of the derivation of the fourier coefficients is provided at the bottom of this page   Derivation of fourier coefficients

It is important to note, from the above notes that

a0 = mean value of f( x ) between x = -π and π
an = 2 x mean value of f( x ).cos( kx ) between x = -π and π
bn = 2 x mean value of f( x ).sin( kx ) between x = -π and π

Even and Odd Functions

Note: If the function is symmetric with respect to the y axis it is an even function.  A function is an odd function if f(-x) is = -f( x ).    A pure sine function is a periodic odd function and a pure cosine is an even periodic function. Even function ...f( x ) = f(-x)
Odd function ...f( x ) = - f(-x)

If f( x ) is an even function then If f( x ) is an odd function then Note: cos( x ) is case of even function for which this is true but this is not general

If a function is even so that f( x ) = f( -x ) , then f( x ).sin( nx ) is odd.     (an even function times an odd function is an odd function.)    Therefore b n = 0, for all values of n.     Similarly, if a function is odd so that f( x ) = - f( x ), then f( x ).cos(nx) is odd.    cos( nx ) is even and an even function times an odd function is an odd function.)    Therefore a n = 0 for all values of n.

A) The Fourier series of an even periodic function f( x ) having a period 2π is a Fourier cosine series

B) The Fourier series of an odd periodic function f( x ) having a period 2π is a Fourier sine series

Functions Having Arbitrary Period

When considering functions which have a period T which is not equal to 2 π it is only necessary to modify the horizontal scale.   Consider a function f( t ) which has a period T then to represent it in terms of a function f( x ) which has a period 2π by setting. The Fourier series with x expressed in terms of t is shown as follows The relevant Euler formulas are as follows A) The Fourier series of an even periodic function f( t ) having a period T is a Fourier cosine series B) The Fourier series of an odd periodic function f( t ) having a period T is a Fourier sine series Half-Range Fourier Series

Sometimes it is convenient to obtain a Fourier expansion of a function to hold for a range which is half the period of the Fourier Series i.e. to expand f( x ) in the range (0,π) in a Fourier Series of period 2π or more generally in the range ( 0 , L ) in a Fourier Series of period T = 2 L .

Such an expansion can be in a series of sines only, or in a series of cosines only.    Each series will represent the function for the half-period.    However the values of each of the two series for the remaining half-period will be different .

If f( t ) is considered even then a fourier cosine series results If f( t ) is considered odd then a fourier sine series results The figure below illustrates a trace of function f(t) for t = 0 to l.
The differences between function as even with period 2.L and odd with period 2.L are illustrated Rough proof of coefficient equations

 First a number standard trigonometric relationships are listed A number of easily proved integrations are now shown The Kronecker delta function δkn is often used for these equation .δkn = 1 if k = n and = 0 otherwise Integrating both sides of trigonometric equation (A) from -π to +π results in Term by term integration results in the following The first term on the RHS = 2.π.a0 while all other terms = zero.    Therefore To determine a1, a2, a3 ... multiply (A) by cos(kx) and integrate the parts from -π to +π The first integral term on the RHS is zero and the last integral term on the RHS is also zero.     The second integral term is π when k = n. Therefore. To determine b1, b2, b3 ... multiply (A) by sin(kx) and integrate the parts from -π to +π The first integral term on the RHS is zero and the second integral on the RHS term is also zero. The third integral term on the RHS is π when k =n. Therefore. Plots of fourier series with associated plots of spectrum comprising the fourier series Useful Related Links Fourier Series ...Very clear cand comprehensive Notes Fourier Series Applet .. Applet Illustrating fourier series for different waveforms. Fourier Series .. A very easy to understand set of notes Gallery Mathods online .. A very accessible applet based tutorial providing qualitative understanding.
Maths Index
Fourier index