Half Range Fourier Series
Rough Proof of Coefficient Equations
Example plot of Spectrum of F.S.
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.
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.
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.
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
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)
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
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 .
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.
Rough proof of coefficient equations
Plots of fourier series with associated plots of spectrum comprising the fourier series
|Useful Related Links||