Fourier SeriesFloyd Maseda

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What is a Fourier Series?

A Fourier series is an approximate representation of any periodic function in terms of simpler functions like sine and cosine.

The series has many applications in electrical engineering, acoustics, optics, processing, quantum mechanics, etc.

an and bn are called “Fourier coefficients” and are defined by the following equations:

where L is half the length of the segment being periodically (L=λ/2) repeated and f(x) is a function describing the segment.

There are two types of triangle waves: Odd waves (such as the one pictured above) and even waves. While both waves are basically the same thing, they differ slightly in what is being repeated.

Even waves repeat a single line while odd waves repeat a “bent” line.

Example: Triangle Wave

EVEN WAVE ODD WAVE

The ProgramsEVEN WAVE ODD WAVE

... DO II=1,100 DO JJ=1,50 COEFF = 2.0*CNUM - 1.0 FAC = (4.0/PI)*COS(COEFF*XX)/(COEFF**2) FUNC = FUNC + FAC CNUM = CNUM + 1.0 END DO WRITE(6,*) XX,FUNC FUNC = 0.0 CNUM = 1.0 XX = XX + 0.1 END DO...

... DO II=1,100 DO JJ=1,50 COEFF = 2.0*CNUM - 1.0 FAC = ((-1)**(CNUM+1.0))*(4.0/PI)*SIN(COEFF*XX)/(COEFF**2) FUNC = FUNC + FAC CNUM = CNUM + 1.0 END DO WRITE(6,*) XX,FUNC FUNC = 0.0 CNUM = 1.0 XX = XX + 0.1 END DO...

The OutputEVEN WAVE

The OutputODD WAVE

Another Example: Square Wave

The square wave I attempted to recreate was a representation of an analog-digital conversion of an audio signal.

Alterations

In order to make the square wave work, I had to make some alterations to the way I approached the wave.

Since the wave was neither even nor odd, I thought of it as a translation of another function. If the wave was shifted down 1/2 unit, it would work similarly to the previous two waves.

The Fourier Series

The Output

Gibbs Phenomenon:Approximation encounters

largeoscillations at jump

discontinuitiesin the original function.

Nonlinear Fourier SeriesTo demonstrate the versatility of the Fourier Series, I decided to try a non-linear function. While according to the research I did, anything plugged into the Fourier series that is a function will work, some functions are harder than others to integrate and come up with a sigma representation.

Trying to integrate something with a square root (semicircle, sideways parabola, etc.) is a nightmare even for WolframAlpha!

Example: y=x2

The derived Fourier series

The Output

Summary

Any periodic function can be expressed as a superposition of many simple trigonometric functions

Most of the work involved is actually in integrating the function itself

Some functions are harder to integrate than others