1. Fourier Series
Floyd Maseda

2. 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.

3. 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.

4. Example: Triangle Wave
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.

5. EVEN WAVE
ODD WAVE

6. The Programs EVEN WAVE ODD WAVE
DO II=1, DO JJ=1, COEFF = 2.0*CNUM FAC = (4.0/PI)*COS(COEFF*XX)/(COEFF**2) FUNC = FUNC + FAC CNUM = CNUM END DO WRITE(6,*) XX,FUNC FUNC = CNUM = XX = XX END DO...
...
DO II=1, DO JJ=1, COEFF = 2.0*CNUM FAC = ((-1)**(CNUM+1.0))*(4.0/PI)*SIN(COEFF*XX)/(COEFF**2) FUNC = FUNC + FAC CNUM = CNUM END DO WRITE(6,*) XX,FUNC FUNC = CNUM = XX = XX END DO...

7. The Output
EVEN WAVE

8. The Output
ODD WAVE

9. Another Example: Square Wave
The square wave I attempted to recreate was a representation of an analog-digital conversion of an audio signal.

10. 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.

11. The Fourier Series

12. The Output Gibbs Phenomenon: Approximation encounters large
oscillations at jump discontinuities
in the original function.

13. Nonlinear Fourier Series
To 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!

14. The derived Fourier series
Example: y=x2
The derived Fourier series

15. The Output

16. 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