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Fourier Series

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- Floyd Maseda

- 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.

EVEN WAVE

ODD WAVE

EVEN 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...

EVEN WAVE

ODD WAVE

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

- 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.

Gibbs Phenomenon:

Approximation encounters large

oscillations at jump discontinuities

in the original function.

- 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!

The derived Fourier series

- 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