User:Jstapko/personal/Fourier

= Introduction =

This page is to describe my own personal journey to understanding Fourier Synthesis, or the Fourier Transform. In retrospect, this journey must have started when I was quite young, watching the waves of the ocean, the ripples on the pond interfere, the mechanisms of steam locomotives that combine several mechanally moving parts into one composite motion to control a steam valve. It was only quite recently, however, that the pieces started falling into place. Yet, every time I think I have put the last edge piece into the puzzle, some new information comes along that makes me realize that at best, I have merely scratched the surface of the ice berg. But before we get too lost in details, the basic Fourier principle is this:

1) Any wave, periodic or not, can be created by combining individual sine waves (or ideed, ANY type of wave) of appropriate frequencies and amplitudes 2) All waves ACTUALLY CONTAN component waveforms consisting of sinusoids, or any other type of wave, and can be broken up into them with suitable filters.

It also turns out that pretty much the entire universe is made out of waves, so this principle has very far reaching implications.

Part I: Early Years (mechanical manifestations)

 * perhaps my earliest direct introduction to the concept of combining waves, or motions, into a single composite motion was when my mom got a copy of Henry T. Brown's Five Hundred and Seven Mechanical Movements. I was always interested in mechanical things, so this book absorbed me for hours.  Particularly relevant to Fourier synthesis are mechanisms 122 and 125 (page 34).  These consist of gear trains of different sizes each driving an end of a combination lever.  By changing the gear sizes, combination lever lengths, and the distance of the drive pins from the gear centers, a single output shaft could be made to undertake a very complex, but repeatable, pattern of oscillation.  A simpler version, with a paragraph explaining the principle of the combination lever, is the steam enging valve gear described in mechanism no. 185.
 * While a high school student, I took a course called "Conceptual Physics," in which I had to, by hand, draw two wave forms on graph paper, and also by hand, calculate and draw the waveform resulting from their interference. This gave me a very intimate understanding of how simple waves interfere to produce complex forms.
 * One time in the early 2000's, I went to Philadelphia's Franklin Institute Science museum with my parents. There they had a machine that they called a "harmonic integrator" which was a set of cams driving combination levers, the ends of which were brought out to a display and painted orange to make them very visible. Two banks of lever ends ,oved up and down, forming sinusoid patterns of different wavelength.  One was also changing its phase very slowly.  The third bank of lever ends moved in a pattern which was the sum of the other two.  This device showed not only the effect of adding two waves, but also how changing the phase of one affected the combination.

Part 2: the electronics age
Around the age of 20, I got interested in electronics and read everything I could find on electronics technology between 1890 and 1960. Principles introduced in this period included:
 * when radio signals are modulated or passed through nonlinear amplifiers, "harmonics" are produced, which must be removed from the output to prevent unwanted interference with other signals
 * band stop filters pass most frequencies, but block the frequency for which they are designed.
 * band pass filters pass the frequency for which they are deigned, and stop all others
 * There are MANY differnt ways to build band pass/stop filters including with crystal resonators, LC circuits, magnetostriction transducers, reverb tubes, tuning forks, optical filters, and many varieties of computer software based filters
 * buried in one engineering book, title now forgotten, was a reference to fourier transforms and my formal introduction to the principle that any wave can be made out of a collection of individual sine waves
 * the phasitron, a vacuum tube for putting audio signals onto an FM radio wave, uses magnetic fields from the audio signal to vary the phase, not the freqency, of the rf signal at the tube's input, but this has the same effect as frequency modulation from the receiver's perspective

part 3: The mathematical perspective

 * in 2011, I took precalculus II, which introduced threeconcepts that would prove critical to understanding the Fourier transform:
 * 1) the De Moivre theorem
 * 2) what a series is, and ways to work with it
 * 3) what sigma notation is and how to work with it
 * The sigma symbol means to generate the all the numbers from the one below to the one above, then apply the formula following the symbol to every number resulting, and then add the results of all the applications of the formula
 * around the time of taking calculus I, I figured out that by using sigma notiation, including a sine function in the formula with some other functions, I could generate a set of sine waves with different frequencies and wavelengths, from a single formula, add them up, and plot them on a graphing calculator. Using this technique, I was able to generate a square wave on the graphing calculator from its component sine waves, with any desired number of harmonics included.
 * watching the square wave form on the calculator felt like dropping the last piece of a decade plus long effort fall into place. I couldn't have been more wrong.
 * in summer of 2011, I took Calculus I, which would prove critical to understanding the Fourier transform, described below

part 4: Quantum Reality
In 2013, I read Nick Herbert's Quantum Reality. Major new concepts introduced to me by this book included:
 * any wave can be considered to be made up of combinations of ANY TYPE of wave, not simply as a combination of sine waves
 * the type of wave it appears to be made of depends on the filter used to take it apart into components
 * some indication of a wave's true nature may be found by looking at a characteristic called "spectral bandwidth," which is the number of individual component waves that a given filter breaks a wave into
 * Bell's interconnectedness theorem proves that either quantum theory is wrong, or something must travel faster than light
 * the Einstein-Podolsky-Rosen paradox and related experiments prove that either quantum theory is wrong, or something must be traveling faster than light

Really, there was so much more to this book, anyon with a serious interest in this should just find a copy and read it

part 5: Fourier synthesis to Fourier transform

 * I started out with the knowledge described above
 * I checked Wikipedia for information, it was not helpful
 * I watched this series of videos:
 * Fourier Transform, part I
 * Fourier Transform, Part II
 * Part III

This series showed how the fourier transform actually worked, and explained the "w"(omega) symbol in the big formula
 * I then watched this video, which introduced the concept of mathematical orthogonality, saying that in some way, the sine and cosine are at right angles to each other, like the way any vector can be represented by component vectors at right angles to each other, pointing in any pair of directions
 * I then read this website, which introduced the concept that the 2pi in the exponent are there to cause the signal to be periodic
 * the website above had a link to this website, which showed how the Fourier transform works in polar coordinates

Part 6: my current understanding

 * 1) All wave forms can be represented by any kind of wave
 * 2) sine and cosine wave are mathematically orthogonal, which means it is very convenient to use them as "component vectors" of a wave, or that they work in directions at right angles to each other
 * 3) you represent the wave as a complex pair of sines and cosines, putting it in a form ready for use with the De Moivre theorem
 * 4) using Euler's Formula, you can condense the cos(x) +i sin(x) down to e^ix (the rest of that stuff is there to compensate for wavelengths and the fact that a circle = 2pi rads)
 * 5) the e^is is great for working with the higher level formulae, but when you actually want to integrate, it's easier to convert back to cos(x) +isin(x), then integrate each part
 * 6) to find out how much of a particular frequency is in a given wave, adjust omega (frequency), generate a cos wave form and an i sin wave form with omega, multiply the original function by the cos and sine waves to make new functions, then integrate the new function on its domain (see the 3 part video series, above)

I still don't understand:
 * what is mathematical orthogonality, and how does it work?
 * how does multiplying and integrating show how much of a given frequency is present?

Next Steps
my mom suggested wacthing the following videos to get a greater understanding:


 * Stanford video lectures on the Fourier Transform
 * MIT Open Courseware on Fourier basics
 * do Saylor.org's ma102 free online calculus II course