Fourier Series Tutorial

Hi,

I have created a tutorial for the fourier series.

It is located at

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It has audio lectures and interactive flash programs that let you see how the fourier series actually works.

Reply to
bulegoge
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Nice. Are you interested in suggestions, or is this done?

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Regards,

John Popelish
Reply to
John Popelish

Excellent, only gone through the first few minutes.

Nice graphics!

martin

Reply to
Martin Griffith

I would welcome suggestions, especially blatant errors.

Thanks,

Brent

PS I am working on doing the Fourier series in complex notation now. These things take a while to do however, and this is kind of a hobby at the moment.

Reply to
bulegoge

Reply to
Martin Griffith

(snip)

No blatant errors, just some little things that jarred my mental model of a beginner student taking your course. As it is, the course is a nice review for people who have already had an earlier course, and covers many weaknesses in other versions.

My concerns are all about starting out with clear, but not overly detailed and complete definitions.

I am bothered about your use of words like basis vectors, orthogonal and use them without first giving a simple definition of what those words mean in this application. I am sure they are so familiar to you that they seem obvious, but that is not necessarily the case for the student.

I think you might mention that Fourier analysis decomposes a repeating waveform into a sum of component parts, those parts being sine and cosine waves of all frequencies that are integer multiples (including zero) of the period of the repeating waveform. You work your way around this definition, but it is strung out through a lot of words.

I like the part of the discussion of Fourier analysis where you mention that it is a tool to go from thinking of a wave as a repeating sequence of variations as time passes to thinking of that same wave as a combination of frequencies that are present throughout the wave... switching the viewpoint from the time domain to the frequency domain.

I think a nice addition would be a bit about how the X and Y axes are orthogonal because they are oriented at 90 degrees with respect to each other, so a point can be moved vertically without changing its X component, or moved horizontally without changing its Y component. This independence of movement vertically and horizontally is what makes the X-Y coordinate system a set of basis vectors that uniquely identify any point location as a unique combination of an X and Y displacement from the origin.

And that the points distance from the origin is the square root of the sum of the squares of its X and Y components.

Then show how similar to this two dimensional location of a point is to a two dimensional description of an arbitrary amplitude and phase shifted sinusoidal wave with sine and cosine components. Note how the sine and cosine are also phase shifted 90 degrees with respect to each other, similar to the way the X and Y axes are rotated with respect to each other. This 90 degree shift or rotation in a cyclic sense is what makes these two components orthogonal to each other and thus, usable as basis vectors that can locate the amplitude and phase of the arbitrary sinusoid.

And like the points distance from the origin being the square root of the sum of the squares of its two orthogonal components, the magnitude of the arbitrary sinusoidal wave is the square root of the sum of the squares of the magnitudes of two wave orthogonal wave component magnitudes, (sine and cosine).

In other words, I think you might spend a bit more time extending what might be most obvious to the beginner about the Cartesian plane to the new concept of sine and cosine as orthogonal another kind of two dimensional way to capture a different kind of information. You do some of this, but it didn't seem like you were trying to build on an existing mental concept (the X-Y plane) as much as mention some similarities it has with the sine cosine basis vectors.

I love your Java applets that allow waves to be built up from components, but if you could add a live numerical match score, it would help the user make sure a given change took him closer to the ideal solution, rather than further away, teaching him to visually recognize a real improvement, rather than guess what that looks like.

Given what you already have, I am guessing that coming up with a live match score wouldn't add much. But ignorance is always bliss.

I think you might also eliminate the frequency adjustment in the cases where it changes all frequencies together, and has no effect on the result. Just pick, say, two cycles of the wave.

I haven't gotten past 4.5, so I have no comments past the first part.

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Regards,

John Popelish
Reply to
John Popelish

One other thing I thought of and forgot.

You spend a lot of words on even and odd symmetry, using the word symmetry many times without being very clear.

Try bringing in the concepts of mirror and rotational symmetry, which can be illustrated with a simple series of flip frames (folding a wave across a mirror fold, or rotating a wave around a center, by a half rotation).

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Regards,

John Popelish
Reply to
John Popelish

Martin , Thank you for your insightful comments. I did put a back button in on some of the later lectures , an I need to learn how to control the audio better, as you have suggested.

Brent

John,

Thanks for your detailed thoughts regarding this. I agree that I should consider updating the introduction, and I like your suggestion about even/odd symmetry.

I think you are probably correct that the tutorial seems to target the person that has already had some exposure to the Fourier Series. I am not sure if I want to change that or not, however.

I appreciate your taking th time to make these detailed comments.

Brent

Reply to
bulegoge

I got here a bit later. Tame down the colors. Avoid "outlined" fonts. Introduce phase shift versus sine/cosine coefficients earlier; the idea is that you may get all sine or all cosine results more often. Clean up the navigation. Add many more examples, i know it is hard, but please. Even though this is a relatively introductory explanation, please consider adding windowing issues. I also hope you will add many student exercises but require "site registration (tell us a little about you)" to access the answers.

Reply to
Joseph2k

Joseph2k wrote: (snip)

(snip)

I disagree. Windowing does not apply to Fourier analysis (of perfectly and infinitely) repeating waveforms.

Windowing applies to the approximate extension of Fourier analysis, applied to non-repeating waveforms.

Different topic.

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Regards,

John Popelish
Reply to
John Popelish

Well, I've got a headache so it must be good :)

Apart from the already mentioned "back" button I'd like to see a "pause" for the audio.

For the flash I agree with an earlier poster's comments about loosing the frequency slider where it doesn't really change anything.

I'd also like to see some "canned" waveforms in the final animation, A few "silly-simple" ones that the results will be so obvious even I can see where there coming from would be appreciated.

H.

Reply to
Howard Eisenhauer

Thanks for your comments. As far as introducing phase shift and sine/ cosine coefficients: I think that was the emphasis of the first flash program, so it is the first thing I tried to cover.

The windowing stuff would come much later. My next goal is to get some stuff up regarding the complex representation of the Fourier Series.

Regarding getting your comments on getting all cosine or sine results more often:

One of the things that really drove me to do this was the confusion caused by books that present all cosine or all sine results in a textbook example. That was a point of great confusion for me. So I really am trying to emphasize that all cosine or all sine results are not the normal thing, but a contrived result in most cases.

I want to get some problems in the tutorial soon, but my problems will not be hard. Gawd, how many hours did I spend doing incredibly hard problems that in the end I knew nothing more than when I started( This was years ago as an undergrad - recently I said to myself - "I want to really learn this stuff"). I would rather drive home simple points and "build" the students "self esteem" (ha ha).

thanks for your comments.

Brent

Reply to
bulegoge

It is ok. You need to reduce the use of garish colors, get rid of the outline fonts they both degrade readability. I recommend introducing time / phase shifting as preferred to having so many mixed sine / cosine functions. I see that the site is still incomplete, not having much in the way of examples or student problems yet. Most of the flash animations are pretty decent. Keep at least the first ten harmonics for the demos. I suggest adding some stuff on windowing side effects. All in all a worthy effort.

Reply to
Joseph2k

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