Re: Fourier Series Tutorial

On Jan 2, 3:53 am, buleg...@xxxxxxxxxxxxxxx wrote:
On Jan 1, 8:38 pm, John Popelish <jpopel...@xxxxxxxx> wrote:



buleg...@xxxxxxxxxxxxxxx wrote:
On Dec 31 2007, 11:03 pm, John Popelish <jpopel...@xxxxxxxx> wrote:
buleg...@xxxxxxxxxxxxxxx wrote:
Hi,
I have created a tutorial for thefourierseries.
It is located atwww.fourier-series.com
(snip)
Nice. Are you interested in suggestions, or is this done?
I would welcome suggestions, especially blatant errors.

Thanks,

Brent

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 thatFourieranalysis 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 ofFourieranalysis 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.

--
Regards,

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 theFourierSeries. 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

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.
.