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Fourier Plane

DFTBuildup

Is a notebooks on the discrete Fourier Transform. It is my feeling that this subject

  • is essential for the understanding of technology since many internal routines use this;
  • has more applications than any other math technique;
  • can be learned to 12 year - olds (age twelve or grade 6)

This I explained in the first half of my talk "Computer screens and toys" 6 ATCM, Melbourne.

I am joining a copy of this talk, but It should NOT to be included in the material for students.

The next notebooks studies this content in more detail.

Directions

First Directions in the plane are introduced intuitively and color-coded.It is a natural sequel to the UGroup text and prepares for later (non-discrete) Fourier series and transforms. Directions will be used again later in polar coordinates, linear algebra, dynamical systems, ...

Directions2

For young pupils the Euler notation should be made invisible and replaced by e.g. unit[j]. Directions2 shows a possible way to hide complex notations by using white text.

For later use in linear algebra, unit vectors are modified by a 2 x 2 matrix in the Directions text.

The DFTBuildup notebook details the construction of DFT by hanging weights on spoke wheels.

Showing, for a input data list, all harmonic redistributions (together with each sum vector) would make a nice Web animation.

The inverse DFT theorem is proved geometrically.

DFTBuildup2

Is a document on the use of simulations in DFT, and how the DFT can smooth data corrupted with noise.

This is offered separate from the DFTBuildup notebook since it would get too long.

SpiroFourier

The next notebook is about polynomial images of the unit circle (see also my Melbourne ATCM paper: "Computer screens and toys"). It is a prerequisite for the fortcoming notebook on rational approximations for periodic phenomena. It does cover Fourier cosine and sine series, and this is obtained without burdening students with the usual full trigonometry course. Conversely, one can say that showing how simple it is, is a good way to motivate students to continue in this field.

I have tried to give this notebook a gradual buildup, starting from simple examples into the historic ellipse construction and cycloids before handling arbtirary polynomial images. Lecturers may select the most interesting partagraphs for their students. This is also the reason why definitions are repeated often in this text. It is recommended however to evaluate paragraphs in sequence to avoid old definitions interfering.

From the behavior of polynomial images of the unit circle it is possible to continue into other topics in Fourier analysis and into complex analysis (winding numbers, Rouche theorem, conformal mappings, ...). I have material available for those interested, but this is more suited for university juniors (third year, final year bachelor or beyond).

SquareHole

Gives an example of an unusual application of this technique.

Shapes

This is about approximation of periodic phenomena by rational functions. It is simple as soon as rational images of the unit circle (similar to the ellipse construction) are well understood.

I want to tell something about its origins. Two facts made me think about such approximations.

1. When I heard Hirotaka Ebisui's first talk on ovals (defined by long trigonometric expressions) I realised that his ovals - and many of his other drawings - were just rational images of the circle. Many shapes (ellipses, cardioids, ovals, cycloids, ...) can be modeled as exact (finite) rational images, and it is interesting to investigate the degree (highest - lowest power in the rational function) for each shape. For some shapes however only approximations are possible. An example of the latter is the square, and a reasonable approximation is in "squarehole". It made me think about approximating curves defined by selecting points. The first example in shapes is a square drawn by hand using the mouse. Approximating this is actually very simple once one realises that the conversion back from polar coordinates corresponds to a multiplication of the polynomial/rational function by the complex variable z :

x = r cos t

y = r sin t

-> x + i y = r z , with z a point on the circle.

Defining the polar angles may still be a little slow, but Mathematica will convert large sets of points in a minute.

[In the meantime colleague Zhang from Shanghai has presented exact Schwarz-Christoffel like formulas for obtaining starlike domains with piecewise smooth boundary as mappings of the unit disk in the case the expression of each piece of the boundary is given exactly. His work is related, but there is no overlap]

2. Another reason is more fundamental. In considering the DFT of some values, the values are suspended on spokes evenly distributed from the origin. Suppose now the angles are distorted, and the original vectors are pointing in random directions. What can we recover? What values hanging from evenly distributed spokes would correspond best to the given values? One has to find an interpolation that is very much non-linear. Smooth images of the circle are appropriate here.

[It corresponds to the following problem. Suppose a sound is defined by Amplitudes and Frequencies, i.e. by a sum of coefficient(j) * sin( j t ) ,

except that j is replaced by j +/- a small error in the sine argument. I have generated such sounds in Mathematica and they sound awful (not unlike instruments that are badly off-key). To have a little more melody, we have to find new coefficients that allow us to write a combination of trig functions with exact multipliers j . Doing this, we get much nicer sounds. One of my university collaborators , Tom Deneckere who is a professional musician, pianist and conductor, fully agrees.]

As possible applications. I see:

  • sound recognition,
  • compression of periodic data (iris scans?),
  • building reinforced concrete structures,
  • hydrodynamics (flow lines around shapes) and aerodynamic designs,

 

StatisticsShapes

Most surprising to me was the fact that the method also works for sets of data that are nowhere near the boundary of a domain, but clutter (around a mean value) in 2 dimensions. It opens new perspectives in statistical analysis, as you may see in this notebook.

Spiroids

This notebook is a sequel to the spirograph material that I have given at the ATCM conferences. It shows the unexpected beauty of such spirograph images if they are "rotated" in space. Along the same lines I can generate generalised toruses (an example is the almost "square" torus I included in SquareHole). Spherical approximation in 3D offers a lot of potential applications.

Examples