ClassPad Report

Discrete Fourier Transformation

Caution!

This is an activity for ClassPad Manager Ver2.0 only because of the memory capacity. This activity requires lots of memory for using the CAS within the Spreadsheet application. If you tried this activity on the ClassPad 300 (Handheld), you would find "Insufficient System Memory to Run…". We are hoping that someday we will have a Super ClassPad 300 with big memory.

Discrete Fourier Transformation

1st Part
2nd Part
3rd Part
4th Part

What is Discrete Fourier Transformation (DFT)?
We already have an eActivity for Fourier series and can study it with CAS and Spreadsheet.

When you know the original function f(x), you can have Fourier series. It is described by the sine and cosine functions.
But, if you only have the data like below, how can you guess the equation?

 

DFT enables you to have the equation when the data is of a periodic form.

This is the eActivity to explore DFT. The ClassPad 300 does not currently have enough memory to calculate lots of data for DFT, but it's enough to understand the algorithm. In the real world, we use the Fast Fourier Transformation (FFT) instead of DFT. Oh! The EA200 has an FFT function! We should try to correct the data for FFT with EA200. It will be my next activity.

Let's try to study DFT.

1^st Step

 

Open eActivity:   Open "Data Table & Plot" strip: 

Select cells and then click on the scatter plot icon to plot the data:

 

We know that Spreadsheet has the ability to draw curves, such as sin(x), on the Spreadsheet graph window. All we need to do is input the equation in an empty cell, and then drag & drop it the Spreadsheet graph window.

Type the equation you guessed into an empty cell in the Spreadsheet and then drag & drop it into the plot graph window. Try changing your curve to make it fit better and then drag & drop it again.

When you have your own equation, go to 2nd step.

2^nd step

You will find the definition of DFT and the transformation to use it on Spreadsheet.

 

The strips in the eActivity, "Transformation", "Review Euler", and "Review De Moivre" are helpful in understanding the transformation.

   

3^rd Step

Open the strip "DFT by Spreadsheet". The data in columns B and C, x and f(x), is the data we will explore.

 

When you scroll to the right, you will find nk, f_re, f_im, sum(re) and sum(im).
Tap the cell to see how it is defined.

  

  

Note: Cell F3 is the number of K.

As you scroll the window, you find the same definitions for k=0 to k=9.

Select rows 15 and 16, and then draw a bar graph.  

4^th Step

Tap the bar graph to find the cell.

   

We find that the sum(re) is 5 at J15 and AH15. J is the column for k=1. AH is the column for k=9.

 

We find that the sum(im) is -5 at K16 and 5 at AI16. K is the column for k=1. AI is the column for k=9

 

From the definition,

Where

We have

And

Now we will conjecture what the equation is.

When k=1 and k=9 we have a conjugate relation using 10 data.
The sum(f_re) is 5 when k=1. This means that the function contains cos(1*x) and that the coefficient is 5/5=1.
The sum(f_im) is 5 when k=1. This means that the function contains sin(1*x) and that the coefficient is 5/5=1.

Finally we guess that the equation is y= cos(x) + sin(x).
Go back to the original data table and try dropping our equation to the plot graph. Does it fit?

I will try to use the EA200 to give you an example to use FFT.
See you later.

Canyon