Fourier Series: Several functions

Second post dedicated to the Fourier series [Link] with several examples of classical periodic functions.


1- Triangle wave

The formula is:

$f\left(x\right)=\frac{4}{\pi }(\frac{1}{9}\mathrm{cos(3x)}+\frac{1}{25}\mathrm{cos(5x)}+...+\frac{1}{n^2}\mathrm{cos(}nx))$

Or can be written like this...

$f\left(x\right)=\frac{4}{\pi }\sum _{k=1}^{\mathrm{\infty }}\frac{\mathrm{cos(2k+1)}.x}{{\mathrm{(2k+1)}}^2}$

and the corresponding macro...

+++ IJ snippet: Triangle_wave_fourier_series.ijm +++

	// Triangle Wave for ImageJ
	// Jean-Christophe Taveau
	// http://crazybiocomputing.blogspot.com
	w=200;
	h=80;
	newImage("FourierSeries", "32-bit Black", w, h, 1);
	run("Macro...", "code=[v= 4/PI * cos((2*y+1)*x/w*8*PI) / (2*y+1)/ (2*y+1)]");
	exit();

view raw gistfile1.js hosted with ❤ by GitHub

+++ End of IJ snippet +++

 To create the triangle wave, select the whole image (Ctrl+A) then compute the profile (Analyze > Plot Profile or Ctrl +K).

Fig. 1: Triangle wave obtained from the sum of each row of the Fourier Series.

2- Sawtooth wave

Another function defined by ...

The following video shows the evolution of the resulting curve when we add more components.

3- JavaScript

Here is a small script containing all the various functions previously described. For sake of convenience, this script is written in JavaScript but uses exactly the same function Process > Math > Macro...

+++ IJ JavaScript snippet +++

	// Fourier Series of several periodic functions
	// Jean-Christophe Taveau
	// Nov. 2013
	// http://crazybiocomputing.blogspot.com
	var w=256;
	var h=80;
	var names=["Square","Triangle","Sawtooth"];
	if (!showDialog())
	throw "End of script";
	// M A I N
	imp = IJ.createImage("Fourier Series of "+names[type]+" Wave", "32-bit black", w, h, 1);
	if (type==1) // Triangle
	IJ.run(imp, "Macro...", "code=[if (y%2==1) v= 4/PI * cos(y*x/w*8*PI) / y/y]");
	else if (type==2) // Sawtooth
	IJ.run(imp, "Macro...", "code=[v= 2/PI * pow(-1,y+1)*sin(y*x/w*8*PI) / y]");
	else // 0 = Square
	IJ.run(imp, "Macro...", "code=[if (y%2==1) v= 4/PI * sin(y*x/w*8*PI) / y]");
	imp.setRoi(0,0,w,h);
	IJ.run(imp, "Plot Profile", "");
	imp.show();
	// F U N C T I O N S
	function showDialog() {
	gd = new GenericDialog("Fourier Series");
	gd.addChoice("Functions:", names, names[0]);
	gd.showDialog();
	if (gd.wasCanceled())
	return false;
	type = gd.getNextChoiceIndex();
	return true;
	}

view raw FourierSeries.js hosted with ❤ by GitHub

+++ End of IJ JavaScript snippet +++

4- Links

Square wave image [Link]