# How much of a problem is the Gibbs phenomenon and would this solve it?

By iterating or nesting a sine curve like in this question

I get curves like these:

that seem to tend to a square wave.

The eight case of these looks like:

which here on purpose was chosen for it's roundness.

The Mathematica code for these plots can be found here: http://pastebin.com/6UK1u1uX

I don't know much about signal processing but I recalled the Gibbs phenomenon in square waves after I saw these curves.

Would they solve the problem with the Gibbs phenomenon in the case of square waves?

In the Fourier transform this kind of function is not of any use though I understand.

Edit 13.1.2013:

Sawtooth wave: http://pastebin.com/JNg7bzzB

Triangular wave (partial sums instead of integrals): http://pastebin.com/wRCBV7NF

Dirac comb http://pastebin.com/QMSMQf26

• Sorry, can you clarify? What specifically is your question? What is the problem you are trying to solve? – Hilmar Jan 12 '13 at 18:04
• @Hilmar I just happened to find this kind of curve and was hoping that there would some use for it since it is so smooth. But I don't know the Gibbs phenomenon in practice well enough to tell if a new kind of trigonometry based curve would help it. – Mats Granvik Jan 12 '13 at 18:17
• In wikipedia I find this part of a paragraph: "In signal processing, the Gibbs phenomenon is undesirable because it causes artifacts, namely clipping from the overshoot and undershoot, and ringing artifacts from the oscillations. " – Mats Granvik Jan 12 '13 at 18:41
• I see. While it sure is interesting, I can't think of a useful application. While it avoids the Gibbs phenomon of Fourier synthesis, Square waves can be easily made directly in the time domain. If you need a smoothly deformed sine wave this can be easily done with by running a sine wave a "soft clipper", i.e. a static non-linearity. – Hilmar Jan 12 '13 at 19:31
• can your recursive sine waves be used to decompose other waveforms like sawtooth or triangle waves or arbitrary functions? – endolith Jan 12 '13 at 20:26

Recursive sinusoids is the basic principle of FM synthesis (used in famous Yamaha DX7 etc.) : with such synthesis, oscillators (named "operators") can be added but also embedded like this : sin(sin(t+sin(...))+...)