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By iterating or nesting a sine curve like in this question

I get curves like these: partial sums

that seem to tend to a square wave.

The eight case of these looks like:

degree 8 curve

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 sawtoothwave

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

Dirac comb http://pastebin.com/QMSMQf26 Dirac comb

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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
1  
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

2 Answers 2

up vote 5 down vote accepted

I would say that this is interesting. There has been a lot of work done regarding the study of Gibbs Phenomenon. You should check out the following document to get a better understanding of how it comes up in practical DSP applications:

http://people.clarkson.edu/~ajerri/books/examples/Gibbs_Book.pdf

The typical way to manage Gibbs Phenomenon is to use time domain window functions that taper data at the start and finish. The window functions reduce spurious contributions to frequency domain information that come from discontinuities at the edges of data sequences.

I haven't seen much application of generating signals by composing them with individual sine waves. Generally signal generation is done directly in the time domain. I'm not sure how the function constructions you've documented can be employed to solve a practical problem, but perhaps there is application if you can identify an appropriate problem.

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@B Z : Generating signals by composing them with individual sine waves was used a lot in FM synthesis : good old Sound Blaster MIDI sounds (OPL3 chip), Yamaha DX7 famous synthesier, etc. –  Basj Nov 8 '13 at 20:02

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(...))+...)

enter image description here

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