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 Posted: Mon Oct 05, 2015 9:05 pm

Joined: Thu Aug 28, 2008 1:17 am
Posts: 591
I'm thinking in terms of audio. Is there a significance in the relationship of the derivative to the main function/waveform?
Something like sin^3(x) and its derivative 3sin^2(x)cos(x). Maybe in a situation where the derivative is the modulator?

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 Posted: Tue Oct 06, 2015 1:32 am

Joined: Sun Mar 19, 2006 9:44 pm
Posts: 953
Location: Japan
I have no idea about the main question, since I'm not a trig guy.

However, if you do a little mental visualization, it's easy to see that the derivative of a triangle wave is a square wave. And the derivative of a sawtooth wave is probably some mutant pulse wave with 99% duty cycle and a horrendous drop. And anything in between is probably covered in Casio's keyboard patents

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 Posted: Tue Oct 06, 2015 7:15 am

Joined: Sun Sep 19, 2004 11:12 pm
Posts: 20261
Location: NE Indiana, USA (NTSC)
ccovell wrote:
And the derivative of a sawtooth wave is probably some mutant pulse wave with 99% duty cycle and a horrendous drop.

Which is called a Dirac comb.

Derivative is a high pass filter at 6 dB/octave.
Integral is a low pass filter at 6 dB/octave.

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 Posted: Tue Oct 06, 2015 7:25 am

Joined: Sun Jan 22, 2012 12:03 pm
Posts: 6403
I suppose the question is whether there are audio applications of calculus? The answer is of course yes, there are tons.

Derivatives are often used when interpolating samples. You might want to create an interpolator by defining a curve that is smooth at the first or second derivative, etc.

The idea of DPCM is essentially recovering a PCM signal from it's discrete derivative, so in that case it's being used for compression.

When doing a pitch bend or time stretch that varies over time (e.g. slowly slowing down), you would take an integral of the function that controls speed to determine the resulting recording length.

The Fourier Transform is defined as a series of integrals. If you're working with samples you do this in discrete steps, normally, but if you were doing theoretical work on an ideal waveform, you can do it algebraically and come up with with a function that describes the frequency content of that waveform.

Similarly there's all sorts of applications of calculus in filter design, etc.

I don't know how to begin to answer this question properly, because the applications are endless and pervasive of the field. If you've done any audio programming you've probably already used various discrete calculus solutions, knowingly or unknowingly.

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 Posted: Wed Oct 07, 2015 4:45 pm

Joined: Mon Sep 27, 2004 2:57 pm
Posts: 1249
A sinc pulse is the derivative of an edge of a square wave. The best available anti-aliasing for square waves is achieved by using windowed sinc pulses. You have a large table of sinc pulses that are sampled at various subsample offsets, and you have the output buffer. To create the square wave's edge, you add/subtract the appropriate sinc pulse to/from the output buffer. Remember that square wave edges are nearly always between two samples, so that's why you need the sinc pulses sampled at different subsample offsets.

Sinc pulses are several samples long, and in a square wave, they'll often overlap each other. If you were just copying a square wave edge into the output, you'd be overwriting other samples, so that's why using the sinc pulse (which, again, is the derivative of a square wave's edge) and just adding/subtracting it into the output (which is basically integration) is the better way to do this type of anti-aliasing, you're inserting the edge into the output without erasing the ripples of other edges in the output.

There you go, that's one practical use of sound wave derivatives.

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 Posted: Wed Oct 07, 2015 7:52 pm

Joined: Thu Aug 28, 2008 1:17 am
Posts: 591
Thanks for the replies. I was actually thinking more along the lines of synthesis type sound for instruments and such. The harmonic relationship between certain functions and their derivatives. Something like a timbre relationship or such. I.e. one being a modulator of the other, or some other synthesis blend/type.

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