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# mathmatically modeled audio

use regression to model sound as math functions.
 (+2, -4) [vote for, against]

a traditional "lossless" audio format such as wav is comprised of a 1 dimensional sample (volume) refreshed thousands of time per second (up to 192.1) . Currently 24 bit audio allows 16 million possible loudness levels which generally gives more than enough dynamic range.

The problem is if you zoom in extremely close it looks like a stair step instead of a fluid sine shape. This also creates large files as you as 24 bit 96khz sample rate wav file will be 4 mb/sec of data.

I suggest using trigonometry and physics to model sound as a sine based function. physics modeling already exists for drums and other instruments and the effects can be eerily life like. These programs model how the an object vibrates a surface to make sound. Drums are "easy" (but still hard to figure out the math)

Basically one would use a regression analysis to find the best fit curve of an audio signal. You would use a variable "function rate" instead of a variable bit rate, to determine how long each sine wave function would be. You would also use addition to add multiple sine waves together instead of using many shorter sine waves to describe the sound.

The payoff? Extremely small bit rate for lossless audio and high sound quality, plus dynamic range and frequency response can be arbitrary as you can define a waveform as high frequency or as loud as you want (limited by hardware). Using a raster based approach is a brute force method of describing sound.

the savings come because a single waveform function on general would be much longer than a .wav bit rate. I'm guessing you would only need a function rate of around 100-1Khz to successfully design an audio signal. The sine wave function could probably be parsed down to a few hundred bits, it would comprise of a sine function or linear function of say 3 digits of say 2 digits of precision then a multiplier to define loudness a multiplier to determine frequency and a marker to define the stop. As long as the sine wave function size x sample rate was lower than a similar lossless audio format file size could be remarkbly smaller.

The trade off would be increased complexity in encoding. (however sound processing is no longer considered computationally intense)

The alternative method would be to describe the sound at a fixed sample rate some fraction of the maximum desired hearable frequency (22khz). For example if you picked a 11khz frequency rate. Each function would describe 1/2 of a sine wave at 22 hz. Which might allow for less information.

 — metarinka, Jan 19 2011

Stair-steps in digital audio http://www.tomshard...nyl-digital#t351127
A good post about the stair-step myth. The rest of the thread has things to say about it too. [Wrongfellow, Jan 19 2011]

Wavelets http://en.wikipedia.org/wiki/Wavelet
[Jinbish, Jan 19 2011]

Audiophiles http://xkcd.com/841/
//The problem is if you zoom in extremely close it looks like a stair step instead of a fluid sine shape.// Well don't look so bloody close! [Jinbish, Jan 19 2011]

vector-mapped_20waveforms A nearly identical idea from 2003, but specifically aimed at software synth. music. [spidermother, Apr 15 2011]

Steps http://en.wikipedia...i/Steps_%28group%29
Unlistentoable? [pocmloc, Apr 15 2011]

Audio Reconstruction filter http://en.wikipedia...construction_filter
After a DAC [csea, Apr 16 2011]

What you're referring to is called 'resynthesis'.
 — FlyingToaster, Jan 19 2011

 //The problem is if you zoom in extremely close it looks like a stair step instead of a fluid sine shape.//

 Only if the reconstruction filter is broken (or if you're looking at the signal prior to filtering).

 The idea that sampled audio is made up of "stair-steps" is one of the simplifying myths that gets used to avoid exposing people to more mathematical detail than they need.

 Did you get taught in school that electrons moved around the nucleus like planets orbiting the sun? Did they then reveal to you a couple of years later that this was actually a complete lie?

Stair-steps in sampled waveforms are the same kind of lie.
 — Wrongfellow, Jan 19 2011

The usual challenge for lossless codecs is pink noise (equal energy per octave.) As described, it's highly unlikely that "regression" would reduce bit rate. [-]
 — csea, Jan 19 2011

I grantee that your speakers cannot reproduce that stairway.
 — WcW, Jan 19 2011

 Would this not actually be quite complicated mathematics? The regression stuff would be non-linear, Shirley?

Where's Fourier when you need him?
 — Jinbish, Jan 19 2011

 my faux pas about the stair steps aside.

Would there not be gains to reprsenting sound (waveforms) mathmatically as wave forms. instead of representing them as raster images of millions of discrete points? at the very least I think there would be file size savings.
 — metarinka, Jan 20 2011

Well, one issue is periodicity. For example, any periodic waveform can by represented by a series of sinusoids. As soon as you lose the periodicity, you (theoretically) need infinite sinusoids.
 — Jinbish, Jan 20 2011

 Yes, there are the advantages you've mentioned... mind you, I'm talking about music synthesizers, not... whatever it is you're referring to.

They may not be stairsteps but D/A output is hardly smooth: depends on what the slew rate is between positions.
 — FlyingToaster, Jan 20 2011

There are entire fields of mathematics dedicated to baking this. Look up compression (lossless), compression (lossy), filter theory, sampling theory, signal processing, and psychoacoustics.
 — AntiQuark, Jan 20 2011

the air that sound travels through cannot transmit "steps".
 — WcW, Apr 15 2011

It can, but then your speaker would move reactionlessly across the floor.
 — MaxwellBuchanan, Apr 15 2011

It can, but they're called 'shockwaves'.
 — spidermother, Apr 15 2011

it can but [Vernon] would call it AAARRRTTTTP, or some such...
 — 4whom, Apr 15 2011

Any practical continuous signal DAC (digital to analog converter) will be followed by a reconstruction filter to remove the stairsteps. [link]
 — csea, Apr 16 2011

 [annotate]

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