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Human hearing beats FFT
ncdrawl
post Feb 9 2013, 15:20
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http://phys.org/news/2013-02-human-fourier...-principle.html
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Yaakov Gringeler
post Apr 2 2013, 01:03
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EST is a new transform that can explain the results of the article.

Fourier-related transforms, like FFT, are just one way to find frequencies, and clearly not the best possible.

EST derives frequencies from samples and is unrelated to Fourier/FFT.
The process of EST is deterministic, does not use non-linear equations, and can handle noise.

In the ideal case of a noiseless signal composed of n sinusoids, the frequencies, amplitudes and phases are precisely recovered from 3n
equally spaced real samples.

A noisy signal will require more samples, depending on noise level.

Other than the minimum for the ideal case, accuracy does not depend on the number of samples (time). The additional samples for a noisy signal
are needed to handle noise.

EST can also transform samples into increasing/decreasing sinusoids, which is a better way to model audio. In such a case, for a noiseless
signal, 4 samples are required per increasing/decreasing sinusoid, and more for a noisy signal.

EST can be evaluated using a demo program that implements it. There is also a paper that details the transform and its mathematical basis.

Those interested to see the paper and/or the demo program, can email me at gringya atsign gmail dot com.
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Woodinville
post Apr 2 2013, 23:47
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QUOTE (Yaakov Gringeler @ Apr 1 2013, 17:03) *
Fourier-related transforms, like FFT, are just one way to find frequencies, and clearly not the best possible.

Which, of course, depends entirely on your definition of "Frequency", something that itself is trickier than some seem to realize.
QUOTE
EST derives frequencies from samples and is unrelated to Fourier/FFT.

What does "EST" stand for, in the first place. Does it use a complex exponential or a representation of a complex exponential?

QUOTE
The process of EST is deterministic, does not use non-linear equations, and can handle noise.

Which is true of the Fourier Transform, as well.
QUOTE
In the ideal case of a noiseless signal composed of n sinusoids, the frequencies, amplitudes and phases are precisely recovered from 3n
equally spaced real samples.

Sounds pretty good. What's the basis set you're using? Sounds a lot like a * sin (b *t +c) where a,b,c are the 3 samples. Not sure what "equally spaced" means here, unless you're referring to the fact you can characterize a sine wave with 3 non-degenerate points.
QUOTE
A noisy signal will require more samples, depending on noise level.

No surprise.
QUOTE
Other than the minimum for the ideal case, accuracy does not depend on the number of samples (time). The additional samples for a noisy signal
are needed to handle noise.

EST can also transform samples into increasing/decreasing sinusoids, which is a better way to model audio. In such a case, for a noiseless
signal, 4 samples are required per increasing/decreasing sinusoid, and more for a noisy signal.

So it's Laplace-based instead of Fourier based, then?

Instead of bombarding us with a bunch of not-very-specific qualities, why not just tell us what the basis set is, and how the analysis works?

I am aware of approximately infinite (well, literally infinite but obviously I haven't generated them all!) numbers of basis sets, many of which this could describe.


--------------------
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J. D. (jj) Johnston
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Yaakov Gringeler
post Apr 3 2013, 01:42
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EST stands for Exponential Sum Transform and it uses complex exponentials.

The basis is sigma(c*b^t) where b and c are non-zero complex numbers and the set of b is distinct. If all b are on the unit circle, then it is simply a spectrum.

When all b are on the unit circle and the samples are real, this becomes sigma(a*cos(b*t+c))

The samples must be equally space, not just non-degenerate.

It clearly looks more like Laplace than Fourier, but a specific relation, if exists, is not known to me.

As for describing the analysis, I offered to send the detailed paper. Do you prefer an informal description?

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Posts in this topic
- ncdrawl   Human hearing beats FFT   Feb 9 2013, 15:20
- - probedb   QUOTE (ncdrawl @ Feb 9 2013, 14:20) http:...   Feb 9 2013, 15:51
|- - ncdrawl   QUOTE (probedb @ Feb 9 2013, 09:51) QUOTE...   Feb 9 2013, 16:02
|- - probedb   QUOTE (ncdrawl @ Feb 9 2013, 15:02) it is...   Feb 11 2013, 09:26
- - lvqcl   From the comments: QUOTE The Fourier uncertainty ...   Feb 9 2013, 16:10
- - greynol   That title? What about the title of this discussi...   Feb 9 2013, 18:00
- - Alexey Lukin   Love the title   Feb 9 2013, 20:13
- - greynol   ...with a blunt stick.   Feb 9 2013, 20:17
- - Woodinville   Ad the Gabor limit does not address known frequenc...   Feb 10 2013, 04:41
|- - Kees de Visser   QUOTE (Woodinville @ Feb 10 2013, 04:41) ...   Feb 10 2013, 08:31
|- - Woodinville   QUOTE (Kees de Visser @ Feb 9 2013, 23:31...   Feb 10 2013, 12:51
|- - Porcus   QUOTE (Kees de Visser @ Feb 10 2013, 08:3...   Feb 10 2013, 17:15
- - 2Bdecided   Is it worth spending $25 to read this in full...   Feb 11 2013, 10:25
|- - Kees de Visser   I just stumbled upon this blog. Are we heading tow...   Feb 11 2013, 10:56
||- - Porcus   QUOTE More pointedly: until scientists devise and ...   Feb 11 2013, 11:46
||- - 2Bdecided   QUOTE (Porcus @ Feb 11 2013, 10:46) QUOTE...   Feb 11 2013, 12:21
|- - ojdo   QUOTE (2Bdecided @ Feb 11 2013, 17:25) Is...   Feb 11 2013, 17:13
- - db1989   Given that the writer misspelled mathematical and ...   Feb 11 2013, 16:39
|- - Ethan Winer   QUOTE (db1989 @ Feb 11 2013, 10:39) Given...   Feb 11 2013, 23:05
- - greynol   Would someone mind correlating the article with th...   Feb 11 2013, 16:48
- - benski   Off the top of my head. I might be wrong. But pa...   Feb 11 2013, 18:32
- - 2Bdecided   By taking an FFT, you are performing an (approxima...   Feb 11 2013, 19:02
- - krabapple   Here's what the audiophilosphere reliably take...   Feb 12 2013, 07:18
- - knutinh   Is not some wavelet/filterbank transform more rele...   Feb 12 2013, 09:59
|- - Garf   QUOTE (knutinh @ Feb 12 2013, 09:59) Is n...   Feb 19 2013, 17:27
|- - jmvalin   QUOTE (Garf @ Feb 19 2013, 11:27) Yes, th...   Feb 19 2013, 21:07
- - scuttle   I thought it would be fun and informative to poke ...   Feb 17 2013, 17:33
- - Willakan   Good God, I have been reading/getting PMed on othe...   Feb 18 2013, 20:03
|- - Porcus   QUOTE (Willakan @ Feb 18 2013, 20:03) or ...   Feb 18 2013, 20:22
|- - Woodinville   QUOTE (Willakan @ Feb 18 2013, 11:03) Goo...   Feb 19 2013, 00:44
- - 2Bdecided   It's amazing what people conclude, given that ...   Feb 19 2013, 10:53
- - lvqcl   I didn't find any mention of FFT in this artic...   Feb 19 2013, 15:51
|- - greynol   QUOTE (lvqcl @ Feb 19 2013, 06:51) I didn...   Feb 19 2013, 16:56
- - Canar   The first thing that comes to my mind is: Now how ...   Feb 19 2013, 16:46
|- - Garf   QUOTE (Canar @ Feb 19 2013, 16:46) The fi...   Feb 19 2013, 17:32
- - Paulhoff   For those in the know.......... The Princess an...   Feb 20 2013, 19:51
- - Woodinville   It's still a confused headline. Recognzing one...   Feb 21 2013, 04:55
|- - Garf   QUOTE (Woodinville @ Feb 21 2013, 04:55) ...   Feb 21 2013, 07:54
|- - Woodinville   QUOTE (Garf @ Feb 20 2013, 22:54) QUOTE (...   Feb 22 2013, 04:22
- - dhromed   Yeah, we don't want to turn HA into an extensi...   Feb 21 2013, 10:02
- - 2Bdecided   Never mind the title, I still don't find a sat...   Feb 25 2013, 13:14
|- - knutinh   QUOTE (2Bdecided @ Feb 25 2013, 13:14) Ne...   Feb 25 2013, 15:32
||- - 2Bdecided   QUOTE (knutinh @ Feb 25 2013, 14:32) I gu...   Feb 25 2013, 17:24
|- - jmvalin   QUOTE (2Bdecided @ Feb 25 2013, 07:14) Ne...   Feb 26 2013, 00:10
- - Woodinville   There are a number of issues confused in this thre...   Feb 26 2013, 04:30
|- - Alexey Lukin   QUOTE (Woodinville @ Feb 25 2013, 23:30) ...   Mar 14 2013, 17:39
- - 2Bdecided   Thank you JJ.   Feb 26 2013, 10:57
- - krabapple   http://arstechnica.com/science/2013/02/hum...3s-so...   Feb 27 2013, 01:51
- - greynol   Well the quality of the comments look pretty encou...   Feb 27 2013, 02:44
|- - Woodinville   QUOTE (greynol @ Feb 26 2013, 17:44) Well...   Feb 27 2013, 05:55
- - 2Bdecided   Interesting paper, thank you.   Mar 15 2013, 12:15
- - Yaakov Gringeler   EST is a new transform that can explain the result...   Apr 2 2013, 01:03
|- - Woodinville   QUOTE (Yaakov Gringeler @ Apr 1 2013, 17...   Apr 2 2013, 23:47
|- - Yaakov Gringeler   EST stands for Exponential Sum Transform and it us...   Apr 3 2013, 01:42
|- - Porcus   I think I could very well use a formula or two ......   Apr 3 2013, 20:34
- - Alexey Lukin   Yaakov, also check out the Reassigned spectrogram ...   Apr 2 2013, 23:59
- - Canar   I think a lot of us here would be interested in a ...   Apr 3 2013, 05:27
|- - Yaakov Gringeler   QUOTE (Canar @ Apr 3 2013, 05:27) I think...   Apr 3 2013, 18:14
- - db1989   If I may guess, I think he means that this site ha...   Apr 3 2013, 18:31
- - Yaakov Gringeler   The following link: http://www.mediafire.com/view...   Apr 3 2013, 22:10
- - Woodinville   Hm. Define "noiseless". Most instrumen...   Apr 11 2013, 11:09
- - Yaakov Gringeler   The paper described the mathematical basis of EST,...   Apr 11 2013, 19:33
|- - Woodinville   QUOTE (Yaakov Gringeler @ Apr 11 2013, 11...   Apr 11 2013, 20:36
|- - Yaakov Gringeler   QUOTE (Woodinville @ Apr 11 2013, 20:36) ...   Apr 11 2013, 21:32
|- - Woodinville   QUOTE (Yaakov Gringeler @ Apr 11 2013, 13...   Jun 4 2013, 01:51
- - Specy   Over 10 years ago, for my master thesis, I wrote a...   Aug 17 2013, 11:52
- - Yaakov Gringeler   Several months ago, in posts in this topic, I prov...   Nov 4 2013, 20:15


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