Human hearing beats FFT 
Human hearing beats FFT 
Feb 9 2013, 15:20
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#1


Group: Members Posts: 37 Joined: 5October 08 Member No.: 59436 



Apr 2 2013, 01:03
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#2


Group: Members Posts: 7 Joined: 1April 13 Member No.: 107483 
EST is a new transform that can explain the results of the article. Fourierrelated 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 nonlinear 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. 


Apr 2 2013, 23:47
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#3


Group: Members Posts: 1403 Joined: 9January 05 From: JJ's office. Member No.: 18957 
Fourierrelated 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 nonlinear 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 nondegenerate 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 Laplacebased instead of Fourier based, then? Instead of bombarding us with a bunch of notveryspecific 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.  
J. D. (jj) Johnston 


Apr 3 2013, 01:42
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#4


Group: Members Posts: 7 Joined: 1April 13 Member No.: 107483 
EST stands for Exponential Sum Transform and it uses complex exponentials.
The basis is sigma(c*b^t) where b and c are nonzero 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 nondegenerate. 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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