P
Phil Hobbs
Guest
whit3rd wrote:
Bollocks. An FFT is an information-preserving operation, unlike
least-squares fits.
> there\'s zero difference between the transform\'s inversion and the original data,
Right, i.e. it\'s not a least-squares fit, it\'s exact.
> which (zero) is obviously the minimum of sum-of-squares-of-differences.
You\'re maybe thinking of a continuous-time orthonormal-function
expansion, e.g. a Fourier or Bessel or Chebyshev series. In that case,
_truncating_ the series leads to the least-squares optimum for that
order. Least squares optima tend not to be that useful--the infamous
\"Gibbs phenomenon\" being a typical example.
But there are a lot, a lot of ways of producing a finite expansion that
don\'t have that problem, just as there are all sorts of ways of
controlling noise gain in deconvolution.
Dividing by an inverse function is precisely a weighting operation.
What makes that the One True Algorithm? Why assume that all frequencies
have to have the same SNR? That\'s not at all common in real life, and
it\'s often a win to sacrifice a significant amount of SNR for improved
resolution, data rate, or what have you. It\'s horses for courses.
And none of that has anything to do with how you perform the actual
deconvolution.
Cheers
Phil Hobbs
--
Dr Philip C D Hobbs
Principal Consultant
ElectroOptical Innovations LLC / Hobbs ElectroOptics
Optics, Electro-optics, Photonics, Analog Electronics
Briarcliff Manor NY 10510
http://electrooptical.net
http://hobbs-eo.com
On Thursday, January 5, 2023 at 1:41:38 PM UTC-8, Phil Hobbs wrote:
whit3rd wrote:
[about FFT/divide/inverseFFT deconvolution]
...the FFT algorithm has no mechanism to
accept data with non-constant signficance, which is what, obviously,
happens with a divide-by-almost-zero step in the data processing.
It\'s gonna give you what the \'signal\' says, not what the \'signal\' and known
signal/noise ratio, tell you. That means using an FFT for the inverse is
excessively noise-sensitive. There\'s OTHER ways to do a Fourier inversion
that do allow the noise estimate its due influence.
The problem has nothing to do with the FFT, and everything to do with
what you\'re trying to do with it. Dividing transforms is a perfectly
rational way to deconvolve, provided you take into account the
finite-length effects and prepare the denominator correctly.
Think again; an FFT algorithm implements least-squares fitting, essentially;
Bollocks. An FFT is an information-preserving operation, unlike
least-squares fits.
> there\'s zero difference between the transform\'s inversion and the original data,
Right, i.e. it\'s not a least-squares fit, it\'s exact.
> which (zero) is obviously the minimum of sum-of-squares-of-differences.
You\'re maybe thinking of a continuous-time orthonormal-function
expansion, e.g. a Fourier or Bessel or Chebyshev series. In that case,
_truncating_ the series leads to the least-squares optimum for that
order. Least squares optima tend not to be that useful--the infamous
\"Gibbs phenomenon\" being a typical example.
But there are a lot, a lot of ways of producing a finite expansion that
don\'t have that problem, just as there are all sorts of ways of
controlling noise gain in deconvolution.
Dividing by an inverse function is precisely a weighting operation.
But, it\'s not correct if the standard deviations of the elements are not identical,
because it IS minimizing sum-of-squares of differences, rather than the
(correct) sum of (squares-of-differences/sigma-squared-of-this-element).
What makes that the One True Algorithm? Why assume that all frequencies
have to have the same SNR? That\'s not at all common in real life, and
it\'s often a win to sacrifice a significant amount of SNR for improved
resolution, data rate, or what have you. It\'s horses for courses.
And none of that has anything to do with how you perform the actual
deconvolution.
Cheers
Phil Hobbs
--
Dr Philip C D Hobbs
Principal Consultant
ElectroOptical Innovations LLC / Hobbs ElectroOptics
Optics, Electro-optics, Photonics, Analog Electronics
Briarcliff Manor NY 10510
http://electrooptical.net
http://hobbs-eo.com