From: Chuck Simmons
Organization: You jest.
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Subject: Re: How does a mixer work?
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Date: Fri, 25 Oct 2002 15:28:21 GMT
NNTP-Posting-Date: Fri, 25 Oct 2002 08:28:21 PDT
Chuck Simmons wrote:
> Kevin Aylward wrote:
> > Yes, the Fourier transform is well defined, i.e. F(f(t)).
> > The standard deviation is well defined Sd(x)
> By standard deviation, do you mean L2 metric distance rather than the
> usual meaning from probability? If so, is there any particular reason
> why you could not say that you mean L2 metric distance? I don't
> personally refer to L2 metric distance as standard deviation. It
> probably has to do with the company I keep.
Thanks for a mention of D. Gabor. I looked again at the section in
"Principles of Optics" by Born and Wolf on image reconstruction from
diffraction. Although the inequalty you give is not mentioned in Born
and Wolf, Gabor's motivation in seeking it is quite clear. In electron
microscopy of the time (circa 1955), there there were resolution
problems. Gabor looked at the possibility of viewing an electron
diffraction pattern in visible light and reconstructing an image
magnified by the ratio of wavelengths. He realized that it was
impossible to capture the entire diffraction pattern and since long
period artifacts in diffraction correspond to short period artifacts in
the image, he needed some sort of idea of how much resolution he would
lose given the limits on the diffraction pattern he could capture.
Diffraction is approximately modeled with the Fourier transform in this
case and he was able to obtain a very gross estimate of loss of
resolution due to the limits on capturing the diffraction pattern. Since
the selection of delta t and delta omega are completely arbitrary, the
estimate is gross.
Gabor's work, while interesting, hardly qualifies as fundamental in
Fourier analysis. The work was very important, however, in other ways.
... The times have been,
That, when the brains were out,
the man would die. ... Macbeth
Chuck Simmons email@example.com