"glen herrmannsfeldt" <gah@ugcs.caltech.edu> wrote in message news:hN2dnTnho6-UvkTZnZ2dnUVZ_rGdnZ2d@comcast.com... ...........................> > In the case of the Fourier transform the boundary conditions > are periodic by definition.I think you mean: Discrete Fourier Transform. Even if that is the transform one has in mind to use, there remain conditions for it to be applicable. The *operation*, e.g. DFT, does not define the data. It's usually the other way around. The data defines what operations are suitable. I don't think it's reasonable to flat out deny continuous transforms of infinite extent. Fred
Strange Discrete Fourier Transform question
Started by ●July 30, 2006
Reply by ●August 9, 20062006-08-09
Reply by ●August 9, 20062006-08-09
Fred Marshall wrote: (I wrote)>>In the case of the Fourier transform the boundary conditions >>are periodic by definition.> I think you mean: Discrete Fourier Transform.Actually, I was going to write Fourier series, but that doesn't fit with DFT. Why isn't it called DFS, anyway?> Even if that is the transform one has in mind to use, there remain > conditions for it to be applicable. The *operation*, e.g. DFT, does not > define the data. It's usually the other way around. The data defines what > operations are suitable.> I don't think it's reasonable to flat out deny continuous transforms of > infinite extent.If your source has infinite extent, I agree. What I don't agree with is people with a finite extent wanting to zero extend the signal to +/- infinity. -- glen
Reply by ●August 9, 20062006-08-09
Fred Marshall wrote: ...> I don't think it's reasonable to flat out deny continuous transforms of > infinite extent.Nor is it reasonable to bring them up in a sampled context. :-) Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●August 9, 20062006-08-09
glen herrmannsfeldt wrote: ...> fit with DFT. Why isn't it called DFS, anyway?Because thinking about these distinctions has been sloppy from the outset. Not only should it be DFS, but FFS as well. Every Fourier series with a line spectrum is periodic. In the limit, as line spacing --> zero, a Fourier transform results. We can't do that with present computers. ... Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●August 9, 20062006-08-09
Jerry Avins wrote:> glen herrmannsfeldt wrote:>> Why isn't it called DFS, anyway?> Because thinking about these distinctions has been sloppy from the > outset. Not only should it be DFS, but FFS as well.I wonder what Gauss called it?> Every Fourier series with a line spectrum is periodic. In the limit, as > line spacing --> zero, a Fourier transform results. We can't do that > with present computers.Algorithms are often described in O() notation, with the limit as N goes to infinity, even though it never does on real machines. -- glen
Reply by ●August 9, 20062006-08-09
glen herrmannsfeldt wrote:> Jerry Avins wrote:...>> Every Fourier series with a line spectrum is periodic. In the limit, >> as line spacing --> zero, a Fourier transform results. We can't do >> that with present computers. > > Algorithms are often described in O() notation, with the limit > as N goes to infinity, even though it never does on real > machines.There you nailed down the treatment: unless the algorithm is O(1), we can't make N infinite on present computers. That's not to say that an asymptotic approximation easily used for large but finite N is useless. For large distances from the origin, XY = <constant> may well be treated as XY = 0, which is often a simplification. Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●August 10, 20062006-08-10
"Jerry Avins" <jya@ieee.org> wrote in message news:z4edncOTiNcFZETZnZ2dnUVZ_radnZ2d@rcn.net...> Fred Marshall wrote: > > ... > >> I don't think it's reasonable to flat out deny continuous transforms of >> infinite extent. > > Nor is it reasonable to bring them up in a sampled context. :-) > > Jerry > --Jerry, It is if one needs to go through a constructive description of possibilities that are related. I find no problem relating continuous and sampled / finite and infinite in the proper discussion. Fred
Reply by ●August 10, 20062006-08-10
Fred Marshall wrote:> "Jerry Avins" <jya@ieee.org> wrote in message > news:z4edncOTiNcFZETZnZ2dnUVZ_radnZ2d@rcn.net... >> Fred Marshall wrote: >> >> ... >> >>> I don't think it's reasonable to flat out deny continuous transforms of >>> infinite extent. >> Nor is it reasonable to bring them up in a sampled context. :-) >> >> Jerry >> -- > > Jerry, > > It is if one needs to go through a constructive description of possibilities > that are related. I find no problem relating continuous and sampled / > finite and infinite in the proper discussion.You're right that "never say never" is correct here, but the continuous properties don't illuminate the samples cases one for one. Too often we -- at least I -- have to fight to keep the distinctions in mind. Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●August 10, 20062006-08-10
"Jerry Avins" <jya@ieee.org> wrote in message news:4LudnbtfnrNBqkbZnZ2dnUVZ_s6dnZ2d@rcn.net...> Fred Marshall wrote: >> "Jerry Avins" <jya@ieee.org> wrote in message >> news:z4edncOTiNcFZETZnZ2dnUVZ_radnZ2d@rcn.net... >>> Fred Marshall wrote: >>> >>> ... >>> >>>> I don't think it's reasonable to flat out deny continuous transforms of >>>> infinite extent. >>> Nor is it reasonable to bring them up in a sampled context. :-) >>> >>> Jerry >>> -- >> >> Jerry, >> >> It is if one needs to go through a constructive description of >> possibilities that are related. I find no problem relating continuous >> and sampled / finite and infinite in the proper discussion. > > You're right that "never say never" is correct here, but the continuous > properties don't illuminate the samples cases one for one. Too often we -- > at least I -- have to fight to keep the distinctions in mind. > > Jerry > --Yes. My favorite is sinc vs. Dirichlet ..... infinite vs. periodic. Then, periodic in one domain leads to discrete in the other .... So, discrete in one domain doesn't imply discrete in the other! Seems folks often forget that little factoid - or perhaps don't even realize it what with computers and FFTs all over the place. Fred
