On May 4, 7:09�pm, robert bristow-johnson <r...@audioimagination.com> wrote:> On May 3, 5:09�pm, dbd <d...@ieee.org> wrote: > > > > > > > On May 3, 1:09�pm, robert bristow-johnson <r...@audioimagination.com> > > wrote: > > ... > > > > and the mapping is invertible. �really the DFT and iDFT are sorta the > > > same thing because they can be redefined to be "unitary" by adjusting > > > the constant in front of the summation: > > ... > > > > and +j and -j are qualitatively the same thing. �both have equal claim > > > to be the sqrt(-1). again n0 and k0 can be whatever integer you like. > > > ... > > > r b-j > > > It isn't clear to me from what you have posted: Do you intend the > > concepts of "sorta the same thing" and "qualitatively the same thing" > > to apply to items that are functions of terms that are the square > > roots of any real number or only the square roots of the same negative > > real numbers? (I couldn't find either term in wikipedia or in the new > > or old testaments of O&S.) > > this is what i mean in Wikipedia: > > �http://en.wikipedia.org/wiki/Imaginary_unit#i_and_.E2.88.92i > > +j and -j are qualitatively the same thing. �everything you say about > +j is something you can say about -j. �but they are quantitatively > *not* the same thing. �they are non-zero and are negatives of each > other. �they are not the same number. �but their roles can be > interchanged. �this qualitative equivalence is not true for the real > unit +1 and its negative -1. �this is why imaginary numbers are more > different than real numbers than just a matter of convention (the real > axis is qualitatively different than the imaginary axis). �but the > difference between +j and -j is merely one of convention. �and that > means that (with scaling made so that the transforms are "unitary") > the only difference between the forward and inverse Fourier Transform > (and the DFT vs. iDFT) is merely a matter of convention. > > On May 3, 6:54 pm, glen herrmannsfeldt <g...@ugcs.caltech.edu> wrote: > > > robert bristow-johnson <r...@audioimagination.com> wrote: > > > (snip) > > > > and +j and -j are qualitatively the same thing. �both have equal claim > > > to be the sqrt(-1). > > ... > > > You can change all +j to -j, but some is not enough. > > yup. �if all textbooks and literature in math, physics, EE or whatever > science or discipline, where changed so that +j and -j �(and +i and - > i) were switched, every fact and theorem would continue to be just as > valid. > > r b-j- Hide quoted text - > > - Show quoted text -It is wierd though that nature often prefers one way over another. Even humans who in general have bilateral symmetry has some asymmetries that happen mostly one way. For example the heart being left of center and the left lung having only 2 lobes unlike the right lung's 3. But every now and then nature will mess up and about 1 in 60000 individuals (wiki reports fewer than 1 in 10000) will have their internal organs mirror reversed. You can read about it here: http://en.wikipedia.org/wiki/Situs_inversus The trick is how can nature code for one way over another. I've read some interesting articles on how genetic coding picks one direction over another. Even many basic life molecules have stereo isomers (mirror image arrangments of their atoms) which behave chemically very different. The amino acids making up the proteans in living organisms are all the "left hand" variety. The handedness stems from which way they rotate polarized light - CW or CCW. Often a bening left hand molecule becomes "drug like" when it is converted over to a right hand version and introduced into a world whose life forms use mostly left hand molecules. Glen has already mentioned the CP and CPT (Charge, Parity, Time) symmetries and how CP is not always conserved. That discovery really shook up things. Another example of ambiguity is the difference between electron and conventional current and its direction of flow. Common symbols on schematic diagrams (diodes, transistors, etc) have arrows pointing in the direction of conventional current. But a simple Hall Effect measurement will reveal the polarity of the moving charges, and a right Crookes tube will show their direction of motion. But yes you can usually swap "j" and "-j" and as long as you consistantly do it, your math tends to work out. We see this in the argument of the exponential kernal in defintions of Fourier and inverse Fourier transforms. Clay
simple questions about the DFT and its limits of integration
Started by ●May 3, 2010
Reply by ●May 5, 20102010-05-05
Reply by ●May 5, 20102010-05-05
Clay <clay@claysturner.com> wrote: (snip)> It is wierd though that nature often prefers one way over another. > Even humans who in general have bilateral symmetry has some > asymmetries that happen mostly one way. For example the heart being > left of center and the left lung having only 2 lobes unlike the right > lung's 3. But every now and then nature will mess up and about 1 in > 60000 individuals (wiki reports fewer than 1 in 10000) will have their > internal organs mirror reversed. You can read about it here: > http://en.wikipedia.org/wiki/Situs_inversusAs I remember it (first from some murder mystery show) it is most common in twins. The handedness starts out with a chemical concentration gradient. In the case of twins, the source from the other twin is often enough to reverse the original handedness. (snip)> Another example of ambiguity is the difference between electron and > conventional current and its direction of flow. Common symbols on > schematic diagrams (diodes, transistors, etc) have arrows pointing in > the direction of conventional current. But a simple Hall Effect > measurement will reveal the polarity of the moving charges, and a > right Crookes tube will show their direction of motion.Hall effect will give you the average sign of the flowing current carriers. Many metals, aluminum being a prime example, have large hole and electron bands. At high magnetic fields, aluminum has more holes than electrons as current carriers. At lower fields, the electrons are in a slim majority. As aluminum is used for most long distance power lines, positive charge carriers (holes) account for a large fraction of distributed power. As it is usually at higher voltage and lower current, I am not so in the case of current. Though with the overall length of long distance power lines, integrating over volume should be enough to swamp the current carrying copper.> But yes you can usually swap "j" and "-j" and as long as you > consistantly do it, your math tends to work out. We see this in the > argument of the exponential kernal in defintions of Fourier and > inverse Fourier transforms.-- glen
