Martin Eisenberg wrote:> Jerry Avins wrote: > >> I hit /send/ as is instead of (as last and this time /send in >> RTF-8/). How is this: Ω (capital omega); ω (small omega); ° >> (degree); µ (mu). > > Degree and mu might just be safe because they are in the ANSI > charset, which Unicode keeps in place.Unicode has 2 µ (mu) symbols. The one in the ASCII subset is called "micro sign"; this μ after lambda is called "Greek Small Letter Mu". Jerry -- Engineering is the art of making what you want from things you can get. ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
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Duality in continous Fourier transform
Started by ●October 28, 2008
Reply by ●October 29, 20082008-10-29
Reply by ●October 29, 20082008-10-29
Jerry Avins <jya@ieee.org> writes:> > Unicode has 2 µ (mu) symbols. The one in the ASCII subset is called > "micro sign"; this μ after lambda is called "Greek Small Letter Mu". >The first one (I'm assuming micro sign) renders for me (gnus/emacs under Solaris 2.8) as a substantially smaller glyph than the second one. Just another data point. --L
Reply by ●October 29, 20082008-10-29
On Wed, 29 Oct 2008 15:07:02 -0400, Jerry Avins <jya@ieee.org> wrote:>Eric Jacobsen wrote: >> On Wed, 29 Oct 2008 08:51:05 -0700 (PDT), robert bristow-johnson >> <rbj@audioimagination.com> wrote: >> >>> On Oct 29, 9:33 am, Jerry Avins <j...@ieee.org> wrote: >>>> paulz wrote: >>>>> Let's say Fourier Transform (FT) of x(t) is X(�) and duality property says >>>>> FT of X(t) is 2�x(-�). >>>>> We know FT of dx(t)/dt is j�X(�). From above duality property, we can >>>>> get FT of tX(t) is 2�jdx(�)/d�. However, from this property, we can't >>>>> get FT of tx(t), which may sometimes cause confusion. >>>> Does everyone see those lovely omegas? I'd much rather write � than w, >>>> but I've been concerned that it would come through as '?' or worse. >>>> >>> i can see the omega � and pi � (reading with Google Groups), but i >>> don't know what key sequence to hit to get it. here i just copied and >>> pasted. maybe someday we'll be able to use LaTeX paste-up on USENET >>> to *really* present our meaning clearly. >>> >>> i still think (paulz) that representing the FT with "f" instead of "�" >>> in it results in a much simpler representation of the Duality and >>> Convolution properties and Parseval's theorem. >>> >>> r b-j >> >> Bleah. In Jerry's post I'm seeing question marks where there's >> evidently supposed to be an omega. In rb-j's post I'm seeing some >> funky foreign (unidentifiable) characeters. >> >> I'm using Free Agent to read newgroups. > >I hit /send/ as is instead of (as last and this time /send in RTF-8/). >How is this: ? (capital omega); ? (small omega); � (degree); � (mu). > >JerryBoth omegas look like question marks to me, but the degree and mu are fine. Eric Jacobsen Minister of Algorithms Abineau Communications http://www.ericjacobsen.org Blog: http://www.dsprelated.com/blogs-1/hf/Eric_Jacobsen.php
Reply by ●October 29, 20082008-10-29
On Oct 29, 1:23�pm, Eric Jacobsen <eric.jacob...@ieee.org> wrote:> On Wed, 29 Oct 2008 08:51:05 -0700 (PDT), robert bristow-johnson > > > > <r...@audioimagination.com> wrote: > >On Oct 29, 9:33�am, Jerry Avins <j...@ieee.org> wrote: > >> paulz wrote: > >> > Let's say Fourier Transform (FT) of x(t) is X(�) and duality property says > >> > FT of X(t) is 2�x(-�). > > >> > We know FT of dx(t)/dt is j�X(�). From above duality property, we can > >> > get FT of tX(t) is 2�jdx(�)/d�. However, from this property, we can't > >> > get FT of tx(t), which may sometimes cause confusion. > > >> Does everyone see those lovely omegas? I'd much rather write � than w, > >> but I've been concerned that it would come through as '?' or worse. > > >i can see the omega � and pi � (reading with Google Groups), but i > >don't know what key sequence to hit to get it. �here i just copied and > >pasted. �maybe someday we'll be able to use LaTeX paste-up on USENET > >to *really* present our meaning clearly. > > >i still think (paulz) that representing the FT with "f" instead of "�" > >in it results in a much simpler representation of the Duality and > >Convolution properties and Parseval's theorem. > > >r b-j > > Bleah. � In Jerry's post I'm seeing question marks where there's > evidently supposed to be an omega. � In rb-j's post I'm seeing some > funky foreign (unidentifiable) characeters.SOB! that's what i see, too, but when i typed it (in Google Groups), i hit reply, got an editable text box that quoted Jerry's post, the omegas and the pi looked just fine in the edit box, i typed what i wanted to say while copying and pasting the omega and pi, and pressed send. i didn't even bother to look at it (lately Google has been so slow in showing my posts). and you're right, Eric. it looks wrong even from Google. i don't know what happened. maybe we better just stick with 7-bit ASCII, guys. r b-j
Reply by ●October 29, 20082008-10-29
On Oct 29, 4:12�pm, Scott Hemphill <hemph...@hemphills.net> wrote:> robert bristow-johnson <r...@audioimagination.com> writes: > > On Oct 29, 9:33�am, Jerry Avins <j...@ieee.org> wrote: > >> paulz wrote: > >> > Let's say Fourier Transform (FT) of x(t) is X(�) and duality property says > >> > FT of X(t) is 2�x(-�). > > >> > We know FT of dx(t)/dt is j�X(�). From above duality property, we can > >> > get FT of tX(t) is 2�jdx(�)/d�. However, from this property, we can't > >> > get FT of tx(t), which may sometimes cause confusion. > > >> Does everyone see those lovely omegas? I'd much rather write � than w, > >> but I've been concerned that it would come through as '?' or worse. > > > i can see the omega � and pi � (reading with Google Groups), but i > > don't know what key sequence to hit to get it. �here i just copied and > > pasted. �maybe someday we'll be able to use LaTeX paste-up on USENET > > to *really* present our meaning clearly. > > > i still think (paulz) that representing the FT with "f" instead of "�" > > in it results in a much simpler representation of the Duality and > > Convolution properties and Parseval's theorem. > > This is interesting. �In Jerry's posts, I saw open rectangles where > all the omegas should have been. �I could cut and paste the open > rectangles into a text window, and they would appear as omegas in that > text window. �However, all of the quoted omegas, as well as the omegas > and pi in your post appear perfectly. > > I can also cut and paste them: omega � and pi �. > > I notice that Jerry's post has a header: > > � Content-Type: text/plain; charset=UTF-8; format=flowed > > while yours has: > > � Content-Type: text/plain; charset=ISO-8859-7 >so, as a usenet client and newsserver, Google Groups does not copy that Content-Type header. that explains it. sorta. r b-j
Reply by ●October 29, 20082008-10-29
robert bristow-johnson wrote:> On Oct 29, 1:23 pm, Eric Jacobsen <eric.jacob...@ieee.org> wrote: > >>On Wed, 29 Oct 2008 08:51:05 -0700 (PDT), robert bristow-johnson >> >> >> >><r...@audioimagination.com> wrote: >> >>>On Oct 29, 9:33 am, Jerry Avins <j...@ieee.org> wrote: >>> >>>>paulz wrote: >>>> >>>>>Let's say Fourier Transform (FT) of x(t) is X(�) and duality property says >>>>>FT of X(t) is 2�x(-�). >> >>>>>We know FT of dx(t)/dt is j�X(�). From above duality property, we can >>>>>get FT of tX(t) is 2�jdx(�)/d�. However, from this property, we can't >>>>>get FT of tx(t), which may sometimes cause confusion. >> >>>>Does everyone see those lovely omegas? I'd much rather write � than w, >>>>but I've been concerned that it would come through as '?' or worse. >> >>>i can see the omega � and pi � (reading with Google Groups), but i >>>don't know what key sequence to hit to get it. here i just copied and >>>pasted. maybe someday we'll be able to use LaTeX paste-up on USENET >>>to *really* present our meaning clearly. >> >>>i still think (paulz) that representing the FT with "f" instead of "�" >>>in it results in a much simpler representation of the Duality and >>>Convolution properties and Parseval's theorem. >> >>>r b-j >> >>Bleah. In Jerry's post I'm seeing question marks where there's >>evidently supposed to be an omega. In rb-j's post I'm seeing some >>funky foreign (unidentifiable) characeters. > > > SOB! that's what i see, too, but when i typed it (in Google Groups), > i hit reply, got an editable text box that quoted Jerry's post, the > omegas and the pi looked just fine in the edit box, i typed what i > wanted to say while copying and pasting the omega and pi, and pressed > send. i didn't even bother to look at it (lately Google has been so > slow in showing my posts). and you're right, Eric. it looks wrong > even from Google. i don't know what happened. > > maybe we better just stick with 7-bit ASCII, guys. > > r b-j >Your header says text/plain; charset=ISO-8859-1 Can you set up to use Unicode?
Reply by ●October 29, 20082008-10-29
On Oct 29, 2:45�pm, Glen Herrmannsfeldt <g...@ugcs.caltech.edu> wrote:> robert bristow-johnson wrote: > > (snip) > > > i still think (paulz) that representing the FT with "f" instead of "w" > > in it results in a much simpler representation of the Duality and > > Convolution properties and Parseval's theorem. > > You can move the 2*pi around but you can't get rid of it.it's an issue of minimizing the places where it appears and other related stuff.> I like the "w" form better because exp(-iwt) looks nice. > Physics is usually done with "w" because, in addition to the > Fourier transform, there are many other places where it is used, > (such as circular motion). > > With f you get an extra 2*pi with every derivative. > > Engineers like f because that is the way frequency counters > work. �If frequency counters and frequency standards were > in radians/second it would be easier to use "�". >Glen, the real reason i like it better is that, with the exception of the derivative/integral theorems of the FT, all the other ones are simpler to write and remember. for me, it's easier then to remember (in the rare occasions that i do a derivative or integral of a signal in these digital times) the 2*pi factors (as well as the "j") for the derivative/integral and have all the other theorems simpler. most particularly the Duality property, the FT of a convolution (or the product) of two signals, and Parseval's theorem. those are a bitch to remember the proper scaling which is different depending on which direction (time to frequency domain or frequency to time domain) that you're going. the "f" definition is perfectly unitary or isomorphic (or whatever is the term that means a mapping and it's inverse are sorta identical, like f(x) = 1/x) because +j and -j have identical properties. this can be done with the "w" definition of the FT, but then you have to put 1/sqrt(2*pi) in front of both the FT and iFT. but then you lose the nice properties that X(0) is the integral of x(t) and x(0) is the integral of X(f). just that property is so useful to me to be able to derive the correct scale factor of some transform that i partially remember. anyway, with mathematics, unlike politics, there is more than one "right" answer. r b-j
Reply by ●October 30, 20082008-10-30
Jerry Avins wrote: (snip)> And the 60Hz power line would be approximately 377 radians per second. > The earth would turn approximately .2618 radians per hour. I prefer 15°.Reminds me of one I thought of recently. I was taking apart a PC power supply (to add more disk drive connectors) and thinking about 20kHz transformers. Also, the amazing transition of the US to digital TV in a short time. The question is, what would be the advantages and disadvantages to switching to 20kHz power from 60Hz? I suppose that is somewhat harder than the DTV conversion, but maybe there are some advantages. How much more efficient are ferrite transformers at 20kHz than our usual silicon steel power line transformers? Assuming one uses superconducting DC power lines, how much easier is it to convert from DC to 20kHz? How much more efficient are various SCR based power controllers at 20kHz? How much would 20kHz to 60Hz converters cost? Most likely not practical, but there will have to be big changes in energy use in the coming years to reduce the global warming contributions. More efficient use of electrical power would help. -- glen
Reply by ●October 30, 20082008-10-30
Glen Herrmannsfeldt wrote:> Jerry Avins wrote: > (snip) > >> And the 60Hz power line would be approximately 377 radians per second. >> The earth would turn approximately .2618 radians per hour. I prefer 15°. > > Reminds me of one I thought of recently. > > I was taking apart a PC power supply (to add more disk > drive connectors) and thinking about 20kHz transformers. > Also, the amazing transition of the US to digital TV > in a short time. > > The question is, what would be the advantages and disadvantages > to switching to 20kHz power from 60Hz? I suppose that is > somewhat harder than the DTV conversion, but maybe there are > some advantages. > > How much more efficient are ferrite transformers at 20kHz than > our usual silicon steel power line transformers? Assuming > one uses superconducting DC power lines, how much easier is it > to convert from DC to 20kHz? How much more efficient are various > SCR based power controllers at 20kHz? How much would 20kHz to 60Hz > converters cost? > > Most likely not practical, but there will have to be big > changes in energy use in the coming years to reduce the > global warming contributions. More efficient use of electrical > power would help.The biggest problem would be in distribution. Radiation would become much more of a problem, and dielectric losses would overwhelm the energy budget. Even with 60 (or 50) Hz, transmission losses are high enough to provide a strong incentive for high-voltage DC transmission. Power is generated at AC, then stepped up to 400 KV and rectified. Semiconductor devices convert back to AC at the receiving end. The inevitable losses and the cost of the equipment are amply repaid by the savings. I suppose that when all signaling lines are optical, inductive coordination won't be an issue, but it would be as long as we use copper telephone circuits. Bad idea. :-( Jerry -- Engineering is the art of making what you want from things you can get. ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
Reply by ●October 30, 20082008-10-30
Jerry Avins wrote: (snip on 20kHz power distribution)> The biggest problem would be in distribution. Radiation would become > much more of a problem, and dielectric losses would overwhelm the energy > budget. Even with 60 (or 50) Hz, transmission losses are high enough to > provide a strong incentive for high-voltage DC transmission. Power is > generated at AC, then stepped up to 400 KV and rectified. Semiconductor > devices convert back to AC at the receiving end. The inevitable losses > and the cost of the equipment are amply repaid by the savings.Isn't it easier to convert DC to 20kHz than to 60Hz? (Nice small ferrite transformers instead of big iron ones.)> I suppose that when all signaling lines are optical, inductive > coordination won't be an issue, but it would be as long as we use copper > telephone circuits.> Bad idea. :-(Dielectric loss in air should still be low, but radiation would have to be considered. So far it doesn't look good, but I am not sure everything has been considered yet. -- glen
