Richard Owlett wrote:> Jerry Avins wrote: >> paulz wrote: >> >>> Let's say Fourier Transform (FT) of x(t) is X(ω) and duality property >>> says >>> FT of X(t) is 2πx(-ω). >>> We know FT of dx(t)/dt is jωX(ω). From above duality property, we can >>> get FT of tX(t) is 2πjdx(ω)/dω. 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 ω than w, >> but I've been concerned that it would come through as '?' or worse. >> >> Jerry > > I see omega(ω). Is there a simple general way of getting those in a > post. I did it this time with cut-n-paste.There are character tables in Windows and Macs. In windows, it's called "Character Map" (On XP, the program is C:\WINDOWS\system32\charmap.exe). I keep a shortcut on the desktop. You can also start->run->charmap. On a Mac, it's called Keycaps. The character map gives the 4-digit character number in hex; ω is 03C9. Converting to decimal, that's 0969. Holding the right Alt key on a PC keyboard while typing those digits on the keypad produces an ω in most fonts -- courier, e.g., -- but unfortunately not in the UTF-8 I use for email. 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
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-jBleah. 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. 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
Jerry Avins wrote:> Does everyone see those lovely omegas? I'd much rather write ω > than w, but I've been concerned that it would come through as > '?' or worse.And with reason. Here, your UTF-8 pi becomes the pair (double-dotted I, Euro sign) and omega (ddI, box) or (ddI, permille) depending on the font. And in Robert's ISO-8859-7 post they come out as eth and u- grave, respectively. Martin -- Quidquid latine scriptum est, altum videtur.
Reply by ●October 29, 20082008-10-29
robert bristow-johnson wrote: (snip)> 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.You can move the 2� around but you can't get rid of it. I like the "�" form better because exp(-i�t) looks nice. Physics is usually done with "�" 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� 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
Reply by ●October 29, 20082008-10-29
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(ù) and duality property says >>>> FT of X(t) is 2ðx(-ù). >>>> We know FT of dx(t)/dt is jùX(ù). From above duality property, we can >>>> get FT of tX(t) is 2ðjdx(ù)/dù. 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 ù than w, >>> but I've been concerned that it would come through as '?' or worse. >>> >> i can see the omega ù and pi ð (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 "ù" >> 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: Ω (capital omega); ω (small omega); ° (degree); µ (mu). Jerry -- Engineering is the art of making what you want from things you can get. ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
Reply by ●October 29, 20082008-10-29
Glen Herrmannsfeldt wrote:> robert bristow-johnson wrote: > (snip) > >> i still think (paulz) that representing the FT with "f" instead of "ω" >> in it results in a much simpler representation of the Duality and >> Convolution properties and Parseval's theorem. > > You can move the 2π around but you can't get rid of it. > > I like the "ω" form better because exp(-iωt) looks nice. > Physics is usually done with "ω" 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 2π 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 "ω".And the 60Hz power line would be approximately 377 radians per second. The earth would turn approximately .2618 radians per hour. I prefer 15°. Jerry -- Engineering is the art of making what you want from things you can get. ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
Reply by ●October 29, 20082008-10-29
Martin Eisenberg wrote:> Jerry Avins wrote: > >> Does everyone see those lovely omegas? I'd much rather write ω >> than w, but I've been concerned that it would come through as >> '?' or worse. > > And with reason. Here, your UTF-8 pi becomes the pair (double-dotted > I, Euro sign) and omega (ddI, box) or (ddI, permille) depending on > the font. And in Robert's ISO-8859-7 post they come out as eth and u- > grave, respectively.Oh, well! Jerry -- Engineering is the art of making what you want from things you can get. ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
Reply by ●October 29, 20082008-10-29
On 2008-10-29 15:45:32 -0300, Glen Herrmannsfeldt <gah@ugcs.caltech.edu> said:> robert bristow-johnson wrote: > (snip) > >> 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. > > You can move the 2� around but you can't get rid of it. > > I like the "�" form better because exp(-i�t) looks nice. > Physics is usually done with "�" because, in addition to the > Fourier transform, there are many other places where it is used, > (such as circular motion).If you take the second course called Abstract Harmonic Analysis where the period is defined to 1 the two pi is built into the character function which is the abstract version exp(ix). The two pi returns in derivatives. It all boils down to whether you define the derivative of sine at 0 to be 1, i.e. calculus, and then work to figure out the period of two pi or if you define the period to be 1 and then work to figure out the scaling constant for the derivative which turns out to be two pi.> With f you get an extra 2� 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
Reply by ●October 29, 20082008-10-29
robert bristow-johnson <rbj@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(ω) and duality property says >> > FT of X(t) is 2πx(-ω). >> >> > We know FT of dx(t)/dt is jωX(ω). From above duality property, we can >> > get FT of tX(t) is 2πjdx(ω)/dω. 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 ω than w, >> but I've been concerned that it would come through as '?' or worse. >> > > i can see the omega ω and pi π (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 "ω" > 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 ω and pi π. 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 Scott -- Scott Hemphill hemphill@alumni.caltech.edu "This isn't flying. This is falling, with style." -- Buzz Lightyear
Reply by ●October 29, 20082008-10-29
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. Martin -- Quidquid latine scriptum est, altum videtur.
