Eckard Blumschein wrote:> Just to those who incidentally read my reply, I will in brief explain > why we need continuation at all. > It's a pity, definition of Fourier transform nurtures an illusion since > it is as an integral over time from -infty to + infty: the illusion that > time always means physical time in that case. Actually, half of the time > axis is fictitious. It would not make sense to include past as well as > future time. This fact is called causality. All physical relationship > could be completely expressed within past time. However, use of complex > calculus requires negative values of time and frequency, too.The connection between causality, real and imaginary components, and times of plus and minus infinity, all come out in the Kramers-Kronig relations in physics. It does seem that you can't do without them. Among other things that can be shown is that it is physically impossible to make a perfect (infinitely narrow) bandpass filter. I think it also shows that it is impossible to make infinitely sharp filters of any shape, but then we probably knew that already. (big snip) -- glen
simultaneous frequence and phase estimation
Started by ●April 21, 2004
Reply by ●May 7, 20042004-05-07
Reply by ●May 10, 20042004-05-10
glen herrmannsfeldt wrote:> Eckard Blumschein wrote: > >> It's a pity, definition of Fourier transform nurtures an illusion >> since it is as an integral over time from -infty to + infty: the >> illusion that time always means physical time in that case. Actually, >> half of the time axis is fictitious. It would not make sense to >> include past as well as future time. This fact is called causality. >> All physical relationship could be completely expressed within past >> time. However, use of complex calculus requires negative values of >> time and frequency, too. > > > The connection between causality, real and imaginary components, > and times of plus and minus infinity, all come out in the Kramers-Kronig > relations in physics. It does seem that you can't do without them.Why not? You are sharing your problem with Moslems who cannot imagine abandoning their Koran or with Christians to whose the Bible is indispensable. I do not deny that complex numbers are more general than real numbers, and real numbers are more general than positive numbers. However, there is no reason for taking complex Fourier transform a gospel. On the contrary, nobody is entitled to define causality. Kramers-Kronig relations reveals the link between real and imaginary part while Bode relation reveals the link between magnitude and phase on condition of causality. Look into L. J. Wang: 'Causal "All-pass" Filters do not Satisfy Kramers-Kronig Relations' in order to realize that sometimes zeros prohibit the application of the Cauchy's theorem. I would like to evade much effort to understand 'Causality and negative group delay in a simple bandpass filter' by M.W. Mitchell and R. Y Chiao. It is so far just a guess of mine that the whole superluminality stuff (with a spectral window near DC causing negative group velocity of evanescent modes) is an artifact of complex Fourier transform. Pretty please do not disdain IR+ as stubbornly does John Baez. H. A. Kramers's (1922) and R. Kronig's (1926) relations as well as H. W. Bode's (1945) relations are based on the inherent redundancy of zero continuation based complex Fourier transform. They are nothing but crutches to walk on.> Among other things that can be shown is that it is physically impossible > to make a perfect (infinitely narrow) bandpass filter.This is also obvious in IR+. Thank you for reminding me of filters. Who is stupid enough as to imagine bandwidth extending from a negative to a positive frequency? Who is stupid enough as to imagine complex modulation with zero carrier frequency? Well, I do not intend objecting against use of negative frequency. However, I found cases of misinterpretation even in widely used books. Eckard Blumschein
Reply by ●May 10, 20042004-05-10
Eckard Blumschein wrote: ...> ... On the contrary, nobody is entitled to define causality."Causality" is well defined. Breiefly, effects do not precede their causes.> Kramers-Kronig relations reveals the link between real and imaginary > part while Bode relation reveals the link between magnitude and phase on > condition of causality. > Look into L. J. Wang: 'Causal "All-pass" Filters do not Satisfy > Kramers-Kronig Relations' in order to realize that sometimes zeros > prohibit the application of the Cauchy's theorem. I would like to evade > much effort to understand 'Causality and negative group delay in a > simple bandpass filter' by M.W. Mitchell and R. Y Chiao. It is so far > just a guess of mine that the whole superluminality stuff (with a > spectral window near DC causing negative group velocity of evanescent > modes) is an artifact of complex Fourier transform.You will find that superluminality applies only to group velocity and not to phase velocity. Google for "anomalous dispersion". There's no need to invoke the supernatural.> Pretty please do not disdain IR+ as stubbornly does John Baez. > H. A. Kramers's (1922) and R. Kronig's (1926) relations as well as H. W. > Bode's (1945) relations are based on the inherent redundancy of zero > continuation based complex Fourier transform. They are nothing but > crutches to walk on. > >> Among other things that can be shown is that it is physically impossible >> to make a perfect (infinitely narrow) bandpass filter. > > > This is also obvious in IR+. > Thank you for reminding me of filters. Who is stupid enough as to > imagine bandwidth extending from a negative to a positive frequency? > Who is stupid enough as to imagine complex modulation with zero carrier > frequency? > Well, I do not intend objecting against use of negative frequency. > However, I found cases of misinterpretation even in widely used books.And who confuses the mathematical convenience of exponential notation with the reality of sinusoids? [Randy! Don't answer that!] Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●May 10, 20042004-05-10
Jerry Avins wrote:> Eckard Blumschein wrote: > >> ... On the contrary, nobody is entitled to define causality. > > "Causality" is well defined. Briefly, effects do not precede their > causes.One can well describe rather than define real things like you, like me and perhaps causality.>> Kramers-Kronig relations reveals the link between real and imaginary >> part while Bode relation reveals the link between magnitude and phase >> on condition of causality. >> Look into L. J. Wang: 'Causal "All-pass" Filters do not Satisfy >> Kramers-Kronig Relations' in order to realize that sometimes zeros >> prohibit the application of the Cauchy's theorem. I would like to >> evade much effort to understand 'Causality and negative group delay in >> a simple bandpass filter' by M.W. Mitchell and R. Y Chiao. It is so >> far just a guess of mine that the whole superluminality stuff (with a >> spectral window near DC causing negative group velocity of evanescent >> modes) is an artifact of complex Fourier transform. > > > You will find that superluminality applies only to group velocity and > not to phase velocity. Google for "anomalous dispersion". There's no > need to invoke the supernatural.I did not invoke the supernatural. I simply guess that the whole confusion is based on zero continuation. Eckard
Reply by ●May 10, 20042004-05-10
Jerry Avins <jya@ieee.org> wrote in message news:<409f92ea$0$3034$61fed72c@news.rcn.com>...> > Well, I do not intend objecting against use of negative frequency. > > However, I found cases of misinterpretation even in widely used books. > > And who confuses the mathematical convenience of exponential notation > with the reality of sinusoids? [Randy! Don't answer that!]Eh... you wouldn't by any chance be referring to my recent post on Euler's equations? Granted, there *was* some confusion but only because I forgot to include the "-" sign in the second exponential term of Euler's equations. It was merely a typo... ;) Rune
Reply by ●May 10, 20042004-05-10
Jerry Avins <jya@ieee.org> writes:> And who confuses the mathematical convenience of exponential notation > with the reality of sinusoids?That is tantamount to saying that the real numbers are just a mathematical convenience of notation - all we really have are the positive integers and 0. Until you recognize that the complex numbers are indeed a unique and significantly more capable arithmetic system that, e.g., the reals, neither one of us is going anywhere on this point.> [Randy! Don't answer that!]Umm, sorry - you don't have that option. -- Randy Yates Sony Ericsson Mobile Communications Research Triangle Park, NC, USA randy.yates@sonyericsson.com, 919-472-1124
Reply by ●May 10, 20042004-05-10
Rune Allnor wrote:> Jerry Avins <jya@ieee.org> wrote in message news:<409f92ea$0$3034$61fed72c@news.rcn.com>... > >>>Well, I do not intend objecting against use of negative frequency. >>>However, I found cases of misinterpretation even in widely used books. >> >>And who confuses the mathematical convenience of exponential notation >>with the reality of sinusoids? [Randy! Don't answer that!] > > > Eh... you wouldn't by any chance be referring to my recent post > on Euler's equations? > > Granted, there *was* some confusion but only because I forgot to > include the "-" sign in the second exponential term of Euler's > equations. It was merely a typo... ;) > > RuneI wasn't addressing you at all. Negative frequencies arise when replacing xin(wt) and cos(wt) with e^+jwt and e^-jwt, which it is mathematically convenient to do. The price we pay for that convenience is either using negative frequency or imagining that time marches backward as well as forward. (Almost invariably, we chose the former.) As for my plea to Randy, he assigns physical meaning to negative frequencies. I don't, but I can if I want to, just as I can compute with them. Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●May 10, 20042004-05-10
Randy Yates wrote:> Jerry Avins <jya@ieee.org> writes: > > >>And who confuses the mathematical convenience of exponential notation >>with the reality of sinusoids? > > > That is tantamount to saying that the real numbers are just a mathematical > convenience of notation - all we really have are the positive integers and > 0. > > Until you recognize that the complex numbers are indeed a unique and > significantly more capable arithmetic system that, e.g., the reals, > neither one of us is going anywhere on this point. > > >>[Randy! Don't answer that!] > > > Umm, sorry - you don't have that option.All numbers are complex. It's just that some of them are special cases. Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●May 10, 20042004-05-10
Jerry Avins <jya@ieee.org> writes:> Randy Yates wrote: > > > Jerry Avins <jya@ieee.org> writes: > > > > >>And who confuses the mathematical convenience of exponential notation > >> with the reality of sinusoids? > > > That is tantamount to saying that the real numbers are just a > > mathematical > > > convenience of notation - all we really have are the positive integers and > > 0. Until you recognize that the complex numbers are indeed a unique > > and > > > significantly more capable arithmetic system that, e.g., the reals, > > neither one of us is going anywhere on this point. > > > > >>[Randy! Don't answer that!] > > Umm, sorry - you don't have that option. > > > All numbers are complex. It's just that some of them are special cases.That's not what my coding theory class taught me. (Can you say GF(p^m)?) -- Randy Yates Sony Ericsson Mobile Communications Research Triangle Park, NC, USA randy.yates@sonyericsson.com, 919-472-1124
Reply by ●May 10, 20042004-05-10
Randy Yates wrote:> Jerry Avins <jya@ieee.org> writes: > > >>Randy Yates wrote: >> >> >>>Jerry Avins <jya@ieee.org> writes: >>> >> >>>>And who confuses the mathematical convenience of exponential notation >>>>with the reality of sinusoids? >> >>>That is tantamount to saying that the real numbers are just a >>>mathematical >> >>>convenience of notation - all we really have are the positive integers and >>>0. Until you recognize that the complex numbers are indeed a unique >>>and >> >>>significantly more capable arithmetic system that, e.g., the reals, >>>neither one of us is going anywhere on this point. >>> >> >>>>[Randy! Don't answer that!] >>> >>>Umm, sorry - you don't have that option. >> >> >>All numbers are complex. It's just that some of them are special cases. > > > That's not what my coding theory class taught me. (Can you say GF(p^m)?)I overstated. Cardinal numbers excepted. Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
