Any justifications? ------------------------------------------- One of the interesting properties implied by causality is that the imaginary part of Fourier transform is completely determineted by a knowledge of its real part and vice-versa.
why can causality imply the following property?
Started by ●November 11, 2004
Reply by ●November 11, 20042004-11-11
kiki wrote:> Any justifications? > > ------------------------------------------- > > One of the interesting properties implied by causality is that the imaginary > part of Fourier transform is completely determineted by a knowledge of its > real part and vice-versa. > > >First, I think you mean the continuous Fourier transform of the impulse response of a LTI system, i.e. a signal which is zero for all t < 0? This constraint gives you a proof: First assume an all-real signal. In order for the original signal to be zero for all time t < 0 it must be the sum of an even signal and an odd signal who's values for t > 0 are equal to 1/2 the original signal (so the t < 0 parts will cancel). The transform of the original will then contain an even real part that is the transform of the even component of the original, and an odd imaginary part that is the transform of the odd component of the original. So to find the corresponding imaginary part one only needs to find the inverse transform of the real part, multiply it by i times the signum function (x = -1 for all x < 0, +1 for all x > 0 and zero for x = 0), and you have your answer. Because I am lazy cleaning up, finding errors in, and extending this proof to a complex signal will be left as an exercise to the reader. -- Tim Wescott Wescott Design Services http://www.wescottdesign.com
Reply by ●November 12, 20042004-11-12
"kiki" <lunaliu3@yahoo.com> wrote in message news:<cn0f0u$k9d$1@news.Stanford.EDU>...> Any justifications? > > ------------------------------------------- > > One of the interesting properties implied by causality is that the imaginary > part of Fourier transform is completely determineted by a knowledge of its > real part and vice-versa.Well, the idea is based on that any real function f(t) can be separated into a sum of an even function f_e(t) and an odd function f_o(t): f(t)=f_e(t)+f_o(t) [1] If you require f(t) to be causal, this means that f(t) = 0 for all t<0 (the t=0 case introduces some cumbersome technical difficulties that I prefer to omit in the following). Define this causal function f(t) as / f'(t), t>0 [2a] f(t) = | \ 0, t<0 [2b] (the apostrophe "'" is just a tag to identify the causal part of f(t)). In this case, one can construct an even and an odd function according to [1] and [2], as / f'(t), t>0 [3a] f_e(t) = | \ f'(-t), t<0 [3b] / f'(t), t>0 [3c] f_o(t) = | \ -f'(-t), t<0. [3d] A causal f(t) can then be represented as / (f'(-t)-f'(-t))/2 = 0 t<0 [4a] f(t) = | \ (f'(t)+f'(t))/2 = f'(t) t>0. [4b] You can set up similar relationships between the real and imaginary parts of spectrum of a real signal (remember, F(-w)= conj(F(w)) where F(w) is the Fourier transform of some real f(t)), so what you end up with, is a set of interconnected even and odd functions in both time and frequency domains. The justification you are looking for is based on the analysis of these general relations. An electrical engineer would describe the results of this analysis in terms along the lines of "The real and imaginary parts of the spectrum of a real signal form a Hilbert transform pair", while a mathematician might say something like "this property relates to the Poisson kernel". If you want to see the theory from a DSP point of view, check out chapter 7 of Oppenheim & Schafer: "Digital Signal Processing" Prentice-Hall, 1975. If you want to look into the maths of the Poisson kernel, check out chapter 12 of Churchill & Brown: "Complex Variables and Applications" 6th ed., McGraw-Hill, 1996 (according to the preface of the 6th edition, the chapter order of the 6th edition may be different from earlier editions). Be warned, these things are not the easiest to understand or interpret in terms of physical concepts. Most of it is "merely" algebraic manipulations of analytic functions. I don't know more about complex numbers than what is covered in university intro cources on general maths, and I found it very difficult to follow the manipulations in both Oppenheim & Scahfer and in Churchill & Brown. Rune
Reply by ●November 12, 20042004-11-12
Rune Allnor wrote: ...> (the apostrophe "'" is just a tag to identify the causal part of f(t)).... I was taught to read f' as "f prime" and f" as "f double prime". Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●November 12, 20042004-11-12
Rune Allnor wrote: (someone wrote)>>One of the interesting properties implied by causality is that the imaginary >>part of Fourier transform is completely determineted by a knowledge of its >>real part and vice-versa.(snip)> An electrical engineer would describe the results of this analysis > in terms along the lines of "The real and imaginary parts of the > spectrum of a real signal form a Hilbert transform pair", while a > mathematician might say something like "this property relates to > the Poisson kernel".In physics, it is the Kramers-Kronig relation. Note that many quantities can be considered complex that are often considered separate real quantities. The imaginary part of the index of refraction describes absorption. Also, the imaginary part of the dielectric constant is related to conductivity, and the index of refraction is the square root of the dielectric constant. (Which isn't that constant, as it usually depends on frequency.) The Kramers-Kronig relation, then, describes the connection between index of refraction and absorption, or between conductivity and dielectric constant. This is true even without a Fourier transform. -- glen
Reply by ●November 13, 20042004-11-13
Jerry Avins <jya@ieee.org> wrote in message news:<2vk4gtF2mjdioU1@uni-berlin.de>...> Rune Allnor wrote: > > ... > > > (the apostrophe "'" is just a tag to identify the causal part of f(t)). > > ... > > I was taught to read f' as "f prime" and f" as "f double prime".I won't disagree with that. For the purposes of this thread, the important point is to NOT read it as df/dt or something like that. Rune
Reply by ●November 13, 20042004-11-13
Rune Allnor wrote:> Jerry Avins <jya@ieee.org> wrote in message news:<2vk4gtF2mjdioU1@uni-berlin.de>... > >>Rune Allnor wrote: >> >> ... >> >> >>>(the apostrophe "'" is just a tag to identify the causal part of f(t)). >> >> ... >> >>I was taught to read f' as "f prime" and f" as "f double prime". > > > I won't disagree with that. For the purposes of this thread, the > important point is to NOT read it as df/dt or something like that. > > RuneFrom my experience the "prime" in f' is just a ticky mark; it acquires whatever meaning the author wants it to acquire -- it may mean df/dt, it may mean "the best estimate of f", it may mean "the other f" -- whatever the author means, that's what it means. And if the author means anything other than something that's already well-established in that particular field then he'd better point it out! -- Tim Wescott Wescott Design Services http://www.wescottdesign.com
Reply by ●November 13, 20042004-11-13
Tim Wescott wrote:> Rune Allnor wrote: > >> Jerry Avins <jya@ieee.org> wrote in message >> news:<2vk4gtF2mjdioU1@uni-berlin.de>... >> >>> Rune Allnor wrote: >>> >>> ... >>> >>> >>>> (the apostrophe "'" is just a tag to identify the causal part of f(t)). >>> >>> >>> ... >>> >>> I was taught to read f' as "f prime" and f" as "f double prime". >> >> >> >> I won't disagree with that. For the purposes of this thread, the >> important point is to NOT read it as df/dt or something like that. >> >> Rune > > > From my experience the "prime" in f' is just a ticky mark; it acquires > whatever meaning the author wants it to acquire -- it may mean df/dt, it > may mean "the best estimate of f", it may mean "the other f" -- whatever > the author means, that's what it means. > > And if the author means anything other than something that's already > well-established in that particular field then he'd better point it out!Long ago, kiddies, a dot above the variable was sometimes used to indicate the derivative (Newton's notation?). The author of my undergrad differential-equations text disparaged the practice with the statement, "Flies have been known to produce unwanted differentiations." Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●November 13, 20042004-11-13
Jerry Avins wrote:> > Long ago, kiddies, a dot above the variable was sometimes used to > indicate the derivative (Newton's notation?). The author of my undergrad > differential-equations text disparaged the practice with the statement, > "Flies have been known to produce unwanted differentiations." > > JerryIt's still quite popular in physics and control systems analysis, specifically indicating differentiation with respect to time. I generally prefer dx/dt over x-dot, though. -- Tim Wescott Wescott Design Services http://www.wescottdesign.com
Reply by ●November 13, 20042004-11-13
Tim Wescott wrote:> Jerry Avins wrote: > > >> >> Long ago, kiddies, a dot above the variable was sometimes used to >> indicate the derivative (Newton's notation?). The author of my undergrad >> differential-equations text disparaged the practice with the statement, >> "Flies have been known to produce unwanted differentiations." >> >> Jerry > > > It's still quite popular in physics and control systems analysis ...Some people never learn! Pesky flies!> I generally prefer dx/dt over x-dot, though.Good man! :-) Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
