Gibbs phenomenon and spectral leakage. What is the relation between them? As a cause,which lead to which? Regards HyeeWang
Gibbs phenomenon and spectral leakage.
Started by ●May 6, 2009
Reply by ●May 6, 20092009-05-06
On 6 Mai, 07:34, HyeeWang <hyeew...@gmail.com> wrote:> Gibbs phenomenon and spectral leakage. > > What is the relation between them?I'm inclined to say that there is no relation between them. Gibbs phenomenon has to do with the FT representing a discontinuity by a continuous function. The Fourier basis vectors can represent continuous signals, but only approximate discontinuous signals. Spectral leakage has to do with how the vectors that represent signals are being mapped onto the Fourier basis vectors. Unless the signal happens to match up with a small subset of the basis vectors, one needs contributions from all the basis vectors to represent the signal vector. Now, both Gibbs' phenomenon and spectral leakage have to do with signal representations in vector space, so *maybe* (just *maybe*) there might be some subtle similarites that might be seen as 'relations', deep, deep down. But don't waste any time looking for that kind of stuff. At least wait till you have taken a few classes in real analysis, before you do.> As a cause,which lead to which?There are no cause-effect relations between them. Spectral leakage occurs without Gibbs' phenomenon. Try to compute the N-pt DFT of a sinusoidal with frequency Fs(2N-1)/2N*4. Spectrum leakage without Gibbs phenomenon. Gibbs' phenomenon exists without spectral leakage. Compute (analytically) the frequency response of an ideal square pulse, x(t) = 1, |t| < 1, 0 otherwise. Then IFT the spectrum to compute the time response from the frequency response. Gibbs' phenomenon is present. Rune
Reply by ●May 6, 20092009-05-06
HyeeWang <hyeewang@gmail.com> writes:> Gibbs phenomenon and spectral leakage. > > What is the relation between them? As a cause,which lead to which?Gibbs phenomenom occurs when the analysis bandwidth is less than the signal bandwidth. Spectral leakage occurs when the analysis time is less than the signal time. -- % Randy Yates % "Maybe one day I'll feel her cold embrace, %% Fuquay-Varina, NC % and kiss her interface, %%% 919-577-9882 % til then, I'll leave her alone." %%%% <yates@ieee.org> % 'Yours Truly, 2095', *Time*, ELO http://www.digitalsignallabs.com
Reply by ●May 6, 20092009-05-06
On 2009-05-06 06:20:37 -0300, Rune Allnor <allnor@tele.ntnu.no> said:> On 6 Mai, 07:34, HyeeWang <hyeew...@gmail.com> wrote: >> Gibbs phenomenon and spectral leakage. >> >> What is the relation between them? > > I'm inclined to say that there is no relation > between them.Others would say they are the same phenomena. Gibb's is the result of truncating freuencies and looking at the effects in time. Spectral leakage is the result of truncating time and looking at the effects in frequency. They are Fourier duals. The various schemes for dealing with them are the same formulaes with one using t and the other f. The formal names for these schmemes are various weights. The informal names are often called windows in less than formal settings. It can be confusing if you only see one formally and the other informally.> Gibbs phenomenon has to do with the FT representing > a discontinuity by a continuous function. The Fourier > basis vectors can represent continuous signals, > but only approximate discontinuous signals.The main attention in Gibb's is the poor approximation at points of discontinuity. Just look at the convolution of the original with the transform of the frequency truncation window. This quickly shows that the relative overshoot at a discontinuity will remain the same as more terms are added, just the peak moves closer to the break.> Spectral leakage has to do with how the vectors that > represent signals are being mapped onto the Fourier > basis vectors. Unless the signal happens to match up > with a small subset of the basis vectors, one needs > contributions from all the basis vectors to represent > the signal vector. > > Now, both Gibbs' phenomenon and spectral leakage > have to do with signal representations in vector > space, so *maybe* (just *maybe*) there might be > some subtle similarites that might be seen as > 'relations', deep, deep down.Two sides of an isomorphism is not very subtle. And it is isomorphism that is the bread and butter of signal processing.> But don't waste any time looking for that kind of > stuff. At least wait till you have taken a few > classes in real analysis, before you do.You do not need measure theory, which is the usual content of real analysis, when advanced (i.e. a second course) in calculus will do the job.>> As a cause,which lead to which? > > There are no cause-effect relations between them. > > Spectral leakage occurs without Gibbs' phenomenon. > Try to compute the N-pt DFT of a sinusoidal with > frequency Fs(2N-1)/2N*4. Spectrum leakage without > Gibbs phenomenon. > > Gibbs' phenomenon exists without spectral leakage. > Compute (analytically) the frequency response of an > ideal square pulse, x(t) = 1, |t| < 1, 0 otherwise. > Then IFT the spectrum to compute the time response > from the frequency response. Gibbs' phenomenon > is present. > > Rune
Reply by ●May 6, 20092009-05-06
Randy Yates wrote:> HyeeWang <hyeewang@gmail.com> writes: > >> Gibbs phenomenon and spectral leakage. >> >> What is the relation between them? As a cause,which lead to which? > > Gibbs phenomenom occurs when the analysis bandwidth is less than the > signal bandwidth. > > Spectral leakage occurs when the analysis time is less than the > signal time.Neat, memorable, but a bit off the mark. The Gibbs phenomenon occurs whenever a discontinuity in an otherwise (piecewise) continuous function is approximated by its component basis functions, however numerous those might be. Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●May 6, 20092009-05-06
On 6 Mai, 14:50, Gordon Sande <g.sa...@worldnet.att.net> wrote:> On 2009-05-06 06:20:37 -0300, Rune Allnor <all...@tele.ntnu.no> said: > > > On 6 Mai, 07:34, HyeeWang <hyeew...@gmail.com> wrote: > >> Gibbs phenomenon and spectral leakage. > > >> What is the relation between them? > > > I'm inclined to say that there is no relation > > between them. > > Others would say they are the same phenomena.Then 'others' would be wrong.> Gibb's > is the result of truncating freuencies and looking > at the effects in time.Not according to my references. Papoulis' "The Fourier integral and its applications" discuss Gibbs' phenomenon in terms of discontinuities of the signal to be FT'ed (p. 30). In particular, 2-93 shows that in the limit w-> inf, the Fourier representation of x(t) has the value (f(0+)-f(0-))/2 when f(t) has a discontinuity at t=0. Dym & McKean's "Fourier series and integrals" state that Gibbs' name was associated with the effect after a debate in Nature between him and a Michaelson (as in Michaelson-Moreley?) over the FT's validity near discontinuities: "Michaelson was reportedly put out with mathemathics because the output of his machine for computing the first 80 terms of a Fourier series was not 'close enough' to f at the jumps" (p. 44). Rune
Reply by ●May 6, 20092009-05-06
On 2009-05-06 10:27:50 -0300, Rune Allnor <allnor@tele.ntnu.no> said:> On 6 Mai, 14:50, Gordon Sande <g.sa...@worldnet.att.net> wrote: >> On 2009-05-06 06:20:37 -0300, Rune Allnor <all...@tele.ntnu.no> said: >> >>> On 6 Mai, 07:34, HyeeWang <hyeew...@gmail.com> wrote: >>>> Gibbs phenomenon and spectral leakage. >> >>>> What is the relation between them? >> >>> I'm inclined to say that there is no relation >>> between them. >> >> Others would say they are the same phenomena. > > Then 'others' would be wrong. > >> Gibb's >> is the result of truncating freuencies and looking >> at the effects in time. > > Not according to my references. > > Papoulis' "The Fourier integral and its applications" > discuss Gibbs' phenomenon in terms of discontinuities > of the signal to be FT'ed (p. 30). In particular, > 2-93 shows that in the limit w-> inf, the Fourier > representation of x(t) has the value (f(0+)-f(0-))/2 > when f(t) has a discontinuity at t=0.When w is not infinite that is called truncating in frequency. At the point of discontinuity it is the average of the right continuous and left continuous versions of the function. If one makes that "correction" for discontinuity first then there is no problem. Sets of measure zero and all that stuff! The issue is when you are only adjacent to the discontinuity rather than exactly at it. Then you get oscillations. The Gibb's phenomema is about the approximation adjacent to the discontinuity rather than the fact that it weights the the left and right continuous versions equally. Of course the weighting is used to keep students awake in a class at 8:00AM that they would rather not be attending.> Dym & McKean's "Fourier series and integrals" state > that Gibbs' name was associated with the effect after > a debate in Nature between him and a Michaelson > (as in Michaelson-Moreley?) over the FT's validity > near discontinuities: "Michaelson was reportedly > put out with mathemathics because the output of his > machine for computing the first 80 terms of a Fourier > series was not 'close enough' to f at the jumps" (p. 44).Notice the number of terms. Notice the discussion is "near discontinuities". Which source are you citing? The one that is concerned with exactly the point of discontiuity or the one concerned with being near the discontuity? Both in fact seem to be concerned with truncating the frequencies. One as part of a limit and the other explicitly at a finite number.> Rune
Reply by ●May 6, 20092009-05-06
Here is my 2 cents worth (having worked on this very stuff with some concentration): Mind you, I'm an engineer and not so much a mathematician - so please forgive my lack of precise terminology. I do believe that nothing is lost. I agree pretty much with Gordon but have my own twist on it: The Gibb's phenomenon is seen at discontinuities when using a truncated Fourier Series. It gets no better as the limiting frequency is increased. Leakage is seen across the range - resulting from truncation in the opposite transform space. Notice that I didn't really mention time or frequency or space. You pick... it doesn't matter. Some observations: 1) Do a normal (i.e. continuous) FT on an infinitely long function whose FT has a discontinuity. Multiply the FT by a gate function - thus truncating it. This latter operation has the result that the original function is convolved with a sinc. It's the convolution with the sinc that causes the Gibb's phenomenon - getting narrower and narrower as the gate gets wider. A step function effectively integrates the sinc - so the Gibb's phenomenon is nicely viewed as caused by integrating a sinc. 2) Do a normal (i.e. continuous) FT on an infinitely long function. Here, it doesn't matter if it has a discontinuity or not. Multiply the FT by a gate function - thus truncating it. This latter operation has the result that the original function is convolved with a sinc. It's the convolution with the sinc that causes leakage. An impulse response effectively replicates the sinc - so leakage is nicely viewed as ... well .. convolving with a sinc. So, the two things are the same except that the observation is made on a different part of the original function. In one case we study the effects around discontinuities like step functions as the best example. In the other case we study the effects around impulses. In the first case, as above the convolution integral yields what looks like the integral of the sinc. In the second case, the convolution integral yields a replica of the sinc. That is the relationship between the two. Now, of course, we *usually* truncate in time to create *spectral* spreading. And, we *ususally* truncate in frequency to create the Gibb's phenomenon. But, what we usually do and what we observe has nothing to do with that. What we observe are two different outcomes (in terms of our focus) on having done exactly the same thing in one domain. There's no need for duality here. We just sort of introduce duality when we do what we *usually* do as above. Fred
Reply by ●May 6, 20092009-05-06
Rune Allnor wrote: ...> (as in Michaelson-Moreley?)Yes. (It's Morley.) ... Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●May 6, 20092009-05-06
Fred Marshall wrote:> Here is my 2 cents worth (having worked on this very stuff with some > concentration): > > Mind you, I'm an engineer and not so much a mathematician - so please > forgive my lack of precise terminology. I do believe that nothing is lost. > > I agree pretty much with Gordon but have my own twist on it: > > The Gibb's phenomenon is seen at discontinuities when using a truncated > Fourier Series. It gets no better as the limiting frequency is increased.So, what has truncation to do with it then? Truncation the spectrum changes the spacing of the oscillations for the obvious reason that they can't be faster than the highest frequencies we allow. In the limit, as the bandwidth becomes infinite, a calculation of the peak overshoot amplitude remains the same as at low bandwidth. (But the duration vanishes, do is overshoot really there?) ... Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
