This is just a question about Gibb's effect. Is Gibb's Effect is present in stationary data?
Gibb's Effect
Started by ●February 26, 2009
Reply by ●February 26, 20092009-02-26
On Feb 26, 2:54�pm, "tkremund98" <tkremun...@hotmail.com> wrote:> This is just a question about Gibb's effect. Is Gibb's Effect is present in > stationary data?yes
Reply by ●February 26, 20092009-02-26
>On Feb 26, 2:54=A0pm, "tkremund98" <tkremun...@hotmail.com> wrote: >> This is just a question about Gibb's effect. Is Gibb's Effect ispresent =>in >> stationary data? > >yes >Do you believe that the Gibb's is significant in comparison with the general spread of the data?
Reply by ●February 26, 20092009-02-26
On Feb 26, 3:26�pm, "tkremund98" <tkremun...@hotmail.com> wrote:> >On Feb 26, 2:54=A0pm, "tkremund98" <tkremun...@hotmail.com> wrote: > >> This is just a question about Gibb's effect. Is Gibb's Effect is > present = > >in > >> stationary data? > > >yes > > Do you believe that the Gibb's is significant in comparison with the > general spread of the data?I'm not trying to be glib, but if you try to make a very narrow bandpass filter using a brickwall approach ( for example), the ringing is very apparent. Will the ringing be a problem - maybe - depends on the app. Which aspect of Gibb's is a concern for you? Are you interested in the Wilbrahem constant? Are you concerned with trying to approximate a piecewise continuous function with a sum of uniformly continuous ones? The far field distribution of E-M radiation is the Fourier transform of the near field illumination current - are the side lobes an issue? If you can add some more detail to the question, I feel that I and others can better answer your question. In optics you can see Gibb's phenomina quite well. Look up Fresnel Integrals and their use in Cornu's spiral to figure the strength of the diffracted (spread) signal. This holds for radio waves as well. Clay
Reply by ●February 26, 20092009-02-26
�The far field distribution of E-M radiation is the Fourier> transform of the near field illumination current -THANKS!!! I never made that mental connection before .. Wow, it REALLY IS all connected.. Mark
Reply by ●February 26, 20092009-02-26
>On Feb 26, 3:26=A0pm, "tkremund98" <tkremun...@hotmail.com> wrote: >> >On Feb 26, 2:54=3DA0pm, "tkremund98" <tkremun...@hotmail.com> wrote: >> >> This is just a question about Gibb's effect. Is Gibb's Effect is >> present =3D >> >in >> >> stationary data? >> >> >yes >> >> Do you believe that the Gibb's is significant in comparison with the >> general spread of the data? > >I'm not trying to be glib, but if you try to make a very narrow >bandpass filter using a brickwall approach ( for example), the ringing >is very apparent. Will the ringing be a problem - maybe - depends on >the app. > >Which aspect of Gibb's is a concern for you? Are you interested in the >Wilbrahem constant? Are you concerned with trying to approximate a >piecewise continuous function with a sum of uniformly continuous >ones? The far field distribution of E-M radiation is the Fourier >transform of the near field illumination current - are the side lobes >an issue? > >If you can add some more detail to the question, I feel that I and >others can better answer your question. In optics you can see Gibb's >phenomina quite well. Look up Fresnel Integrals and their use in >Cornu's spiral to figure the strength of the diffracted (spread) >signal. This holds for radio waves as well. > >Clay >I'm actually somewhat crosstraining into DSP methods after receiving a degree in statistics. In other words, I'm somewhat wet behind the ears still. I notice that after having filtering data with a windowed-sinc filter I have the usual ring in the neighborhood of a step function but it dies out. Seeing this prodded some investigation. I tried some simulations using a signal simulated using user specified amplitudes for sine and cosine waves and combining using a synthesis equation. Then I superimposed this on a hard step function. When I vary the magnitude of the step function from large to small the ring disappears into the data and is seemingly indistiguishable from the data after having filtered it. I really appreciate the help, thanks!
Reply by ●February 26, 20092009-02-26
tkremund98 wrote:>> On Feb 26, 3:26=A0pm, "tkremund98" <tkremun...@hotmail.com> wrote: >>>> On Feb 26, 2:54=3DA0pm, "tkremund98" <tkremun...@hotmail.com> wrote: >>>>> This is just a question about Gibb's effect. Is Gibb's Effect is >>> present =3D >>>> in >>>>> stationary data? >>>> yes >>> Do you believe that the Gibb's is significant in comparison with the >>> general spread of the data? >> I'm not trying to be glib, but if you try to make a very narrow >> bandpass filter using a brickwall approach ( for example), the ringing >> is very apparent. Will the ringing be a problem - maybe - depends on >> the app. >> >> Which aspect of Gibb's is a concern for you? Are you interested in the >> Wilbrahem constant? Are you concerned with trying to approximate a >> piecewise continuous function with a sum of uniformly continuous >> ones? The far field distribution of E-M radiation is the Fourier >> transform of the near field illumination current - are the side lobes >> an issue? >> >> If you can add some more detail to the question, I feel that I and >> others can better answer your question. In optics you can see Gibb's >> phenomina quite well. Look up Fresnel Integrals and their use in >> Cornu's spiral to figure the strength of the diffracted (spread) >> signal. This holds for radio waves as well. >> >> Clay >> > > I'm actually somewhat crosstraining into DSP methods after receiving a > degree in statistics. In other words, I'm somewhat wet behind the ears > still. I notice that after having filtering data with a windowed-sinc > filter I have the usual ring in the neighborhood of a step function but it > dies out. Seeing this prodded some investigation. I tried some > simulations using a signal simulated using user specified amplitudes for > sine and cosine waves and combining using a synthesis equation. Then I > superimposed this on a hard step function. When I vary the magnitude of > the step function from large to small the ring disappears into the data and > is seemingly indistiguishable from the data after having filtered it. > > I really appreciate the help, thanks!The ringing amplitude is proportional to the step amplitude. If you smack a bell with a hammer, there will be a much louder clang than if you hit it with a water hose. Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●February 26, 20092009-02-26
On 26 Feb, 20:54, "tkremund98" <tkremun...@hotmail.com> wrote:> This is just a question about Gibb's effect. Is Gibb's Effect is present in > stationary data?Gibbs' effect has nothing to do with whether data are stationary, but with properties of the Fourier transform around discontinuities. The problem is that the Fourier transform converges in the mean square sense, i.e. that the L2 reconstruction error (view with fixed-width font) n-1 e_n^2 = integral |f(t) - sum A_n exp(j w_n t)|^2 dt k=0 vanishes as n -> infinity. However, this convergence in the mean does not ensure *pointwise* convergence, since pointwise convergence is expressed in terms of the L1 norm: n-1 E_n = |f(t) - sum A_n exp(j w_n t)| k=0 This discrepancy between pointwise and mean convergence becomes particularly important at dicontinuities in f(t), which is what causes Gibbs' phenomenon. Rune
Reply by ●February 27, 20092009-02-27
On 26 Feb, 23:38, Rune Allnor <all...@tele.ntnu.no> wrote:> On 26 Feb, 20:54, "tkremund98" <tkremun...@hotmail.com> wrote: > > > This is just a question about Gibb's effect. Is Gibb's Effect is present in > > stationary data? > > Gibbs' effect has nothing to do with whether data are > stationary, but with properties of the Fourier transform > around discontinuities. > > The problem is that the Fourier transform converges > in the mean square sense, i.e. that the L2 reconstruction > error (view with fixed-width font) > > � � � � � � � � � � � � �n-1 > e_n^2 = integral |f(t) - sum A_n exp(j w_n t)|^2 dt > � � � � � � � � � � � � �k=0 > > vanishes as n -> infinity. However, this convergence > in the mean does not ensure *pointwise* convergence, > since pointwise convergence is expressed in terms of > the L1 norm:This should be the L_inf norm: n-1 E_n = max { |f(t) - sum A_n exp(j w_n t)| } k=0> This discrepancy between pointwise and mean convergence > becomes particularly important at dicontinuities in f(t), > which is what causes Gibbs' phenomenon. > > Rune
Reply by ●February 27, 20092009-02-27
>On 26 Feb, 23:38, Rune Allnor <all...@tele.ntnu.no> wrote: >> On 26 Feb, 20:54, "tkremund98" <tkremun...@hotmail.com> wrote: >> >> > This is just a question about Gibb's effect. Is Gibb's Effect ispresen=>t in >> > stationary data? >> >> Gibbs' effect has nothing to do with whether data are >> stationary, but with properties of the Fourier transform >> around discontinuities. >> >> The problem is that the Fourier transform converges >> in the mean square sense, i.e. that the L2 reconstruction >> error (view with fixed-width font) >> >> =A0 =A0 =A0 =A0 =A0 =A0 =A0 =A0 =A0 =A0 =A0 =A0 =A0n-1 >> e_n^2 =3D integral |f(t) - sum A_n exp(j w_n t)|^2 dt >> =A0 =A0 =A0 =A0 =A0 =A0 =A0 =A0 =A0 =A0 =A0 =A0 =A0k=3D0 >> >> vanishes as n -> infinity. However, this convergence >> in the mean does not ensure *pointwise* convergence, >> since pointwise convergence is expressed in terms of >> the L1 norm: > >This should be the L_inf norm: > > n-1 >E_n =3D max { |f(t) - sum A_n exp(j w_n t)| } > k=3D0 > >> This discrepancy between pointwise and mean convergence >> becomes particularly important at dicontinuities in f(t), >> which is what causes Gibbs' phenomenon. >> >> Rune > >Thanks for the help understanding this. I guess the only reason I mentioned the stationarity, or near stationarity, of the data is that if there are any events such as step functions or impulses that are not naturally cyclic. There will surely be some sort of ripple induced. However, I have thought that the max possible ripple would be so small that it would be insignificant compared with the variance of the data.
