>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 i
presen=
>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
mentioned the stationarity, or near stationarity, of the data is that i
there are any events such as step functions or impulses that are no
naturally cyclic. There will surely be some sort of ripple induced.
However, I have thought that the max possible ripple would be so small tha
it would be insignificant compared with the variance of the data.