Second moment of filter
 

What does it imply if two filters have same second order moments?
For eg . box filter and Gaussian filter ? (Pope 2000, p.563)
Thanks
SM

Really good question; as usual, the answer is "it depends", from what you are using the filter for.

Just from the math point of view, it means that they have the same coefficient for the second order term in a Taylor expansion, either in physical space or in spectral space. Thus, if you plot them you get a similar behaviour (up to 4th order in this case).

If you go more specific, this also means that if you apply these two filters, you actually get the same behaviour for the filtered field at the largest scales. More importantly, if you try to approximate the unclosed terms by approximately inverting the filters, the first two terms of the reconstruction are also equal.

If you instead try to estimate the filter cut-off length by the second moment, you also get the same answer.

Long story short: the filters are similar at low order, hence some answers you try to get from them are similar too.

Thank you for the answer.

I understand that I can conclude -
If a data is filtered with these two filters they should
produce same mean and variance?

Further can you suggest a function that has analytical solution
available for checking out these filters in 1D?
I know filtering step function with Gaussian has analytical solution
(but not sure about top-hat).
Regards
SM

in 1D you can easily construct a function summing up to any frequency you want, C(k) * exp(i*k*x)