[SciPy-User] OT: Stats Q (educate me please)

David Goldsmith d.l.goldsmith@gmail....
Thu Jun 10 20:51:32 CDT 2010

On Thu, Jun 10, 2010 at 5:48 PM, Charles R Harris <charlesr.harris@gmail.com
> wrote:

> On Thu, Jun 10, 2010 at 3:41 PM, Anne Archibald <
> aarchiba@physics.mcgill.ca> wrote:
>> On 10 June 2010 16:52, David Goldsmith <d.l.goldsmith@gmail.com> wrote:
>> > I recently had cause to ponder moving averages, and given my general
>> > interest in noise theory, it got me wondering: relative to the PS of the
>> > signal, what's the PS of a n-width moving average.  After unsuccessfully
>> > (though far from exhaustively) looking for some results in the
>> literature, I
>> > just started thinking about it myself, and came to realize, both based
>> on
>> > what the graphs are saying about the situation and then in light of
>> that, in
>> > retrospect, conceptually as well, since the moving average is a
>> smoothing of
>> > the signal, it's some kind of low-pass filter (removing power at higher
>> > frequencies), which begs the question: what kind of low-pass filter?  In
>> > particular, is it a truncation filter, completely removing any power
>> from
>> > windows smaller than n (the intuitive, though far from obvious,
>> conclusion),
>> > or is it an attenuation filter, applying some monotonically decreasing
>> > envelope to the PS for frequencies corresponding to windows smaller than
>> n?
>> > (Or does it somehow influence even the power of frequencies
>> corresponding to
>> > windows larger than n?)  Reference/proof?  Thanks for the education.
>> An n-width moving average is (I'm assuming equally-spaced data points)
>> convolution by a boxcar of width n.  So its effect on the power
>> spectrum is multiplication by a sinc function whose first zero is at a
>> period of n samples and whose amplitude at zero frequency is 1. (If
>> you have a finite-length data set and are doing circular convolution,
>> for "sinc" read the Dirichlet kernel.)
> The sinc needs to be squared for the power spectrum... Chuck

Right, thanks.  (I was wondering 'bout that: I didn't immediately run and
check my Abromowitz, but I didn't _think_ sinc was strictly non-negative, as
a PS envelope must be if the result is to remain a PS.)


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