[vorbis-dev] Some random thoughts

[Marshall Eubanks]

Segher Boessenkool wrote:

> > This is actually not true & Kjetil is correct - in the limit, as the
> > number of
> > data points -> infinity, a truely randomly sampled time series
> > will not be band-limited (i.e., will have a spectrum extending up to
>
> You mean measuring over an infinite interval or infinite points in a finite
> interval? Second case is trivial, and evly sampled does it as well; first
> one surprises me. Need to think some more.

I meant the first.

The way to think about it is :

in the uniformly sampled case, there are only so many intervals between data points.
If the sampling rate is del_t (say), then there are no points with a spacing 0.1 x del_t (for example), and
so no visibility to frequencies near 1 / (2 x 0.1 x del_t), or to any freqency > 1 / (2 x del_t)
In the randomly sampled case, then there will be (with probability one) pairs of
points with ANY arbitrary interval as the number goes to infinity.
(In fact, there will be infinitely many of them!). So, even
if the AVERAGE data spacing is del_t, in the random case,
there will be pairs of points with a spacing (close to) 0.1 x del_t ,
and these will constrain the power around the frequency 1 / (2 x 0.1 x del_t), and in every other
possible frequency. (How well depends on the amount of data and the luck of the draw in
practice.)

> > infinite frequency). Points to note :
>
> [snip]
>
> > 3.) (The killer) : This only works if the data are truely SAMPLED
> > (i.e., if instantaneous estimates are available). This is a
>
> This is true for uniform sampling as well, although less devastating
> (it's not a killer).
>
> > mathematical abstraction. Real data are based on time averages over some
> > interval. If data are sampled at some Nyquist rate, good engineering
> > would dictate that the "samples" are actually averages over
> > ~ 1 / Nyquist rate (which will minimize the measurement noise). This will
> > band-limit the spectrum, so that the random sampled spectrum really
>
> It will give heavy aliasing into the higher frequency regions, so the
> high-frequency part is unusable; to get the continuous signal back from
> our sampled signal, we will have to do the band-limiting ourselves.
>
> > has  little or no extra information over the uniformly sampled one.(It
> > is also common for this averaging to occur both in the analogue
> > part of the recording equipment, and in the A to D process.)
>
> Unavoidable, I'm afraid.
>
> > Frequently in astronomy, observations (of some star's pulsations, say), really
> > are much shorter in duration (minutes to hours) than the periods between
> > measurements (days to weeks), so that these idea really are useful there.
>
> I see your point. Sounds like fun stuff to work with :-)
>
> > I think that their utility in this case would be basically nil.
>
> Agreed. You can't hear the higher sounds, anyway.
>
> Could it be of any use in the future Ogg video codec?
>
> Dagdag,
>
> Segher


--
                                 Regards
                                 Marshall Eubanks

T.M. Eubanks
Multicast Technologies, Inc
10301 Democracy Lane, Suite 410
Fairfax, Virginia 22030
Phone : 703-293-9624
Fax     : 703-293-9609
e-mail : tme at on-the-i.com     tme at multicasttech.com

http://www.on-the-i.com http://www.buzzwaves.com

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