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'[OT] "Real" Random sequence sought'
1999\10\15@163845 by

> Once recorded, does a random sequence cease to be random as can now be
> repeated at will, and is surely now - by definition - deterministic? Can
> randomness be quantified?  Any mathematicians out there care to enlighten
> me?

For a sequence to be random, all that's really required is that each entry
in the sequence is completely independant of the preceeding entries. Now, if
I take a finite-length sequence of random numbers, and then play this
sequence end-to-end repeatedly, then the longer sequence that I end up with
no longer obeys this requirement.

The easiest way to consider this is from the point of view of an "observer"
(eg. a PIC), who runs through the list one entry at a time. As long as they
run through the list only once, then the observer is unable to predict any
future entries, because they're independant of the preceeding ones.  If the
list has been precomputed, then from *our* point of view, yes, it's
deterministic, but only because we have knowledge of the whole list.

Cheers,
Ben

Another property of random number sequences is that there are no long
autocorrelations. A 64k snippet, or any other snippet for that matter will
eventually have a 100% overall autocorrelation. You may rethink your reasons
for needing random numbers and decide that pseudorandom numbers will do.
They are easy to generate algorithmically as you already have discovered.
The most economical solution that I have seen so far is the diode scheme.
Once that is generated then sampling it with an analog to digital converter
to the desired resolution is all that is needed. This should result in a
totally noncorrelatable sequence. Of course the catch-22 is that to
determine if a sequence has 0 autocerrelation you have to look at it
forever.

At 09:30 PM 10/15/99 -0700, Keith Causey wrote:
>Another property of random number sequences is that there are no long
>autocorrelations. A 64k snippet, or any other snippet for that matter will
>eventually have a 100% overall autocorrelation. You may rethink your reasons
>for needing random numbers and decide that pseudorandom numbers will do.
>They are easy to generate algorithmically as you already have discovered.
>The most economical solution that I have seen so far is the diode scheme.
>Once that is generated then sampling it with an analog to digital converter
>to the desired resolution is all that is needed. This should result in a
>totally noncorrelatable sequence. Of course the catch-22 is that to
>determine if a sequence has 0 autocerrelation you have to look at it
>forever.

We are apparently discussing two different types of randomness: 1) random
'numbers', which have some finite bit length, and 2) random bit streams,
which are sampled at some underlying rate.

The Platonic ideal of 'randomness' is useful in defining tests for
practical sequences. The fact that infinite bandwidth is impossible is
irrelevant in practical problems, since bandwidth is always limited
intentionally.

Random sequences are infinite-bandwidth and 'white' in the same way ideal
op-amps are infinite-bandwidth and gain. Real-world implementations, of
course, only approach the ideal.

In the case of random sequences, the upper limit on bandwidth required is
1/2 the sampling rate. The 'white' bandwidth required is usually much less
than this. I.e., one is much more concerned with proximate correlation than
distal correlation.

'Whiteness' implies two principal properties:

1. 'Equi-distribution', i.e., any subset is equivalent to any other subset.
2. 'Uniformity', i.e., correct DC component.

Since random numbers/digits are used in n-tuples (singles, pairs, triples,
etc.) with 'n' a small number (usually < 10), then having zero low-order (<
10) autocorrelations usually suffices.

================================================================
Robert A. LaBudde, PhD, PAS, Dpl. ACAFS  e-mail: rallcfltd.com
Least Cost Formulations, Ltd.                   URL: http://lcfltd.com/
824 Timberlake Drive                            Tel: 757-467-0954
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