Re: FFT test with few kbits
From: Ernst Lippe (ernstl-at-planet-dot-nl_at_ignore.this)
Date: 02/23/04
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Date: Mon, 23 Feb 2004 19:15:54 +0100
On Mon, 23 Feb 2004 12:58:07 +0000, Cristiano wrote:
> Cristiano wrote:
>> Bill Unruh wrote:
>>> Yes, they are independent.
>>> <c(m)c(n)>= sum_k sum_l <b(k)b(l)> cos(2 pi m k) cos( 2 pi n l)
>>> = sum_k <b^2(k)> cos(2 pi m k)cos(2 pi n k)
>>> =<b^2> sum_k [cos(2 pi (m+n) k) +cos(2 pi (m-n) k) ]/2
>>> =0 unless m=n or m=-n
>>> Similarly for the sine and teh cross ones.
>>> I used that the bits are different positions are independent, and
>>> that all of the positions have the same mean square (1/2 in your
>>> system). Thus the correlations between different values are zero.
>>
>> As you know, my math skill is poor, so I sent your answer to the
>> Italian mathematician to hear his thought.
>
> I got his answer; hard translation...
>
> To be honest, when I said they are not independent I had a doubt. But
> strictly speaking I'm right. The question is a bit subtle.
Yes, it is. Bill was referring to the fact that they are not linearly
correlated, which is, as you correctly noted, a different notion.
> Showing that the mean value of the product of two random variate is the
> product of the mean values is just a needed condition for the independence,
(The standard english mathematical term is "necessary" condition)
> but not sufficient.
OK!
> You can easily see that they are not independent taking a small n, e.g. n=2.
> For big n's the c[m] tend to be normal and for random variate normally
> distributed the above condition is also sufficient.
> Thus they are _not_ independent, but they are independent as n get bigger;
> in other word, increasing n, the independence is stronger.
I am feeling a bit uncomfortable with this sentence.
First of all, the standard statistical definition of
independence is a pure black-and-white one, two random variables
are either independendent or they are not independent. As
far as I know there is not really a standard measure for the
"degree of independence".
In one of my previous post I more or less suggested an
ad-hoc measure of the "degree of independence" based on
entropy, and I argued that according to this measure
two different Fourier components are actually what
you would call highly dependent. Your statement at
least suggests that when N becomes very big the
dependence between two Fourier components becomes
small. In the example that I gave when N is a prime
the degree of dependence will be high even when
N becomes very large (and I expect that you will find
similar results for the case that N is not prime, they
are just somewhat harder to analyze).
So, in order to give a mathematical interpretation to
your notion of "independence" we should first have
some way to quantify it, and even when you select
such a measure, I am not all that certain that it
is true that the "independence" become really larger
when N gets larger.
Ernst Lippe
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