"SJ (James) Kim" <[EMAIL PROTECTED]> writes:
> The Chi^2 distribution is well used to analysis the performance of the
> wireless communication systems with Chi^2 distributed SNR. Similarly, we
> need the Chi distribution to know its gain, while Chi^2 denotes its
> power.
Yes. In communication systems we are nearly always interested in
power (as opposed to amplitude). Chi^2 is easier to work with
mathematically, e.g., the central chi-square with 2 degrees of freedom
-- chi_2^2(0) -- is also the exponential distribution. If you really
need amplitude, it is often simpler to work with the energy and
convert to amplitude at the very end.
> In this article, I support Johan�s method with the reference of
> the related internet site, and ask one dimplr question to know the pdf
> of the proposed random variable.
>From Tomas S Ferguson _Mathematical Statistics_ p 101 and 103:
Central chi^2 distribution is a subset of the gamma.
Gamma density G(a,b) is
f(x) = exp(-x/b)*x^(a-1) for x > 0
= 0 otherwise
the chi_n^2 is G(n/2,2)
The non-central chi^2 is
f(x) = \sum_{j=0}^\infty p_{\gamma^2/2}(j)f_{n+2j}(x)
where p_{\gamma^2/2}(j) is Poisson with parameter \gamma^2/2 and
f_{n+2j}(x) is G(n/2+j,2) = chi_{n+2j}^2.
> Johan Kullstam wrote:
> > The chi is just the square-root of the chi^2.
>
> You're very right!
>
> I found the site, where same explanation to your one is presented.
> Site address is
> "http://mathworld.wolfram.com/ChiDistribution.html.";
>
> In addtion, Chi pdf is also called as 'Generalized Rayleigh pdf', which
> is described in Proakis book.
I am not fond of Proakis' book. For example, his treatment of fading
spends an inordinate amount of time on distributions of Rayleigh and
Rician amplitudes without fully developing
* chi^2 energy,
* complex representation as 2-D gaussian,
* time dependency,
which are mostly what you want.
> However, please let me know this: what is this pdf, which I said in
> upper article?
> X(1,n) = sqrt(X(2,n))*exp(j*2*Pi*theta),
> where theta is uniform random variable between 0 and 1.
>
> In summary, Chi random variable was found to be denoted as sqrt of Chi^2
> one, and one simple question to ask the pdf of the proposed variable is
> dictated.
Let the chi^2 distribution be F(u)
then the chi distribution G(a) is
G(a) = F(a^2) or G(sqrt(u)) = F(u) for a,u >= 0.
Take derivatives to get density.
--
Johan KULLSTAM <[EMAIL PROTECTED]> sysengr
.
.
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