tricky problem defining function

From: Col Brown (cb135_at_hotmail.com)
Date: 01/26/05 

Date: 26 Jan 2005 08:30:54 -0800

All,

This is the bare bones of my problem:

I have a variable

X_n = E{B_n} for n=0,1,...M-1

M is an arbitrary finite value, and E{x} represents the energy of x
defined as E{x} = sum ^{N-1}_n=0 |x_n|^2

Now, onto my problem, i would like to generate a function, f(X_n),
such that it gives me a cdf of C_n = f(X_n):

(1) C_n = n, for X_n ~= K, n=0,1,...M-1

where "~=" means approx. equal to, and K is a constant.

(2) C_n = 0, for n = 0,1,..L

    C_n = 1, for n = L+1, L+2,...M-1

where L is some arbitrary "cut off" point.

So i would like a function f(X_n) that satisfies both conditions
above.

Perhaps an example to qualify my poor attempt to explain myself.

If X_n = [1 1.1 0.9 0.95 1.3], then i want to be able to select the
values X_0, X_1,...X_{M-1}, with approx. equal probability.

if X_n = [1 2 3 4 5], then i want to be able to select the values X0
and X1 with a high probability and X2..X4 with a low probability.

One more example i think...

If X_n = [1 1 1 2 2 2 3 3 10], then i want to select the values X0..X2
with high probability, X3..X5 with a medium probability (medium
probability sounds really ill defined!) and X6..X8 with a low
probability.

The "cut-off" point {L} that i mentioned is where i would like to set
the transition from high to low probability. The value that seems to
work well is when the transition occurs after 10-15% of the length of
the vector X_n.

My method for selecting the values {X_n} is to sample the cdf C_n with
a uniform random number generator, and select the value X_n that falls
closest to the random number. So, i roll the dice of the random
generator (y-axis of my cdf function), and read off the value of X_n
on the x-axis that falls closest to my number. I hope all of that is
clear!

The way i see it so far, is that i have 2 degrees of freedom to play
around with, one is obviously the function f(x). However, i have also
some degree of freedom to condition the X_n data when i calculate the
energy E{x}!

Can anyone help with this problem?

Thanks,

col

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