Re: prob 4.17

Ed Casas (edc@ece.ubc.ca) Thu, 3 Feb 2000 16:11:08 -0800 

Date: Thu, 3 Feb 2000 16:11:08 -0800
From: Ed Casas <edc@ece.ubc.ca>
Subject: Re: prob 4.17



On Thu, Feb 03, 2000 at 03:08:20PM -0500, Maria Lei wrote:

> I values I got for sigma & A are extremely small, and the
> resulting cdf does not addup to -80dBm.
> 
> This is what I did:
> 
> r_median = 1.177 sigma = 10^-7   (= -70dBm)

This is true for the Rayleigh distribution but it's not the case
for a Ricean distribution.  For a Ricean distribution the median
will be a function of both A and sigma (as noted in problem
4.16).  For large values of K (A>>sigma), the Ricean distribution
approaches a Gaussian distribution and the mean and median both
approach A (you can use this, and the fact that A=0 corresponds
to a Rayleigh distribution to check your results).

You can find the median value from a CDF plot by finding the
value for which the CDF is 0.5 (this is the definition of the
median).

If you move to a point 10 dB below the median you can read the
CDF for a value 10 dB below the median.  The CDF is the
probability that the RV will be less than the given value (while
the question asks for the probability that the signal will be
*greater* than the given value).

Note that the Ricean distribution is the distribution of the
magnitude of the complex number (the voltage) not
magnitude-squared (the power) so convert to/from dB accordingly.

While the values along the abscissa of the CDF will be a function
of the values you choose for sigma and A, the shape of the curve
will be fixed for any given K, so it shouldn't matter what value
you use for sigma as long as you scale A accordingly (I used
sigma=1 to keep the numbers manageable).

If you use a different sigma you'll have to do the integration
over a different range of values to get an accurate plot.  Also
remember that you have to multiply the cumulative sum by the
integration interval.

But wait!  You shouldn't need to compute anything (if you don't
want to) you can read all the numbers you need off the curves
that I put on the course web page.

-- 
Ed Casas   edc@ece.ubc.ca   http://casas.ece.ubc.ca   +1 604 822-2592