JPL's Wireless Communication Reference Website

Chapter: Wireless Channels
Section: Multipath fading, Rayleigh fading, PDF

Exercise

Let z = x + jy with x and y i.i.d. Gaussian with zero mean and variance s2.
  We initially compute the distribution of received power p. From this the PDF of received power and the PDF of the amplitude can be derived.

The probability that (x2+y2)/2 < p, with p some power value is

           1                         x2+y2
F (p) =  ----       INT     exp(-   ------)  dxdy
 p       2p s2    x2+y2< 2p             2s2

Using the cartesian to polar transform dxdy = r dr dq, we get

           1       2p       SQRT(2p)        r2
F (p) =  ----     INT   dq   INT  r exp(-   ---) dr
 p       2p s2       0         0             2s2
Thus,
Fp(p) = 1 - exp(- p/s2).
This is the (exponential) distribution of received power p.

The PDF is found by taking the derivative

        1         p
fp(p) = --- exp(- --)
        s2        s2

The instantaneous power p thus has the above exponential pdf. Conversion between the probability density of amplitude and that of power involves

|fr(r) dr| = |fp(p) dp|
and p = r2/2, so dp = rdr. We get:

The pdf of the amplitude is

        r         r2
fr(r) = --- exp(- --)
        s2        2s2
for r > 0.
 



JPL's Wireless Communication Reference Website © Jean-Paul M.G. Linnartz, 1993, 1995.