JPL's Wireless Communication Reference WebsiteChapter: Wireless Propagation Channels

for the most general case, i.e., with nonzero time difference t = t_{2}  t_{1}, we have
and
where J_{0} is the Bessel function of first kind of order 0, and T_{RMS} is the rms delay spread. Note that the Bessel function J_{0}(0) = 1 for zero time differences. We now have to convert I and Q to amplitudes R.
f(r_{1},f_{1},r_{2},f_{2}) = r_{1}r_{2} f(i_{1}=r_{1}cos f_{1}, q_{1}=r_{1}sinf_{1}, i_{2}=r_{2}cosf_{2}, q_{2}=r_{2}sinf_{2}).
Integrating over f_{1} and f_{2} gives
where the Bessel function I_{0} occurs due to ò exp{cosf} df
The normalized correlation coefficient r is
Inserting the PDF (with Bessel function) gives the Hypergeometric integral
This integral can be expanded as
Mostly, only the first two terms are considered
It is defined as
Here we have that
Localmean value:  ER_{1} = Ö(p/ 2) 
Variance:  VARR_{1} = SIG^{2} R_{1} = (2  p / 2) 
Correlation:  ER_{1}R_{2}
» p
/2 [1 + r^{2
}/ 4] 