This Appendix derives the approximate spatial autocovariance function for small-scale Rayleigh fading signals.

The spatial autocovariance function for received envelope is defined as follows [Jak74], [Tur95]:

Equation D-1. 

where is a position in the plane of the horizon (arbitrary if the fading process is considered to be wide-sense stationary) and is a unit vector pointing in the direction of receiver travel, θ. To develop an approximate expression for the autocovariance of multipath fields, first expand the function, ρ(r), into a Maclaurin series:

Equation D-2. 

Equation (D.2) contains only even powers of r, since any real autocovariance function is an even function. The differentiation of an autocovariance function satisfies the following relationship [Pap91]:

Equation D-3. 

and is useful for re-expressing the Maclaurin series:

Equation D-4. 

Now consider ρ(r) as being approximated by an arbitrary Gaussian function represented by a Maclaurin expansion:

Equation D-5. 

A Gaussian function is chosen as a generic approximation to the true autocovariance, since it is a convenient and well-behaved correlation function. The appropriate constant a is chosen by setting equal the second terms of Equations (D.4) and (D.5), ensuring that the behavior of both autocovariance functions is identical for small r:

Equation D-6. 

Therefore, using Equations (D.5) and (D.6), the approximate spatial autocovariance depends only on the three multipath shape factors, as shown in Equation (5.115).