The thin Gaussian antenna pattern given by Eq. (6.69) is

so that from Eq. (6.19), the complex antenna pattern p( x) is

in which . Then, from Eq. (6.50),

To analytically evaluate this integral, we require a quadratic approximation for the range r about r ₀, the radial distance to the beam center on the ground. For this, we have

First add and subtract and then factor out

We now use the approximation

(for which the error is less than 1 percent for y < 0.326) to obtain the approximation

Our approximation thus is

over the range for which s( x) differs significantly from zero. This range is chosen to be x ? x ₀ < 3 a. Outside this range, s( x) is less than 1.111 percent of its maximum value so the error obtained by using Eq. (A.4) to evaluate the integral Eq. (A.3) is small.

The inequality in Eq. (A.5) can be expressed more meaningfully in terms of the forward-look angle ? ₀ and the angular antenna beamwidth ? as shown in Fig. A.1. This is done by noting that the maximum value of is obtained for x = x ₀ + 3 a so that

Fig. A.1: Geometry for converting to antenna angular parameters.

Thus the maximum value of the ratio in Eq. (A.5) is

Now, from Fig. A.1 we obtain the relations

Substituting these relations in...