Following a standard approach we can expand 
in a
Fourier series of the type (we drop the explicit dependence on 
and 
on 
, for the sake of simplicity):
where the coefficients of the Fourier series are defined as:
If we now assume that 
can be approximated by the expansion
to first order shown in Eq. (7 (click here)), it can be easily shown that:
Hence, can also be written as:
where all other coefficients of the Fourier series, 
and
, are 0 for 
, as a natural consequence
of the special form of Eq. (7 (click here)).
We now briefly discuss the case of an elliptical antenna-beam, and
what useful information can be obtained through the Fourier analysis.
If the beam is non-circular, then 
can no longer be represented by
Eqs. (1 (click here)) or (6 (click here)). Instead, the normalized Gaussian
antenna power pattern becomes:
where both x and y are expressed in units of the full beamwidth
at half maximum along the X direction, FWHMx. We then define
the ratio, f, of the full beamwidths at half maximum along the X and Y
directions:
hence one obtains 
and 
. For comparison
with Sect. 2 (click here) we then calculate the quantity 
,
where 
and 
:
where we have used Eqs. (2 (click here)) and (8 (click here)). From
Eq. (A7 (click here)) we can then expand to first order,
analogously to what we did in Eq. (7 (click here)):
If f=1 (circular beam) we see that Eqs. (A7 (click here)) and
(A8 (click here)) reduce to Eqs. (6 (click here)) and (7 (click here)),
respectively.
We can now calculate, from Eqs. (A2 (click here)), the first five coefficients
in the Fourier series:
where we have defined
If f=1 Eqs. (A9 (click here)) reduce to Eqs. (11 (click here)) and (A3 (click here)).
We see that the coefficient 
is a measure of the
ellipticity in the antenna-beam, as one can calculate f
from :
We note that c1 contains the unknown, ; this potential
problem has already been discussed in Sect. 3.1 (click here).