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[Beam Convolution] [2005-11-11]
iamet 发表在 Ω〖物理〗
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A normalized Gaussian beam pattern for an antenna is given by
 (1)
where  is the radial offset from beam center and  is the Gaussian standard deviation. Since it is normalized, it satisfies
 (2)
Let the beam center be pointed a distance R away from the center of the field of view (of diameter  ). In polar coordinates, the law of cosines relates R and to the center of the field of view by
 (3)
Therefore
 (4)
The response of the antenna to a point source at R is then
  (5)
But
 (6)
(Arfken 1985, p. 615), so
 (7)
and
 (8)
Therefore, plugging (8) into (5) gives
 (9)
which unfortunately cannot be integrated analytically.
In the case when R = 0, the integral (9) simplifies to
 (10)
Make the substitution  , so  and (10) becomes
    (11)
As  (infinite field of view), all the power in the point source is seen by the beam, so we expe 
 (12)
Make the substitution  , so  , and
 (13)
Now define
   (14)
  (15)
  (16)
so (12) becomes
 (17)
From Gradshteyn and Ryzhik (1980),
 (18)
where  is a Whittaker function. We therefore have
 
 (19)
But
 (20)
so
 (21)
and (19) becomes
 (22)
In order to find the half-power beamwidth  , set  equal to half of the peak beam response, as given by equation (9).
 (23)
 (24)
This can be solved by looking for the roots of
 (25)
To solve using Newton-Raphson iteration, we also need the derivative  , which is
  (26)
But
 (27)
so
  (28)
so we have finally
 (29)
where
 (30)
Unfortunately, the  and  functions blow up for  . However, by using an asymptotic form, the integrals can be done acceptably. The asymptotic expansion for modified Bessel functions is
 (31)
for . Here,
 (32)
Therefore, the integrand of F becomes
 
 
 (33)
Similarly,
 (34) |
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