a Matlab package for phased array beam shape inspection

20 4 A PHASED ARRAY
From Eq. (57) we Eq. (58) we get antenna’s “power integral” PI as
PI = 1 ∫
G E |AF| 2 dΩ . (59)
4π
4π
The array factor can always be evaluated numerically directly from the defining sum
Eq. (41); or by computing the 2D Fourier transform Eq. (42) by FFT; or, as we mostly
do in this package, where we take the excitation field as constant, even from the closed
expression Eq. (50). For many purposes, like when inspecting pointing accuracy, this
is already enough, for only the relative gain, that is, the gain relative to the maximum
level, the beam center, may be of interest. But when the attainable signal levels are of
concern, the absolutely normalized gain of Eq. (58) is required. The problem then is to
evaluate the power integral.
4.8. The power integral for an array with isotropic elements
For the array with isotropic elements, G E = 1, the power integral over the full solid
angle can readily be evaluated. The squared magnitude of the array factor is
|AF| 2 = ∑ a m a m ′e i2πu 0D(m−m ′) e i2πuD(m−m′ )
(60)
The only non-trivial step is to evaluate the integral
∫
I = e i2πuD(m−m′) dΩ(û), (61)
4π
Noting that the orientation of the coordinate system does not matter when integrating
over the full solid angle, we will orient the z-axis of our 3D spherical coordinate system
along the direction of the vector D(m − m ′ ), so that θ is the polar angle of û = (u, u z )
from that direction, and hence the integrand in I becomes
e i2π cos θ‖D(m−m′ )‖ sin θdθdφ .
The integral can be immediately evaluated over θ (and trivially,φ), to give
The power integral thus is
I = 4π sinc(2π‖D(m − m ′ )‖) . (62)
PI = ∑ a m a m ′ sinc(2π‖D(m − m ′ )‖) e i2πu 0D(m−m ′) . (63)
It may be observed that G = |AF| 2 /PI, with AF taken from Eq. (60) and PI from
Eq. (63), has the expected special cases. For instance, when D x = 0 and D y = 0, we
have only one, isotropic, element, so we expect G → 1 when D x → 0, D x → 0, and this
clearly happens. For the case of two elements along the x-axis, a 00 = 1 and a 01 = ±1,
with D x = D, D y = 0, the absolutely normalized gain is
G =
1 ± cos[2πD(u x − u 0 x)]
1 ± sinc(2πD) · cos 2πDu 0 , (64)
x
where u x = sin θ, u 0 x = sin θ 0 . Here, as in general, the normalization depends both on
the array spacing, the excitation pattern, and the beam direction.