Graphics Programs Reference
In-Depth Information
8.4.2. Computation of the Radiation Pattern via the DFT
Fig. 8.8
shows a linear array of size , element spacing , and wavelength
. The radiators are circular dishes of diameter . Let and ,
respectively, denote the tapering and phase shifting sequences. The normalized
electric field at a far field point in the direction-sine
N
d
λ
d
w
() Φ()
sin
ψ
is
N
∑
1
N
2
1
j
∆φ
n
-------------
E
(
sin
ψ
)
=
w
()
e
(8.45)
n
=
0
where in this case the phase reference is taken as the physical center of the
array, and
2π
d
λ
----------
∆φ
=
sin
ψ
(8.46)
Expanding Eq. (8.45) and factoring the common phase term
yield
exp
[
jN
1
(
) φ 2
⁄
]
e
jN
1
(
) φ 2
⁄
jN
1
(
) φ
jN
2
(
) φ
E
(
sin
ψ
)
=
{
w
()
e
+
w
()
e
(8.47)
++
…
wN
1
(
)
}
By using the symmetry property of a window sequence (remember that a win-
dow must be symmetrical about its central point), we can rewrite Eq. (8.47) as
e
j
φ
0
jN
1
(
) φ
jN
2
(
) φ
(8.48)
E
(
sin
ψ
)
=
{
wN
1
(
)
e
+
wN
2
(
)
e
++
…
w
()
}
d
Φ ()
Φ ()
Φ ()
Φ ()
Φ ()
w
()
w
()
w
()
w
()
w
()
d
Figure 8.8. Linear array of size 5, with tapering and phase shifting hardware.
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