a Matlab package for phased array beam shape inspection

6 2 TIME STEERING AND PHASE STEERING
We will first show that as a function of û, z(t) is ”maximized“ in the direction of
û 0 so that the parameter û 0 indeed is the beam-steering direction as we have been
saying. And we verify that in that direction there is no dispersion, that is, all the
frequency components of the pulse yield exactly the same, equally phased, contribution
to the sum, and hence the temporal shape of the incoming signal is recovered without
distortion. On the other hand, there is dispersion in non-center-of-the-beam directions.
The effect hopefully is insignificant for narrow beams, but perhaps we should make sure.
(Question: Does GJ’s system simulation already do this?) Expanding Eq. (6) we get
z(t, û; û 0 ) = ∑ ∫
a m dω Sû(ω)e iω[(û−û 0)· Rm c −t] , (7)
m
so that in the direction û = û 0 we have
z(t, û 0 ; û 0 ) = ∑ ∫
( ∑
a m dω Sû0 (ω)e −iωt =
m
m
a m
)
s(t, 0; û 0 ) . (8)
In the case that the element amplification factors a m are all unity, the beamformed
signal from direction û 0 is just the number of elements times the signal due to the field
at coordinate origin, and in general, is intact in shape and is amplified by ∑ a m .
We verify that the energy
‖z‖ 2 =
∫ +∞
−∞
dt |z(t)| 2 (9)
of the beam-formed signal is maximum when û = û 0 . As an underlying experimental
setup, we are imagining here that the array steering has been fixed to û 0 , and the
direction but not the distance to the point-target is varied, implying that the plane wave
packet of Eq. (3) comes from various directions û, but is otherwise always the same so
that Sû(ω) = Sû0 (ω). We write Eq. (7) in the form
∫
} ∫
z(t, û; û 0 ) = dω a m Sû(ω)e i ω c [(û−û 0)·R m]
e iωt ≡ dωZ(ω)e iωt , (10)
{ ∑
m
where
Z(ω) = ∑ m
a m Sû(ω)e i ω c (û−û 0)·R m
(11)
is, in fact, the Fourier-transform of the beam-formed signal. According to Parseval’s
theorem,
∫
‖z‖ 2 = dω|Z(ω)| 2 . (12)
Because, with a m ≥ 0 and Sû(ω) = Sû0 (ω), Eq. (11) implies that
|Z(ω)| ≤ ∑ ( ) ( )
∑ ∑
a m |Sû(ω)| = a m |Sû(ω)| = a m |Sû0 (ω)| (13)
m
m
m
it follows from Eq. (12) that
‖z(t, û; û 0 )‖ 2 ≤
( ∑
m
a m
) 2 ∫
dω|Sû0 (ω)| 2 =
a m
) 2
‖s‖ 2 , (14)
( ∑
m
where ‖s‖ 2 is the energy of s(t, 0; û 0 ). From Eq. (7) and Eq. (14) we see that, indeed,
the array gain in the direction û 0 is maximal: ‖z(t, û; û 0 )‖ ≤ ‖z(t, û 0 ; û 0 )‖.