a Matlab package for phased array beam shape inspection

2.2 Monochromatic signals—time-steering equals phase steering 7
2.2. Monochromatic signals—time-steering equals phase steering
We will now show that for a monochromatic wave (continuous, not pulsed, sinusoidal
wave), time-steering reduces to distortion-free phase steering. Assume that instead of the
wave packet of Eq. (3), the wave illuminating the time-steered array is a monochromatic
plane wave with angular frequency ω 0 coming coming from direction û,
s(t, r; û) = S 0 e i( ω 0
c û·r−ω 0 t) . (15)
This has spectrum S(ω) = S 0 δ(ω − ω 0 ). Inserting the spectrum to Eq. (7) gives the
time-steered beam-former output as
We regroup Eq. (16) as
z(t, û; û 0 ) = ∑ a m S 0 e i[ ω 0
c (û−û 0 )·R m−ω 0 t] . (16)
z(t, û; û 0 ) = ∑ [
s(t, Rm ; û) a m e −iΨm] , (17)
where
Ψ m = ω 0
c û0 · R m . (18)
Equation (17) shows that for a monochromatic wave, the end result of the time-delaybased
beam-forming can be also achieved by using the non-delayed signal from each
element, but instead applying an element-dependent phase offset Ψ m across the array.
Note incidentally that Eq. (18) shows why we need to take the amplification factors a m
real-valued: if not real-valued, they would directly affect the steering phase.
For the monochromatic wave, phase-steering and time-steering are equivalent, and
Eq. (8) therefore says that in the beam maximum direction neither method distorts the
(now sinusoidal) signal shape. Actually, there is no distortion in any other direction
either for the sinusoidal signals, for Eq. (16) says that the beam-formed signal is just
the wave’s time-form measured at the array origin, multiplied by the array factor AF:
z(t, û; û 0 ) = S 0 e −iωot × AF(ω 0 , û − û 0 ) , (19)
where
AF(ω, U) ≡ ∑ m
a m e i ω c U·Rm . (20)
2.3. Phase steering
For any incoming wave, monochromatic of not, phase-steering beam-forming is defined
via Eq. (17) and Eq. (18), by using some representative value for ω 0 in Eq. (18) when
computing the element phasing angles.
For the non-monochromatic case, phase-steering has the considerably drawback compared
to time-steering that it is dispersive even in the steering direction û 0 , so that the
beam-formed signal cannot be factored as in Eq. (8). Moreover, there is no guarantee
that the û 0 used in Eq. (18) is really precisely the direction of the maximum gain. In
the general case, the phase-steered beam-formed signal for a wave coming from direction
û can be written as
z = ∑ ∫
a m dωSû(ω)e i( ω c û− ω 0
c û 0 )·R m
e −iωt . (21)