a Matlab package for phased array beam shape inspection

30 7 SNR ESTIMATE FOR THE TEST ARRAY
6.2. Results
b) Histogram of the beam direction change around the value θ 0
∆θ = θ 0 trial − θ0 .
The mean value and standard deviation of ∆θ over the trials is shown in line
three of the plot header.
c) Histogram of the beam width change from the computed value W 0 of the
undistorted beam,
∆W = W trial − W 0 .
The mean value and standard deviation of ∆W over the trials is shown as
the last item in line three of the plot header.
Figures 21-24 show the beam shape jitter characteristics for a 50 × 20 element array
when the timing jitter at the elements varies from 100 ps to 1000 ps. In all cases, we
assumed amplitude jitter 0.2 around the undistorted value of 1.0, (and normalized the
amplitude power sum to unity). The beam width, in the xz-plane in this vertical pointing
is W = 0.68 ◦ . This means that if we can accept 0.1 × W errors in pointing, in essentially
no simulation run, should ∆W exceed 0.07 ◦ . The panel (b) of the figures show that
this requirement clearly is fulfilled for timing jitter up to 500 ps, but starts to become
questionable at the timing jitter of 1000 ps. The changes in beam width are similarly
small up to about 500 ps.
What is, perhaps, more troublesome is the loss of gain. At ∆t = 100 ps, the mean
loss of gain in the main beam is about 10% (0.5 dB), at ∆t = 300 ps the loss is about
25% (1.3 dB) and at 500 ps about 50% (3.0 dB).
We emphasize that the loss of gain is mostly due to the “incohence” caused by the
phase jitter. If we keep in the above four runs the input parameters otherwise intact but
set the element amplitude jitter to zero, the beam jitter distributions change very little
from those shown in Fig. 21-24. For example, the gain loss at 500 ps reduces from the
50% to about 44%.
Unless there is some fundamental problem with these simulation, it appears very clear
that to get, say 0.1×beamwidth pointing accuracy, it is definitely enough to be able
to control the element phase with something like 10–20 ◦ accuracy (clock timing with
100–300 pc accuracy at 225 MHz), as long as the phase errors at individual elements are
uncorrelated. The large number of elements (1000 in this simulation) then takes care
about averaging the distortions so that the main effect is just some loss of gain. The
randomness of the timing/phasing errors at the elements is so important that it perhaps
should be enforced.
7. SNR estimate for the test array
To get a rough estimate of SNR for the Kiruna test array, assume that the test array is
oriented towards the VHF antenna, long size (the x-direction) towards the VHF antenna,
and that the VHF antenna is perpendicular to that direction, at x = L = 200 km in
the test array coordinate system, and assume that the line from the test array to the
VHF antenna is perpendicular to the VHF vertical direction (“flat Earth”), see Fig. 25.
Also, assume optimal polarization matching and assume that the angle χ between the