a Matlab package for phased array beam shape inspection

28 6 JITTER CALCULATIONS
direction would be if the start time error would vary systematically across the array, say
increase with a constant step from element to element.
It is possible to imagine the DDS start timing to be implemented in such a way that
systematic start time gradient would occur naturally. For instance, if we were using
a trigger that spreads from the origin across the array. Or if we were having a part
DDS, part analog up-conversion to RF, the analog LOs, even though obviously being
phase-locked, could still have a systematically increasing initial phase across the array.
To be avoided.
Instead, all the DDSs should (conceptually) have their own clock, which starts the
DDS using only information that is independent of the pointing direction. Say, starts
after a hardwired, calibrated-once, element-depending, delay after receiving a centrally
generated trigger pulse. Or starts directly based on some pre-programmed instant of
time at the element’s local clock.
Even small random timing errors will to some degree resemble an overall phasing
gradient, and thus change the beam direction. And even if not changing the beam
direction, the timing jitter would cause incoherence to the array’s radiation, and this
would reduce the available level of constructive interference that is the basic of the
beam-forming, and thus would reduce the maximum available gain. Simulation seems to
indicate that the actual required timing accuracy is not anywhere near the 8 ps. Rather,
it appears that someting like 100 ps would already be enough.
6. Jitter calculations
6.1. Method
To allow both amplitude and phase jitter at the antenna elements, we write the array
factor as
AF(u x , u y ) = | ∑ m x
∑
m y
a mxm y
e iψmxmy e i2πDx(ux−u0 x )mx e i2πDy(uy−u0 y )my | , (89)
where u 0 x = δ x /2πD x and u 0 y = δ y /2πD y are the direction cosines of the phase-steered
beam. The amplitudes a are taken to jitter around unity with a distribution that we
will take to normal with variance ∆a
a = 1 + N(0, ∆a). (90)
The errors ψ of the steering phase steps δ x and δ y will be normally distributed around
zeros with standard deviation ∆ψ
ψ = N(0, ∆ψ). (91)
Without any jitter, the maximum value of AF, the (relative) gain in direction (u 0 x, u 0 y),
is equal to the number N = M x M y of array elements. The first observation is that, with
only phase errors, but not amplitude errors,
AF(u x , u y )