Arrays
An array can be thought of as a ‘sampled aperture’. Hence the concepts
associated with sampling for digital signal processing apply to array antennas
with a change of terminology.
In signal processing, the signal is sampled in the time domain. The baseband
sampling theorem requires that the sampling frequency must be greater than
twice the highest signal frequency (or the sample spacing must not be greater
than half the period of the highest frequency). The process of sampling
causes the sampled spectrum to contain repeats of the baseband spectrum,
spaced at multiples of the sampling frequency. The consequence of sampling
at too low a rate is aliasing, where the repeat spectra fold into the required
baseband spectrum and corrupt the signal.
For arrays it is the field within an aperture that is sampled across the aperture
space. The process of sampling causes potential repeat radiation beams at
angles other than the desired angle. If the sample spacing is too large, these
‘grating lobes’ appear in the radiation pattern (equivalent to aliasing). To avoid
grating lobes, the sample spacing must be:
d 
d
for an unsteered array.

for a steered array. (sampling theorem)
2
The beam can be steered by introducing a progressive phase shift between
array elements, given by:

2d sin 
where  is the angle off boresight.

This relationship applies to a linear array, in which all of the array elements lie
along one axis. For a planar array there will be two relationships, one for each
axis of the array aperture. A linear array is illustrated below showing the
difference in path length to a point in the far field between adjacent elements.
Converting this to a phase shift gives the steering expression quoted above.
d sin 

d
Arrays.doc
To analyse or design an array of antenna elements, linking the radiation
pattern to the array structure, the ‘Pattern Multiplication Theorem’ is useful. To
use this:
1. Obtain the pattern of a single element (Pelement)
2. Obtain the pattern of the array assuming all elements are ispotropic (Piso)
3. Obtain the pattern of the array using Parray = Piso x Pelement
This can be explained using Fourier transforms because convolution in the
aperture plane is equivalent to multiplication in the radiation pattern. (An
isotopic element will be represented as an impulse in the aperture plane.
Convolution of an impulse with the element field distribution will re-generate
that field distribution at the sample point.)
For simple arrays of two isotropic radiators, the resulting radiation pattern can
be found by considering the relative phase associated with signals arriving at
the far field from each radiator and sketching the resulting interference pattern
as a function of angle off axis. This is a good way to become familiar with the
way in which radiation patterns are formed.
A simple example is shown below where the elements are spaced by /2 and
both are fed in phase. Check that you can work out the patterns shown in the
two principal planes.
z
z
z
/2
x
1
2
y
x
y
Consider the effect on the radiation pattern of the following changes:





Increase spacing to 
Keep spacing at /2 but feed element 2 with phase +180o relative to
element 1.
Reduce spacing to /4
Reduce spacing to /4 and feed element 2 with phase +90o relative to
element 1.
Replace both elements with half wave dipoles, oriented parallel to the zaxis.
Arrays.doc