IEEE Geoscience and Remote Sensing Magazine - September 2021 - 18

which is assumed to remain constant; i.e., Nact
is independent
of time. (Due to the geometry, a constant 1 km of
ground range is mapped to a decreasing angular segment
from the near range to the far one. Further, reflector antennas
exhibit a defocusing effect for large angle-of-arrival
deviations with respect to the boresight. Both effects cause
the number of active channels to change. Strictly speaking,
this effect should be considered, but it is ignored here for
the sake of simplicity.) However, the elements that are active
change while the echo signal is received from different
directions. This is accounted for by including the time
dependency in
N (),tact
nt0 (), and ().nt1
With reference to
Figure 6, the activation matrix can be understood as a mask
placed on the data of the feed elements to select only the
samples of relevant signal energy.
From a theoretical perspective, it is always beneficial to
extend the beamforming to the maximum number of elements
that can, for example, be used for sidelobe reduction
and ambiguity suppression. However, the weight of the nonactive
elements is very small, and the elements' utilization
requires knowledge of their complex-valued secondary patterns.
In cases where knowledge of the antenna pattern accuracy
is limited, additional elements might contribute only
as noise, which makes them useless for reflector beamforming.
Part 2 of the tutorial elaborates on the proper feed element
selection approach and provides simulated examples
of the effect on beamforming performance.
Thus, from a beamforming perspective, it is perfectly
valid to set the weights of noisy data streams to zero; thus,
()
wt 0 6ntNact (), avoiding noise amplification. From
n
=
"
a hardware architecture point of view, however, it does not
make sense to waste processing resources by multiplying
inactive channels by zero weights. An " economical " design
would take this aspect into account by reducing the number
of multiplications, as all but Nact
multiplications are considered
unnecessary. With reference to Figure 7, this would
mean limiting the summation to a (changing) set of active
elements such that the output data vector is given by
nt
==
!
nwt
nt
nn
nn t
()
= 0
1()
bs wts
()
// (19)
N ()act
n().
On the other hand, reducing the number of multipliers may
not be worthwhile, as it requires the implementation of an
elaborate data routing procedure.
Mathematically, the selection of the set of active antenna elements
Nact can be represented through an NN binary
[; ,, ,, = f ,, ,],
activation matrix A == f
where NN .
++ -
ij,
actel
#
ij ij
,,
ment unit vector ,ei
actel
ai 12 N jNact
el # act
12
The activation matrix is a Toeplitz matrix
,
elenn
,0 ++f
as the set of selected elements N ()tact
= ,;
changes through
nn N11
00 act
-
aa a11 ij== the first column of which is an Nel
i.e., a vector with zeros in all its entries
except the ith entry, which is set to one. The activation matrix
thus " selects " antenna elements
time, the activation matrix itself becomes a function of
time. With this,
18
where cn is the Nrb element vector of channel n; the implementation
would then sum up the individual channel data
streams (see Figure 7) to generate the output beam:
bc .c//n
Nact
==
=
nn
n
1
n
basis at a rate of :fs
These operations are carried out on a sample-by-sample
one element of each cn
vector is computed
every /econdfs1s , requiring N4 act real-valued multiplications
and N2 act
sln
wtn () are complex numbers); at the same time, Nact
act
-
real-valued additions (both [] and
complex
samples are added to yield one element (sample) of the
output vector ,b which requires ()N21
real additions.
Note, again, the issue of the matrix notation in (21), which
was resolved in the " Clarification of the Correct Usage of
Matrix Notation " section. Similar to before, a correct notation
requires rewriting the signal vector sn
a block matrix S pn
of channel n as
vector [[ ], [],, []]
ww wM01
f
nn nw-1
and multiplying by the complex weights
T of that channel.
DISPERSIVE SCORE
An effect that should be considered when applying weights
is the pulse extent mentioned previously. For large reflector
antennas, the secondary beamwidth in the elevation
may be smaller than the angular pulse extent. This causes
IEEE GEOSCIENCE AND REMOTE SENSING MAGAZINE SEPTEMBER 2021
(22)
where the term in parentheses is an NN matrix and
the modified Nact element weight vector wl now includes
only nonzero entries.
rb # act
HARDWARE REALIZATION VERSUS
MATHEMATICAL DESCRIPTIONS
In the following, the difference between mathematical
expressions and the computations required for the hardware
to implement the expressions are considered. The
expressions in (8) and (20) suggest that SCORE output
data are computed in a single-shot matrix-vector multiplication.
The underlying hardware implementation
would rather generate individual outputs for each channel
[] []·( )
cl sl wtnn n lN which, when
=
for =-f ,, ,,
01
written in vector notation, becomes
(),
cs wtnn n
=
rb 1
(21)
◗ Parameter values: Usually, the echo signal of each
antenna element is digitized; i.e., no combination
of the RF signals is applied. The number of active
elements,
N ,act
can be as small as two to four in
basic (single) SCORE. Applying beamforming for
sidelobe suppression requires activating elements
outside the signal focus, in which case Nact
as large as eight to 14.
may be
act
l l ,
bS tAw Sw==l (( ))
(20)
IEEE Geoscience and Remote Sensing Magazine - September 2021

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