IEEE Geoscience and Remote Sensing Magazine - September 2021 - 16

extent is the minimum look angle subtracted from the maximum
look angle; in the slant range, this corresponds to
(),RR
farnear
-
and the temporal swath extent is T ]echo
Mw
m |j
= jj
-
a
D - j
near
|jj
farnear
,
divide by the step size to yield the total number of steps:
()
near
where the step size is fixed to its minimum value across
the swath, i.e., taking ()
for M .w Finally, the maximum update rate of the weights
is given by the inverse of the time during which the beam
coefficients remain fixed:
TTw
==
=
w
jj
-
farnear
echo
T
f M
T
1
echo
w
a am |jnear k
D -
j () .
(13)
◗ Parameter values: Consider the reference SAR system
with a
D m15-=
tion of xn=p 29 s , the angular pulse extent ()
applied, which are updated at a rate of f 22 kHz .
It should be emphasized that the preceding values
and
(12)
are obtained based on one possible criterion for the
beam transition; the aim is to show how simple models
may be formed to estimate the number of beam transitions.
Accounting for aspects such as ground topography,
antenna pointing errors, the impact of phase jumps on
interferometric applications, and the pattern calibration
will lead to other criteria and may result in rather
different values.
to arrive at an upper limit
reflector diameter. The swath limits
are set such that farnearjj-= 4°. For a pulse dura|jj
varies
from 0.5 to 0.4°, while the pattern beamwidth
is 1.1°. Then
M .67w = distinct weights are
w .
CLARIFICATION OF THE CORRECT
USAGE OF MATRIX NOTATION
The weight vector ()tw is explicitly written as a function
of time to facilitate a simpler mathematical representation
and to indicate that the weights change more slowly
than the signal sampling. However, a straightforward
application of matrix multiplication in (8) is ambiguous
and would probably not yield the right result. To explain
this, considering an arbitrary column of the matrix ,S
the direct application of (8) would cause all Nrb
to be multiplied by the same weight (),wtn
of sn
elements
which
clearly neglects the fact that the weights change with the
time sample index l of [].sln
A correct matrix notation
is stated here for completeness and as a justification to
later maintain the simpler time function representation,
which is believed to give better insight into the beamforming
process.
To arrive at the correct matrix equation corresponding
to (8), the signal matrix S is rewritten as an ()rb #
block (or partitioned) matrix S .p A block matrix can be inNN
M·
el
Φ
w
terpreted as being broken into sections called blocks or submatrices;
here, all off-diagonal blocks are zero matrices. The
signal matrix is thus given by
Angle
(a)
S =
p
RSp1
S
S
S
S
S
S
T
Angle
(b)
i 1
=
h
h
Sp2
j
j
g
0 g
Sp3
j
g
where each Spi for ,,
i =
g
j
j
Sp4
h
h
V
S Mp w
W
W
W
W
W
W
X
such that R wM mNrb . The elements of Spi
iM1= f w is an mNi # el submatrix
are then given
by mi complex signal samples of all channels. Note
Φ -
(c)
FIGURE 9. An (a) antenna receive radiation pattern and the pulse
angular extent versus the elevation angle; (b) the pulse moves
across the ground, and, when it reaches the half-power point of the
radiation pattern, (c) the weights are updated, generating a new
SCORE beam shifted by
U | .j16
Angle
that
the number of rows and, by this, the size of the submatrices
may change across the range, taking the typical
SAR geometry and antenna beamwidth into account;
this is because the slope of the angle-to-range mapping
function and the antenna beamwidth change across
the off-nadir angle. The structure of the block matrix is
shown in Figure 10(a) for the case of N 3el
= elevation
channels and M 4w = unique sets of weights. Each row
includes Nel
nonzero elements, which are the signal
samples for a single time instance l, while all other elements
are zero.
Consider the number Mw
of unique weights within
the echo window time, which is smaller than the number
IEEE GEOSCIENCE AND REMOTE SENSING MAGAZINE SEPTEMBER 2021
,
(14)
IEEE Geoscience and Remote Sensing Magazine - September 2021

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