IEEE - Aerospace and Electronic Systems - May 2022 - Tutorial XV - 50

A Rationale for Backprojection in Spotlight Synthetic Aperture Radar Image Formation
Figure 9.
Scatterers perpendicular to the wave direction are reflected at the
same time.
To get the receiver signal for a scatterer located off the
rotated x0 axis, consider Figure 9 in particular rs. Let ^x0 be
a unit vector pointing in the direction of the x0 axis at
angle u relative to the x axis, ^x0 ¼ cos u; sin uðÞ. Then,
x0
s ¼ ^x0 rs ¼ rsjj cos us uðÞ. The scaled-x0
accumulated from the origin to x0
s
x x0ðÞ ¼ p kx0 2k rsðÞ:
is kx0
cos us uðÞ¼ k rs and thus, recalling (9)
r0
MULTIPLE AND CONTINUOUS SCATTERERS
We now turn to having multiple point scatterers and then a
continuously reflecting ground patch. Considering again
Figure 9, let there be multiple scatterers that happen to lie
along the line ^x0 rs ¼ x0
s, call them rnfgNs
construction
x^0 r1 ¼ ^x0 r2 ¼ ¼ ^x0 rNs
and also
k r1 ¼ k r2 ¼ ¼ k rNs
¼ k rs:
Each reflects pðÞ with its own weight or reflection coefficient
gnfgNs
n¼1, and each reflected gnpðÞ adds to the
others.5 The reflection coefficients gn are complex-valued
to allow for both amplitude and phase modification of the
reflected signal. Assuming linearity, the receiver signal
due to all the reflections is
r0
x x0ðÞ ¼ pkx0 2k rsðÞ gn:
XNs
n¼1
(11)
5There are two assumptions at work here. The first is linearity, that
reflections sum in proportion to the reflection coefficients. The second
is the Born approximation which assumes that secondary reflections
between scatterers are weak and can be ignored. This is
potentially a severe limitation of SAR in some situations and it is
rarely addressed. An exception is [35] in imaging the interiors of
comets.
50
¼ ^x0 rs
n¼1. Then by
units
s ¼ kjj rsjj
(10)
Figure 10.
Multiple scatterers aligned on the x0 axis.
If there is instead a continuum of reflectors g y0ðÞ
along the same line then the receiver signal takes
the form
r0
x x0ðÞ ¼ pkx0 2k rsðÞ g y0ðÞdy0
Z
(12)
where the integration limits are usually considered to be
set by the ground patch as it is illuminated by the main
beam of the antenna.
Now consider a ground patch, which is comprised of a
series of reflectors arranged along the x0 axis, as shown in
Figure 10. Let the scatterers be rmfgMs
rm ¼ x00
m; 0
ficients gmfgMs
ity, the receiver signal is
r0
x x0ðÞ ¼
m¼1 such that each
and again associate a set of reflection coefm¼1.
From (9) and the assumption of linearXMs
m¼1
gmp
k x0 2x0
m :
An important difference with (11) is that here, copies of
pðÞ arrive at different time due to scatterers being different
distances from the receiver and thus must therefore
appear under the summation. In order to pass to a continuous
distribution of scatterers along x0,Figure 11 shows a
division of the axis approximating a continuous reflectivity
g x0ðÞ as a series ofM equal-width rectangles according to a
standard method ofcalculus. Allowing for a dummy variable
under the summation results in
r0
x x0ðÞ
MX1
m¼0
g m DðÞp kx0 2km DðÞD:
As M!1and D ! 0 we get the integral
r0
x x0ðÞ ¼
Z
gðÞp k x0 2ðÞ
½d:
(13)
Wewould now like to combine the results (12) and (13).
Toward this end, introduce a general ground patch gx; yðÞ
illuminated by the antenna. Recall the earlier discussion of
the slant plane and that the ground patch as used here is the
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IEEE - Aerospace and Electronic Systems - May 2022 - Tutorial XV

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