IEEE - Aerospace and Electronic Systems - May 2022 - Tutorial XV - 51

Bauck
Figure 11.
A continuous 1-D scattering function g x0ðÞ in the x0 direction
quantized by Dx.
where dðÞ is the 1-D Dirac delta generalized function or
simply 1-D impulse. Both of these express the fact that
scatterers on a line parallel to the y0 axis arrive back at the
receiver at the same time. In applications, this set of line
integrals through the ground patch is called a projection of
gx; yðÞ at angle u or u-projection as shown in Figure 12.
Generally, this function evaluated over all u and perhaps
more clearly denoted with the redundant-in our notation-
g0 u;x0ðÞ, is known as the Radon transform of gx; yðÞ.6
Despite the appearance of a function ofpolar coordinates-
x0 in radius and u in angle-this function should not be considered
as an expression in polar coordinates because it is
multivalued at x0 ¼ 0. It is, however, periodic in u with
period 2p. Examples ofRadon transforms computed from a
simple test function can be found in [20] and [27] among
many other places. With this understanding of g0 x0ðÞ and in
view of (13), we can replace the integration of the continuous
line ofscatterers along the x0 axis gðÞ with the line integral
g0 x0ðÞ and write
r0
x x0ðÞ ¼
Z
g0ðÞp k x0 2ðÞ
½d ¼ r0
x u;x0ðÞ:
(15)
Figure 12.
A projection g0 x0ðÞ as a collection of line integrals through a
ground patch gx; yðÞ at an angle u.
actual area on the ground that is illuminated, projected onto
a plane containing the radar and the center of the actual
ground patch. Equation (12) contains continuous scatterers
that were artificially constrained to lie on a line parallel to
the y0 axis and an integration was calculated along this line.
With gx; yðÞ, this operation properly becomes a line integral
along the same line but cutting through the ground patch.
This line integral can be expressed in several ways, including
[36], by solving (4) for x and y,
g0 x0ðÞ ¼
and [20]
g0 x0ðÞ ¼
ZZ
or more compactly,
g0 x0ðÞ ¼
MAY 2022
gx; yðÞd x0 ^x0 xðÞdxdy
Z
gx0 cos u y0 sin u;x0 sin u þ y0 cos uðÞdy0
To recap: this is the spatial version of the signal at the
receiver for a general ground patch gx; yðÞ after adjusting
the phase reference xT ¼ 0. g0 x0ðÞ is the projection,
a set of line integrals through gx; yðÞ at angle u.The
received signal is the convolution of a possibly scaled
version of the basic signal with the projection of the
ground patch. This result compares to a central result
of [1] wherein a mixed spatial-temporal signal format
is preferred.
To demonstrate the use of (15), consider again the
receiver response to an off-center scatterer when the radar
is on the u-rotated x0-y0 system. The scatterer is modeled
as an impulse at rs, that is, gx; yðÞ¼ d2 x rsðÞ where
d2ðÞ is the 2-D Dirac delta generalized function or simply
the 2-D impulse. If rs ¼ xs;ysðÞ, then gx; yðÞ¼
d2 x rsðÞ¼ d x xsðÞd y ysðÞ. The projection of the
ground patch is
g0 x0ðÞ ¼ g0 u;x0ðÞ¼ d x0 x0
d x0 ^x0 rsðÞ¼ d x0 rsjj cos u usðÞ
s ¼
ðÞ. This Radon
transform of the off-center 2-D impulse is an impulsive
ridge tracing a sinusoidal pattern in the u-x0 Radon plane
with x0-amplitude rsjj and phase us. In a more complicated
ground patch, the Radon transform plotted this way will
appear to have many sinusoidal patterns ofvarious intensities
(height of ridge or other feature), amplitudes in x0, and
phases in u superimposed; for this reason such graphical
displays are sometimes called sinograms. Placing this into
(15) gives
Z
g xðÞd x0 ^x0 xðÞdx
(14)
6The Radon transform is closely related to the Hough transform
which is used to detect shapes, commonly lines, in digital images
[20], [37]. In SAR, the Radon transform is of the ground patch
which is not digital.
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IEEE - Aerospace and Electronic Systems - May 2022 - Tutorial XV

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