IEEE - Aerospace and Electronic Systems - May 2022 - Tutorial XV - 48

A Rationale for Backprojection in Spotlight Synthetic Aperture Radar Image Formation
is the same as the nonrotated system so we can immediately
write
r0 x0;y0;t
ðÞ¼ p vt þ k x0 þ x0
T 2x0
s
Figure 6.
A wave reflecting from a point scatterer at xs.
:
(3)
Before proceeding with this more useful case, first consider
the simplified system where r0 x0;y0;t
ðÞ¼
p vt þ kx0½. The rotated coordinates, in terms of the unrotated
coordinates, are
x0 ¼ x cos u þ y sin u
y0 ¼y sin u þ x cos u
(4)
so that the simplified wave expressed in the x-y coordinates
is
rx; y; tðÞ¼ p vt þ k x cos u þ y sin uðÞ (5)½:
Let r ¼ x; yðÞ be a generic point in the x-y system and
define a vector4 k ¼ k cos u; k sin uðÞ¼ kx; ky
Figure 7.
A propagating wave approaching a point scatterer off the x axis,
at rs.
interesting, rx; y; 2tTðÞ¼ p k xð½
xT 2xsÞ, that is, the
" bulk delay " of2tT having elapsed, the signal is 2xs spatial
units in front of the receiver. This mental picture of the spatial
signal or several different copies of it " stacked up " in
front of the receiver will be useful later. In double the total
round-trip travel time, rx; y; 2tT þ 2tsðÞ¼ p k x xTðÞ
½
and the wave is back at the receiver.
A further generalization is shown in Figure 7, which is
once again displayed as an x-y plane but with an overhead
view. Here, a scatterer is shown at position rs ¼ xs;ysðÞ.
Since the wave is planar and its wavefront is perpendicular
to the x axis, it reflects from the scatterer at the same time
as a scatterer at xs; 0ðÞ3, so the previous result applies, as
we insert the geometrical form for xs;
rx; y; tðÞ¼ p vt þ k x þ xT 2 rsjj cos usðÞ
½:
Yet another generalization for the propagation model
will be to consider waves traveling not parallel to the x
axis but at an angle u relative to the x axis. Figure 8 shows
this situation, employing a coordinate system x0;y0ðÞ0,
which is rotated by an angle u relative to the x; yðÞ coordinates.
Concerning notation, the prime mark 0 will be used
on rotated coordinate variables, functions associated with
those variables, and when denoting coordinate pairs in the
rotated system,;ðÞ0. Unless stated otherwise there will
always be the assumption of a rotation by an angle u. For
a radar at x0
0
T; 0
and a scatterer at x00
s; 0 , the situation
3This is the crux of the plane wave approximation. With a spherical
wave, this point would reflect later than a point on the x axis at the
same xs distance.
48
0
x0
s;y0
s
This result deserves two notes. First, we defined r0
, not r00
s ¼ x0
s ¼
s; 0 , that is, the scatterer was not
constrained to lie on the x0 axis and so the result is more
general. The quantity k r0
s ¼ constant defines a line of
constant wave amplitude that is perpendicular to the x0
axis. All reflections from reflectors that lie on this line
return to the receiver at the same time. Second, the
receiver in practice will start recording only around the
time that the signal of interest, reflected from the ground
patch, arrives. There is no purpose to recording sooner
than that because there will be little or no reflected signal
or the reflected signal that might exist will be of no
4As we introduce vectors and dot products, recall that a convenient
property of nonzero vectors a and b is that a b ¼ ajj bjj cos a
where a is the included angle.
IEEE A&E SYSTEMS MAGAZINE
MAY 2022
T and x0
denoting
a wave traveling in the u direction with scale factor k ¼
jjk . Now the generic, simplified wave (5) can be expressed
as
rx; y; tðÞ¼ p vt þ k rðÞ:
(6)
But instead of this simplified setup, we have (3) in which
x0
Assuming the notation
r ¼ x; yðÞ rT ¼ xT;yTðÞ rs ¼ xs;ysðÞ
r0 ¼ x0;y0ðÞ0
r00
T ¼ x0
T;y0
T
r00
s ¼ x0
s;y0
s
for respectively a generic point, the transmitter location,
and a scatterer location in each of the coordinate systems,
we can use the same process above to get the compact
representation for the wave reflected from a point scatterer
on the x0 axis:
rx; y; tðÞ¼ p vt þ k r þ rT 2rsðÞ
½:
s are also expressible using forms similar to (4).
IEEE - Aerospace and Electronic Systems - May 2022 - Tutorial XV

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