IEEE - Aerospace and Electronic Systems - May 2022 - Tutorial XV - 52

A Rationale for Backprojection in Spotlight Synthetic Aperture Radar Image Formation
r0
x x0ðÞ ¼
Z
which evaluates to
r0
d ^x0 rsðÞp k x0 2ðÞ
½d
r0
x x0ðÞ ¼
1
2
Z
g0
2
px0ðÞd
x x0ðÞ ¼ p k x0 2^x0 rsðÞ
½¼ p kx0 2k rsðÞ (16)
which is reassuringly the same as (10). For a centered scatterer,
rs ¼ 0; 0ðÞ0, the result is just
r0
x x0ðÞ ¼ p kx0ðÞ:
IMAGE RECONSTRUCTION
We now specialize pðÞ to a particular form, introduce 2-D
Fourier transforms, and show how one or more scatterers-impulses
in the ground patch-can be reconstructed
from several r0
x x0ðÞ as measured from different us.
WAVE NOTATION
So far, the formofpðÞ has been completely general, unspecified.
In the development, it has been portrayed graphically
as a generic pulse but that was only for convenience and to
allow us to think of a " leading edge " as a time reference. In
fact, any other feature ofany other signal could have served
as a time reference. Indeed, there has been so far no functional
limitation onpðÞ whatsoever.
Before we proceed further, we need to introduce a bit
ofnew notation. Up to now, we have used k as a scale factor
in a generic, prototype function, and a vector version k
that connotes both a scale factor and a direction of propagation.
As useful as this is, we will introduce a new vector
which will subsume k if we want it to. With k real, define
a new vector k ¼ kk so that kjj ¼ kjj kjj and so that k
points in the same or opposite direction as k depending on
the sign of k. This setup provides a two-level scaling for
the spatial aspect of the wave, a convenience in some
cases but a nuisance in others. Therefore, consider a usage
whereby k is a unit vector, k ¼ 1; and thus kjj ¼ kjj. This
effectively disables the scaling due to k for instances,
where pðÞ requires no scaling but leaves it as an option
when needed while offering a standard wave notation for
what follows. That kjj ¼ 1 will bear on everything we do
in the sequel unless stated otherwise. An effect of this
choice is that (15) simplifies a little
r0
x x0ðÞ ¼
Z
g0ðÞpx0 2ðÞd ¼ r0
¼ g0 x0ðÞ p 2x0ðÞ
where the second line expresses the relationship with convolution
notation. An alternate form of (17), found by
changing variables under the integral is
52
x u;x0ðÞ
(17)
7Care should be taken when computing with delta generalized functions.
For example, d x=2ðÞ¼ 2d xðÞ which result is found by writing
the delta as its defining limiting integral and changing variables.
8Obviously, a radar signal has to have a beginning and an ending, a
pulse. Here we shall push that detail aside for the time being and
revisit it later in this section.
IEEE A&E SYSTEMS MAGAZINE
MAY 2022
MONOCHROMATIC WAVES
Now, consider a specific form for pðÞ, a monochromatic
( " one color " ) plane wave, the 2-D traveling wave spatial
analog ofa sinusoidal function oftime. The desired form is8
p vt kx
ðÞ¼ ej vtkx
ðÞ:
The right-hand side is a periodic function of its argument.
For example, hold x constant and consider a time-based
phase function ft tðÞ ¼ vt. Define T such that
ft t þ TðÞ¼ ft tðÞ þ 2p ¼ v t þ TðÞ implying that the
period T ¼ 2p=v. Similarly, define a space-based phase
function fx xðÞ ¼ kx while holding t constant. Then,
define such that fx x þðÞ¼ fx xðÞ þ 2p, implying that
¼ 2p=k;, the spatial period, is called the wavelength.
For a scatterer at rs, from (7), the reflected field is
rx; y; tðÞ¼ ej vtþk r2rsðÞ
½¼ ej vtþkx x2xsðÞþky y2ysðÞ
½
(18)
which, as set up before, propagates backward along the x0
axis, which is rotated from the x axis by u. The wavenumber
vector is k ¼ k cos u;k sin uðÞ¼ kx;ky
which can ease computations in some cases.7 Notably, for
a particular angle u; ifpðÞ is an impulse, the projection of
the ground patch at angle u is returned directly as the
receiver signal but with a scaling by 1=2 in amplitude and
distance-stretched in space. Alternately, if the ground
path g xðÞ ¼ d2 xðÞ, the transmitted pulse is returned
directly, r0
x x0ðÞ ¼ px0ðÞ as we have seen earlier. Otherwise,
the scaled projection is filtered by a filter which has
an impulse response ofpx
ðÞ.
with k ¼ kjj
and indicates the direction opposite the direction of propagation;
k is the wavenumber or spatial frequency in radians=m
along the x0 axis, and kx
and ky are the
wavenumbers in the x and y directions, respectively. For
example, if u ¼ 0, then kx ¼ k and ky ¼ 0; there is no spatial
variation in the y direction. v is the temporal frequency
in radians=s and as stated earlier the propagation
speed is implied by the relationship c ¼ v=k. The monochromatic
receiver signal from a scatterer, from either
(10) or (16), is
IEEE - Aerospace and Electronic Systems - May 2022 - Tutorial XV

Table of Contents for the Digital Edition of IEEE - Aerospace and Electronic Systems - May 2022 - Tutorial XV
