IEEE - Aerospace and Electronic Systems - May 2022 - Tutorial XV - 45

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references to backprojection will be mentioned as needed;
backprojection is covered more robustly in the literature
than what is indicated here. Backprojection borrows heavily
from CT image reconstruction although SAR requires modifications
of CT methods as it is a fundamentally different
sensing modality. Nonetheless, a wealth of relevant knowledge
resides in these and related texts: [20]-[24]. Finally, an
extensive index ofSAR research is kept at [25].
Common, but certainly not universal, assumptions
include using the slant plane, which is the plane described
by the path of the radar and the center of the ground patch,
a flat Earth, straight or circular flight paths, the absence of
multipath and multiple-time-around echoes, and plane
waves. The latter will be discussed shortly but while the
others are important in applications they are not germane
to the present discussion.
The purpose of this note is mainly pedagogical as the
concepts described are known from other sources and
even widely known in the SAR community, and the mathematics
required to understand those sources is usually
not terribly advanced-Fourier transforms, calculus, some
signal processing, and communications theory, and a little
geometry. However, we shall take a rather leisurely path
through the details and ultimately arrive at backprojection
by physical reasoning that might provide a different
perspective and perhaps a modicum of insight that other
writings tend to obscure for the sake of efficiency of
exposition.
The note begins with a section which develops a
detailed reflection model starting from first principles-
the wave equation and a general solution thereto. There is
no prejudice as to the particular form of the transmitted
signal. This section is very basic and derives results readily
stated elsewhere but will serve other purposes in the
next section. The next section motivates backprojection
by showing that it is the plane wave spectrum reconstruction
of a single point scatterer and relates the method to
restoring spatial information that is lost by the radar
receiver. Throughout, simple cases are developed before
more general cases, and at times then the general case will
be specialized back to a specific case as regards geometries,
signals, and ground patch models.
In this work, plane waves are assumed throughout even
though the signal emitted by a radar is not planar, even in the
MAY 2022
far field where it is spherical.1 While strictly speaking, the
plane wave assumption is wrong, it is commonly made with
a great advantage in analytical simplicity. In practice, the
plane wave assumption is supported by the fact that the
antenna main beam limits the angular extent of the radiated
ground patch so that the error caused by actual wavefront
curvature over that narrow angle remains small and at least
approximately known over the illuminated area, where
" small " is related to angular and range resolution, and wavelength.
But always, the approximation and its effects are recognized
so that results maintain their validity within
carefully stated limits of operating parameters. Countless
books, for example [14], and papers have been written detailing
methods to correct, counteract, focus, and otherwise
adjust for the approximation. Alternately, spherical waves,
represented by circular intersections with a flat Earth, also
lend themselves to backprojection methods [27]-[32] including
elliptical backprojections in the case of bistatic modes.
We shall not be concerned with these issues since they are
not relevant to the present topic, the plane wave assumption
being acceptable for present purposes. Even so, the principles
discussed herein can be adapted to the spherical wave
assumption but the presentation becomes more complicated.
PROPAGATION AND REFLECTION MODELS
In this section, a simple but general solution to the wave equation
is developed for the radar application and then a reflection
model is developed. Some readers might find the early
part ofthis section too elementary and can skip it but the later
result feeds immediately into section " The Receiver Signal. "
SIGNAL SCALING AND SHIFTING
In the following, signal and wave descriptions will appear
with various scale factors and various amounts of shift.
1Some authors describe, without qualification, the far field of a radiating
source as being characterized by plane waves and the near
field as spherical waves. Both are incorrect; the far field is characterized
by spherical waves or by waves approaching sphericity and
include a 1=r amplitude dependence. The details are more technical
and include noting that higher order powers of 1=r are increasingly
ignorable for large r [26]. Plane waves, like sine waves, are a convenient
fiction.
IEEE A&E SYSTEMS MAGAZINE
45
IEEE - Aerospace and Electronic Systems - May 2022 - Tutorial XV

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