IEEE - Aerospace and Electronic Systems - May 2022 - Tutorial XV - 63

Bauck
SUMMARY AND COMMENTS
An intuitive explanation and interpretation of the use of
backprojection in forming spotlight synthetic aperture
radar images has been presented. A radar signal propagation
model incorporating the spotlight geometry under the
plane wave assumption was developed from first principles,
the wave equation and some simple solutions in the
radar context. This model does not specify the form of the
transmitted signal and so is suitable for any signal. The
propagation model was solved to describe both the timedomain
and space-domain signal that arrives back at the
radar. The model was applied first to a single point scatterer
and then to specialized collections of point scatterers
arranged along the rotated coordinate axes, finally being
generalized to a continuous collection of scatterers, the
ground patch reflectivity. This resulted in the important
result that the reflected signal is the convolution of the
transmitted signal with a projection of the ground patch
reflectivity. Next, a specific signal was selected, the monochromatic
plane wave. The intimate connection between
monochromatic plane waves and the Fourier transform
was demonstrated and the concept of backprojection was
introduced with the satisfying result that the operation
restores the spatial information, the full plane wave, that
was lost by sampling the reflected wave at only one point,
the receiving antenna. It was shown that highly focused
impulses could be reconstructed by backprojecting a variety
of monochromatic received signals covering a wide
range in the 2-D frequency domain in both rectangular
and polar formatted data, and how less well-resolved
images could be constructed from a limited range of spatial
frequency information. The all-important Projection
Slice Theorem was derived, connecting projections and
Fourier transforms even more closely. Due to the general
signal model, it was shown how to correct the undesirable
effects of a general pulse shape including but not limited
to the popular linear frequency modulation (LFM). After
this lengthy introduction, the full backprojection method
for a general pulse and general ground patch was introduced,
by now allowing a good comprehension of how
and why backprojection works. This led naturally into a
discussion ofcomputational considerations and a comparison
of convolution-backprojection and direct Fourier
inversion, the polar format algorithm. These methods
were shown to be the same in their theoretical underpinning
but differing in implementation details.
Chirp signals [40] wherein the phase of a sinusoid is
made to change quadratically with time so that the frequency
changes linearly with time, usually with an offset
frequency due to a RF carrier, are commonly used. The
theory of LFM and demodulation in the context of spotlight
SAR is thoroughly covered in [1]. These signals are
compatible with the presentation herein but due to their
special nature are especially attractive for this application.
MAY 2022
Typically, the demodulation results in a signal which can
be cast directly in polar format form in the k-domain and
thus requires an inverse Fourier transform before being
suitable for backprojection.
By default, so far, it has been assumed that the signal
which is to be backprojected is the bandpass signal which
is carried on a radio frequency (RF) carrier. This would be
an arduous task not the least because of the difficulty of
sampling the RF signal directly. Radar receivers normally
convert the bandpass signals centered on the RF carrier frequency
to baseband via time-domain processing before
digitizing and processing for myriad reasons. While the
per-pulse demodulation is done by the receiver, it is convenient
to envision that, in the case of direct Fourier inversion,
a 2-D demodulation is carried out as follows. The data
are conceptually arrayed in the k-plane as in Figure 16.
The correct demodulation to baseband takes place by simply
translating the entire constellation intact toward the origin
so that it is more or less centered there. This is really a
conceptual translation and nothing needs to be done as long
as the data in computer memory are interpreted as though
they are arrayed near the origin. The translation affects
only the phase of the image, not the magnitude, so the
direct Fourier inversion by FFT yields the same magnitude
image but with a dramatically increased efficiency since a
raft of zero-value samples need not be processed [1]. This
bulk demodulation is distinctly different from a process of
demodulating each return signal separately in one dimension
and sliding it toward the origin of the k-plane which
will give the wrong result, a star-like or wedge-like constellation
centered on the origin, not the desired shape. In the
case of backprojection, the receiver demodulation implies
the same pulse-by-pulse translation along the respective k0
x
axes toward the origin. In order to maintain the advantage
ofthe projection-slice theorem, a spatial remodulation outward
along the k0
compensating for the temporal modulation which was
removed in the receiver. Remember that spatial frequencies
are c=2 times smaller than temporal frequencies.
At the outset we stated that one of the assumptions is a
2-D geometry, in effect, a flat Earth. Most synthetic aperture
radars operate with this assumption to good effect. The mapping
from the slant plane to the ground plane is straightforward,
mostlyjust a scaling ofthe range dimension. However,
in the presence of significant deviations from flatness, image
distortions called layover will appear which can be challenging
for human image interpreters to understand. Examples
are hills and mountains, valleys and canyons, trees, open
mine pits, and tall buildings. Other distortions can appear if
the radar moves out ofthe slant plane. There is a signal processing
solution to this situation but it comes at a significant
cost: expand the geometry to three dimensions [15]. Under
the plane wave assumption, the equal time-of-flight loci
which are the basis for the mathematical concept of projections
are then planes rather than lines as studied earlier.
IEEE A&E SYSTEMS MAGAZINE
63
x axes is usually implemented [1], [50],
IEEE - Aerospace and Electronic Systems - May 2022 - Tutorial XV

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