IEEE - Aerospace and Electronic Systems - May 2022 - Tutorial XV - 56

A Rationale for Backprojection in Spotlight Synthetic Aperture Radar Image Formation
Figure 14.
Rectangular grid of 2-D impulses in the wavenumber domain,
each represented by a dot. Each impulse represents a plane wave
ofwavenumber k propagating in direction u.
impulse appear at a distance of 1=2 Dkx or y=2p ¼
51:25, just outside the plotted area; their near sidelobes
can be seen appearing near the edges on the coordinate
axes. In the preceding calculation, the two in the denominator
accounts for two-way propagation. Since any but a
nearly central scatterer will cause at least one alias to
appear within the plot, we see that the frequency sampling
interval is too large by a factor of two in this illustrative
example; halving both Dkx and Dky would push the aliases
out far enough that they would not appear on this particular
example of a ground patch for any scatterer placement.
The preceding discussion has concentrated on image
reconstruction based on a rectangular sampling grid in the
wavenumber plane. While it is possible to make such a spotlight
SAR in which the waveform generation is probably
tractable, the design ofthe receiver becomes rather impractical,
a laboratory or turntable setting notwithstanding. Consider
a radar platform steadily sweeping through an angle u
while staring at a grid such as Figure 14 but which in reality
would have far more impulses to fill in. The waveform synthesizer
would be very busy generating sine waves of various
frequencies and at various times as u sweeps through the
various known wavenumber domain impulse positions; the
program would be deterministic but very challenging. However,
the receiver would need to be matched to each reflection
of the many transmitted sines, each at the appropriate
time. In many instances, the large number of narrow bandpass
filters required to be generated simultaneously, along
with the fact that the center frequencies ofthe filters in many
cases would need to be very close, make this receiver design
most likely impractical. However, there is one advantage:
the backprojections required can be implemented en masse
at a great computational savings, without literal backprojections,
by using an inverse Fast Fourier Transform algorithm
operating on the whole batch ofcollected complex reflection
amplitudes. The related process, called direct Fourier inversion,
will be elaborated upon later but in a different context.
This imaging concept was in fact described in a turntable
setting in [39].
56
SPOTLIGHT SAR
The results of the section " Backprojection and Impulse
Reconstruction " are interesting but not immediately applicable
to the usual spotlight SAR scenario because that
kind of radar naturally has data in the well-known polar
format. A spotlight SAR operates typically by transmitting
a signal, frequently a linearly frequency modulated (LFM)
chirp [40], from each of many positions as the radar flies
past the ground patch while the antenna dwells on the
patch as in Figure 1. Almost universally assumed is that
the radar is stationary at a fixed u while each signal is
transmitted and the reflections are received, then advances
to the next transmit-receive position, the so-called stopand-go
or stop-and-hop model [11, 41, Ch. 21].10 The general
reflection model of the section " The Receiver
Signal " can handle any kind of signal including a chirp,
but the monochromatic discussion of section " Image
Reconstruction " is potentially limiting. For present purposes,
as we saw in making Figure 15, we can somewhat
work around this limitation by considering a series of
monochromatic transmissions and combine the various
backprojections by depending on linearity to get a polychromatic
result. Working in this direction, consider that a
series of monochromatic waves can be transmitted from
each location u where the temporal frequency is an initial
value and then increases by a constant amount for each
10This model is mostly good in many applications but should always
be considered carefully, especially in space-borne systems. In
some cases a bistatic solution helps, with the transmitter and
receiver in slightly different positions since the platform travels
while the pulse is in flight. In others, a quasi-fan-beam geometry is
helpful to understand and process the effects of a fast-moving platform.
Both of these situations are considered to different degrees
in [27] and bistatic systems generally are discussed widely. An
application of relaxing the stop-and-go assumption in the case of
frequency-modulated continuous wave SAR is [42].
IEEE A&E SYSTEMS MAGAZINE
MAY 2022
Figure 15.
Centered impulse reconstruction from plane waves represented as
a biuniform grid of 124
124 points in the Fourier plane ranging
over k values of2p 0:6 to þ2p 0:6 in both dimensions.
IEEE - Aerospace and Electronic Systems - May 2022 - Tutorial XV

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