IEEE - Aerospace and Electronic Systems - May 2022 - Tutorial XV - 54

A Rationale for Backprojection in Spotlight Synthetic Aperture Radar Image Formation
obtained by producing signals with a range of frequencies
in v.
For the second useful transform, we show simply the
Fourier transform of a plane wave ejk0x so that
F ejk0x
¼
Z
ej kk0ðÞxdx ¼ d k k0ðÞ:
(27)
Thus, a plane wave with wavenumber vector k0 is
expressed simply in the wavenumber domain as
d k k0ðÞ.
PROJECTIONS AND SLICES
We now have the pieces we need to derive an important
result. Compute the 1-D Fourier transform of the projection
(14)
F g0 x0ðÞ
fg¼ G0 kðÞ ¼
Z
¼
ZZ
Z
g0 x0ðÞejkx0
dx0:
Substitute (14) and reverse the order of integration,
G0 kðÞ ¼
g xðÞd x0 ^x0 xðÞejkx0
dxdx0
¼ Gkx;ky ¼ G kðÞ
g xðÞejk^x0xdx ¼ Gk^x0ðÞ ¼ Gkcos u;k sin uðÞ
where G kðÞ is the 2-D Fourier transform of the ground
patch reflectivity g xðÞ. The salient version, for our purposes,
is
G0 kðÞ ¼ Gk^x0ðÞ
(28)
which is often referred to as the Projection Slice Theorem.
It states that the 1-D Fourier transform of a u-projection is
the same as a slice at angle u through the origin of the 2-D
Fourier transform of the ground patch reflectivity. Even
though the argument of the right-hand side of (28)
accesses the k-plane generally, the notation Gk^x0ðÞ allows
a convenient parameterization in polar coordinates with
the radius expressed as k and the angle u expressed within
the unit vector ^x0 ¼ cos u; sin uðÞ.
This result contains nothing of the transmitted signal.
But this was worked out earlier in (24) so now we can also
write
R0
x kðÞ ¼ G 2k^x0ðÞPk
ðÞ ¼ R0
x u;k
ðÞ (29)
and, at least in principle, we can recover the effect of a
transmitted signal that has a nonflat spectrum as
G 2k^x0ðÞ¼
R0
Pk
x kðÞ
ðÞ
:
Divisions of this sort must always be done cautiously;
obviously it will fail where Pk
expect failure when Pk
54
ðÞ ¼ 0. One should also
ðÞ is small due to noise spoiling
the calculation. And one should never divide by a
We have seen something like this before in converting kx0
to the unrotated coordinate system in (4)-(6). With this
conversion, we can express the backprojection of the
receiver function from a centered scatterer in the x-y system
as
ru x; yðÞ¼ ejkx
where we have adopted a u subscript notation to explicitly
indicate the direction ofthe backprojection in the unrotated
x-y system. This result is the same as (20) after converting
IEEE A&E SYSTEMS MAGAZINE
MAY 2022
spectrum calculated by a discrete Fourier transform (DFT)
without first taking steps to deal with the periodicity; to
see what the problem is, calculate and plot the inverse
DFT of the reciprocal of the discrete Fourier transform
without zero padding-this is what is being convolved
with R0
x kðÞ. There are methods of dealing with this deconvolution
problem but those will not enter here. Some signals
with approximately flat spectra include the linear
frequency modulation (LFM) pulse, the windowed sinc
pulse, and any sufficiently short pulse even though the latter
can be expected to suffer from poor signal to noise
ratio except possibly at short ranges-that is why LFM
and pulse compression were developed for radar. As
always, the bandwidth of R0
x kðÞ is inversely proportional
to range resolution.
We shall revisit this important result (28) in due
course. In the meantime, notice that in the case of a transmitted
sinusoid Pk
from (29), R0
ðÞ ¼ d k k0ðÞ, the returned signal is,
x kðÞ ¼ G 2k^x0ðÞd k k0ðÞ¼ G 2k0^x0ðÞ; that
is, the Fourier transform of the ground patch is sampled at
the location of the sinusoid in the 2k-plane, at a radius of
2k0 and an angle of u.
BACKPROJECTION AND IMPULSE RECONSTRUCTION
We now examine the relationship between the monochromatic
receiver signal (21) and the Fourier kernel
in (26).
A backprojection at an angle u or u-backprojection is
a 2-D signal, which has been formed from a 1-D signal
simply by positioning the 1-D signal in a 2-D space, oriented
at angle u in the space, then smearing or extruding
the 1-D signal at a right angle to u through the 2-D space.
Think of the pattern made by a serrated knife dragged
sideways across peanut butter. Figure 13 demonstrates
this principle with a sinusoid backprojected across the x-y
plane at u ¼ 30	. Mathematically, for some function
f0 x0ðÞ, simply define a 2-D function f0
signal for a centered impulsive scatterer under monochromatic
illumination (21), we get
r0
b x0;y0ðÞ0¼ ejkx0
:
b x0;y0ðÞ0¼ f0 x0ðÞ,
indicating no variation in the y0 direction.
Applying the backprojection principle to the receiver
IEEE - Aerospace and Electronic Systems - May 2022 - Tutorial XV

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