IEEE - Aerospace and Electronic Systems - May 2022 - Tutorial XV - 55

Bauck
1=2 to attain the correct size. This is easily done by relabeling
the coordinate axes.
The reconstruction formula (30), being an integral, is
linear and as such applies to any collection of impulsive
scatterers g xðÞ ¼
P
n gnd x xnðÞ as a kind of discretescatterer
case for any n. Since any ground patch can be
written using the sifting property of the generalized Dirac
function as g xðÞ ¼
R
g yðÞd y xðÞdy as a continuous colFigure
13.
A 1-D sinusoidal function shown backprojected at an angle of
u ¼ 30	.
(20) to phasor form.9 Thus, the backprojection operation
reconstructs the spatial field, restoring the spatial information
that was lost when the receiver recorded only a slice of
the wave field. An important observation here is that a
backprojection is aplane wave. This is the same expression
as the inverse Fourier kernel in (26). (Any concerns that we
were using k or k for two purposes should now be
assuaged.) We might as well now write (26) as
d xðÞ ¼
1
4p2
Z
ru x; yðÞdk:
(30)
Arriving at this relationship is a central purpose of this
article. When viewing this result, remember that the context
defining r0
lection of impulses, we arrive at the important conclusion
that any ground patch image can be reconstructed by backprojecting
and summing all u-projections of all frequencies.
We shall consider this in more detail in the section
" Backprojection Generally. "
The preceding has shown two points. First, u-backprojection
reconstructs the 2-D field that is due to a single
transmission but which was lost due to sampling the field
at only one point, and second, ground patch reflectivity
functions can be reconstructed by summing these perpulse
reconstructed fields.
The form of (26) and (30) containing dk ¼ dkxdky
inspires a discretization in the manner of
d xðÞ ¼
1
4p2
X
n
X
m
ejkx;mxþky;nyðÞDkx Dky
(32)
where kx;m ¼ m Dkx, ky;n ¼ n Dky, m and n are integers,
and where d xðÞ is taken as an approximate reconstruction
of the impulse and should not be confused with the differential
quantity dx. Using (27), d xðÞ has a Fourier transform
D kðÞ ¼
b x0;y0ðÞ0 here is that of a monochromatic
wave and that k carries both the necessary direction and
frequency information. However, (30) indicates a summation
of many monochromatic waves of different frequencies
which is of course a polychromatic result. There are
more adjustments to be made before we can consider this
a result suitable for our radar problem.
Applying the same backprojection principle to the offset
impulse receiver function under monochromatic illumination
(19) results in
ru x; yðÞ¼ ejk x2rsðÞ
(31)
which is the same as the kernel of the inverse Fourier
transform (25) but with x0 ¼ 2rs. Thus, a reconstruction
formula identical to (30) also applies here but with
ru x; yðÞ defined for the offset scatterer as above and with
the additional step that an image reconstructed from these
ground patch measurements must be scaled by a factor of
9There are conceptually two x-y coordinate systems: one defines the
ground patch and the other is where the image is imagined to be
reconstructed, either an abstract mathematical construct or as memory
in a computer. We leave it to the reader to discern which is
intended, as necessary.
MAY 2022
Dkx Dky
4p2
X
m
X
n
d2 kx kx;m;ky ky;n
a 2-D series of impulses in the k-plane, which at several
points samples the unit function that is the Fourier transform
of the centered impulse. A " textbook " discretization
might assume Dkx ¼ Dky, leading to a biuniform sampling
of the plane as depicted in Figure 14. However, one
could well specify Dkx 6¼ Dky indicating a mapping from
the biuniform case that indicates uniformity in each direction
separately, or even a different style of discretization
using different values for both Dkx and Dky. In the case of
two different choices of area, say Dkx Dky
1
and
Dkx Dky 2 related by a differentiable mapping, the ratio
of the areas reflects the Jacobian determinant of the mapping.
Of course, the usual rules of maximum allowable
sampling interval must be observed to avoid spatial
aliasing.
Figure 15 shows the magnitude of a centered-scatterer
backprojection reconstruction made from a rectangular
grid of 124
124 plane waves in the manner of Figure 14
centered on the k-plane over a range in each direction of
1:2p, making Dkx ¼ Dky 0:0613; the extent of the
image is50
50 spatial units plotted with 256
256
pixels. The sampling intervals Dkx and Dky
have been
chosen such that the first of the periodic aliases of the
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IEEE - Aerospace and Electronic Systems - May 2022 - Tutorial XV

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