IEEE - Aerospace and Electronic Systems - May 2022 - Tutorial XV - 46

A Rationale for Backprojection in Spotlight Synthetic Aperture Radar Image Formation
Figure 2.
Examples of shifting and scaling a function. In this example, shift
v> 0 and scale a> 1.
Figure 1.
SAR geometry showing a zero-degree orientation of the radar and
two rotated orientations.
These signals can be either time-dependent, space-dependent
in one or several dimensions, or both, that is, spatiotemporal
signals. It will be useful to review a kind of
canonical form for these signal variations. Consider a
generic signal pu
ðÞ and versions thereof that are shifted
and scaled by various amounts. Figure 2 shows various
shifted and scaled versions ofpu
form pa u vðÞ
½. In the figure only, it is assumed that
scale factor a and shift amount v are positive and further
that a> 1. Many times the argument of a function that
has simple amounts of scaling and shifting is expressed in
various forms but usually the argument can be rewritten in
this canonical form that eases interpretation. In the form
pa u vðÞ
½, u is the independent variable, a is the scale
factor, and v is the shift amount. With a and v both positive,
the interpretation is: first, scale p by a, then shift right
by v. A negative a will cause the function to be flipped
around the origin as well as scaled, and a negative v will
cause a leftward shift.
WAVE EQUATION AND SOLUTIONS
Although electromagnetic waves generally have a vector
amplitude, the scalar wave equation [33], [34] suffices for
many radar problems including this one. Generally, it can
be expressed as
r2s ¼
1
c2
In one dimension, this becomes
@2s
@x2 ¼
@2s
@x2 þ
@2s
@y2 þ
1
c2
@2s
@t2
and in three Cartesian dimensions, it is
@2s
@z2 ¼
1
c2
@2s
@t2
with s the wave amplitude which is sometimes considered
to be complex-valued; x, y, and z are spatial coordinates, t
46
@2s
@t2 :
ðÞ based on the canonical
Figure 3.
Wave propagating to the right at the transmitter position xT and at
time t ¼ 0.
is the time variable, and c is the constant propagation
speed.
We start with a prototype function pðÞ and convert it
into a traveling wave that satisfies a wave equation. In one
dimension, two solutions are sx; tðÞ ¼ p vt kx þ fðÞ
and sx; tðÞ ¼ p vt þ kx þ fðÞ2. That these satisfy the
wave equation can be shown by direct substitution as long
as we constrain c ¼ v=k: in the first case, @s=@x ¼kp0,
@2s=@x2 ¼ k2p00, @s=@t ¼ vp0, @2s=@t2 ¼ v2p00. The former
propagates in the þx direction and the latter in the
x direction as t increases according to the interpretation
of section " Signal Scaling and Shifting. " The usefulness
of having two constrained scale factors instead of just c
will become apparent when we study monochromatic solutions
in section " Monochromatic Waves. " We require a
solution to exist in at least two spatial dimensions (the
slant plane) as well as time. This is easily done, for a wave
traveling in the þx direction, by simply defining the wave
s as
sx; y; tðÞ¼ p vt kx þ fsðÞ (1)
where fs is to be determined. That y does not appear in the
right-hand side of this expression simply expresses the
fact that the wave amplitude does not depend on y, even
2More traditional solutions are pct x þ fðÞ. In the sequel, at least
up to the point where monochromatic solutions are considered in
the section " Monochromatic Waves, " the reader might want to
mentally substitute v ¼ c and k ¼ 1.
IEEE A&E SYSTEMS MAGAZINE
MAY 2022
IEEE - Aerospace and Electronic Systems - May 2022 - Tutorial XV

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