IEEE - Aerospace and Electronic Systems - May 2022 - Tutorial XV - 53

Bauck
r0
x x0ðÞ ¼ ejkx02krs
ðÞ:
equations become, for the field,
rx; y; tðÞ¼ ej vtþkrðÞ
and
r0
x x0ðÞ ¼ ejkx0
for the receiver signal.
FOURIER TRANSFORMS
The 2-D Fourier transform and its inverse with space and
wavenumber as the conjugate variables are in the traditional
analysis and synthesis interpretations, respectively,
Skx;ky ¼
and
sx; yðÞ¼
1
4p2
ZZ
Skx;ky ejkxxþkyyðÞdkxdky
(22)
where the limits of integration are assumed to be from
1 to 1 unless stated otherwise for some special case.
(This s is different than the s of the section " Wave Equation
and Solutions " ). A more compact notation suitable
for any number of dimensions is
S kðÞ ¼
and
s xðÞ ¼
1
ðÞ2p n
Z
S kðÞejkxdk
where the multiplicity of the integrals n is the same as the
dimensionality of the vectors. When n ¼ 2 or n ¼ 3, S kðÞ
is sometimes called the plane wave spectrum or angular
spectrum of the field s xðÞ. In the above, we have used the
convention that lower case and upper case versions of the
same letter symbol represent Fourier transform pairs, i.e.,
S kðÞ ¼ F s xðÞ
fg with Ffg representing the Fourier
transform operator. These transform definitions could be
increased in dimensionality by one by including the time
component since all of the basic functional forms that we
have dealt with include an vt component of the argument
[38]. However, we have eliminated this factor in the
receiver signal by freezing time at t ¼ 2tT. For the general
field for monochromatic waves such as (18), it is common
practice by most authors to remove, cancel, or simply
ignore ejvt term. In the first case, the remaining spatial
term is called a phasor, indicating the complex amplitude
of the v-oscillating field at each position x; yðÞ.
MAY 2022
Z
s xðÞejkxdx
ZZ
sx; yðÞejkxxþkyyðÞdxdy
(21)
(19)
If the scatterer rs is at the origin then the previous two
(20)
The phasor concept is common in circuit theory but
works just as well here. In the second case, ignoring
the time term, some authors simply state that the time
component is " understood " ; the difference is really
one of style but it seems more satisfying to specify
the phasor concept. It is in this way that the 2-D
spatial transform works for these monochromatic field
problems.
With the Fourier transform now defined, it is possible
to state the Fourier transforms of (15) and (17) which will
be useful soon. The Fourier transform of the doubly scaled
(15) with respect to x0 is
R0
x kðÞ ¼ G0 2kðÞP
1
k
k
k
¼ R0
x u;k
ðÞ (23)
whereas the Fourier transform of the simplified version
(17) is
R0
x kðÞ ¼ G0 2kðÞPk
ðÞ ¼ R0
x u;k
ðÞ:
first the Fourier transform ofan impulse at x0,
F d x x0ðÞ
fg¼
Z
(24)
Two transforms will be particularly useful. Consider
d x x0ðÞejkxdx ¼ ejkx0
which is a stationary plane-wave-like function in the
wavenumber domain oriented at an angle equal to the
angle between the vector x0 and the x axis, with period in
that same direction of2p= x0jj. Also,
F1 ejkx0
1
¼
4p2
Z
ejk xx0ðÞdk ¼ d x x0ðÞ: (25)
If the impulse is centered, x0 ¼ 0; 0ðÞ, then
F d xðÞ
fg¼ 1
and then the trivial but important result, for our purposes,
F1 1fg ¼
1
4p2
Z
1 ejkxdk ¼ d xðÞ:
(26)
This well-known transform says that an impulse can be
constructed by summing equal-amplitude plane waves of
all wavenumbers and all orientations. Of course, it is
impossible to collect this much data from any system. We
shall take the position that any amount of data will provide
some sort of useful result and as such we consider measuring
or sampling parts of the wavenumber domain. Thinking
in the kx-ky plane, one can imagine, for example,
measuring some rectangular region kx1
kx kx2
and
ky1 ky ky2 or an annular region kmin k kmax or a
fgI
collection of impulses d k kiðÞ i¼1. As is well known
and as we shall see soon, spotlight SAR collects data from
a truncated annulus defined by kmin k kmax and
umin u umax. The u-variation is obtained by moving
the radar around the ground patch and the kjj variation is
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IEEE - Aerospace and Electronic Systems - May 2022 - Tutorial XV

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