IEEE - Aerospace and Electronic Systems - May 2022 - Tutorial XV - 49

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ofan actual ground patch that consists ofmore than a single
point scatterer.
POINT SCATTERERS
The radar receiver does not record the entire field rx; y; tðÞ
or r0 x0;y0;t
each transmission ofpðÞ. For a monostatic radar that sample
is r0 x0; 0;t
Figure 8.
Coordinates x0;y0ðÞrotated u with the radar atx0
ðÞ0 but only a slice or sample of that field, per
ðÞ0-the y0 aspect has been lost, having been samT.
interest.
In practice, there will be many scatterers or a continuum
of scatterers and the receiver will begin recording
only when the earliest reflection of interest arrives. We
have seen that the round-trip delay from a scatterer at the
center of the coordinate systems is 2tT ¼2xT=c. xT is
now seen as a rather arbitrary reference point for the wave
and is not necessarily the most useful reference since it
describes a lot ofdelay that does not carry any information
since it is common to all transmitted pulses. Therefore, we
will eliminate it, or rather, set it to zero, as doing this better
models the receiver at around the time that it is recording
the signal of interest. (More precisely, we are shifting
the data by this much.) We note that setting it to zero
along with the likelihood that the center of the coordinate
systems will be the center of the ground patch will result
in a noncausal signal. This is not a problem if we consider
that we are ultimately desiring to reconstruct an image of
the ground reflectivity and as such, noncausality for spatial
signals is not an existential question as it is for temporal
signals. In the latter case, the reference can be explicitly
adjusted to avoid the problem or, more practically, the
way that an array in computer memory is interpreted can
simply be modified. Thus, the expression for the reflected
wave field can be written
rx; y; tðÞ¼ p vt þ k r 2rsðÞ
½:
(7)
If rs ¼ 0; 0ðÞ, i.e., if the scatterer is at the origin, then we
get back the familiar form rx; y; tðÞ¼ p vt þ k rðÞand if
k ¼ k; 0ðÞ then we get back the earlier familiar form
rx; y; tðÞ¼ p vt þ kxðÞsince k; 0ðÞ x; yðÞ¼ kx.
THE RECEIVER SIGNAL
The previous work has derived the field due to a radar signal
reflection from a single scatterer. There remain more tasks:
find the signal that is actually recorded by the radar receiver,
show how to form-reconstruct, in radar terminology-the
image ofa single scatterer, and show how to form the image
MAY 2022
pled at a single y0 location, y0 ¼ 0. Samples from different
angles are recorded for different orientations ofthe x0-y0 system,
that is, for different u, and the various recorded signals
are processed as a group to form the ground patch image.
The sampled or sliced return r0 x0; 0;t
a single spatial variable and time and these variables are
linked by the propagating nature ofthe wave, namely, by the
propagation speed c. In what follows both a time-domain
version and a space-domain version of the receiver signal
will be allowed, and these are notated, respectively, r0
xðÞ where the latter subscript is intended to denote
tðÞ
and r0
" space " and not a particular coordinate system.
Examining first the case of a point scatterer on the x0
axis, the result (3) can be readily converted to both a temporal
and a spatial receiver function. To get the time function
at the receiver, fix the spatial variable to be at the
receiver by letting x0 ¼ x0
p vt þ k 2x0 T; then r0 x0
r0 x0 T 2x0
s
¼ p v t 2tT 2tsðÞ
T; 0;t ¼
. With xs ¼ cts and xT ¼ctT,
T; 0;t ½. Setting the phase reference
to zero as before and adopting the receiver signal
notation for time domain, then
r0
t ½:
ðÞt ¼ p v t 2tsðÞ
(8)
The time signal is delayed by the amount of delay
coded in the position of the scatterer relative to the
x0-y0 system.
Next, examine the case of the point scatterer at the
same position but derive the spatial signal at the receiver.
Since we are hoping to eventually constitute a spatial picture,
this is potentially a more useful function. It is helpful
to imagine the reflected signal at t ¼ 2tT as frozen in
time, somewhere in the vicinity of the receiver. Substituting
t ¼ 2tT into (3), r0 x0; 0; 2tTðÞ¼ p 2vtTþð
T 2kx0
kx0 þ
kx0
r0 x0; 0; 2tTðÞ¼ p k x0 x0
sÞ. Recalling that tT ¼xT=c, we get
T 2x0
s
and we see that the
signal is 2xs spatial units in front ofthe receiver, the factor
two appearing because two-way travel doubles the distance.
Finally, using the zeroed phase reference and invoking
the 1-D spatial signal notation
r0
x x0ðÞ ¼ p k x0 2x0
s
:
(9)
By now we recognize that some " vt " quantities can be
replaced with " kx " quantities and we could have arrived
at (9) more directly with this replacement in (8).
IEEE A&E SYSTEMS MAGAZINE
49
ðÞ0 is a function only of
IEEE - Aerospace and Electronic Systems - May 2022 - Tutorial XV

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