IEEE - Aerospace and Electronic Systems - May 2022 - Tutorial XV - 47

Bauck
Figure 4.
Side view of a wave propagating to the right. The prototype wave
is shown in dashed line.
though the wave exists in the x; yðÞ plane as it propagates-the
wave value is constant as a function of y. This
setup will later allow waves propagating in other
directions.
Assume that the radar is situated on the x-axis at position
xT which is negative. See Figure 3 for a rendering of
a hypothetical prototype 2-D wave traveling to the right
but just leaving the radar when t ¼ 0. For simplicity,
Figure 4 displays Figure 3 as viewed from they axis.
We need to determine the term fs. The figure shows the
prototype px
ðÞ in dashed form. We desire to scale p spatially
by some amount to be determined so we include k.
We also include the time scale factor v. Referring to the
figure, we see that we need p to come out reversed, so that
the leading edge in time is also the leading edge in space.
This is because of the differing signs on the vt term and
the kx term in (1). With t ¼ 0 and x ¼ xT, we write
sxT;y; 0ðÞ¼ pkxT þ fsðÞ¼ p 0ðÞ, resulting in fs ¼
kxT and
sx; y; tðÞ¼ p vt kx þ kxTðÞ:
Checking the result at t ¼ 0 we see that sx; y; 0ðÞ¼
pk x xTðÞ
ðÞ, flipped, scaled by k, and shifted left by
jjxT since xT is negative. This wave will propagate rightwards
toward the origin for a time tT where there is
assumed to be a point scatterer, a small object which
reflects part of the wave back toward the radar. (Later,
point scatterers will be modeled as Dirac delta generalized
functions.) Assuming for now that all of the wave's amplitude
is reflected, we write sx; y; tTðÞ¼ p vtT kxþ
ð
kxTÞ¼ pkxðÞ, the latter requirement deriving from
examining Figure 5. Equating the arguments, we find that
tT ¼xT=c. This result could have been written by
examination by knowing a little basic physics but here we
have derived it fully.
FIELD DUE TO ONE SCATTERER
When t ¼ tT, a backward-traveling wave is created,
rx; y; tðÞ¼ p vt þ kx þ frðÞ. From Figure 5, the requirements
for a scatterer at x ¼ 0 are that
s 0;y; tTðÞ¼ r 0;y;tTðÞ
MAY 2022
Figure 5.
A wave beginning the process of reflecting. The dashed form is
the scaled prototype and from this point on can be considered to
be the reflected wave for x< 0, traveling to the left.
thus
p vtT þ kxTðÞ¼ p vtT þ frðÞ
leaving fr ¼ kxT and
rx; y; tðÞ¼ p vt þ kx þ kxTðÞ:
(2)
(Some function names will be recycled many times to
mean different things so the actual meaning should be
determined by the context of its use.) We can check a couple
of cases. When t ¼ tT, rx; y; tTðÞ¼ p k xþð½
xT þ
ctTÞ ¼ p kxðÞ because ctT ¼xT. Also, rx; y; 0ðÞ¼
p k x þ xTðÞ
½; although it does not fit the physical model
at times before the instant of reflection such as this one,
the " prereflected " signal appears to the right of the central
scatterer position. And back at the receiver, rx; y; 2tTðÞ¼
p k x xTðÞ
ðÞ¼ p v t 2tTðÞ
waveform delayed by twice the one-way propagation
time.
The above results can be generalized by allowing a
scatterer that is not at the origin of the x-y system. For a
simple extension, let the scatterer be at x; yðÞ¼ xs; 0ðÞ.
Using the same approach as before, the form of the
reflected, backward-traveling wave is the same,
rx; y; tðÞ¼ p vt þ kx þ frðÞ, but now fr will be different
because the conditions have changed. The propagation
speed c was found earlier as the proportionality between
time and space so that now xs ¼ cts. Examining Figure 6,
apparently the required conditions are
sxs;y;tT þ tsðÞ¼ rxs;y;tT þ tsðÞ
thus
p vtT þ vts kxs þ kxTðÞ¼ p vtT þ vts þ kxs þ frðÞ
resulting in fr ¼ k xT 2xsðÞand
rx; y; tðÞ¼ p vt þ k x þ xT 2xsðÞ
½:
When t ¼ tT þ 2ts, rx; y; tT þ 2tsðÞ¼ p kxðÞ; the reflected
wave has progressed back to the origin, the forward wave
taking tT þ ts seconds to travel to the scatterer plus an additional
ts seconds to reflect back to the origin. Perhaps more
IEEE A&E SYSTEMS MAGAZINE
47
½; as expected, the signal is about to enter the
receiver. Similarly, the time waveform at the receiver can
be found as rxT;y;t
½, the original time
IEEE - Aerospace and Electronic Systems - May 2022 - Tutorial XV

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