IEEE Geoscience and Remote Sensing Magazine - June 2020 - 51

SIGNAL DISTRIBUTION FOR RANDOM
TARGET MODELS
The target response is assumed a random variable with a
known distribution. A commonly adopted distribution is
the Gaussian one, which is motivated by the fact that a larger-size target consists of many identical elementary scatterers so that the central limit theorem applies. This model allows for the description of the target RCS values in terms of
target variance, and then of the SCR, and applies it to complex or extended targets. In this case, vector A is assumed
a Gaussian random vector with zero mean and covariance
matrix C A = v 2A 11 T , which corresponds to the target covariance matrix in the stationary case, that is, obtained for
a radial velocity equal to zero. As previously noted for the
clutter signal, in ATI applications the target is viewed by the
SAR antennas with the same view angle and a significantly
small time delay. The result is that random vector Z is a
circularly symmetric Gaussian complex vector with mean
vector n and covariance matrix C H1, assuming the expressions reported in Table 2 [39]. In this case, SCR = v 2A /v C2 .
To show the dependence of the received signal model
(1) on the involved target parameters, a phasor diagram
of the interferometric signals for the case N = 2 is shown
in Figure 3. The phasors of the measured data Z 1 and Z 2
are represented in red. These data are a result of the sum
of the target phasors (in blue) and the clutter phasors (in
green). The target phasors exhibit the same amplitude A,
the same initial phase offset z A, and different phases due
to the radial velocity phase term z 2. Two different values
of z A are detailed in Figure 3(a) and (b). Instead, the clutter phasors, due to their extremely high clutter correlation,
do not change from one acquisition to the other (thermal
noise has not been considered, as its effect is equivalent
to the production of a negligible change of the clutter
phasors that can be neglected for high CNR values). It is
evident that both the signal amplitude and phase change
from acquisition 1 to acquisition 2 depending on the target
velocity. For low values of SCR, the target phase z A also
influences the value of the acquired signals. Figure 3(b) displays the phasors that would be obtained from an estimation procedure not involving the estimation of the phase
offset z A, assuming a real-valued target (z A = 0). The solution obtained is different from the real one [reported
in Figure 3(a)] and leads to a wrong estimate of the phase
related to the radial velocity. In any event, for considerably high SCR values, when clutter can be neglected with
respect to the target signal, the velocity estimate becomes
independent of the target phase z A , as expected.
PHASE-ONLY METHODS
Consider an ATI-SAR system with only two antennas (N = 2).
From the ATI signal model defined in the "Along-Track Interferometric for Ground-Moving Target Indication" section,
it is easy to see that, when the target power is much greater
than the clutter plus noise power, the ATI phase estimate
zt = arg (Z 1 Z 2) ) is strictly related to the target velocity through
JUNE 2020

IEEE GEOSCIENCE AND REMOTE SENSING MAGAZINE

(3). For this reason, conventional ATI techniques use the interferometric phase as data and can detect moving targets with
high RCS and estimate their velocities just by computing the
interferometric phase and comparing it with a proper threshold. Due to the phase wrapping effect, we note that, in this
case, we can define the maximum unambiguous radial velocity value v a = v a2, with v a2 given by (4), as the value that produces an interferometric phase equal to r. We have
m
v a = 4b v p ,

(11)

where b = b 2 is the baseline between the two acquisitions.
Then, to obtain a sufficiently large range of velocity values
that can be unambiguously estimated, small baselines and
a fast platform are preferred alongside a lower radio frequency (RF).
When detecting a moving target with a radial velocity
different from zero, not only must the ambiguity issue be
taken into account, but also the fact that, in conventionally
processed SAR images, targets moving along the radial direction appear displaced in azimuth from their original position. In particular, the azimuth shift T az amounts to [12]
vr
,
vp

T az = - R

(12)

Im
ZC 1
ZT 1

ZC 2

Z2 φ2
ZT 2

φA

Z1
Re

(a)
Im
ZC2
Z2

ZT 2

φ2
φA = 0

ZT1
Z1

ZC 1
Re

(b)
FIGURE 3. A phasor diagram of the interferometric signals with the
measured data phasors in red, the target phasors in blue, and the
clutter phasors in green. (a) z A ! 0 and (b) z A = 0.

IEEE Geoscience and Remote Sensing Magazine - June 2020

Table of Contents for the Digital Edition of IEEE Geoscience and Remote Sensing Magazine - June 2020