IEEE Geoscience and Remote Sensing Magazine - June 2020 - 58

The statistical test (28) depends on the ML estimates of
three unknown parameters H D = (SCR, v r, z A), where SCR
is related to ; A ;. In the Gaussian clutter assumption, it is
shown that the ML estimate of the complex amplitude
A = ; A ;e jz A can be expressed in closed form [53] and substituted in (28), so that, after certain algebraic manipulations,
the GLRT can be written as
H1

{ H C H-10 Z 2 2
KlGLRT ^Z h = max
dl.
vr
{ H C -H10 { 1

(29)

H0

The two probabilities PFA and PD, are then defined by
PFA = Pr " KlGLRT ^Z h $ dl ; H 0 , =
PD = Pr " KlGLRT ^Zh $ dl ; H 1 , =

3

#

dl
3

#

dl

fKlGLRT ^ x; H 0h dx,
fKlGLRT ^ x; H 1h dx.

(30)

Unfortunately, both the threshold dl corresponding to
a given PFA and the receiver operating characteristic (ROC)

1
0.9

PD

0.8
0.7
Single Baseline SCR = 3 dB
Single Baseline SCR = 5 dB
Single Baseline SCR = 7 dB
Dual Baseline SCR = -3.4 dB
Dual Baseline SCR = -1.4 dB
Dual Baseline SCR = 0.6 dB

0.6
0.5
0.4
10-4

10-3

10-2
PFA

10-1

100

FIGURE 13. The ROC (PD versus PFA in log scale) obtained using

the GLRT (29) for the TerraSAR-X parameters, target radial velocity
v r = 50 km/h, CNR = 10 dB, varying SCR, and single baseline
(b 2 = 1.2 m) and dual baselines (b 2 = 1.2 m and b 3 = 3.1m).

1

PD

0.8
0.6
0.4
LRT
GLRT

0.2
0

-3

-2.5

-2
-1.5
-1
Phase (rad)

-0.5

0

FIGURE 14. The PD of GLRT (29) versus a target phase for RDS-2
parameters, L = 1, c H 0 = 0.98, SCR = 3 dB, and PFA = 10 -4 .

58

cannot be expressed in closed form, because the statistic
of (29) is not of a known type. But they can be empirically
derived via Monte Carlo simulations.
In [27], it is noted that, to find an accurate estimate of
the target radial velocity, the initial phase offset z A of the
target signal has to be taken into account.
GENERALIZED LIKELIHOOD RATIO
TESTS FOR RANDOM TARGET MODELS
In this case, H G = (SCR, v r), with SCR = v 2A /v C2 , and v 2A is
the target signal power. It can be shown [53] that, also in
this case, the GLRT can be written as (29) and both threshold and ROC cannot be expressed in closed form but are
empirically derived via Monte Carlo simulations. Then,
the GLRT obtained by using a random target model with a
zero-mean Gaussian distribution is the same as that which
is obtained for the deterministic model.
In Figure 13, the ROC curves are shown for GLRT (29)
using TerraSAR-X parameters for the single-baseline case
(b 2 = 1.2 m, v a = 49 m/s = 176 km/h) a n d f o r t h e d u a l baseline case (the baseline b 3 = 3.1m has been added) [27].
Homogeneous clutter and a deterministic target model
have been assumed, the velocity value is 50 km/h = 14 m/s,
CNR = 10 dB, and different SCR values for the two cases are
considered. Note that, in the case of a single-baseline system, for detecting a target moving with a radial velocity of
50 km/h and a PD greater than 80% and PFA = 10 -4, an SCR
greater than 5 dB is required. For a dual-baseline system, an
SCR greater than −1.4 dB is sufficient. It has been shown
[39] that increasing the velocity and reaching its ambiguity
value allow for a PD greater than 80% with a lower SCR. In
summary, a single-baseline ATI system can easily detect fast
and high SCR targets, while slow targets are detected with a
high probability if the SCR is sufficiently high. For a dualbaseline ATI system, the requirements for target velocity
and SCR can be significantly relaxed, while still detecting
slow and weak targets.
In [27], the authors discuss how the single-baseline
ATI detector outperforms the DPCA detector for low SCR,
while the difference decreases when the SCR increases.
When the velocity increases, the DPCA detector approaches the ATI one; as expected, DPCA is not well suited to detect slow targets.
Regardless of which detection approach is used, phase
only or amplitude and phase, it is important to realize that,
to achieve any meaningful detection performance, the correlation coefficient between the channels must be as close
to 1 as possible. In other words, as a prerequisite, the channels must be as equal as possible in range (receiver transfer
function) and azimuth (complex antenna pattern). As this
equality is virtually impossible to achieve in hardware, an
adaptive digital balancing step is typically required prior to
computing of the detection statistic [54].
For the detection of moving targets, the use of complex data results in performance improvement [19], [55].
Figure 14 shows the PD in the case of homogeneous
IEEE GEOSCIENCE AND REMOTE SENSING MAGAZINE

JUNE 2020
IEEE Geoscience and Remote Sensing Magazine - June 2020

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