Aerospace and Electronic Systems Magazine January 2018 - 30

Malanowski et al.

Figure 13.

History of target detections.

sulting from range and velocity resolution cells (see (5) and (7))
are fulfilled for this particular scenario. The rocket echo is visible
on three range-velocity planes. Since Tx2 transmits two signals
in adjacent bands, both of which were used jointly, the bandwidth
of the signal is two times higher than in the case of Tx1 and Tx3.
This is visible on the crossambiguity function as increased range
resolution. The lighter area for small absolute values of the bistatic
velocity results from the clutter removal procedure, which in this
case was an adaptive lattice filter [13]. The adaptive filter is fed
with a reference signal and an echo signal. The filter tries to remove delayed copies of the reference signal from the echo signal,
thus removing the direct path interference and clutter reflections.
Target detection was performed by thresholding the absolute
value of the crossambiguity function. Typically, the constant false
alarm rate (CFAR ) algorithm is used for this purpose. CFAR algorithm compares the value of the signal in a resolution cell, the
so-called Cell Under Test (CUT), with the average level of signals in reference cells, surrounding the CUT [14]. In this way,
the threshold level is adjusted to the local noise and interference
level. In the considered case, the noise-induced values of the crossambiguity function were constant over the entire range-velocity
plane. Thus, a constant threshold can be applied. Resolution cells
in which the detection threshold is exceeded indicate potential target echoes. The individual detections are clustered into groups. For
each group, the precise bistatic range and velocity is estimated by
fitting a parabola to values surrounding the crossambiguity function peak [15]. In this way, a bistatic plot is created, representing
a potential target. The history of plots over the entire flight of the
rocket is shown in Figure 13. The maximum bistatic velocity of
the rocket is ca. 600 m/s. Plots with bistatic velocity less than 50
m/s were rejected, which results in a detection-free area for small
absolute values of velocity.
As can be seen, apart from detections corresponding to the real
target, clearly following a trajectory, there are numerous false detections appearing randomly on the range-velocity surface. This
situation is typical for any radar, and the false plots are routinely
eliminated by the tracking algorithm, which analyzes the dynamics
of targets, and rejects plots which do not follow physically feasible
trajectory. There are also some false plots which are placed in a nonrandom fashion, for example in Figure 13b visible as vertical lines
for constant velocities. These false plots were caused by the residue
from the clutter removal procedure, which produces spurious peaks
30

in the crossambiguity function (visible also in Figure 12b). They can
also be eliminated by the tracking algorithm, as their behavior will
not be consistent with the physical model of the target motion.
The tracking was performed in two phases [16], [17]. First, targets are tracked in bistatic coordinates, i.e. bistatic range and velocity. Bistatic tracking is simple, but efficient at the same time, as it
can easily eliminate most false plots. In the second phase targets are
tracked in Cartesian coordinates, which is more complicated due to
the nonlinear relationship between measurement and state space.
The tracking process starts by feeding plots to the bistatic
tracker. The tracker is based on a linear Kalman filter operating in
bistatic coordinates with the measurement vector:
Z =  Rˆ ,Vˆ 



(8)

where Rˆ is the measured bistatic range, Vˆ is the measured bistatic
velocity, and superscript T denotes vector transpose. The state vector of the Kalman filter is:
X = [ R,V , A]T

(9)

where R, V, A are the estimated bistatic range, velocity, and acceleration, respectively. Tracks are initialized with the standard
"M/N" logic [18], [19]. This means that each unassigned plot generates a tentative track. In the next N iterations, plots are assigned
to this track based on the so-called association gate [18]. If a plot
is assigned to this track at least M times during N time instants, the
track is confirmed.
Once bistatic tracks are established, they are used by the localization algorithm [20]. Based on bistatic range measurements from
at least three transmitter-receiver pairs, the Cartesian position, i.e.
x, y, and z coordinates are calculated, as visually shown in Figure
2. The Cartesian position combined with bistatic velocity measurements allows the Cartesian velocity vector to be calculated.
Cartesian parameters (position and velocity) are used to establish
Cartesian tracks. The Cartesian tracker uses the Extended Kalman
Filter [18], with the state vector:
Xc = [ x,Vx , Ax , y,Vy , Ay , z ,Vz , Az ]T

(10)

where x, y, z are target position components, Vx, Vy, Vz are target
velocity components, and Ax, Ay, Az are target acceleration compo-

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JANUARY 2018

Table of Contents for the Digital Edition of Aerospace and Electronic Systems Magazine January 2018