IEEE - Aerospace and Electronic Systems - November 2022 - 9

Chalise et al.
where g is the unknown threshold, which can be chosen to
satisfy a certain global requirement of PFA. It is worth
noting that each node achieves the global GLRT value
despite communicating with only its neighboring nodes,
which is a significant outcome, especially in a large size
network, wherein the maximum numbers of neighboring
nodes are much smaller than the total number of the
nodes. Calculating convolution of independent random
variables fwigQ
i¼1 under the hypothesis H0, we obtain the
Figure 2.
Sharing GLRT values, fwig, between neighboring nodes.
reflectivity coefficient as well as signal power attenuation
according to radar range equation (RRE) [1]. In this case,
the test statistics of the GLRT detector at the ith node and
the corresponding local detection rule for determining the
presence or absence of the target, are respectively, given
by [41]
Li ¼ log
xixH
2
6
6
4
yH
yiyH
i
i IL jjxijj2 yi
i
3
7
7
5¼)Li 6
H0
H1
gi
(2)
where gi is the threshold value that is typically determined
to satisfy a certain requirement on probability of false
alarm (PFA). It is worth noting that the decision statistics
in (2) does not depend on unknown ai and s2
n;i, which
implies that the considered GLRT has the property of a
constant false alarm rate (CFAR) detector. Substituting
yi ¼ ni into (2), we can determine the closed-form expression
for the local PFA. From this expression, we determine
the local threshold value, gi,as gi ¼
ðL1Þ log ðPFAÞ, where
1
PFA is the required PFA at the ith node [41].
We assume that the bit depth used for quantization is
sufficient to ensure that bit error rate (BER) is negligible
so that the estimates of the GLRT values fwj ,Lj;j ¼
1; ... ;Nig at the ith node are the same as their actual values.
After receiving the estimates of these GLRT values
from its neighboring nodes, the ith node runs a distributed
consensus algorithm. The objective of this algorithm is to
minimize average divergence of the GLRT values, which
can be solved with the steepest descent method [15]. This
method allows the ith node to update its GLRT value at
the nth update step, wi½n, with the knowledge of corresponding
values, fwj½ngj2Qi
, from its neighboring nodes.
The step-size for the steepest descent method can be determined
in closed form to ensure its convergence [15]. Upon
convergence, each node achieves the global GLRT value
w, which is the average of the individual GLRT values of
all nodes in the network. The corresponding detection
probability PD is given by
1
w ¼
NOVEMBER 2022
Q
XQ
i¼1
wi ¼)PD ¼ Prf
XQ
i¼1
wi gg
(3)
following expression for the global PFA:
PFA;g ¼
GðQ; ðL 1ÞgÞ
GðQÞ
s
(4)
Rwhere GðzÞ is the gamma function and GðQ; sÞ¼
1
tQ1etdt is the incomplete upper Gamma function
[42]. Using the expression of (4), g can be determined
for a given global PFA;g. When the global and local PFA
requirements are set to be equal, the relation between the
local and global threshold values can be established by
equating PFA;g and PFA.
To illustrate the effectiveness of the proposed distributed
method, we consider three scenarios: ideal, nonideal,
andpractical. The target is assumed to be a nonfluctuating
point scatterer, yielding associated target SNR (as
seen by the ith node) of
a2
i
s2
n;i
nodes observe similar high SNRs, and consequently high
GLRT values. In the non-ideal scenario, all nodes observe
similar low SNRs, and therefore corresponding low
GLRT values. In the practical scenario, a majority of the
nodes (three in our example) observe high GLRT values,
while a minority (one in this case) observes low GLRT
values. It is important to note that if all nodes determine
no target within the observed space, PD approaches PFA.
To determine SNR at each node, we calculate the thermal
noise power as s2
n ¼ kT0BF, where k is the Boltzmann
constant, B is the noise bandwidth, and F is the receiver
noise figure. We choose B as the radar pulse bandwidth,
T0 ¼ 290oK, and F ¼ 5 dB. We calculate ai using the
RRE [1], ai ¼
power, GT;i is the transmit antenna gain, GR;i is the
receive antenna gain,i is the transmit wavelength, Ri is
the distance between target and the ith node, and s is the
target radar cross section (RCS). We choose PT;i ¼ PT ¼
1 W, GR;i ¼ GR ¼ 0 dBi, GT;i ¼ GT for all i, and change
GT from 0 to 16 dBi so that different values of ai would
lead to different SNRs at the nodes, as shown in Table 1.
The stationary target's ½x; y; zT
r
location, in meters, is
chosen at ½150; 150; 1T. In the ideal and non-ideal scenarios,
the node locations are chosen such that Ri ¼ 170 m
and Ri ¼ 400 m for all i, respectively. For the practical
scenario, the coordinates of the nodes from 1 to 4 (in
meters) are, respectively, chosen as ½130;130; 1T,
½10; 10; 1T, ½40; 40; 1T, and ½40; 10; 1T. The target RCS
IEEE A&E SYSTEMS MAGAZINE
9
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
PT;iGT;iGR;is2
ð4pÞ3R4
i
i , where PTi is the transmit
. In the ideal scenario, all
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