IEEE - Aerospace and Electronic Systems - November 2022 - 12

Detection, Mode Selection, and Parameter Estimation in Distributed Radar Networks
Figure 5.
Distributed parameter estimation and coprime array configuration for DoA estimation.
in (7) is an open problem. In the case of a small number of
nodes, each permutation of
K nodes can be tested with
enumeration. However, the complexity of such approach
will increase exponentially with increased number of
nodes. Therefore, suboptimum approaches with low-complexity
methods are desired. A detailed investigation of
such approaches is beyond the scope of this article.
The nodes assigned to the passive mode of operation
can limit their tasks to passive source localization and
direction finding to acquire information about RF emitters.
By sharing this information, the passive nodes can assist
the active nodes in obtaining improved radar target parameters.
In the following section, we will present distributed
approach for estimating target/source azimuth angles,
which is known as distributed DoA estimation approach.
DISTRIBUTED DoA ESTIMATION
Consider a network ofQ nodes cooperatively working in a
distributed manner to estimate DoAs of K targets (sources).
Without loss of generality, we consider that each
node is equipped with Ne antenna elements, where Ne
K. The ith (i 2Q) node communicates with only its
neighboring nodes and shares parameters, such as sparse
vectors, ri, and dual vectors, di
configurations can be easily considered. The coprime array at
each node is formed from two uniform linear arrays (ULAs)
as shown in Figure 5(b). The first ULA has N antennas with
an interelement spacing ofMd, whereas the second ULA has
2 M elements with an interelement spacing ofNd.Here, M
and N are coprime integers with M< N, d ¼
2 ,and is
the wavelength. When two ULAs are aligned, the coprime
array has sensors located at the positions given by the set
SI ¼fðNdm; 0 m 2M 1Þ[ ðMdn; 0 n
N 1Þg. The first element ofeach ULA is the reference element
and forms the first element of the resulting aligned
coprime array, which hasNe ¼ 2MþN 1 sensors.
Let q ¼½q1; ... ;qNe
T denote the sensor positions at
each node, where the first sensor is assumed to be the reference,
i.e., q1 ¼ 0. At a given time instant, l ðl ¼
1; ... ;LÞ, where L denotes the number of samples in an
observation period, the signal, xiðlÞ2CNe1, is transmitted
by the ith node from its Ne antennas. The signal is
reflected by K targets located at angles ui ¼½ui
1; .. . ; ui
KT
as seen by the ith node. The received signal vector of
length Ne 1 at the ith node can be expressed as
yiðlÞ¼
2 as shown in Figure 5(a)3.
In an ideal case, each sparse vector will have K large nonzero
values and the remaining values close to zero. The
indices of these K values correspond to DoAs ofK sources
in a given angular grid. We assume that all nodes are
monitoring the same scene and sense the same number of
targets in a given observation time. As in the case of the
distributed detection, the underlying network forms a connected
graph.
We assume that each node in Figure 5(a) employs the
same coprime array configuration, although other sparse array
2The dual vectors are auxiliary variables that arise while estimating
sparse vectors with the ADMM approach.
3In the distributed DoA estimation, sparse vectors are defined over a
grid ofazimuth angles, whereas in the distributed delay and Doppler
estimation of section " Distributed Delay-Doppler Estimation, " they
will be defined over a 2-D grid ofdelay and Doppler.
12
where ai
XK
k¼1
ai
kaðui
kÞaTðui
kÞxiðlÞþ niðlÞ
(8)
k includes the effects of target reflectivity coefficient
and attenuation (associated with the kth target and
observed by the ith node) quantified with the RRE. The
target angles, as seen by the transmitter and receiver of
the ith node, are assumed to be identical due to monostatic
radar configuration, and niðlÞ2 CNe1 is additive noise.
aðui
kÞ¼½1; ej2pq2
sin ðui
kÞ; ... ; ej
2pqNe
sin ðui
kÞT denotes the
steering vector for both transmit and receive sparse arrays.
The covariance matrix of yiðlÞ is given by Ryy;i ¼
EfyiyH
PL
1
L
l¼1 yiðlÞyH
i g. In practice Ryy;i is estimated as R^yy;i ¼
i ðlÞ. A virtual array can be formed by vectorizing
Ryy;i as zi ¼ vecðRyy;iÞ, which can be equivalently
expressed as a received data vector for an array with an
extended coarray aperture. For the case of a Swerling-I target
model [1], we can consider ai
Gaussian random variable with a variance ofs2
IEEE A&E SYSTEMS MAGAZINE
k as zero-mean complex
k;i.Similarly,
assuming that xiðlÞ corresponds to samples of orthogonal
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