IEEE - Aerospace and Electronic Systems - November 2022 - 13

Chalise et al.
frequency division multiplexing (OFDM) waveforms from
all antennas at a time instant l, we can show that EfxixH
i IL. Therefore, zi can be expressed as
i g¼
s2
zi ¼ aðui
where ~i ¼ vecðINe
and s2
Þ, pi ¼½Nes2
1Þ aðui
1Þ; .. . ; aðui
KÞ aðui
1;is2
i ; .. . ;Nes2
KÞ;~i pi ,AðuiÞpi
n;iT,
K;is2
i ; s2
n;i is the noise variance. The vectorization converts
the problem into direction finding of coherent sources,
and as such, subspace-based DoA estimation techniques
(e.g., MUSIC [21]) cannot be directly applied to zi.In
order to decorrelate the input and recover the rank of the
coarray covariance matrix, Rzz;i ¼ zizH
i , spatial smoothing
must be applied. Since this approach only utilizes consecutive
lags, DoA estimation based on CS [24] is
employed for effective utilization of all lags. We utilize zi
to form a CS-based DoA estimation algorithm. To this
end, let ug ¼½u1; ... ; uGT be the vector of u's defined
over a search grid of length G, and let Fi ¼ AðugÞ2
CN2
eG be the dictionary matrix formed using this grid.
Assuming a column vector ri of length G as a sparse vector,
the DoA estimation problem at the ith node can be formulated
as the distributed sparse recovery problem and
expressed as the following optimization problem:
minrifo ,
XQ
i¼1
jjziFirijj2 þ mjjrijj1
s:t: ri ¼ ~rj;j 2Qi;i 2½1; ... ;Q
(9)
where ~rj denotes the jth neighboring node's sparse vector
compensated with respect to the ith node's location and
orientation, and m > 0 is a regularization parameter. For
this compensation, we assume that each node has the
knowledge of locations and orientations of other nodes.
The equality constraints on the sparse vectors force neighboring
nodes, i and j, to achieve consensus on their estimates
of the sparse vectors. The optimization problem (9)
can be solved with the ADMM approach [31], [46]. At the
nth step of the ADMM algorithm, the dual vector, di½n,
and the sparse vector, ri½n, are, respectively, updated with a
linear equation and l1-norm minimization problem as
detailed in [46]. This process is summarized in Algorithm 1.
Computational Complexity ofAlgorithm 1: The update
ofdual vectors, di½n, in Algorithm 1 is linear, and therefore
its computational complexity is negligible compared to
the update of ri½n, which is done by solving, (9), a problem
similar to the constrained l1-norm minimization with
MATLAB's least absolute shrinkage and selection operator
(LASSO) algorithm. Since Fi has a dimension of N2
(N2
eG
e rows and G columns), the computational complexity of
LASSO is given by OðG3 þG2N2
e Þ[47]. Since this LASSO
problem is solved by each node until convergence, the overall
computational complexity of the iterative Algorithm 1 is
given by OððG3 þG2N2
iterations required to achieve a given convergence accuracy.
NOVEMBER 2022
e ÞQNcÞ,whereNc is the number of
However, (9) can be alternatively solved with the coordinate
descent approach [46], thereby further reducing the computational
complexity.
Algorithm 1.
Choose n ¼ 1, fri½n¼ 0gQ
initial objective function, fo;I
Choose maximum number of iterations, Nit, and convergence
accuracy,
Loop over n, where n ¼ 1: Nit
-Loop over i, where i ¼ 1: Q
* Update di½n according to [46]
* Update ri½n according to [46]
- End loop over i
Ifn ¼ Nit or jfo;n fo;Ij
- Exit and return fri½ngQ
tors, frigQ
i¼1
- Otherwise fo;I ¼ fo;n
End loop over n
We evaluate the performance of the proposed algorithm
in a network with Q ¼ 4, wherein each node has a co-prime
array configuration with M ¼ 3 and N ¼ 5 (i.e.,
Ne ¼ 10) [26]. We consider that the distance between the
two farthest nodes is much smaller than the target ranges. In
this case, ui
k
uk and ~rj½n
rj½n. This approximation
will be relaxed in section " Compensation With LSTM
Network, " where we will employ deep learning approach
for compensating rj½n with respect to the ith node's location
and orientation. Each node employs an angular grid that
ranges from90 to 90 with an increment of 0:25. Each
node is capable of estimating DoAs of sources more than
Ne, but as will be shown with simulations, the distributed
approach results in improved DoA estimation capabilities of
the nodes that observe low SNR. We set fs2
and change s2
n;i to have different values ofSNRs 1
s2
k;i ¼ 1; 8i; kg
for different
nodes. Each node estimates the covariance matrix
from its own observations using L ¼ 200 snapshots. We
consider estimation of DoAs of 17 targets that have the
same angular spacing between adjacent targets. More specifically,
we simulate three scenarios: i) Scenario 1 - SNR = [0,
0, 0, -10] dB, where targets span from50 to 50 with an
intertarget angular spacing of 6:25, ii) Scenario 2 - SNR =
[0, 0, 0, -15] dB, where target angular span and angular spacing
are the same as in Scenario 1, and iii) Scenario 3 - SNR=
[5, 5, 5, -15] dB, where targets span from42 to 42 with
an intertarget angular spacing of 5:25. The convergence
accuracy,, for the proposed distributed algorithm, Algorithm
1, is set to 0.002. For brevity and conciseness, we
show the results for only the node with the smallest SNR
because we observe better performance for all other nodes
in the considered simulation scenarios.
The amplitudes of the spatial spectra of the estimated
n;i
DoAs, for the fourth node, are shown in Figure 6(a) and (b)
at first iteration and upon convergence, respectively, for
IEEE A&E SYSTEMS MAGAZINE
13
i¼1 as estimated sparse veci¼1,
fdi½n¼ 0gQ
i¼1, and
IEEE - Aerospace and Electronic Systems - November 2022

Table of Contents for the Digital Edition of IEEE - Aerospace and Electronic Systems - November 2022
