Sparse Source Localization in Presence of Co-Array
Perturbations
Ali Koochakzadeh
Piya Pal
Dept. of Electrical and Computer Engineering
University of Maryland, College Park
Email: alik@umd.edu
Dept. of Electrical and Computer Engineering
University of Maryland, College Park
Email: ppal@umd.edu
Abstract—New spatial sampling geometries such as nested
and coprime arrays have recently been shown to be capable
of localizing O(M 2 ) sources using only M sensors. However,
these results are based on the assumption that the sampling
locations are exactly known and they obey the specific geometry
exactly. In contrast, this paper considers an array with perturbed
sensor locations, and studies how such perturbation affects source
localization using the co-array of a nested or coprime array. An
iterative algorithm is proposed in order to jointly estimate the
directions of arrival along with the perturbations. The directions
are recovered by solving a sparse representation problem in each
iteration. Identifiability issues are addressed by deriving Cramér
Rao lower bound for this problem. Numerical simulations reveal
successful performance of our algorithm in recovering of the
source directions in the presence of sensor location errors. 1
I. I NTRODUCTION
Uniform spatial sampling geometries such as the Uniform
Linear Array (ULA) have been studied for decades and they
are well known to be capable of resolving O(M ) sources
using M sensors [1], [2]. Recently, new non-uniform array
geometries, namely nested [3], and coprime [4] arrays, have
been proposed which can dramatically increase degrees of
freedom from O(M ) to O(M 2 ) leading to capability of
estimating O(M 2 ) source directions. In these techniques, the
vectorized form of the covariance matrix of received signals is
studied, which leads to the concept of virtual co-arrays. Two
main approaches have been proposed to use this enhanced
degrees of freedom in order to improve the DOA (directions
of arrival) estimation. The first approach is the subspace
methods, where the typical subspace methods for arrays, such
as MUSIC, are performed on the spatially smoothed co-array
manifold [5]. In the second approach, sparsity is used as the
key to recover the directions of sources. In this case, the range
of all possible directions is discretized into a grid, and the
DOA estimation problem is then reformulated as a sparse
representation problem [6]–[9]. We review this approach in
more details in Section II-A.
The results established so far are based on an unperturbed
array geometry, one in which the sensor locations exactly
satisfy the geometrical constraints. Array imperfections, however, are known to severely deteriorate the performance of
DOA estimation algorithms [10], [11]. This is mainly due to
strong dependence of these algorithms to the availability of an
accurately known array manifold. One of the most common
forms of these imperfections is uncertainty about the accurate
location of the sensors. To address the array imperfections
and suppress their effect on DOA estimation, there are many
1 This
work was supported by the University of Maryland, College Park.
c
978-1-4673-7353-1/15/$31.00 2015
IEEE
methods proposed in the literature, which are known as selfcalibration methods. In these methods, the perturbations are
modeled as unknown but deterministic parameters, and these
parameters are estimated jointly with the directions of the
sources. The methods in [11], [12], and [13] resolve the sensor
location uncertainty, using eigenstructure-based methods. [14]
proposes a unified framework to formulate the array imperfections, and renders a sparse Bayesian perspective for array
calibration and DOA estimation. In all of these methods, the
array is assumed to be ULA, and none of the self-calibration
methods in the literature use the concept of co-arrays.
In this paper, we study the effect of random perturbations on
the sensor locations and investigate if it is possible to perform
DOA estimation using a co-array in presence of perturbations.
We use the grid based model to obtain a sparse representation
and propose a sparsity-aware estimation framework to jointly
recover the DOAs and the perturbation errors using an iterative algorithm. We also address identifiability issues for this
framework by explicitly deriving the Cramér Rao bound and
studying conditions under which it exists.
The paper is organized as follows. In Sec II we formulate
our problem. Sec III establishes Cramér Rao based conditions
under which our problem is identifiable. In Sec IV, our proposed approach for jointly estimating the source directions and
perturbations is presented. Sec V elaborates the performance
of our proposed algorithm via numerical simulations. Sec VI
concludes the paper.
Notation: Throughout this paper, matrices are represented
by capital bold letters, vectors by lower case bold letters. xi
denotes the ith entry of a vector x. xa:b represents the (Matlabstyle) subvector of x starting at a and ending at b. Similarly,
X(a:b,c:d) stands for the submatrix of X consisting of the
(a, b)th through (c, d)th elements. (.)∗ , (.)T , (.)H represent
the conjugate, transpose, and hermitian, respectively. ◦, ⊙, ⊗
stand for Hadamard product, column-wise Khatri-Rao product,
and Kronecker product, respectively. k.kF denotes the matrix
Frobenius norm. k.k0 , and k.k denote the ℓ0 pseudo-norm
and ℓ2 norm of a vector, respectively. vec(.) represents the
vectorized form of a matrix. Re{.} returns the real part of a
complex valued number.
II. P ROBLEM F ORMULATION
We consider an array of M sensors impinged by K narrowband sources with unknown directions θ. Let y(t) ∈ CM
represent the vector of the received base-band samples on each
sensor, s(t) ∈ CK be the vector of the emitted signals from
each source, and w(t) be the additive noise on each sensor,
respectively, at the time snapshot t. The sources are assumed
to be zero mean, stationary and pairwise uncorrelated signals.
Further, we assume that the noise vector is zero mean, i.i.d.,
2
white with variance σw
, and uncorrelated from the signal.
Assume the sensors are originally designed to be at locations
introduced by the vector d ∈ RM , however due to deformation
of the sensor array, the sensors are actually located at d + δ,
in which δ ∈ RM is an unknown perturbation vector. The
received samples can be written as
y(t) = A(θ, δ)s(t) + w(t),
(1)
in which A(θ, δ) = [a(θ1 , δ), . . . , a(θK , δ)] denotes the array
manifold, and a(θi , δ) ∈ CM is the steering vector for the kth
source whose mth element is am (θk ) = ej(2π/λ)(dm +δm ) sin θk .
We can further rewrite the array manifold A(θ, δ) as
A(θ, δ) = A0 (θ) ◦ P(θ, δ),
(2)
where A0 (θ) is the array manifold as if the sensors were
located at their originally designed positions dm , P(θ, δ) is
the perturbation matrix introduced by the sensor location errors
δm , and ◦ denotes the Hadamard matrix product. Furthermore,
the (m, k)th components of M × K complex valued matrices
A0 (θ) and P(θ, δ) are ej(2π/λ)dm sin θk , and ej(2π/λ)δm sin θk ,
respectively. Given the problem setting as stated above, the
goal is to jointly estimate both the vectors δ and θ by
observing L snapshots of the received samples on the sensors.
We compute the covariance matrix of the received signals as
Ry = E(yyH ). Assuming that the sources are uncorrelated,
following [3] the vectorized form of the covariance matrix is
2
z = AKR (θ, δ)rss + σw
vec(IB ),
(3)
where AKR (θ, δ) = A(θ, δ)∗ ⊙ A(θ, δ) is the difference
co-array, rss = [σ1 , σ2 , . . . , σK ] is the diagonal of Rs =
E(ssH ) the covariance matrix of the sources, and z =
vec(Ry ). The (m + m′ M, k)-th element of AKR (θ, δ) is
given by ej(2π/λ)sin(θk )(dm +δm −dm′ −δm′ ) . Thus each column
of AKR (θ, δ) is characterized by the difference co-array:
Sca = {dm − dm′ , 1 ≤ m, m′ ≤ M }
Therefore, AKR (θ, δ) acts as a larger virtual array with
sensors located at dm + δm − (dm′ + δm′ ) for 1 ≤ i, j ≤ M .
In the rest of this paper, without loss of generality, we put
the first sensor on the origin. Therefore d1 = δ1 = 0, and
∂
∂δ1 = 0. It can be simply verified that adding a common bias
to the locations of the sensors will not alter the covariance
matrix Ry .
A. A Grid Based Model and Role of Co-Array
In order to estimate the directions of arrival, following the
same approach as in [7], we use a grid based model. In this
setting, the range of all possible directions, [−π/2, π/2], is
quantized into a certain number of grid points, Nθ , in order
to construct a grid based array manifold Agrid (δ) ∈ CM ×Nθ ,
where each column of this matrix is a steering vector corresponding to a particular direction in the grid. The gridded coarray manifold is given by Aca (δ) = Agrid (δ)∗ ⊙ Agrid (δ).
Clearly, this co-array manifold only depends on δ, and the
structure of the array. We can now rewrite (3) as
2
z = Aca (δ)γ + σw
vec(IB ),
where the non-zero elements of γ ∈ CNθ are equal to the corresponding elements of rss . We can further suppress the effect
of noise by only considering the off-diagonal elements of the
covariance matrix Ry , or in vectorized form by removing the
rows in z and Aca (δ) corresponding to i − j = 0. Hence, we
obtain
zu = Auca (δ)γ
We assume that K is sufficiently smaller than Nθ to have a
sparse vector γ. We will further elaborate this assumption in
the next sections. Considering that the value of perturbations
(δ) are given, the directions of the sources can be found by
recovering the sparse support of γ. The size of the recoverable
sparse support (or equivalently, the number of sources, K)
fundamentally depends on the Kruskal Rank of Agrid (δ). The
support can be recovered by solving
min kγk0
γ≥0
s.t.
z = Auca (δ)γ.
In Absence of Perturbation: It is shown in [15, Thm. 8], in
absence of perturbation (δ = 0), the size of the recoverable
support of γ satisfies Supp(γ) ≤ Mca /2, where Mca is the
number of distinct elements in the co-array Sca . For a ULA,
the coarray is given by
U LA
= {ndn = −M, . . . , M }
Sca
and hence Mca = O(M ) whereas for nested array with even
number of sensors and two levels of nesting
M M
N ested
(
+ 1) − 1}
= {ndn = −M ′ , . . . , M ′ ; M ′ =
Sca
2 2
and hence Mca = O(M 2 ).
Perturbation and Size of Support: The presence of perturbation has a non trivial effect on the size of the recoverable support. Of particular interest is the question: Does
the presence of perturbation δ reduce the size of identifiable
support from O(M 2 ) to O(M ) for nested arrays? By a simple
equations-versus-unknown argument, it can be seen that for
nested arrays, the number of distinct equations in (4) is still
O(M 2 ), whereas the number of unknowns is M + K, since
we have M unknown values for the perturbation and K
unknown values for the source directions. Since the number
of unknowns only increases by O(M ), it may be still possible
to identify K = O(M 2 ) sources in presence of perturbations.
The next section presents a more formal Cramér Rao bound
based study of this problem.
III. I DENTIFIABILTY AND C RAM ÉR R AO B OUND
We derive the stochastic Cramér Rao bound (CRB) for
the grid based model under perturbation, using which we
can claim some necessary conditions under which the sparse
support is identifiable for the perturbed model, for a given
perturbation δ. We consider the signals s(t) to be stochastic
with normal distribution N (0, Rs ), in which Rs is a diagonal
matrix with a few non-zero elements corresponding to the
directions on the grid where we have active sources. Assuming
the noise to be zero mean uncorrelated Gaussian with distri2
bution N (0, σw
I), we get
2
y(t) ∼ N (0, Agrid (δ)Rs AH
grid (δ) + σw I)
2
is known. Thus, the
We assume that the power of noise σw
goal is to estimate the parameter set ψ = [δ 2:M , γ] which
completely characterizes the distribution of y(t), where γ =
diag(Rs ), and δ1 = 0.
Remarks on the derivation of CRB: There are two ways
to derive the CRB. Since the model is underdetermined
20
Nested (Ideal)
ULA (Ideal)
Nested (Pertubed)
ULA (Perturbed)
0
−20
−40
i−th singular value (dB)
(Nθ ≫ M ), it may seem that we need priors on our parameter
γ (such as sparsity, or some other form of Bayesian prior)
for the Fisher Information matrix to be invertible. However,
we will show that the FIM continues to be non singular,
i.e. the parameters continue to be identifiable without
incorporating any prior on the parameter as long as Nθ
remains smaller than a certain threshold. We also show that
beyond this threshold, the FIM necessarily becomes rank
deficient.
−60
−80
−100
−120
−140
A. CRB without Sparsity Priors
Theorem 1. The Fisher information matrix of the parameter
set ψ is given by
Jδδ Jδγ
(4)
J = JH J
δγ
γγ
0
50
100
150
i
Fig. 1. The singular values of Aca (δ) for nested and ULA arrays in perturbed
and ideal settings. M = 12, Nθ = 200, δ = 0.3 × δ 0 with δ 0 defined in
Section V.
(2:M +Nθ ,2:M +Nθ )
where
−160
−T
Jγγ = LAH
⊗ Ry −1 Aca (δ)
(5)
ca (δ) Ry
n
−1
−1 T
◦ D δ Rs AH
Jδδ = 2L Re Dδ Rs AH
grid Ry
grid Ry
T o
−1
(6)
+ Ry −1 ◦ Dδ Rs AH
Agrid Rs DH
grid Ry
δ
n
o
T
−1
−1
H
Jδγ = 2L Re Dδ Rs AH
R
A
◦
A
R
y
grid
y
grid
grid
(7)
PM ∂Agrid
in which Dδ =
i=2 ∂δ i , and we temporarily omitted
argument of δ from Agrid (δ) for the simplicity of notation.
The first row and column of the matrix in the RHS of (4) are
eliminated since we assumed that δ1 is known and equal to 0.
Proof. Jγγ is obtained similar to [16], by plugging the perturbed coarray manifold in the equations therein. The rest
of the Fisher information matrix is derived using the same
approach as in [17] (pp. 960-962). The Cramer-Raó lower
bound on the parameter set ψ is then given by J−1 .
Corollary 1. A necessary condition for identifiability of the
DOA estimation problem is Nθ = rank(Aca (δ)).
Proof. For the matrix J to be invertible, it is necessary that
each diagonal block be nonsingular. (However, this is not a
sufficient condition since the Schur complement of J, Jδδ −
H
Jδγ J−1
γγ Jδγ , should also be nonsingular). It is shown [16] that
Jγγ is nonsingular iff Nθ = rank(Aca ) which completes the
proof.
Remark 1. For δ = 0, rank(Aca (δ)) can be determined
using the structure of the array. For instance, for a ULA array
rank(Aca (δ)) = O(M ), and rank(Aca (δ)) = O(M 2 ) for
nested and coprime arrays. However, for non-zero δ, the rank
of the coarray increases and it can be as large as M 2 −M +1,
since the differences δi − δj can given rise to more unique
differences. This is further validated using simulation. For
ULA, the singular values of Aca (δ) start decaying quickly
beyond 2M , whereas, for nested array, they decay only beyond
O(M 2 ). Figure 1 shows the singular values of Aca (δ) for
ULA and nested arrays in the presence of perturbation. We
will investigate into this interesting effect thoroughly in future
and compare their performance in sparse support recovery.
Notice that the sparsity constraint on the vector γ is not used
so far. The next question to ask would be: how to address the
question of identifiability when Nθ > rank(Aca (δ))?.
B. Oracle CRB
For J to be nonsingular, we need Nθ = rank(Aca (δ)).
When Nθ > rank(Aca (δ)), J (computed without assuming
any prior on γ) is singular, indicating that beyond this value
of Nθ , some form of prior on γ is necessary for identifiability.
It has been shown in [16] that γ is still identifiable, as long
as γ is sparse with
k-rank(Aca (δ))
,
(8)
2
where k-rank(.) represents the Kruskal rank of a matrix.
Considering that γ satisfies the sparsity constraint, we can
still define an oracle CRB assuming that the support of γ
(indices of non-zero elements in γ) is known. In this case,
the same expressions for the Fisher information matrix as
in (4) are valid, if we replace Agrid with AΩ , and AKR
with AKR,Ω , where AΩ , and AKR,Ω are constructed by only
picking the columns corresponding to the indices Ω, from
matrices A and AKR , respectively, where Ω is the support
of γ. By inverting the Fisher information matrix, we obtain
an oracle CRB for the problem of estimating the source powers
jointly with the sensor location errors. Although, the support
of γ is not known in the actual DOA estimation problem, this
oracle CRB indicates a fundamental lower bound that every
estimation algorithm must obey.
kγk0 ≤
IV. A N I TERATIVE A LGORITHM F OR J OINT S UPPORT AND
P ERTURBATION E STIMATION
As per the model from Section II-A, in order to estimate the
directions of sources jointly with the perturbation, we need to
solve a sparse representation problem to recover the support of
γ. Since the sensor location errors δ are unknown, the matrix
Agrid (δ) is also unknown. Hence, the sparse representation
problem cannot be solved directly. To overcome this difficulty,
we use a two-stage approach to jointly estimate the sensor
location errors and solve the sparse representation problem.
In the first stage, we assume that the vector δ is fixed, and
then we solve the following sparse representation problem:
s.t. kγk0 ≤ K. (9)
γ̃ = arg min kAgrid (δ)γ − zk2F
γ
This problem can be solved using basic sparse recovery
methods like OMP [18] or LASSO [19]. In the second stage,
the perturbations are updated assuming that the recovered
sparse vector in the first stage is fixed. Using the updated
vector δ, the first stage is again repeated until a convergence
criterion is met, which can be a certain number of iterations.
The algorithm is initialized with δ (0) = 0. A pseudocode for
this algorithm is illustrated in Figure 2.
A. Updating the perturbation vector
In order to update the vector δ, in the second stage, we
minimize the error value E(δ) = kAca (δ)γ̃ − zk2F . This
minimization can be done by running T iterations of gradient
descent
∂E(δ)
,
(10)
δ ←δ−µ
∂δ
in which µ > 0 is the step size parameter and the mth
is obtained by
component of ∂E(δ)
∂δ
∂Pca (δ)
∂E(δ)
= tr −2zγ̃ T + 2A0,ca γ̃ γ̃ T
, (11)
∂δm
∂δm
where A0,ca , and Pca (δ) are matrices constructed by gridding
A0 (θ), and P(θ, δ).
Lemma 1. E(δ) is Lipschitz continuous with constant C,
i.e.k∇E(δ (1) ) − ∇E(δ (2) )k ≤ C(δ (1) − δ (2) ) , where
√
4πM 2M
k − zγ̃ T + A0,ca γ̃ γ̃ T k
(12)
C=
λ
Corollary 2. A sufficient condition for gradient descent to
converge to a local minimum is µ < µ0 , where µ0 = C2 , and
C is given by (12).
Although the gradient descent approach can find the minimum value of E(δ) given the step size parameter µ is smaller
than µ0 given in Corollary 2, it may require relatively large
number of iterations T for gradient descent to converge. Therefore, the updating process may become very time consuming.
Hence, we propose another approach for minimizing E(δ).
B. Updating the perturbation vector: a linearized approach
To simplify the minimization of E(δ) we can assume that
the perturbations are sufficiently small, which is a natural
assumption. We can use first order Taylor expansion of
Agrid (δ) with respect to δ to approximate E(δ), and then
we minimize this approximated function instead. In this case,
the minimization task simplifies to a least squares problem.
We can approximate E(δ) as
E(δ) ≃ kA0,ca γ̃ + Qδ − zk2 ,
(13)
ca (δ) where the ith column of Q is equal to ∂A∂δ
γ̃. The
δ=0
i
minimizer of (13) is then given by δ̄ 1 = 0, δ̄ 2:M =
(Q̄H Q̄)−1 Q̄H (z − A0,ca γ̃), with Q̄ = Q(2:M,2:M ) .
C. Convergence
The proposed algorithm has the same flavour as K-SVD
[20] algorithm for dictionary learning, in the sense that by
iteratively performing sparse recovery and updating the dictionary atoms, an unknown dictionary is recovered. In our
problem, Agrid (δ) is a structured unknown dictionary. Similar
to K-SVD, the convergence here is subject to success of both
stages. For the first stage to succeed, the sparsity of γ should
be high enough so that recovering a unique sparse solution be
guaranteed. For the second stage, if gradient descent is used,
the step-size parameter and the number of iterations should
be tuned to avoid divergence of this stage. If the linearized
approach is used, the perturbations should be small enough
for the linear approximation to be valid. If all these criteria
are met, in each iteration of our algorithm, the positive valued
function E(δ) decays until converging to a local minimum.
1:
2:
3:
4:
5:
6:
7:
δ (0) ← 0, i ← 0
repeat
n
o
γ (i) ← arg minγ kAgrid (δ (i) )γ − zk2F s.t. kγk0 ≤
K
δ (i+1) ← arg minδ kAgrid (δ)γ (i) − zk2F
i←i+1
until Convergence
δ̂ ← δ (i) , γ̂ ← γ (i)
Fig. 2. Algorithm for jointly estimating the perturbations and the source
directions
V. E XPERIMENTAL R ESULTS
We conduct several numerical experiments to study the
effect of perturbation on DOA estimation, and the performance
of our proposed algorithm. The simulations are performed for
the uniform linear array (ULA), and nested arrays. In all of
the cases the number of sensors are fixed to be M = 8.
In each case, we run the experiment for different number of
sources. The sources are assumed to be uncorrelated complex
Gaussian signals. The number of grid points (Nθ ) is either
19 or 139, where − π2 and π2 are not included in the grid.
These choices are to study both cases where Nθ = rank(Aca )
and Nθ > rank(Aca ). In the cases where number of sources
are 12, we assume that sources are located at directions:
[−63◦ −45◦ −27◦ −18◦ −9◦ 0◦ 9◦ 27◦ 36◦ 45◦ 54◦ 72◦ ].
For 9 sources, we consider the same directions except for
−9◦ , 9◦ , 54◦ . For 6 sources, −45◦ , −18◦ , 27◦ are also discarded. In all the simulations, the vector of location errors (perturbations) is δ = αδ 0 , where δ 0 = λ2 ×
[0 −0.25 −0.1 0.1 0.4 0.35 0.45 0.3]T , and α > 0 is
a scalar, and λ > 0 is the wavelength of the impinging
wave. For the ULA, the unperturbed sensor locations are
d = λ2 ×[0, 1, 2, 3, 4, 5, 6, 7]T . For the nested array, the sensors
are located at d = λ2 × [0, 1, 2, 3, 7, 11, 15, 19]T . In all the
simulations, the number of time snapshots is assumed to be
2
= 4 × 10−4 .
L = 800, and the power of noise is set to σw
We use LASSO [19] to perform sparse recovery, and gradient
descent approach for updating the perturbation vector in each
iteration of our algorithm. The number of sources K can
be estimated by looking into the recovered sparse support
(obtained from LASSO) and then pruning the spurious peaks
by a simple thresholding.
In the first experiment, we consider a nested array with 8
sensors and 9 sources, with the sensor and source locations as
mentioned in the previous paragraph, and α = 0.3, Nθ = 139.
Figure 3 illustrates the recovered powers with and without using our proposed algorithm for recovering the sensor location
errors, showing the improvement in recovery of source powers
when our algorithm is used. In 3a the source powers are all
equal to one. However, in 3b they have the following values,
respectively: 0.7, 1, 0.7, 1, 0.6, 0.8, 0.6, 0.9, 1. The plots show
satisfactory performances in both cases.
In the second experiment, we conduct Monte Carlo simulations to study the performance of our algorithm for recovery of the sensor location errors. Figure 4 shows the
Root Mean Square Error (RMSE) of the recovered sensor
location
q errors δ̂ versus α. The RMSE in this case is defined
PNtests kδ̂−δk2
as
i=1
Ntests , where Ntests indicates the number of
Monte-Carlo simulations for each value of α (In all the
simulations Ntests = 100). As α increase (larger perturbations),
1.4
1.4
γ
γ̃
γ̂
γ
γ̃
γ̂
1.2
1.2
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
−100
−80
−60
−40
−20
0
20
40
60
0
100 −100
80
−80
−60
−40
−20
0
20
40
60
80
100
(a) Equal source powers
(b) Unequal source powers
Fig. 3. γ, γ̃, γ̂ indicate the original source powers, the initial guess assuming
δ = 0, and the recovered source powers using our algorithm, respectively.
20
15
15
10
10
5
RMSE, nested, K=6
CRB, nested, K=6
RMSE, nested, K=9
CRB, nested, K=9
RMSE, nested, K=12
CRB, nested, K=12
RMSE, ula, K=6
o-CRB, ula, K=6
RMSE, ula, K=9
o-CRB, ula, K=9
RMSE, ula, K=12
o-CRB, ula, K=12
0
−5
−10
−15
−20
−25
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
RMSE (dB)
RMSE (dB)
5
0
RMSE, nested, K=6
o-CRB, nested, K=6
RMSE, nested, K=9
o-CRB, nested, K=9
RMSE, nested, K=12
o-CRB, nested, K=12
RMSE, ula, K=6
o-CRB, ula, K=6
RMSE, ula, K=9
o-CRB, ula, K=9
RMSE, ula, K=12
o-CRB, ula, K=12
−5
−10
−15
−20
−25
0
0.5
0.05
0.1
0.15
0.2
0.25
α
0.3
0.35
0.4
0.45
0.5
α
(a) Nθ = 19
(b) Nθ = 139
Fig. 4. The RMSE of the recovered δ̂ vs α, for different array structures and
different number of sources.
35
30
30
25
25
RMSE (degrees)
RMSE (degrees)
20
20
15
15
10
RMSE,
RMSE,
RMSE,
RMSE,
RMSE,
10
RMSE, nested, K=6
RMSE, nested, K=9
RMSE, nested, K=12
RMSE, ula, K=6
RMSE, ula, K=9
5
0
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
5
0
0
0.5
0.05
0.1
0.15
0.2
α
0.25
0.3
0.35
nested, K=6
nested, K=9
nested, K=12
ula, K=6
ula, K=9
0.4
0.45
0.5
α
(a) Nθ = 19
(b) Nθ = 139
Fig. 5. The RMSE of the recovered source directions vs α, for different array
structures and different number of sources.
0
−20
−10
−30
−20
RMSE, nested, K=6
CRB, nested, K=6
RMSE, nested, K=9
CRB, nested, K=9
RMSE, nested, K=12
CRB, nested, K=12
RMSE, ula, K=6
o-CRB, ula, K=6
RMSE, ula, K=9
o-CRB, ula, K=9
RMSE, ula, K=12
o-CRB, ula, K=12
−40
−50
−60
−70
−80
0
0.05
0.1
0.15
0.2
0.25
α
0.3
0.35
0.4
0.45
0.5
RMSE (dB)
RMSE (dB)
−40
−30
RMSE, nested, K=6
o-CRB, nested, K=6
RMSE, nested, K=9
o-CRB, nested, K=9
RMSE, nested, K=12
o-CRB, nested, K=12
RMSE, ula, K=6
o-CRB, ula, K=6
RMSE, ula, K=9
o-CRB, ula, K=9
RMSE, ula, K=12
o-CRB, ula, K=12
−50
−60
−70
−80
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
α
(a) Nθ = 19
(b) Nθ = 139
Fig. 6. The RMSE of the recovered source powers vs α, for different array
structures and different number of sources. o-CRB denotes the oracle CRB.
the algorithm begins to fail to find the correct sensor locations.
However, the CRB still suggests the possibility of identifying
the correct sensor positions in those circumstances. Figure
6 shows the RMSE of the recovered source powers. The
experiments are conducted for both Nq
θ = 19 and Nθ = 139.
PNtests kγ̂−γk2
The RMSE in this case is defined by
i=1
Ntests . In all
the plots, CRB of the corresponding parameters is calculated
by taking square-root of the trace of the CRB matrix for
those parameters. For the cases where Nθ > rank(Aca ), the
oracle CRB is calculated, and is shown by o-CRB. Figure 5
depicts the RMSE of source directions, which is calculated
by comparing the support of γ̂ with that of γ. As it can be
observed, for the nested array, our algorithm can successfully
recover the true support even in high perturbations.
VI. C ONCLUSION
In this paper we studied the effect of random perturbations
on the sensor locations in DOA estimation problem using the
concept of co-arrays. We proposed an iterative algorithm to
find the direction of sources along with the perturbations.
We addressed identifiabilty issues by deriving Cramér Rao
lower bounds. The experimental results revealed successful
performance of our algorithm especially for nested arrays. We
can conclude that conclude that co-arrays of nested arrays are
robust to small perturbations in the sensor locations.
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