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IEEE Proof
IEEE COMMUNICATIONS LETTERS 1
Effective Block Sparse Representation Algorithm for DOA Estimation
With Unknown Mutual Coupling
Qing Wang, Member, IEEE, Tongdong Dou, Hua Chen, Weiqing Yan, and Wei Liu, Senior Member, IEEE
Abstract— Unknown mutual coupling effect can degrade the1
performance of a direction of arrival estimation method. In this2
letter, a new method is proposed for uniform linear arrays (ULAs)3
to tackle this problem. Considering the sparse representation4
exploiting the inherent structure of the received data, the effective5
block sparse representation and the convex optimization problem6
are constructed using the steering vector parameterizing method.7
The proposed solution based on the l1- singular value decomposi-8
tion can exploit the information provided by the whole array and9
the Toeplitz structure of the mutual coupling matrix in the ULA.10
Simulation results are provided to demonstrate its performance11
with unknown mutual coupling in comparison with some existing12
methods.13
Index Terms—Direction of arrival (DOA), block sparse14
representation, mutual coupling, l1-SVD.15
I. INTRODUCTION16 MULTIPLE-INPUT Multiple-Output (MIMO) technique17
is more attractive for increasing spectral and energy18
efficiency in the wireless and mobile communications [1].19
Meanwhile, the MIMO system has more degrees of freedom20
and high spatial resolution than other systems in case of the21
direction of arrival (DOA) estimation [2]–[4]. There is an issue22
that must be considered in the MIMO system which the array23
size has been given, increasing the number of antennas will24
lead to the decrease of the array element spacing, and then25
resulting in a stronger mutual coupling effect between the26
antenna elements.27
Mutual coupling can cause severe performance degradation28
for those conventional direction finding methods [5], [6].29
Therefore, various array calibration techniques have been pro-30
posed [7]–[11]. For a uniform linear array (ULA), the coupling31
between neighboring elements is almost the same along the32
array, so the number of parameters can be reduced, and the33
mutual coupling matrix (MCM) can be modelled as a banded34
symmetric Toeplitz matrix [7]. And the method in [8] used35
auxiliary arrays, exploiting the banded symmetric Toeplitz36
matrix model for the mutual coupling effect, based on the37
ESPRIT algorithm. The special structure of the MCM of a38
ULA was also employed to parameterize the steering vector39
for joint estimation of DOAs and MCM in [9]. With the help40
of the auxiliary elements, the effect of mutual coupling can41
Manuscript received August 8, 2017; accepted August 24, 2017. The
associate editor coordinating the review of this paper and approving it for
publication was K. E. Psannis. (Corresponding author: Hua Chen.)
Q. Wang and T. Dou are with the School of Electrical and Information
Engineering, Tianjin University, Tianjin 300072, China.
H. Chen is with the Faculty of Information Science and Engineering, Ningbo
University, Ningbo 315211, China (e-mail: dkchenhua0714@hotmail.com).
W. Yan is with the School of Computer and Control Engineering, Yantai
University, Yantai 264005, China.
W. Liu is with the Department of Electronic and Electrical Engineering,
The University of Sheffield, Sheffield S1 3JD, U.K.
Digital Object Identifier 10.1109/LCOMM.2017.2747547
be eliminated and the MUSIC and ESPRIT method can be 42
utilized directly to the angle estimation in bistatic MIMO 43
radar [10], [11]. 44
Recently, sparse signal representation based methods have 45
been proposed to tackle spectrum estimation and array 46
processing problems [12]–[16], [18], outperforming many tra- 47
ditional direction finding algorithms. To solve the more general 48
source localization problems, the l1-SVD method was derived 49
in [12], which can be used to tackle a wide variety of practical 50
signal processing problems. An efficient direction finding 51
method based on the separable sparse representation is derived 52
in [13], where it utilizes a separable structure for spatial 53
observation matrix to reduce the complexity. And a perturbed 54
sparse Bayesian learning-based algorithm is proposed to solve 55
the DOA estimation for off-grid signals in [15], which is a 56
more general case in practice. By using the sparse signal 57
reconstruction of monostatic MIMO array measurements with 58
an overcomplete basis, the SVD of the received data matrix 59
can be penalties based on the l1-norm [16]. In [17] and [18], 60
the sparse signal reconstruction based method is considered 61
for DOA estimation with a coprime array, the over-complete 62
representation is formulated for convex optimization problem 63
design by reconstructing the virtual uniform linear subarray 64
covariance matrix. In addition, the application of sparse recon- 65
struction can be devoted to the solution of the mutual coupling 66
problem. For example, it was applied in [14] to compensate 67
for the mutual coupling effect with the help of a group of 68
auxiliary sensors in a ULA. 69
In this letter, we propose a new block sparse signal rep- 70
resentation based DOA estimation method in the presence 71
of unknown mutual coupling effect and no auxiliary array 72
elements are required in the process. By constructing a new 73
over-complete block matrix based on the inherent structure of 74
the steering vector with mutual coupling, we can make full 75
use of the received data of the whole array and eliminate 76
the unknown mutual coupling effect. The resultant sparse 77
optimization problem for DOA estimation is transformed to 78
a convex optimization and then solved using the l1–SVD 79
method. 80
Notation: [·]Trepresents the matrix and vector transpose; 81
diag[·] stands for the diagonalization operation of matrix 82
blocks; ||·||pdenotes the p-norm of a matrix; [·]M×Nindicates 83
amatrixofMrows and Ncolumns; the zero vector or zero 84
matrix is denoted by 0.85
II. PROBLEM FORMULATION 86
Consider a ULA with Msensors with Nfar-field narrow- 87
band impinging signals sn(t),n=1,2,··· ,N,wheretis the 88
sample index, with t=1,2,··· ,T. Firstly, we formulate the 89
received data model for an ideal array without mutual coupling 90
1558-2558 © 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.
See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
IEEE Proof
2IEEE COMMUNICATIONS LETTERS
as91
x(t)=As(t)+n(t)(1)92
In (1), x(t)=[x1(t), x2(t), ··· ,xM(t)]Tdenotes93
the Mreceived antenna signals, and the array steer-94
ing matrix A=[a(θ1), a(θ2), ··· ,a(θN)]with a(θn)=95
[1,β (θ
n), ··· ,β(θ
n)M−1]Tdenoting the nth signals ideal96
steering vector, where β(θ
n)=exp (−j2πλ−1dsin θn),97
and θndenotes the angle of arrival of the nth source, dis98
the adjacent sensor spacing, and λis the signal wavelength.99
s(t)=[s1(t), s2(t), ··· ,sN(t)]Tis the source signal vector,100
and n(t)=[n1(t), n2(t), ··· ,nM(t)]Tis the independent and101
identically distributed additive white Gaussian noise vector102
with zero mean and covariance matrix σ2I,whereσ2denotes103
the power of noise and Iis the identity matrix.104
In practice, we have to consider the mutual coupling effect105
between closely spaced antennas. In this case, the received106
data model can be modified as107
x(t)=CAs(t)+n(t)(2)108
where Cis the MCM. For ULAs, the MCM can be modelled109
as a banded symmetric Toeplitz matrix110
C111
=
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
1c1... cP−1
c11c1··· cP−10
.
.
................
cP−1··· c11c1··· cP−1
....
.
................
cP−1··· c11c1··· cP−1
0....
.
...........
.
.
cP−1··· c11c1
cP−1··· c11
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦M×M
112
(3)113
In (3), cpis a complex number and denotes the mutual114
coupling coefficient between the mth and the (m+p)th sensor115
with p=0,1,··· ,P−1, m=1,2,··· ,M.Pgives the116
maximum distance between antennas over which the mutual117
coupling effect cannot be ignored, which means that for the118
mth sensor, it is affected by electromagnetic coupling coming119
from the (m−P+1)th, ··· ,(m−1)th, (m+1)th, ··· ,120
(m+P−1)th sensors. For multiple snapshots, we can define121
ˆ
X=[x(1), x(2), ··· ,x(T)]∈CM×Tand also define Sand N122
in a similar way. Then, we have123
ˆ
X=CAS +N(4)
124
III. THE PROPOSED METHOD125
In this section, the proposed sparse representation method126
for DOA estimation based on parameterization of the steering127
vector and l1-SVD will be introduced.128
A. Parameterization of the Steering Vector With129
Mutual Coupling130
According to the data model in Section II, the ideal steering131
vector a(θ) is distorted by the effect of mutual coupling in132
practice, and it should be modified as133
˜a(θ) =Ca(θ ) (5)134
According to (3), we can rewrite equation (5) as 135
˜a(θ) =H(θ )(θ)a(θ) (6) 136
where 137
H(θ) =
P−1
l=1−P
c|l|β(θ)l(7) 138
(θ) =diag[μ1,··· ,μP−1,1,··· ,1,ν
1,··· ,νP−1]M×M139
(8) 140
and for k=1,2,··· ,P−1, 141
μk=
H(θ) −
P−1
l=k
clβ(θ)−l
H(θ) ,ν
k=
H(θ) −
P−1
l=P−k
clβ(θ)l
H(θ) 142
(9) 143
Since (θ) is a diagonal matrix and a(θ) is a column vector, 144
(6) can be expressed by 145
˜a(θ) =H(θ )J(θ)v(θ) (10) 146
where 147
J(θ) =
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
1β(θ)
...0
β(θ)P−1
.
.
.
0β(θ)M−P
...
β(θ)M−1
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦M×(2P−1)
(11) 148
149
v(θ) =[μ1,··· ,μP−1,1,ν
1,··· ,νP−1]T(12) 150
Hence, (2) can be changed to 151
x(t)=[˜a(θ1), ˜a(θ2), ··· ,˜a(θN)]s(t)+n(t)(13) 152
From (10), we can see that H(θ) is a scalar parameter 153
related to the mutual coupling coefficients and DOAs. It may 154
take a zero value for some very specific cases. However, 155
in general, it is not zero-valued and we assume H(θ ) = 0, 156
θ∈[−90o,90o]in the following discussion. Then, (13) can 157
be further changed to 158
x(t)=AJs(t)+n(t)(14) 159
where 160
AJ=[J(θ1), J(θ2), ··· ,J(θN)] (15) 161
162
=⎡
⎣
H(θ1)v(θ1)H(θ2)v(θ2)0
0...H(θN)v(θN)
⎤
⎦N(2P−1)×N
(16) 163
Now, we can consider the distinct block columns of the matrix 164
AJ,i.e.J(θn)∈CM×Q, as a new steering vector behaving like 165
a(θn),n=1,2,··· ,N,Q=2P−1, thus, AJbecomes the 166
new manifold matrix of the array with mutual coupling. is 167
a block diagonal matrix. 168
IEEE Proof
WANG et al.: EFFECTIVE BLOCK SPARSE REPRESENTATION ALGORITHM FOR DOA ESTIMATION 3
B. Block Sparsity Representatiom Using the l1-SVD Method169
For the case of sparse reconstruction in direction finding170
without mutual coupling, we first construct an over-complete171
representation ˜
A=[a(θ
1), a(θ
2), ··· ,a(θ
G)]∈CM×Gto find172
the sparsest spectrum of the signal vector ˜s∈CG×1to satisfy173
x=˜
A˜swith respect to all possible DOAs ={θ
g,g=174
1,2,··· ,G},wheretheith row of ˜sis nonzero and equal to175
sn(t)if the DOA of signal nis θ
i,Gis the number of all176
possible DOAs and the set constitutes the sampling grid.177
The formulation of the problem with additive white Gaussian178
noise is given as follows179
x=˜
A˜s+n(17)180
An ideal measure of sparsity is the l0-norm constraint,181
but it is a difficult and intractable combinatorial optimization182
problem. According to [12], we use the l1-norm minimization183
principle to relax the constraint, so the DOA estimation184
problem can be formulated as185
min ||˜s||1,subject to ||x−˜
A˜s||2
2≤ξ2(18)186
Now, let us consider the case with mutual coupling. With (2)187
and (17), we can modify (18) as188
min ||˜s||1,subject t o ||x−C˜
A˜s||2
2≤ξ2(19)189
The above representation is no longer a convex optimization190
problem due to the unknown mutual coupling parameter.191
In order to reconstruct the signal spectrum from (19), we need192
to construct a new over-complete matrix ¯
AJin terms of a193
sampling grid of all potential source locations as follows194
¯
AJ=[J(θ
1), J(θ
2), ··· ,J(θ
G)](20)195
where each M×Qblock matrix J(θ
g)has the same structure as196
J(θ). Meanwhile, because of the matrix in (14), the structure197
of the sparse signal vector is modified as below198
¯s=˜s(21)199
where =diag[H(θ
1)v(θ
1), H(θ
2)v(θ
2), ··· ,H(θ
G)v(θ
G)]200
∈CGQ×Gis a block diagonal matrix, and the (Qi −Q+1)th201
to (Qi)th rows of ¯sare of a nonzero value if the ith row of202
˜sis nonzero and H(θ
i)= 0. So the GQ ×1 signal vector203
¯shas only a few nonzero blocks, each consisting of certain204
Qconsecutive rows, i.e., ¯shas a block-based sparse spatial205
spectrum. Considering Tsamples of the received signal,206
we have207
ˆ
X=¯
AJ¯
S+N(22)
208
where ˆ
X∈CM×T,¯
AJ∈CM×GQ,and¯
S=209
[¯s(1), ¯s(2), ··· ,¯s(T)]∈GQ×T.210
As a result, we can apply the l2-norm for all samples and the211
problem can be again transformed into a convex optimization212
problem, as formulated below213
min ||ˆsl2||1,sub ject to || ˆ
X−¯
AJ¯
S||2
2≤ξ2(23)214
where ˆsl2=[sl2
1,sl2
2,··· ,sl2
G]T,andsl2
g=215
||[sg(1), sg(2), ··· ,sg(T)]||2. It is worth noting that sg(t)216
corresponds to the (Qg −Q+1)th to (Qg)th rows of in the217
tth snapshot.218
When the number of data samples is large (T>K),219
the computational complexity of the above optimization 220
process will be very high. To reduce the complexity and 221
also the sensitivity to noise, we can apply singular value 222
decomposition (SVD) to the received data matrix ˆ
Xto reduce 223
its dimension. Denote the SVD of ˆ
Xby ˆ
X=UV,andwe 224
further have 225
ˆ
X=USSVH
S+UNNVH
N(24) 226
where Sand Nare diagonal matrices whose diagonal 227
entries correspond to the Nlargest singular values and the 228
remaining M−Nsingular values, respectively. The unitary 229
matrices USand VScorrespond to the signal subspace, while 230
the unitary matrices UNand VNcorrespond to the noise 231
subspace. Together we have U=[USUN],V=[VSVN]T,232
and =diag[SN].Then ˆ
X.ˆ
Xcan be reduced to 233
ˆ
XR=CASR+NR(25) 234
where ˆ
XR=ˆ
XVS,SR=SVS,andNR=NVS.235
Then, in a similar way, we can define ¯
SR=¯
SVS=236
[ˆsR(1), ˆsR(2), ··· ,ˆsR(N)],ˆsl2
R=[ˆsl2
1,ˆsl2
2,··· ,ˆsl2
G],andˆsl2
g=237
||¯
SR((Qg −Q+1):Qg,:)||2, and arrive at the following 238
formulation with a much reduced dimension 239
min ||ˆsl2
R||1,subject to|| ˆ
XR−¯
AJ¯
SR||2
2≤ξ2(26) 240
According to the knowledge of the distribution, we can 241
apply the l1-SVD method and the upper value of ||NR||2with a 242
99% confidence interval to select the regularization parameter 243
ξas described in [12]. As (26) shows, we have applied the 244
parameterized steering vector operation to the manifold matrix 245
of the array with mutual coupling. Thus, the spatial spectrum 246
of ¯
SRis block sparse, which is related to the constructed 247
over-complete matrix. And the computational complexity of 248
solving (26) through the second-order cone programming is 249
O((NGQ)3).Soweemploytherecursivegridrefinement 250
procedure [14] to reduce the calculation time. 251
Note that in our discussion, we have ignored the case of 252
H(θ)=0. This may happen for some specific combination 253
of angle and coupling coefficient values, which means the 254
array will not be able to receive the signal correctly for those 255
directions and as a result, the proposed method will fail. 256
However, H(θ ) is a continuous function for given coupling 257
coefficients, so the chance of H(θ )=0 has a measure of zero 258
and we can say in general the proposed solution is valid and 259
effective as demonstrated by the following simulation results. 260
IV. SIMULATION RESULTS 261
In this section, simulation results are provided to show the 262
performance of the proposed method. The number of far- 263
field narrowband signals is N=2 with directions θ1and θ2,264
respectively, the number of the ULA elements is M=10, 265
and the number of nonzero mutual coupling coefficients is 266
P=4. The root mean squared error (RMSE) is adopted as a 267
performance index. 268
Firstly, we show the spectrum obtained by our method and 269
the methods in [8], [12], and [14] in Fig. 1, with SNR=5dB, 270
snapshot number T=200, and directions θ1=−15°, 271
and θ2=20°. The mutual coupling coefficients are 272
c1=0.4864 −0.4776 j,c2=0.2325 +0.1914 j,and 273
IEEE Proof
4IEEE COMMUNICATIONS LETTERS
Fig. 1. Spatial spectrum obtained by the proposed algorithm in comparison
with the algorithms in [8], [12], and [14].
Fig. 2. RMSE of DOA versus SNRs with snapshot number T=400.
Fig. 3. RMSE of DOA versus snapshot number with SNR=20dB.
c3=0.1163 −0.1089 j. We can see that only our method274
can identify the directions of the sources correctly, while the275
methods in [8], [12], and [14] exhibit a large deviation from276
the true values. In particular, the method in [12] even led to277
a pseudo peak close to 5° due to the lack of consideration of278
mutual coupling.279
Secondly, the performance of the proposed method is tested280
by comparing with the methods in [8], [12], and [14] at an281
SNR varying from 0dB to 10dB and with 400 snapshots,282
and directions θ1=−12.1°, and θ2=15.9°. Fig. 2 shows the283
RMSE versus SNR curves obtained by averaging 400 Monte-284
Carlo simulations. The mutual coupling coefficients are285
c1=0.43301 −0.351 j,c2=0.2618 +0.2176 j,andc3=286
0.1414 −0.1414 j. And we used the adaptive grid refinement287
approach to improve the measurement accuracy. As shown288
in Fig. 2, the proposed method has the superior resolution289
performance, that is because [12] suffers from lack of effective290
solution to the mutual coupling problem, while the method291
in [8] and [14] has given up the information received by292
(2P−1)sensors located at the two ends of the ULA.293
The third simulation examines the performance of our294
method at a snapshot number varying from 200 to 1000 with295
200 Monte-Carlo experiments, and directions θ1=−12.1°, and296
θ2=15.9°. The SNR is fixed at 20dB, and the mutual coupling297
coefficients are c1=0.5844−0.5476 j,c2=0.2625+0.1414 j,298
and c3=0.1163 −0.1289 j. As shown in Fig. 3, again the 299
proposed method has achieved the best performance. 300
V. CONCLUSION 301
In this letter, a new method based on sparse representation 302
has been proposed to solve the DOA estimation problem in 303
the presence of unknown mutual coupling for a ULA. The 304
proposed algorithm can be considered as a combination of 305
the parameterized steering vector and the l1-SVD method, 306
where the original non-convex problem with unknown mutual 307
coupling parameters was transformed into a block-sparsity 308
based convex problem by exploiting the banded symmetric 309
Toeplitz property of the mutual coupling matrix. As shown in 310
simulations, the proposed method has demonstrated a superior 311
performance over existing solutions. 312
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