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Science in China Ser. F Information Sciences 2004 Vol.47 No.3 348—361
Robust DOA estimation and array calibration in
the presence of mutual coupling for uniform
linear array
WANG Buhong1, 2, WANG Yongliang2, CHEN Hui2 & CHEN Xu3
1. Nanjing Research Institute of Electronics Technology, Nanjing 210013, China;
2. Key Research Laboratory, Air Force Radar Academy, Wuhan 430010, China;
3. PLA Unit 95857, Xiaogan 432105, China
Correspondence should be addressed to Wang Buhong (email: [email protected])
Received July 7, 2003
Abstract The presence of unknown mutual coupling between array elements is known
to significantly degrade the performance of most high-resolution direction of arrival (DOA)
estimation algorithms. In this paper, a robust subspace-based DOA estimation and array
auto-calibration algorithm is proposed for uniformly linear array (ULA), when the array
mutual coupling is present. Based on a banded symmetric Toeplitz matrix model for the
mutual coupling of ULA, the algorithm provides an accurate and high-resolution DOA
estimate without any knowledge of the array mutual couplings. Moreover, a favorable
estimate of mutual coupling matrix can also be achieved simultaneously for array
auto-calibration. The algorithm is realized just via one-dimensional search or polynomial
rooting, with no multidimensional nonlinear search or convergence burden involved. The
problem of parameter ambiguity, statistically consistence and efficiency of the new
estimator are also analyzed. Monte-Carlo simulation results are also provided to
demonstrate the effectiveness and behavior of the proposed algorithm.
Keywords: mutual coupling, DOA estimation, auto-calibration.
DOI: 10.1360/ 02yf0520
The lack of robustness of high resolution direction-of-arrival (DOA) estimation algorithms to the errors in the array manifold has limited their application to real systems
since their introduction in the early 1980s. Compared with other sources of error, the
calibration of mutual coupling between array elements has long been a difficult problem
due to its complexity. In the early contributions[1,2], the common method of estimating
the DOA in the presence of mutual coupling is to modify the DOA estimation algorithm
with a known electromagnetically-calculated or measured mutual coupling matrix.
However the DOA estimation performance may be deteriorated even more when the
mutual coupling is not compensated for exactly. In addition, it is always indispensable to
re-calibrate the mutual coupling periodically to account for changes in local conditions
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Robust DOA estimation & array calibration in presence of mutual coupling for uniform linear array
349
(e.g. a change from a wet spring to a dry autumn). The alternative methods[3,4] for DOA
estimation including mutual coupling have treated mutual coupling calibration as a parameter estimation problem, in which the mutual coupling matrix is estimated along with
other parameters of interest (e.g. source DOAs). An essential drawback associated with
these methods is that they result in a high-dimensional and multimodal nonlinear optimization problems and the global convergence is not guaranteed.
In general, the mutual coupling matrix has no special structure. However, for a
uniformly linear array (ULA), a banded symmetric Toeplitz matrix provides a satisfac—
tory model for the array mutual coupling[3,5 9]. This will result in a substantial reduction
of the unknown parameters to be estimated, with attendant improvement in estimation
accuracy and reduced computation. The existing methods for estimating the array mutual
coupling matrix do not appear to have exploited this property. In this paper, a robust subspace-based DOA estimation and array auto-calibration algorithm is proposed for ULA,
when the array mutual coupling is present. In light of the banded symmetric Toeplitz
matrix modeling for the mutual coupling of the ULA, the new algorithm provides a
favorable DOA estimate with no knowledge of the array mutual couplings. Moreover, an
accurate estimate of mutual coupling matrix can also be achieved simultaneously for
array auto-calibration. The algorithm is realized just via one-dimensional search or
polynomial rooting, with no multidimensional and multimodal nonlinear optimizing
search involved.
1
Array data model including mutual coupling
Consider a uniform linear array of N omnidirectional sensors, separated by a distance d and impinged by M sources from far field with direction at θ = [θ1, …, θM]T.
The signal waveforms are assumed to be narrowband of known center frequency. With
one of the sensors as reference, the complex envelope of the noise-corrupted array
output vector X(t) may be written as (1):
X (t ) = A(θ ) S (t ) + N (t ), t = 1, 2, " , K ,
(1)
where S(t) is a M × 1 signal vector, N(t) is a N × 1 noise vector and K is the number of
snapshots. It is assumed that the signals and noises are stationary, zero mean uncorrelated Gaussian random processes and further, the noises are both spatially and temporally white with variance σ 2. Array manifold matrix A(θ ) is a N × M matrix whose
columns are the steering vectors. In the presence of mutual coupling, the N × 1 steering
vector W(θi) can be modeled as
W (θ i ) = Za (θ i ), i = 1, 2, " , M ,
(2)
where a (θ i) is an ideal steering vector in the case of the omnidirectional sensors with
identical gain and phase response and it can be defined as
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T
a (θ k ) = ⎡⎣1, e j β k ," e j ( N −1) β k ⎤⎦ , k = 1" M .
(3)
In (3) βk denotes the wave number of kth source and is expressed by
βk =
2π
λ0
d sin(θ k ),
(4)
where λ0 is the wavelength of the signal. In order to guarantee a unique parameter estimation, the array manifold {Za (θ ) : − π 2 ≤ θ ≤ π 2} is assumed to be free of rank
N − 1 ambiguity, i.e. any N distinct steering vectors Za (θ j ), j = 1, 2, " , N ,
θ1 ≠ θ 2 ≠ " ≠ θ N are linearly independent.
It has been shown[3,5 7] that, for ULA, mutual coupling matrix Z can be modeled as
a banded symmetric Toeplitz matrix which is expressed as
—
⎧ Zij = Z1, i − j +1 ,
⎪
⎨ Z1, j = 0, j > p ,
⎪ Z = 1,
⎩ 11
(5)
where Zij are used to denote the (i, j) element of the matrix Z and p is the number of
nonzero elements in the first row of Z. Z can thus be uniquely determined by a vector z
with zi = Z1,i, i = 1, 2, …, p.
The array covariance matrix and its eigendecomposition are expressed as follows:
R = E[ X (t ) X H (t )] = ARS AH + σ 2 I ,
M
R = ∑ λi ei eiH +
i =1
N
∑
λi ei eiH = ES ΛS E S + E N ΛN E N ,
(6)
(7)
i = M +1
where RS = E[ S ( k ) S H ( k )] is a M × M source covariance matrix and I is a N × N iden-
tity matrix. {λi ; i = 1, 2," , N ; λi ≥ λi +1} and {ei ; i = 1, 2, " , N } are ordered eigenvalues and corresponding eigenvectors of R respectively. The signal subspace and noise
subspace of R are respectively the ranges of the matrices:
ES = [e1e2e3…eM],
(8)
EN=[eM + 1eM + 2eM + 3…eN].
(9)
In practice, the array covariance matrix R can be consistently estimated from finite
samples
1
Rˆ =
K
Copyright by Science in China Press 2004
K
∑ X (t ) X H (t )
t =1
(10)
Robust DOA estimation & array calibration in presence of mutual coupling for uniform linear array
351
and its corresponding eigendecomposition is defined in a similar fashion as (7)
l=E
lS l
lS + E
lN l
lN.
R
ΛS E
ΛN E
2
(11)
Algorithm description
Since the mutual coupling matrix Z is a banded symmetric Toeplitz matrix, the
steering vector W(θ ) taking into account the effect of mutual coupling can be reformulated as follows:
W(θ ) = Za(θ ) = T[a(θ )]z,
(12)
where the N × p matrix T[a(θ)] is defined by (13)—(15)
T[a(θ)] = T1[a(θ)] + T2[a(θ)],
⎧[a (θ )]i + j −1 for i + j ≤ N + 1,
[T1 ]i, j = ⎪⎨
⎪⎩
0
otherwise,
⎧[a (θ )]i − j +1 for i ≥ j ≥ 2,
[T2 ]i, j = ⎪⎨
⎪⎩
0
otherwise.
(13)
(14)
(15)
The underlying basis for subspace-based DOA estimation algorithms is the orthogonality between the noise subspace and signal subspace of array covariance matrix
R[10], which means that
WH(θi)EN E NH W(θi) = 0, i = 1, 2, …, M,
(16)
aH(θi)ZHEN E NH Za(θi) = 0, i=1, 2, …, M.
(17)
A natural criterion to estimate θ and Z is to find the minimizer of the following cost
function:
l ] = arg min a H (θ ) Z H E
lN E
l HN Za (θ ).
[θ , Z
θ,Z
(18)
Observe that (18) is multidimensional nonlinear minimization problems. A direct minimization will involve a search in a M + 2p − 2 parameter space. This can be computationally prohibitive. However, inserting (12) into (16) yields (19)—(21):
z H T H [a (θ i )]E N E NH T [a (θ i )]z = 0, i = 1, 2, " , M ,
(19)
z H C (θ i )z = 0, i = 1, 2," , M ,
(20)
C (θ ) = T H [a (θ )]E N E NH T [a (θ )],
(21)
where C(θ )is a p × p Hermitian matrix. Since z (θ ) ≠ 0, (20) means that the matrix C(θ )
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Science in China Ser. F Information Sciences 2004 Vol.47 No.3 348—361
is singular, i.e., rank[C(θ )] < p. Note that under the condition of p ≤ N − M and that
the array manifold {W (θ ) : − π 2 ≤ θ ≤ π 2} is unambiguous, the matrix C(θ ) is singular or rank reduction if and only if the θ = θ i, i = 1, 2, …, M, since the dimension of
signal subspace of R is M. Based on this idea, we develop a DOA estimator as (22) or
(23) and a mutual coupling calibration algorithm as (24) or (25):
θ = arg max
1
,
l (θ ) ⎤
⎡
λmin ⎣C
⎦
(22)
θ = arg max
1
,
l (θ ) ⎤
det ⎡C
⎣
⎦
(23)
θ
or
θ
l (θ )] with e (1) = 1,
z = emin [C
min
(24)
or
1
z =
M
M
∑ emin [Cl (θ i )]
with emin (1) = 1,
(25)
i =1
where
l (θ ) = T H [a (θ )]E
lN E
l HN T [a (θ )],
C
(26)
l N denotes the finite sample estimate of noise subspace E
l N . λ [C
l (θ )] is the
E
min
l (θ ) , e [C
l (θ )] is the eigenvector corresponding to
smallest eigenvalue of the matrix C
min
l (θ ) and det[C
l (θ )] is the determinant of the
the smallest eigenvalue of the matrix C
l (θ ) .
matrix C
From (22)—(26), it is observed that the DOA estimation and array calibration can
be completed just using a one-dimensional search over FOV (field of view) of array. It is
more computationally efficient than refs. [3, 4]. Besides, due to the Vandermonde structure of the ideal steering vector a(θ ) of the ULA, a polynomial rooting with degree of
2(N − 1) × p can also be utilized in light of the idea underlying ROOT-MUSIC algorithm[11]. It is also interesting that the new algorithm proposed here will reduce to the
MUSIC with p = 1, since in this case z = [1]T, T [a(θ )] = a(θ ) and the matrix C(θ ) =
a (θ ) E N E NH a (θ ) becomes a scalar.
With the above preparations, the proposed algorithm can be summarized as follows:
l using (10).
Step 1. Compute the sample estimate of the array covariance matrix R
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Robust DOA estimation & array calibration in presence of mutual coupling for uniform linear array
353
l and find E
lN.
Step 2. Compute the eigendecomposition of R
Step 3. Find the DOA estimate θ by searching for M highest peaks of the spatial
spectrum defined by
F (θ ) =
1
,
l (θ )]
[C
(27)
1
.
l
det[C (θ )]
(28)
λmin
or
F (θ ) =
Step 4. Compute mutual coupling estimate z with (24)—(26) using θ obtained
in step 3.
3
3.1
Performance analysis
Proof of statistical consistence
l→R
As the number of samples K → ∞, the sample array covariance matrix R
l N → E . Moreover, the estimate of θ tends to the minimizer of the
and, therefore, E
N
following cost function:
θˆ = arg min λmin [C (θ )],
(29)
C (θ ) = T H [a (θ )]E N E NH T [a (θ )].
(30)
θ
It is obvious that (29) is minimized to zero if and only if the matrix C (θ ) is singular,
i.e. rank[C(θ )] < p. Hence, in order to establish the consistence of the DOA estimate it is
sufficient to prove the following claims:
1) λmin (C (θ i )) = 0, i = 1, 2, " , M where the θ i denotes any one of the real
source direction,
2) λmin (C (θ )) > 0, θ ≠ θ i .
The proof of claim 1 is obtained directly from (20). The second claim is true due to
the assumption that any N distinct steering vectors Za(θ i) i = 1, 2, …, N, θ 1 ≠ θ 2
≠ … ≠ θ N are linearly independent. In fact, if there is a direction θ M+1 other than θ i
that makes C(θ ) singular, i.e.
z H T H [a (θ M +1 )]E N E NH T [a (θ M +1 )]z = 0,
(31)
where z is a p × 1 vector with its first element equal to 1. It follows that the vector
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T [a (θ M +1 )]z = Za (θ M +1 ) must lie in the signal subspace span(ES), or equivalently
L[ Za (θ1 ), Za (θ 2 ), " , Za (θ M )] = Za (θ M +1 ),
(32)
where L [•] denotes a certain linear operator. Clearly, (32) means that there exist M + 1
≤ N steering vectors which are linearly dependent. Therefore, the statistical consistence
of the new DOA estimator is proved by contradiction.
Next, with the consistent estimate of the source DOAs, let us consider the statistical
consistence of the mutual coupling estimation. As the number of samples K goes to infinity, the estimate for vector z reduces to:
z = emin [C (θ i )], with emin (1) = 1,
(33)
C (θ i ) = T H [a (θ i )]E N E NH T [a (θ i )].
(34)
To prove the consistence of the mutual coupling estimate, it is sufficient to prove the
following claims:
1) z H C (θ i ) z = 0,
2) z H C (θ i ) z > 0 z ≠ z where z is an arbitrary P × 1 vector, except for the real
mutual coupling vector z, and the first element of z is equal to 1.
Similarly, the first claim is readily verifiable from (20) and the second one holds
also due to the assumption that the array manifold including mutual coupling is free of
rank N − 1 ambiguity. Indeed, suppose that there exists a vector z , other than z, so that
C(θ i) is singular, i.e.
z H T H [a (θ i )]E N E NH T [a (θ i )]z = 0.
(35)
(θ ) must lie in the signal subspace span(ES),
It follows that the vector T [a (θ i )]z = Za
i
i.e.
(θ ).
L[ Za (θ1 ), Za (θ 2 ), " , Za (θ M )] = Za
i
(36)
(36) implies that there exist M + 1 ≤ N steering vectors which are linearly dependent.
The proof for statistical consistence of the mutual coupling estimate is then obtained by
contradiction.
Thus we have proved that the new estimator for DOA and mutual coupling is statistically consistent.
3.2
Manifold ambiguity analysis and the choice of p
The above proof of consistence is based on the assumption that the array manifold
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Robust DOA estimation & array calibration in presence of mutual coupling for uniform linear array
355
including mutual coupling effect is free of rank N − 1 ambiguity. The array manifold
completely characterizes an array’s spatial response and is defined as the locus of all the
steering vectors of the array over the feasible set of source DOAs. The array manifold is
a key component in the modeling of the antenna array system and its properties are entirely dependent on the array geometry and the electrical characteristics of the sensors.
The ambiguous parameter estimation is a direct result of the manifold ambiguity, independent of the signal processing algorithm adopted and it can be eliminated only through
judicious sensor arrangement including the array geometry and electromagnetic response
of every sensor. It is well known that an N sensor ULA with inter-sensor spacing less
than λ/2 is free of up to rank N − 1 ambiguity. But when taking into account the effect of
mutual coupling, the above conclusion may not hold. A rigorous analysis of the ambiguity of array manifold taking into account the effect of mutual coupling is very difficult
and intractable. Fortunately, a lot of simulations have led to a useful rule of thumb as
follows:
Suppose that there exist q groups of sources that are in mirrored pairs with respect
to the array normal, in order to avoid the ambiguous parameter estimation. The choice of
p should satisfy
⎡N ⎤
p≤⎢ ⎥ −q
⎢2⎥
(37)
where ⎡⎢•⎤⎥ denotes rounding toward infinity.
To illustrate the truth of the above rules, let us consider some examples. Consider a
ULA of 8 sensors with half wavelength inter-element spacing. Three narrowband uncorrelated sources with equal power impinge on the array at distinct directions −10°, 35°
and 40°, the SNR = 10 dB. Figs. 1—4 show the roots distributions of polynomial
Det[C(x)] = 0, when p = 2, 3, 4, 5, respectively. The phases of the roots on the unit circle
are the estimated wavenumbers of sources, defined in (3). It is observed that when p = 2,
3, 4, three zeros of degree 2 corresponding to the true source wavenumbers appear on
the unit circle. But when p = 5, there appear some spurious zeros ( β = [0 ± π 4 ± π 2 ±
3π 2 ± π]) on the unit circle, which correspond to the spurious DOA estimates [−90°,
−48.5904°, −30°, −14.4775°, 0°, 14.4775°, 30°, 48.5904°, 90°]. The ambiguous DOA
estimates here are the direct reflection of the ambiguity of the array manifold. Note that,
in this case, there exists a certain direction θ so that zT [a (θ )] = 0 and z (1) = 1.
Therefore, the vector zT [a (θ )] lies in the signal subspace as its zero elements. For
example, zT [ a (48.5904°)] = 0 and z = [1 0 0 1]T . Next, when we vary the source
DOAs to [−10°, 10°, 40°], the ambiguous estimate of DOA ([−90°, 0°, 90°]) appears
with p = 4. The reason for the ambiguous estimate here is that there exists a certain di-
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Science in China Ser. F Information Sciences 2004 Vol.47 No.3 348—361
rection θ so that L( ZA(θ )) = T ⎢⎣a (θ ) ⎥⎦ z, which means that there exists M + 1
steering vectors that are linearly dependent. For example:
Fig. 1. The roots distribution of the polynomial
Det[C(x)]. P = 2 and source DOAs: −10°, 35°, 40°.
Fig. 3. The roots distribution of the polynomial
Det[C(x)]. P = 4 and source DOAs: −10°, 35°, 40°.
Fig. 2. The roots distribution of the polynomial
Det[C(x)]. P = 3 and source DOAs: −10°, 35°, 40°.
Fig. 4. The roots distribution of the polynomial
Det[C(x)]. P = 5 and source DOAs: −10°, 35°, 40°.
[T[a(−10°)]z T[a(10°)]z T[a(40°)]z]ζ = T[a(0°)] z ,
(38)
z = [1 −0.6459−0.1652i −0.4272−0.1227i 0.0897−0.2824i]T,
(39)
ξ = [ −0.0701 − 0.5194i 0.3800 + 0.3609i 0] ,
(40)
T
z = [1 − 0.2745 + 0.2091i − 0.0552 + 0.0462i 0.1415 − 0.2115i ] .
T
(41)
A rigorous proof for rule (37) is still an open subject of research.
In practice, the choice of p is a compromise of the mutual coupling approximation
and the unambiguous parameter estimation, which implies
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Robust DOA estimation & array calibration in presence of mutual coupling for uniform linear array
⎡
⎤
⎡N ⎤
p = min ⎢ N − M , ⎢ ⎥ − q ⎥ .
2
⎢ ⎥
⎣
⎦
3.3
357
(42)
Cramer-Rao bounds
The CRB provides a lower bound on the covariance matrix of any unbiased estimator. In this section we develop the closed-form expressions for the CRB associated
with the joint estimation of DOAs and mutual coupling matrix. The statistical efficiency
of the new estimator is shown numerically in section 4 by Monte-Carlo simulations.
Assume that the signal covariance matrix RS is known and the noise variance σ 2 is
normalized to 1. We observe that array covariance matrix R can be perfectly described
by M + 2P −2 unknown parameters. These parameters are the M sources DOAs and 2p
− 2 parameters that define the p×1 vector z. We collect all these parameters to form the
parameter vector ρ
ρT = ⎡⎣θ1 , θ 2 , " , θ M , Re( zT (2 : p )), Im( zT (2 : p )) ⎤⎦ ;
(43)
the CRB associated with the joint estimation of DOAs and mutual coupling matrix can
be expressed as follows:
E{( ρ − ρ0 )( ρ − ρ0 )T } ≥ CRB,
(44)
CRB = F −1,
(45)
where the matrix F denotes the Fisher information matrix and it can be partitioned into
blocks (submatrices) associated with the different parts of the unknown parameter vector
ρ:
⎡ Fθθ
F = ⎢⎢ FRθ
⎢⎣ FIθ
Fθ R
FRR
FIR
Fθ I ⎤
FRI ⎥⎥ .
FII ⎥⎦
(46)
Here Fθθ, FRR and FII are the blocks associated with DOAs, real part and imaginary part
of mutual coupling vector z respectively. The remaining blocks are the corresponding
cross terms. The m, nth element of F is given by[3]
⎧⎪ ∂ 2 L ⎫⎪
⎛ −1 ∂R −1 ∂R ⎞
Fmn = − E ⎨
R
⎬ = K ⋅ trace ⎜ R
⎟,
∂ρm
∂ρn ⎠
⎝
⎩⎪ ∂ρm ∂ρn ⎭⎪
{
{
Fmn = 2 K ⋅ Re trace Dm RS AH Z H R −1 ZARS DnH R −1
{
}
(47)
}}
+ trace Dm RS AH Z H R −1 Dn RS AH Z H R −1 ,
(48)
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Science in China Ser. F Information Sciences 2004 Vol.47 No.3 348—361
Dm =
∂ZA
,
∂ρm
(49)
∂ZA
= ( diag [i,ones( N − i,1)] + diag [ −i, ones( N − i,1) ]) A,
∂Ri
(50)
∂ZA
= j ( diag [i,ones( N − i,1)] + diag [ −i, ones( N − i,1)]) A,
∂I i
(51)
where Ri , i = 1, 2, " , p − 1 denotes the real part of z(i + 1) and I i , i = 1, 2," , p − 1
denotes the imaginary part of z(i + 1).
In the case of known mutual coupling, the CRB reduces to
CRBθ = [ Fθθ ]−1.
4
(52)
Simulations
We present several numerical examples to illustrate the performance of the proposed algorithm and compare it with the MUSIC with known mutual coupling and MUSIC without mutual coupling calibration. In the following examples, we consider a uniformly spaced linear array of 8 sensors with half wavelength inter-element spacing. It
was assumed that each array element was coupled with only its two nearest neighbors,
i.e. p = 3, and mutual coupling vector z =[1, 0.5791 + 0.3303i, 0.3566 + 0.2653i].
Example 1. Three narrowband uncorrelated sources with equal power impinge on
the array, from the far field, at distinct directions −10°, 35° and 40° w.r.t. the broadside
of array. The SNR = 20 dB and 500 snapshots are used to estimate the array covariance
matrices. Fig. 6 shows the spatial spectra obtained form MUSIC with known MCM,
MUSIC without mutual coupling calibration, the algorithms (27) and (28).
Fig. 5. The roots distribution of the polynomial
Det[C(x)]. P = 4 and source DOAs: −10°, 10°, 40°.
Copyright by Science in China Press 2004
Fig. 6. The spatial spectra in example 1. 1, MUSIC
without MCM calibration; 2, method of ref. [27]; 3,
MUSIC with known MCM; 4, method of ref. [28].
Robust DOA estimation & array calibration in presence of mutual coupling for uniform linear array
Example 2. Two narrowband uncorrelated sources at distinct directions
35° and 40°, the number of snapshot K =
500. When SNR varies from −10 to 50,
200 times Monte-Carlo simulations are
performed to demonstrate the statistical
efficiency of the proposed algorithm. In
this simulation, due to the effect of mutual
coupling, the MUSIC without mutual
coupling calibration is disabled at all. Figs.
7—9 show resolution probability, estimate
bias and estimate variance comparison
between the proposed algorithm (27) and
the MUSIC with known mutual coupling,
Fig. 8. Estimate bias versus the SNR in example 2. ◇,
Method of (27); △, MUSIC with known MCM.
359
Fig. 7. Resolution probability versus the SNR in example 2. ◇, Method of (27); △, MUSIC with known
MCM.
Fig. 9. Estimation variance versus the SNR in example 2. ○, Method of (27); *, MUSIC with known
MCM.
respectively. In table 1, we also demonstrate the mean value and variance of the estimate
of the z(2). The corresponding CRB are also given.
Table 1
The estimate of z(2) in example 2 (real value:0.5791 + 0.3303i)
SNR(dB)
Mean value
(real part)
Variance
(real part)
CRB
(real part)
Mean value
(imag part)
Variance
(imag part)
CRB
(imag part)
10
12
14
16
20
30
50
0.5776
0.5779
0.5783
0.5787
0.5790
0.5792
0.5791
3.852e-005
2.7415e-005
1.8928e-005
1.2179e-005
4.8131e-006
4.692e-007
5.566e-009
3.6868e-005
2.3027e-005
1.4454e-005
9.1082e-006
3.6480e-006
3.7558e-007
3.7932e-009
0.3351
0.3340
0.3328
0.3318
0.3307
0.3303
0.3303
9.5204e-005
7.0258e-005
3.6876e-005
3.5023e-005
1.1841e-005
9.2288e-007
1.2742e-008
3.5060e-005
2.3402e-005
1.5707e-005
1.0642e-005
5.0433e-006
7.3060e-007
8.0347e-009
Example 3.
Two narrowband uncorrelated sources at distinct directions 35° and
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Science in China Ser. F Information Sciences 2004 Vol.47 No.3 348—361
40°, SNR = 20 dB. When the number of snapshots K varies from 100 to 3000, 200 times
Monte-Carlo simulations are performed to demonstrate the statistical efficiency of the
proposed algorithm. In this simulation, the MUSIC without mutual coupling calibration
is disabled at all. Figs. 10 and 11 show the estimation bias and the estimation variance
comparisons between the proposed algorithm (27) and the MUSIC with known mutual
coupling, respectively. In table 2, we also demonstrate the mean value and variance of
the estimate of the z(2). The corresponding CRB are also given.
Fig. 10. Estimation bias versus the number of snap- Fig. 11. Estimation covariance versus the number of
shots in example 3. ◇, Method of (27); △, MUSIC snapshots in example 3. ○, Method of (27); *, MUSIC
with known MCM.
with known MCM.
Table 2
The estimate of z(2) in example 3 (real value:0.5791 + 0.3303i)
K
Mean value
(real part)
Variance (real
part) (10e-4)
100
500
1100
2100
2900
0.5784
0.5789
0.5793
0.5792
0.5792
0.1864
0.0430
0.0178
0.0129
0.0090
CRB
(real part)
(10e-4)
0.1824
0.0365
0.0166
0.0087
0.0063
Mean value
(imag part)
Variance (imag
part) (10e-4)
0.3341
0.3307
0.3304
0.3305
0.3304
0.4788
0.1040
0.0466
0.0267
0.0172
CRB
(imag part)
(10e-4)
0.2522
0.0504
0.0229
0.0120
0.0087
From the simulation results we can observe that the MUSIC without mutual
coupling calibration has failed at almost all the tests, but the proposed algorithm (27) or
(28) decouples the DOA estimation from the mutual coupling estimation and possesses
almost the same estimation performance as the MUSIC with known mutual coupling for
relatively high SNR and large K, in spite of the small difference in the performance in
the case of low SNR and little K. In addition to a favorable estimate for source DOAs, an
accurate estimate for mutual coupling can be obtained simultaneously.
5
Conclusion
In this paper, a robust subspace-based DOA estimation algorithm in the presence of
mutual coupling is proposed for ULA. Based on the banded symmetric Toeplitz matrix
Copyright by Science in China Press 2004
Robust DOA estimation & array calibration in presence of mutual coupling for uniform linear array
361
model for the mutual coupling of ULA, the new algorithm provides a favorable DOA
estimates and exploits no knowledge of the sensor mutual couplings. Moreover, an accurate estimate of mutual coupling matrix can also be achieved simultaneously for array
calibration. The new algorithm integrates the robust DOA estimation with the
auto-calibration for array mutual coupling, and therefore is of practical significance.
Acknowledgements This work is supported by the National Natural Science Foundation of China (Grant No.
60272370) and the Teaching and Research Award Program for Outstanding Young Teachers in high education institutes of Ministry of Education, China (TRAPOYT).
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