Hindawi Publishing Corporation International Journal of Antennas and Propagation Volume 2016, Article ID 8109013, 7 pages http://dx.doi.org/10.1155/2016/8109013
Research Article Blind Direction-of-Arrival Estimation with Uniform Circular Array in Presence of Mutual Coupling Song Liu, Lisheng Yang, Shizhong Yang, Qingping Jiang, and Haowei Wu Key Laboratory of Aerocraft Tracking Telemetering & Command and Communication, Ministry of Education, Chongqing University, Chongqing 400030, China Correspondence should be addressed to Lisheng Yang;
[email protected] Received 3 November 2015; Revised 11 December 2015; Accepted 15 December 2015 Academic Editor: Shih Yuan Chen Copyright Β© 2016 Song Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. A blind direction-of-arrival (DOA) estimation algorithm based on the estimation of signal parameters via rotational invariance techniques (ESPRIT) is proposed for a uniform circular array (UCA) when strong electromagnetic mutual coupling is present. First, an updated UCA model with mutual coupling in a discrete Fourier transform (DFT) beam space is deduced, and the new manifold matrix is equal to the product of a centrosymmetric diagonal matrix and a Vandermonde-structure matrix. Then we carry out blind DOA estimation through a modified ESPRIT method, thus avoiding the need for spatial angular searching. In addition, two mutual coupling parameter estimation methods are presented after the DOAs have been estimated. Simulation results show that the new algorithm is reliable and effective especially for closely spaced signals.
1. Introduction Uniform circular array based DOA estimation methods are always attractive because the UCA has a special symmetric structure that provides almost the same resolution ability along the 360β azimuth angle domain. Except for conventional DOA estimation methods such as beam space searching or the Capon method [1], many other methods can be applied, such as the commonly used multiple signal classification (MUSIC) [2] or the well-known ESPRIT [3] in phase mode space. More effective methods such as UCA-RBMUSIC and UCA-ESPRIT [4] have been introduced, and the mapping error reducing method was developed [5]. However, the electromagnetic mutual coupling effect cannot be ignored in a real array. Generally, this effect will severely degrade the performance of the above methods [6]. A classic DOA estimation method based on an iterative search technique was proposed to estimate the DOA and the mutual coupling matrix (MCM) parameters jointly [7], but it has a high computation cost. The rank reduction (RARE) method introduced by [7] was further developed to obtain the DOA estimate for a UCA [8β10], but an angular search is still needed. Moreover, angular ambiguities are challenges for
RARE-based blind methods. For example, two closely spaced signals cannot be differentiated according to the UCA-RARE [8] spectrum because there is a spurious peak at the middle of the two true angles. In this paper, we design a new modified ESPRIT method for UCA to estimate the azimuth angle when mutual coupling is present. The proposed method is blind to mutual coupling and can completely avoid angular searching. Reliable DOA estimates can be obtained, especially for closely spaced signals. In addition, we will introduce two methods to estimate the MCM parameters once the DOA values are calculated.
2. UCA Model with Mutual Coupling Suppose π-element UCA has a radius π (Figure 1). All of the antenna elements are identical, and there are π· far-field narrow signals impinging from {ππ , π = 1, . . . , π·} which are the parameters to be estimated. The snapshot can be written as z (π) = CAs (π) + n (π) , π = 1, . . . , πΎ.
(1)
2
International Journal of Antennas and Propagation z
3. Algorithm 3.1. UCA Model in DFT Beam Space. First we introduce the DFT of the MCM for UCA. Lemma 1 (see [11β13]). If C is a circular matrix with its first column vector as c = [π0 , π1 , . . . , ππβ1 ]π and F is a Fourier matrix
O y
πj x
F = [w0 , w1 , . . . , wπβ1 ] , wπ
Figure 1: Uniform circular array.
πΎ is the total sampling number and A is the manifold matrix: A = [a (π1 ) , . . . , a (ππ·)] , (a (ππ )) π
(2)
= π (ππ ) exp (π
s (π) = [π 1 (π) , . . . , π π· (π)] .
[ π1
π2
where π = 0, . . . , π β 1, then Ξ = FCFπ» is a diagonal matrix with entries as Cβs eigenvalues: πβ1
...
. . . ππβ1 . . . ...
.. .
...
...
π0
...
...
.. .
...
. . . ππβ1 . . .
π=0
F = Fπ ,
If C is also symmetric, this means that C = Cπ ππ ππ = ππβπ ,
(4)
π
0 = wπ Cwπ
β
(11)
π =ΜΈ π
and we have π»
ππβπ = (wπβπ ) C (wπβπ ) = wπ π Cwπ β = ππ ,
(12)
where π = 1, . . . , π β 1. From (8) and (12), we get a linear equation which cσΈ and πσΈ should satisfy for even π:
πΈ [s (π)] = 0,
πΈ [n (π) n (π)π»] = ππ€2 I.
(10)
π
ππ = wπ π Cwπ β = (wπ π Cwπ β ) = wπ π»Cwπ ,
ββ
β is the flooring function and π0 is normalized as 1. For ideal dipole antenna array (Figure 1), induced EMF (Electromotive Force) method can give a close form of self-impedance and mutual impedance and thus MCM can be calculated according to Gupta and Ksienskiβs formulation [6]. Suppose the signals are uncorrelated and n(π) is white Gaussian noise:
πΈ [n (π)] = 0,
π = 1, . . . , π β 1.
Then we can rewrite Ξ as
π0 ]
πΈ [s (π) s (π)π»] = Rπ ,
(8)
(9)
Fπ» = Fβ .
π1
] π2 ] ] .. ] ] . ] ] , π = βπβ . 2 ππβ1 ] ] ] ] .. ] . ]
2πππ ), π
in which π = 0, 1, . . . , π β 1. For the Fourier matrix F
(3)
C is the MCM, which is a symmetric circular Toeplitz matrix with the form . . . ππ
2ππ 2ππ (π β 1) π (7) 1 [1, exp (π ) , . . . , exp (π )] , βπ π π
ππ = (Ξ)(π+1)(π+1) = β ππ exp (π
where π = 1, . . . , π. π is the wavelength of the signals, and a(ππ ) is the manifold vector for the πth signal. π(ππ ) is the common directivity factor of the antenna elements. Since all the antenna elements are identical, π(ππ ) can be normalized as 1, and we use effective SNR which has already included the directivity factor. s(π) is the signal sample vector:
π0 π1 [ [ π1 π0 [ [. .. [. . [. [ C=[ [ππ ππβ1 [ [. .. [ .. . [
=
2ππ 2 (π β 1) π cos (ππ β )) , π π
π
(6)
(5)
ΞcσΈ = πσΈ , where
(13)
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3
π
cσΈ fl [π0 , . . . , ππ/2 ] ,
(14)
π
πσΈ fl [π0 , . . . , ππ/2 ] , 1 2 [ 2π [ β
1 β
1) 2 cos ( [1 [ π [ [ 2π [1 2 cos ( β
2 β
1) [ π [ Ξ fl [ . . .. [ .. [ [ [ 2π π β 2 [1 2 cos ( β
β
1) [ π 2 [ [ 2π π 1 2 cos ( β
β
1) [ π 2
β
β
β
β
β
β
β
β
β
β
β
β
β
β
β
β
β
β
Therefore, if we determine the estimate of πσΈ , then we can obtain the estimate of the mutual coupling parameters cσΈ by (13). In addition, there are similar equations where π is odd. According to [4], we set π½=
2ππ , π
2 2π πβ2 2 cos ( β
1β
) π 2 2π πβ2 2 cos ( β
2β
) π 2 .. .
] ] β1] ] ] ] 1] ] ] . .. ] . ] ] ] ] 2π π β 2 π β 2 2 cos ( β
β
) β1] ] π 2 2 ] ] 2π π π β 2 2 cos ( β
β
) 1 ] π 2 2
π½π (β
) is π order first-kind Bessel function. According to its property π½βπ (π½) = (β1)π π½π (π½)
(16)
Ξπ½ = βπ diag ([πΌπΏ , πΌπΏβ1 , . . . , πΌ0 , . . . , πΌπΏβ1 , πΌπΏ ]) ,
FσΈ = [F1 , F2 ] ,
(17)
F1 = [wβπΏ , . . . , wπΏ ] ,
πΌπ = ππ π½π (π½) , π = 0, 1, . . . , πΏ.
exp (βππΏπ1 ) β
β
β
exp (βππΏππ·) ] [ ] .. .. Μ =[ A ]. [ . β
β
β
. ] [ ) β
β
β
exp (ππΏπ ) exp (ππΏπ 1 π· ] [
Then the snapshot in DFT beam space is π»
Μz (π) = FσΈ z (π) = FσΈ CFσΈ FσΈ As (π) + FσΈ n (π) ,
(19)
Μ ΞA Μ (π) Μz (π) = [ σΈ π» ] s (π) + n ΞπΆF2 A
π»
ΞπΆ 0 π» FσΈ CFσΈ = [ ], 0 ΞσΈ πΆ
(20)
ΞπΆ = diag ([ππΏ , . . . , π1 , π0 , π1 , . . . , ππΏ ]) .
π»
(21)
In addition, we can obtain [4] π»
Μ Ξπ½ A ] ] = [ F2 π»A F2 π»A
F1 π»A
(22)
with Ξπ½ = βπ diag ([πβπΏ π½βπΏ (π½) , . . . , ππΏ π½πΏ (π½)]) .
(23)
(27)
(28)
in which Ξ = Ξ πΆ Ξπ½ ,
where (see (11), (12), and (18))
(26)
Finally, we obtain the snapshot in DFT beam space
if we write FσΈ CFσΈ as
FσΈ A = [
(25)
Μ is the updated manifold matrix with a Vandermonde A structure
(18)
F2 = [wπΏ+1 , . . . , wπβπΏβ1 ] .
π»
(24)
where
where πΏ is the number of excited phase modes. We define another Fourier matrix FσΈ , which is different from (6):
π»
(15)
we get
πΏ = βπ½β ,
π»
1
Μ (π) ] = ππ€2 I. πΈ [Μ n (π) n
(29)
3.2. DOA Estimation and MCM Calculation. We carry out the eigendecomposition on the covariance matrix Rπ§Μ and Μ π , which consists of eigenvectors obtain the signal subspace U corresponding to π· maximum eigenvalues. Select the first Μ π 1 and the second πΏ + 1 rows as U Μ π 2 . Μ π as U πΏ + 1 rows of U Μ Μ Select the first πΏ + 1 rows of A as A1 and the second πΏ + 1 rows Μ 2 . Select the first (πΏ + 1) Γ (πΏ + 1) diagonal matrix of Ξ as as A Ξ1 and the second (πΏ + 1) Γ (πΏ + 1) diagonal matrix as Ξ2 . Use
4
International Journal of Antennas and Propagation
the same notations for other matrices Ξπ½1 , Ξπ½2 , ΞπΆ1 , and ΞπΆ2 . Then we have Μ π 1 = Ξ1 A Μ 1 T, U
(30)
Μ π 2 = Ξ2 A Μ 2 T, U
can be cleared by comparing the RARE spectrum [9] values Μ without on πΜπ and πΜπ +π. We mark the estimated vector as d opt π ambiguity. Finally, we can determine the DOA estimates through the Μ Following is the detailed procedure of the eigenvalues of Ξ¨. blind method for DOA estimation of {ππ }: Μ π§ and do (1) Calculate the sample covariance matrix R eigendecomposition. Get the estimated signal subΜπ . Μ π and noise subspace U space U
where T is a nonsingular matrix. Thus we get Μ π 1 Ξ¨ Μ =U Ξ12 U π 2
(31)
Μ Μ π . Select the first πΏ+ (2) Get the updated signal subspace U Μ Μ Μ π 1 and the second πΏ + 1 rows as U Μ π 2 : 1 rows as U
with Ξ12 = Ξ1 Ξ2 β1 = Ξπ½1 ΞπΆ1 Ξπ½2 β1 ΞπΆ2 β1 = diag (d) , π πΌ ππΌ ππΌ π π πΌ d = [ πΏ πΏ , πΏβ1 πΏβ1 , . . . , 1 1 , 0 0 ] , ππΏβ1 πΌπΏβ1 ππΏβ2 πΌπΏβ2 π0 πΌ0 π1 πΌ1
Μ Μ π = FσΈ π»U Μπ . U (32)
Μ and fitting matrix Ξ¨ Μ according (3) Estimate the vector d Μ and to (34)β(36). Carry out eigendecomposition on Ξ¨ Μ get the DOA estimates {ππ }.
Ξ¨ = Tβ1 diag ([exp (ππ1 ) , . . . , exp (πππ·)]) T. Μ Μπ Μ π with U Since we use sampled data, we should replace U and define the object function as σ΅©σ΅© σ΅© Μ Μ σ΅©σ΅©diag (d) U Μ π 2 β U Μ π 1 Ξ¨σ΅©σ΅©σ΅© σ΅©σ΅© σ΅©σ΅©
Μ Ξ¨} Μ = arg min {d, d,Ξ¨
s.t.
(33)
Μ d Μ d πΏ πΏ+1 = 1.
We use the solution to the above equations from [14, 15]. Consider Μ = arg min dπ»Qd d d
(34) s.t.
Μ d Μ d πΏ πΏ+1 = 1 β1
π» π» Μ Μ Μ ) U Μ Μ U Μ π 1 U Μ Μ Μ π 1 diag (d) Μ = (U Ξ¨ π 1 π 2
(38)
(35)
(4) Compare the blind RARE spectral values [9] on πΜπ and πΜπ + π and clear the ambiguity. Output the final DOA estimates. We label the above mentioned method as βBlind-m1-halfβ Μ Μ π . In addition, we can because we only select πΏ + 1 rows of U Μ Μ Μ π as U Μ π 1 and select the first 2πΏ and the second 2πΏ rows from U Μ Μ . This method is labeled βBlind-m1-full.β Furthermore, we U π 2
Μ Μ Μ π 1 Μ π as U can select the first πΏ and the third πΏ rows from U Μ Μ π 2 or select the first 2πΏ β 1 and the third 2πΏ β 1 rows and U Μ and U Μ . We label the two methods as βBlindΜ as U Μ Μ Μ from U π
π 1
π 2
m2-halfβ and βBlind-m2-full,β respectively. We will carry out simulations on these four methods in Section 4. Now we can estimate the MCM parameters once we get Μ π. Method 1. If ππ = 0, π > πΏ, then, according to (32) and (13), we get
with π
π» Μ Μ Μ π 2 U Μ π 2 ) , Q = Pβ₯ΜΜ β (U Uπ 1
(36)
π» π» Μ Μ Μ Μ Μ π 1 (U Μ π 1 U Μ π 1 )β1 U Μ π 1 and β is the Hadamard where Pβ₯ΜΜ = IβU Uπ 1
product. If πmin is the eigenvector of Q corresponding to its minimum eigenvalue, then we have Μ = πmin , d π
(37)
2
π = (πmin )πΏ (πmin )πΏ+1 . Equation (37) indicates that there is a phase π ambiguity for π, and thus it will introduce a π ambiguity to the DOA Μ However, this ambiguity values through the eigenvalues of Ξ¨.
diag ([
Μ ΜπΏ π π Μ ), , . . . , 0 ]) = Ξπ½2 Ξπ½1 β1 diag (d opt Μ πΏβ1 Μ1 π π
(39)
β1
σΈ
Μc =
Μ π ΜσΈ ) (Ξ β1
Μ π ΜσΈ ) (Ξ
,
(40)
1
Μ = Μ σΈ = [Μ Μ1, . . . , π Μ πΏ ]π , and Ξ π0 , π where ΜcσΈ = [1, Μπ1 , . . . , ΜππΏ ]π , π Μ Ξ(1 : πΏ + 1, 1 : πΏ + 1), which means that Ξ consists of the first Μ 0 is a πΏ + 1 rows and πΏ + 1 columns of Ξ (see (15)). In (40), π nonzero parameter, but it will be cleared by the normalization operation. Method 1 can only estimate πΏ mutual coupling parameters.
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5
Method 2. In addition, we can obtain a more accurate estimate of c by the method introduced in [7]. Consider σΈ π»
ΜcσΈ = arg min c Ξ©cσΈ
Table 1: Mutual coupling parameters. π0 1
π1 β0.3204 + 0.1878π
π2 0.1496 + 0.1153π
ππ , π = 3, . . . , 8 0
c
(ΜcσΈ )1 = π0 = 1
(41) Table 2: Mutual coupling estimates (SNR = 30 dB, πΎ = 1000).
Μ U Μ , Μπ U Μ π π»Ξ₯ (a (π)) Ξ© = Ξ₯π» (a (π)) where ΜcσΈ = [1, Μπ1 , . . . , Μππ]π . If πmin is the eigenvector of Ξ© corresponding to its minimum eigenvalue, then ΜcσΈ =
πmin . (πmin )1
(42)
Μ β πΆπΓ(π+1) than We give a simpler expression of Ξ₯(a(π)) that in [7, 9]. Consider
c1 β0.3204 + 0.1878π β0.3114 + 0.2053π β0.3203 + 0.1877π
True value Method 1 Method 2
Spec. value
s.t.
0.4 0.2 0
Μ , Ξ₯ (:, 1) = a (π)
50
100
150 200 Azimuth (β )
c2 0.1496 + 0.1153π 0.1451 + 0.1179π 0.1496 + 0.1154π
250
300
350
150 200 Azimuth (β )
250
300
350
150
250
300
350
250
300
350
MUSIC without MCM
π = 2, 3, . . . , π, Ξ₯ (:, π + 1)
Spec. value
Μ , π β 1) + circshift (a (π) Μ , 1 β π) , Ξ₯ (:, π) = circshift (a (π)
(43)
Μ , π) , for even π {circshift (a (π) ={ Μ , π) + circshift (a (π) Μ , βπ) , for odd π, circshift (a (π) {
Γ104 5 0
50
100
Spec. value
RARE [9]
Μ π) is a function that shifts a(π) Μ by π times where circshift(a(π), Μ in the counter direction. circularly. If π < 0, then it shifts a(π)
Γ104 2 1 0
50
100
200
Azimuth (β )
4. Simulations
UCA-RARE [8] Spec. value
Consider a 16-element half-wave dipole antenna UCA with π0 = 1.032 GHz and π = 0.7 π. Set πΏ = 4. The mutual coupling vector c is listed in Table 1. The theoretical parameters are calculated according to Gupta and Ksienskiβs formulation [6], and small values are treated as ππ = 0, π = 3, . . . , 8. Real parameters should be estimated by array calibration method. Two signals are impinging from 35β and 45β with the same signal noise ratio (SNR). We apply the classic MUSIC, blind RARE [9], blind R-RARE [10], and UCA-RARE [8] methods to obtain azimuth estimates. The results are shown in Figure 2. It shows that the MUSIC method and RARE-based blind methods cannot differentiate these two signals because RARE-based methods will introduce spurious estimates. We should obtain the initial estimates from the MUSIC spectrum to start the iterative method [7], but it is difficult to find two different spectral peaks from the spectrum of βMUSIC without MCM.β We apply the proposed blind method to the above example. The estimated average DOA absolute bias and root mean square error (RMSE) versus SNR are illustrated in Figures 3β6 (πΎ = 10000, 100 trials). Method in [7] (π = 0.01) and the CramΒ΄er-Rao bound (CRB) with a known MCM are also presented [16].
0.02 0.01 0
50
100
150
200
Azimuth (β )
R-RARE [10]
Figure 2: MUSIC and RARE spectrum (SNR = 20 dB, πΎ = 1000, one trial).
This shows that all the four proposed methods can give satisfactory estimates and that the tendency of the RMSE is the same as the CRB with an increase of SNR. Method Blind-m1-full and method Blind-m2-full are more effective Μ Μ π are than the other two methods because more rows of U Μ (see (35)). A involved when we calculate the fitting matrix Ξ¨ comparison of simulations versus sampling number will give similar results. Method in [7] gives a biased estimate. Table 2 lists the MCM parameter estimates based on the two methods introduced in Section 3. This shows that Method 2 can give more accurate results.
International Journal of Antennas and Propagation 103
101
102
100
101
RMSE azimuth sig.1 ( β )
Average abs (bias) sig.1 (β )
6
100 10
β1
10β2
10β2 10β3 10β4
10β3 10β4
10β1
5
10
15
20
25
10β5
30
5
10
SNR (dB) Blind-m1-half Blind-m1-full Blind-m2-half
15 20 SNR (dB)
Blind-m1-half Blind-m1-full Blind-m2-half
Blind-m2-full Iter. method [7]
25
30
Blind-m2-full Iter. method [7] CRB
Figure 5: RMSE versus SNR for signal 1.
Figure 3: Average DOA absolute bias versus SNR for signal 1. 101 100 RMSE azimuth sig.2 ( β )
Average abs (bias) sig.2 (β )
103 102 101 10
0
10β1
10β2 10β3 10β4
10β2
10β5
10β3 10β4
10β1
5
10
15
20
25
30
SNR (dB) 5
10
Blind-m1-half Blind-m1-full Blind-m2-half
15 20 SNR (dB)
25
30
Blind-m2-full Iter. method [7]
Blind-m1-half Blind-m1-full Blind-m2-half
Blind-m2-full Iter. method [7] CRB
Figure 6: RMSE versus SNR for signal 2.
Figure 4: Average DOA absolute bias versus SNR for signal 2.
Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper.
5. Conclusion In DFT beam space, we utilize a modified ESPRIT algorithm to obtain a reliable DOA estimate when there is a severe mutual coupling effect. The new blind method is efficient because it avoids searching for the spectral peaks. For closely spaced signals, neither the classic MUSIC nor RARE-based methods provide a good estimate, while the proposed new method can produce an accurate estimate. Moreover, these direction estimates can be used for further MCM parameter estimation.
Acknowledgments The work is supported by the Chongqing Academician Fund under Grant cstc2014yykfys90001, by the Project of Chinese Academy of Engineering under Grant 2014-XX-05, by the National Natural Science Foundation of China under Grant 61501068, by the Project of Equipment Pre-Research of the Ministry of Education of China under Grant 62501040217, and by the Fundamental Research Funds for the Central Universities under Grant 106112013CDJZR165502.
International Journal of Antennas and Propagation
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Volume 2014
Volume 2014
International Journal of
International Journal of
International Journal of
Modelling & Simulation in Engineering
Volume 2014
Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Shock and Vibration Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Advances in
Acoustics and Vibration Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014