2-D DOA estimation in case of unknown mutual ...

Multidim Syst Sign Process DOI 10.1007/s11045-014-0290-7

2-D DOA estimation in case of unknown mutual coupling for multipath signals Tansu Filik · T. Engin Tuncer

Received: 20 January 2014 / Revised: 14 July 2014 / Accepted: 17 July 2014 © Springer Science+Business Media New York 2014

Abstract In this paper, two dimensional (2-D) direction-of-arrival (DOA) estimation problem in case of unknown mutual coupling and multipath signals is investigated for antenna arrays. A new technique is proposed which uses a special array structure consisting of parallel uniform linear array (PULA). PULA structure is complemented with auxiliary antennas in order to have a structured mutual coupling matrix (MCM). MCM has a symmetric banded Toeplitz structure which allows the application of the ESPRIT algorithm for 2-D paired DOA estimation. The advantage of the PULA structure is exploited by dividing it into overlapping linear sub-arrays (triplets) and spatial smoothing is employed to mitigate multipath signals. Closed form expressions are presented for search-free, paired and unambiguous 2-D DOA estimation. Two algorithms PULA-1 and PULA-2 are proposed to effectively solve the problem. Several simulations are done and the accuracy of the proposed solution is shown. Keywords Antenna array · 2-D direction-of-arrival estimation · Mutual coupling · Multipath · Parallel uniform linear array

1 Introduction In 2-D DOA estimation, search-free fast algorithms are usually desired for real-time problems (Gershman et al. 2010; Tuncer and Friedlander 2009). The main challenge for fast 2-D DOA estimation is the pairing problem (Kikuchi et al. 2006) where multiple source azimuth and elevation angles should be paired accordingly. The proposed method in Filik and Tuncer (2010) use array interpolation techniques and virtual arrays for uniform circular array (UCA), which allows efficient antenna usage and possibility to compensate the T. Filik (B) Anadolu University, Eskisehir, Turkey e-mail: [email protected] T. E. Tuncer Middle East Tecnical University, Ankara, Turkey e-mail: [email protected]

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unknown mutual coupling between antennas for limited number of non-coherent signals. In case of multipath sources, the performances of these methods are not satisfactory due to array interpolation errors and rank deficiency of the covariance matrix. In Goossens and Rogier (2007), a hybrid algorithm is proposed which combines the rank reduction and root-music for 2-D DOA estimation using UCA in the presence of mutual coupling. In Wang and Hui (2010), unambiguous and paired 2-D DOA estimation problem is handled using UCA in the presence of the elevation-dependent mutual coupling. In Liu et al. (2013) a computationally efficient 2-D DOA estimation algorithm which uses sparse uniform rectangular array (URA) for multipath signals is proposed. In Filik and Tuncer (2009), a closed form solution is given for arbitrary arrays for the search-free 2-D DOA estimation. While the method in Filik and Tuncer (2009) is general, the handling of the mutual coupling and multipath signals simultaneously requires some special array geometry. In this paper, a real antenna array structure (parallel uniform linear array—PULA) and its sub-arrays are used for 2-D DOA estimation. This array structure is previously used for the fast 2-D DOA estimation (Xia et al. 2007; Wu et al. 2003; Filik and Tuncer 2011). In Xia et al. (2007), a polynomial root-finding method is presented. In Wu et al. (2003), a propagator method is presented with additional computational complexity for the solution of pairing problem. In Filik and Tuncer (2011), the 2-D DOA estimation performances of real and virtual PULA which is generated by array interpolation are compared without considering the mutual coupling among the antennas and multipath effects. To our knowledge, there is no previous work where the closed form unambiguous paired 2-D DOA estimates are given when mutual coupling between antennas and multipath signals jointly exist. In this paper, the problem is solved in an effective manner using the overlapping linear sub-arrays of the PULA. Accordingly, triple sub uniform linear arrays (ULAs) of the PULA, which is called as triplet, are used in order to apply ESPRIT algorithm (Kailath and Roy 1990) for search-free and paired 2-D DOA estimation. PULA has a symmetric banded Toeplitz structure for its MCM leading to DOA estimation in case of unknown mutual coupling. It is shown in simulations, the proposed method can effectively handle the coherent (multipath) signals in case of unknown mutual coupling among the antenna array elements.

2 Problem formulation There are D narrow-band plane waves impinging on a planar array composed of P = 2(M + K ) + 1 antennas (sensors) located at the positions (xi ,yi ), i = 1, . . . , P as shown in Fig. 1. M is the number of sensors in each ULA. There are eight auxiliary sensors in order to generate approximately the same Toeplitz coupling matrix structure for each ULA. These auxiliary sensors are shown in Fig. 1 as white circles. The array is partitioned into three ULA structures, as ULA-1, ULA-2 and ULA-3, respectively which constitute a triplet. K is the total number of overlapped triplets in the array. The overall array is composed of the shifted triplets in order to deal with the multipath signals by employing spatial smoothing (Shan et al. 1985). Each ULA is composed of M sensors and spatial smoothing is applied for the shifted arrays. The largest shift is by K − 1 sensors. The DOA angles of the sources are Θd = (φd , θd ), d = 1, . . . , D, where φ and θ are the azimuth and elevation angles, respectively. The antennas are assumed to be identical and omni-directional. Far-field assumption is made. The array output of the P sensors which are positioned as in Fig. 1, can be written as, y(t) = A()s(t) + n(t),

123

t = 1, . . . , N

(1)

Multidim Syst Sign Process

Fig. 1 ULA structures in a triplet

where y(t) = [y1 (t) . . . y P (t)]T is a P × 1 vector. N is the number of snapshots and s(t) = [s1 (t), . . . , s D (t)]T is the signal vector which represents a stationary, zero-mean random processes uncorrelated with the noise. It is assumed that the noise, n(t), is both spatially and temporally white with variance σn2 . A() = [a(Θ1 ) . . . a(Θ D )] is the P × D ideal steering matrix. The steering vector a(Θd ) is given as, (2) a(Θd ) = [a1 (φd , θd ) . . . a P (φd , θd )]T , d = 1, . . . , D  2π  where ai (φd , θd ) = exp j λ (xi αd + yi βd ) for i = 1, . . . , P. (xi , yi ) is the position of the ith sensor, λ is the wavelength of the narrow-band source signals, αd and βd are defined for simplicity as cos φd sin θd and sin φd sin θd respectively. In practical arrays, array elements are located close to each other as in Fig. 1 and this results mutual coupling between array elements. Mutual coupling distorts the theoretical steering vector in (2) and this effect can be modeled by the mutual coupling matrix (MCM), C(). The MCM is a P × P matrix with complex coupling coefficients and generally considered to be independent of DOA angles  (Aksoy and Tuncer 2012). It is also assumed that the coupling coefficients are time-invariant and constant in the range of observation interval hence can be expressed as a constant matrix C. In this case, the array output in (1) can be rewritten as, y(t) = CA()s(t) + n(t), t = 1, . . . , N . (3) The function of the auxiliary sensors is to generate a symmetric banded Toeplitz structure for two parallel ULA as in Ye and Liu (2008). It is assumed that the MCM is normalized

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with respect to the diagonal term in the presence of auxiliary sensors. The C matrix for the PULA structure can be written as,   C1 C2 C= (4) C2 C1 P×P where C1 and C2 are (P/2) × (P/2) sub-matrices. C1 and C2 matrices correspond to the mutual coupling coefficients between antennas in each ULA and the cross antennas between the parallel ULAs respectively. The structures of the C1 and C2 matrices are presented in Appendix in (73) and (74), respectively. The output covariance matrix, R, is E{y(t)y(t) H } = R = CA()Rs A() H C H + σ 2 I

(5)

where (.) H denotes the conjugate transpose of a matrix, Rs is the source correlation matrix and I is the identity matrix. If the signal vector is composed of D coherent sources, then s(t) = [b1 s(t), . . . , b D s(t)]T where bi for i = 1, . . . , D are complex coefficients. s(t) can be represented as s(t) = bs(t) where b = [b1 , . . . , b D ]T . In this case, the source correlation matrix becomes Rs = bb H σs2 , which is a singular matrix. It is known that the subspace based techniques can only work when the source covariance matrix, Rs , is non-singular (Shan et al. 1985). The problem is to find the 2-D paired DOA angles () with a search-free algorithm given the array output in (3) and the output covariance matrix in (5) when Rs is a singular matrix and there is unknown mutual coupling between array elements.

3 Closed form 2-D paired DOA estimation In order to solve the problem defined in the previous section, the proposed method uses the triple linear sub-arrays (ULA-1, ULA-2 and ULA-3) which is called as triplet as shown in Fig. 1. The sensor positions for ULA-1 are given in Cartesian coordinates as (xi ,yi ), i = 1, . . . , M. The sensor positions for ULA-2 and ULA-3 are obtained as (xi + dx ,yi + d y ) √ and (xi − dx ,yi + d y ), respectively, where dx = d y = d and 2d is the inter-element distance for each ULA. The indexes for sensor positions in ULA-2 are given as (xi ,yi ), i = 2, . . . , (M + 1) and for ULA-3, (xi ,yi ), i = (M + K ), . . . , (2M + K − 1). The array outputs of ULA-1, ULA-2 and ULA-3 for the base triplet can be represented as, y(1),1 (t) = J(1),1 y(t) = [y1 (t), . . . , y M (t)]T

(6)

y(1),2 (t) = J(1),2 y(t) = [y2 (t), . . . , y M+1 (t)]T y(1),3 (t) = J(1),3 y(t) = [y M+K (t), . . . , y2M+K −1 (t)]

(7) T

(8)

The sub-index in parenthesis corresponds to the triplet number. The second index corresponds to the ULA number. Therefore, y(1),2 (t) is the output vector for ULA-2 in triplet 1. There are K triplets in the array. J(k),m is the selection matrix for the specified triplet and ULA where k = 1, . . . , K is the triplet number and m = 1, 2, 3 is the ULA number. J(k),m matrices are defined as, J(k),1 = [0 M×(k−1) I M×M 0 M×(P−M−k+1) ] M×P

123

(9)

J(k),2 = [0 M×(k) I M×M 0 M×(P−M−k) ] M×P

(10)

J(k),3 = [0 M×(M+K +k−1) I M×M 0 M×(K −k−1) ] M×P

(11)

Multidim Syst Sign Process

where 0 is the rectangular zero matrix and I is the square identity matrix. In this case, the array output for the kth triplet and mth ULA can be written as, y(k),m (t) = J(k),m y(t) = J(k),m (CA()s(t) + n(t)).

(12)

We can define C(k),m = J(k),m C and n(k),m (t) = J(k),m n(t). The array output in (12) can be rewritten as, y(k),m (t) = C(k),m A()s(t) + n(k),m (t) (13) The structure of the coupling matrices of the ULA-1, 2 and 3 for the kth triplet, namely, C(k),1 , C(k),2 and C(k),3 , are presented in Appendix in (75)–(77), respectively. The relation between these matrices can be written as, C(k),2 = C(k),1 U

(14)

C(k),3 = C(k),1 L

(15)

where U is the P × P shifting matrix. The components of U matrix is defined as, Ui, j = δi+1, j where δi, j = 1 if i = j otherwise it is zero. L matrix is the P × P anti-diagonal identity matrix and its components are defined as, L i, j = δi,P+1− j . The structure of these two matrices, U and L, are given in (16) as, ⎤ ⎤ ⎡ ⎡ 0 1 0 0 ··· 0 0 ··· 0 0 1 ⎢0 0 1 0 ··· 0⎥ ⎢0 ··· 0 1 0 ⎥ ⎥ ⎥ ⎢ ⎢ ⎥ ⎥ ⎢ ⎢ . .. U=⎢ L = ⎢ ... . (16) ⎥ ⎥ ⎥ ⎥ ⎢ ⎢ ⎣0 0 0 ··· 0 1⎦ ⎣0 1 0 ··· 0 ⎦ 0 0 0 · · · 0 0 P×P 1 0 0 · · · 0 P×P It is also possible to find the relation between the coupling matrices of the base triplet, C(1),1 , and kth triplet, C(k),1 as, C(k),1 = C(1),1 Uk where

Uk

(17)

is the kth power of the shifting matrix U which is defined as, . . U Uk = U . 

(18)

k times

Using the relations in (14), (15) and (17), the coupling matrices of the kth triplet can be ¯ k , C(k),2 = CU ¯ k U and C(k),3 = CU ¯ k L where C ¯ = C(1),1 . In this rewritten as, C(k),1 = CU case, the array outputs for the kth triplet can be written as, ¯ k A()s(t) + n(k),1 (t) y(k),1 (t) = CU ¯ k UA()s(t) + n(k),2 (t) y(k),2 (t) = CU

¯ k LA()s(t) + n(k),3 (t) y(k),3 (t) = CU

(19) (20) (21)

In order to clarify the relation between these array outputs, the following relations can be considered, i.e, UA() = A()1 ()

(22)

LA() = A()2 ()

(23)

where 1 () and 2 () are the diagonal matrices as,      2π 2π d(α1 + β1 ) , . . . , exp j d(α D + β D ) 1 () = diag exp j λ λ

(24)

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Multidim Syst Sign Process





2 () = diag exp

   2π 2π j d(−α1 + β1 ) , . . . , exp j d(−α D + β D ) , λ λ

(25)

αd and βd for d = 1, . . . , D are defined in (2). Finally using (22) and (23) the subarray outputs in (19)–(21) for the kth triplet can be rewritten as, ¯ k A()s(t) + n(k),1 (t) y(k),1 (t) = CU ¯ k A()1 ()s(t) + n(k),2 (t) y(k),2 (t) = CU ¯ k A()2 ()s(t) + n(k),3 (t) y(k),3 (t) = CU

(26) (27) (28)

Using the sub-array outputs in (26), (27) and (28), it is possible to find 1 () and 2 () matrices by applying the ESPRIT algorithm twice. But in this case, a pairing problem arises since the terms of these diagonal matrices are ordered arbitrary. There are some pairing techniques in the literature (Kikuchi et al. (2006)) which require additional computational complexity. In addition, these pairing methods do not perform well especially at low signal to noise ratio (SNR). In this paper, the ULA outputs of the kth triplet, y(k),2 (t) and y(k),3 (t), are combined in order to obtain a virtual subarray output, y(k),4 (t) as, y(k),4 (t) = y(k),2 (t) + y(k),3 (t) ¯ k A()(1 () + 2 ())s(t) + n(k),2 (t) + n(k),3 (t) = CU

 

 

(29)

¯ A()()s(t) + n(k),4 (t) y(k),4 (t) = CU

(30)

()

k

n(k),4 (t)

() is a diagonal matrix with complex terms which have both magnitude and phase terms as,       2π 2π j 2π dβ1 j 2π dβ D λ λ , . . . , 2 cos () = diag 2 cos dα1 e dα D e λ λ (31) = diag {v1 , . . . , v D } Note that the magnitude components of vi in Eq. (31), 2 cos( 2π λ d cos φi sin θi ) should always be positive in order to avoid the ambiguity, hence the displacement is chosen as, dx = d y = d ≤ λ/4.

(32)

The phase component is arg(vi ) = 2π λ d sin φi sin θi where the displacement choice, d ≤ λ/4, also avoiding angular ambiguity. These terms have DOA information for i = 1, . . . , D sources. Using magnitude and phase terms simultaneously the automatically paired azimuth and elevation DOA angles can be easily obtained as, ⎛   ⎞2 ⎞  ⎞ ⎛ 2 ⎛ |vi |  arccos ⎜ arg(vi ) ⎟ 2 arg(vi ) ⎠ ⎜  ⎠ ⎟. ⎝ ⎝   , θi = arcsin ⎝ φi = arctan + ⎠ 2π 2π |vi | arccos λ d λ d 2

(33) In order to find (), which is defined in (31), for each triplet, we apply the ESPRIT algorithm for the covariance matrix of R(k) = E{y(k),5 (t)y(k),5 (t) H } where y(k),5 (t) is the combined measurement which is defined as,   y (t) y(k),5 (t) = (k),1 (34) y(k),4 (t) 2M×2M

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Multidim Syst Sign Process

In case of multipath signals, the source correlation matrix will be singular and it is not possible to find () matrix using the eigenstructure methods. Spatial smoothing should be employed in order to mitigate the multipath signals. 3.1 Spatial smoothing of the combined measurements In this part, spatial smoothing (Shan et al. 1985) is used for the PULA in the presence of unknown mutual coupling. Note that it is possible to express the steering matrices of the kth triplets in (26) and (30) as, Uk A() = A()k1 () (35) where 1 () is the diagonal matrix which is previously defined in (24) and k1 () is the kth power of the 1 (). 1 () can be written as the multiplication of two matrices for simplicity as 1 () = F = Fx F y where      2π 2π Fx = diag exp j dα1 , . . . , exp j d(α D ) λ λ      2π 2π F y = diag exp j dβ1 , . . . , exp j dβ D λ λ In this case, the combined measurement in (34) can be rewritten as,     k s(t) ¯ n(k),1 (t) CA()F y(k),5 (t) = ¯ + n(k),4 (t) CA()Fk ()s(t)

(36)

Then the output covariance matrix for the kth triplet, R(k) = E{y(k),5 (t)y(k),5 (t) H }, can be written as, R(k) = QEkx Eky S(Eky ) H (Ekx ) H Q H + σ 2 I2M×2M (37) where

 Q=

¯ CA() 0 M×D ¯ 0 M×D CA()

and

 Ekx

=

Fkx 0 D×D



 S= 2M×2D

0 D×D Fkx

Rs Rs  H () ()Rs ()Rs  H () 

 Eky

=

2D×2D

Fky 0 D×D

0 D×D Fky

 (38) 2D×2D

 (39) 2D×2D

¯ can be found as, The spatially smoothed covariance matrix, R,   K K ! ! 1 1 k k k H k H ¯ = R(k) = Q Ex E y S(E y ) (Ex ) QH + σ 2I R K K k=1 k=1

 

(40)

S¯

where S¯ is the smoothed signal covariance, as it is shown in Shan et al. (1985). In the following subsection, the steps of the ESPRIT-based automatically paired 2-D DOA estimation algorithm are presented. This algorithm is named as PULA-1. PULA-1 solves the pairing problem effectively. However, its DOA performance can be improved if a different sub-array selection is done. PULA-2 algorithm is employed for this purpose. PULA-2 uses the paired 2-D DOA estimates obtained from PULA-1 and improves the DOA performance by increasing the baseline. In this respect, PULA-2 is dependent on PULA-1 since it has an unambiguous solution for d ≤ λ4 .

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3.2 Algorithm for PULA-1 The procedure for the proposed algorithm is as follows: Step 1 Obtain the combined measurement data as in (34) for K triplets, y(k),5 (t), k = 1, . . . , K , using (13) and (29). Step 2 Apply spatial smoothing by using the shifted triplets as in (40). Apply the ESPRIT ˆ¯ and find Sˆ and Sˆ . Sˆ and Sˆ are the algorithm to the final sample covariance matrix R 1 4 1 4 ˆ ¯ matrices composed of the eigenvectors of R corresponding to y(1),1 (t), and y(1),4 (t) as in (34) respectively. The relation between Sˆ 1 and Sˆ 4 is ˆ Sˆ 4 = Sˆ 1 

(41)

ˆ is the estimate of the (31). Find the least squares (LS) solution for  ˆ as  ˆ = Sˆ † Sˆ 4 where  1 † H H −1 where Sˆ 1 = (Sˆ 1 Sˆ 1 ) Sˆ 1 is the Moore-Penrose pseudoinverse. ˆ as Step 3 Compute the eigenvalues of  ˆ vˆ = eig{}

(42)

where vˆ is a D × 1 vector. Compute the azimuth and elevation angles by using (33) for i = 1, . . . , D. If we apply the above algorithm, the automatically paired multiple 2-D DOA angles can be easily found by solving the closed form expressions in a fast manner. It is observed in the simulations that the DOA performance of the method is good but not so close to the CRB of the PULA array due to the antenna displacement in (32) for unambiguous DOA solutions. Therefore the array structure in Fig. 2 is used and the PULA-1 results are improved by the PULA-2 algorithm. In this case, different sub-arrays can be selected within the PULA with a larger baseline and 2-D DOA performance can be improved considerably. 3.3 Selecting different sub-arrays within PULA and the PULA-2 algorithm It can be seen from Figs. 2 and 3 that it is possible to select sub-arrays with a larger baseline √ in x and y axes, respectively. In PULA-1 approach, dx = d y = d = λ4 is chosen and 2 λ4 is the distance between each ULA as shown in Fig. 1. In PULA-2 case, the same array as in Fig. 1 is used. However different ULAs are selected in this case and 2dx = 2d y = λ2 is the distances between each ULA as shown in Figs. 2 and 3, respectively. While PULA-2 has ambiguous solution due to above distances and (33), it solves the pairing problem using the results of PULA-1. In return, PULA-2 has better 2-D DOA accuracy. Two arrays are selected as in Fig. 2 for the x-axis shift and the corresponding selection matrices are defined as, J¯ (k),1 = [0 M×(M+K +k−2) I M×M 0 M×(K −k) ] M×P J¯ (k),2 = [0 M×(k) I M×M 0 M×(M+2K −k−2) ] M×P

(43) (44)

Similarly the other two arrays are selected as in Fig. 3 for the y-axis shift and the corresponding selection matrices are defined as, J¯ (k),3 = [0 M×(k−1) I M×M 0 M×(M+2K −k−1) ] M×P J¯ (k),4 = [0 M×(M+K +k−1) I M×M 0 M×(K −k−1) ] M×P

123

(45) (46)

Multidim Syst Sign Process

Fig. 2 Two subarrays within PULA for x-axis shift

Note that the bar, (¯), represents the new arrays in Figs. 2 and 3. In this case, the array output can be defined similar to (12) as, y¯ (k),m (t) = J¯ (k),m y(t) = J¯ (k),m (CA()s(t) + n(t)),

m = 1, 2, 3, 4.

(47)

¯ (k),m = J¯ (k),m C and n¯ (k),m (t) = J¯ (k),m n(t). It is easy to write the relations We can define the C ¯ (k),m and C(k),m , i.e., between coupling matrices C ¯ (k),1 = C(k),1 L C ¯ (k),2 = C(k),2 = C(k),1 U C ¯ (k),3 = C(k),1 C

¯ (k),4 = C(k),2 L = C(k),1 UL C

(48) (49) (50) (51)

We can now express the coupling matrices in (48)–(51) by using the Eq. (17) as, ¯ (k),1 = CU ¯ kL C ¯ kU ¯ (k),2 = CU C ¯ k ¯ (k),3 = CU C

¯ k UL ¯ (k),4 = CU C

(52) (53) (54) (55)

¯ = C(1),1 . The array outputs for the kth triplet of Fig. 2 can rewritten as, where C

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A Se uxil ns iar or y s

Multidim Syst Sign Process

2(M+K-1) y

U LA -4 ,m =4 ,k

=1

2M+K-1

2dy

3, m =3 ,k

M

=1

M+1

U LA -

M+K+1

M+K

M+K-1

2M+K

2dy 2 dx

dy

x-y plane

x

A Se uxil ns iar or y s

1

Fig. 3 Two subarrays within PULA for y-axis shift

¯ k LA()s(t) + n¯ (k),1 (t) y¯ (k),1 (t) = CU ¯ k UA()s(t) + n¯ (k),2 (t). y¯ (k),2 (t) = CU

(56) (57)

By using (22) and (23) the subarray outputs of y¯ (k),1 (t) and y¯ (k),2 (t) can be rewritten as, ¯ k A()2 ()s(t) + n¯ (k),1 (t) y¯ (k),1 (t) = CU ¯ k A()1 ()s(t) + n¯ (k),2 (t). y¯ (k),2 (t) = CU We can write 1 () = 2 ()x () where      2π 2π x () = diag exp j 2dα1 , . . . , exp j 2dα D . λ λ

(58) (59)

(60)

Finally (58) and (59) can be written as, ¯ (k) ()2 ()s(t) + n¯ (k),1 (t) y¯ (k),1 (t) = CA ¯ (k) ()2 ()x ()s(t) + n¯ (k),2 (t) y¯ (k),2 (t) = CA

(61) (62)

where A(k) () = Uk A(). In this case, if we apply the ESPRIT algorithm to the above array outputs, it is possible to find x () matrix. Similarly the array outputs for the kth triplet of Fig. 3 can be written using (54) and (55) as,

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Multidim Syst Sign Process

¯ k A()s(t) + n¯ (k),3 (t) y¯ (k),3 (t) = CU ¯ k ULA()s(t) + n¯ (k),4 (t). y¯ (k),4 (t) = CU

(63) (64)

By using (22) and (23) the subarray output in (64) can be rewritten as, ¯ k A()1 ()2 ()s(t) + n¯ (k),4 (t). y¯ (k),4 (t) = CU We can write 1 ()2 () =  y () where      2π 2π  y () = diag exp j 2dβ1 , . . . , exp j 2dβ D . λ λ

(65)

(66)

Finally (63) and (65) can be written as, ¯ (k) ()s(t) + n¯ (k),3 (t) y¯ (k),3 (t) = CA ¯ (k) () y ()s(t) + n¯ (k),4 (t). y¯ (k),4 (t) = CA

(67) (68)

In this case if we apply the ESPRIT algorithm to the above array outputs, it is possible to find  y () matrix. As we can see, x () and  y () are the diagonal matrices and include angle information as in (60) and (66), respectively. These matrices are not ordered and (φ, θ ) pairs for each source should be identified. We can easily pair these using the automatically paired 2-D DOA angle estimates of the PULA-1 algorithm. The eigenvalues of the x () and  y () can be defined as p = [ p1 , . . . , p D ] and r = [r1 , . . . , r D ], respectively, where pi = exp( j 2π λ 2dx αi ) 2π and ri = exp( j λ 2d y βi ) for i = 1, . . . , D. In order to pair these vectors, angle pairs obtained by PULA-1 algorithm can be used. Using these paired terms we can find azimuth and elevation DOA angles for i = 1, . . . , D as, ⎛ ⎞ 2  2     arg(ri ) arg(ri ) ⎟ arg( pi ) ⎜ + 2π φi = arctan , θi = arcsin ⎝ 2π (69) ⎠. arg( pi ) 2d λ λ 2d PULA-2 Algorithm: Step 1 Apply PULA-1 algorithm and find the paired 2-D DOA angles (φˆ i , θˆi ) for i = 1, . . . , D. Step 2 Obtain the subarray data using (47) for x-axis and y-axis shifts. For x-axis shifts, obtain y¯ (k),1 (t) and y¯ (k),2 (t) for k = 1, . . . , K . For y-axis shifts, obtain y¯ (k),3 (t) and y¯ (k),4 (t) for k = 1, . . . , K . Combine these data for x-axis and y-axis as:     y¯ (k),1 (t) y¯ (k),3 (t) (70) y¯ (k),x (t) = y¯ (k),y (t) = y¯ (k),2 (t) y¯ (k),4 (t) Step 3 Apply spatial smoothing to these " N shifted subarrays. HFind the sample covariˆ (k),x = 1 ˆ (k),y = ¯ (k),x (t)¯y(k),x (t) and y-axis, R ance matrix of x-axis, R i=1 y N 1 "N H ¯ (k),y (t)¯y(k),y (t) for k = 1, . . . , K using (70). Then find the smoothed correlation i=1 y N matrices as in (40) for x and y-axes. Apply the ESPRIT algorithm to these two smoothed covariance matrices separately to estimate the x () and  y () in (60) and (66). ˆ y () as ˆ x () and  Step 4 Compute the eigenvalues of estimated  ˆ x ()}, pˆ = eig{

ˆ y ()} rˆ = eig{

(71)

where pˆ and rˆ are D × 1 vectors.

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Step 5 In this step, we pair rˆ according to pˆ using the D correct angle pairs, (φˆ i , θˆi ), from Step 1. First, obtain all possible order of vector elements of rˆ . There are D! possible permutations where ! stands for factorial. We can indicate these permutations as rˆ j for j = 1 . . . , D!. For each permutation rˆ j , compute the error term as,   D ! e( j) =  ((φˆ j (i) − φˆ i )2 + (θˆ j (i) − θˆi )2 )

(72)

i=1

where φˆ j (i) = arctan



arg(ˆr j (i)) ˆ arg(p(i))



⎛# 

and θˆ j (i) = arcsin ⎝

ˆ arg(p(i)) 2π λ 2d



2 +

arg(ˆr j (i)) 2π λ 2d

2

⎞ ⎠.

Choose the 2-D DOA angles (φˆ j (i), θˆ j (i)) for i = 1, . . . , D which corresponds to the minimum e( j).

4 Simulations In this section, the proposed methods, PULA-1 and PULA-2 are evaluated in order to show the 2-D paired DOA performance in case of unknown mutual coupling and for the coherent signals. In the simulations, dx = d y = λ/4 is selected in order to avoid ambiguity and the √ distance √ between each antenna in ULA is 2λ/4. The distance between parallel ULA is also 2λ/4. There are 500 trials for each experiment and the number of snapshots is 500. The number of sensors in each ULA is M = 7 and the number of triplets is K = 3. There are 8 auxiliary sensors which are not used to collect data. The unknown mutual coupling coefficients of the antenna array which stated in (3) are selected as c1 = 0.2040 + j0.1373, c2 = −0.0736− j0.0634, c3 = 0.1056− j0.0399 and c4 = 0.2483+ j0.2432 User’s Manuel (2008). c3 is the diagonal, c4 is the across term with respect to the first sensor as in Appendix, (73) and (74). In the simulations PULA-1 represents the performances of the triple sub-arrays in Fig. 1 and PULA-2 represents the performances of the sub arrays in Figs. 2 and 3. Figure 4 shows the 2-D DOA performance for the proposed methods when there are three sources and all the sources are coherent with DOA angles (φ1 = 105◦ , θ1 = 20◦ ), (φ2 = 190◦ , θ2 = 40◦ ) and (φ3 = 210◦ , θ3 = 60◦ ) respectively. In this figure, the azimuth DOA performances are given for the PULA-1 and PULA-2. PULA-2 performance is better than PULA-1 and it is close to the Cramer Rao Bound (CRB). The derivation of the CRB expressions for 2-D DOA estimation can be found in Weiss and Friedlander (1993) and Nielsen (1994). Figure 5 shows the elevation performance in the same scenario. In this case, a similar characteristic as in Fig. 4 is seen. The difference between PULA-1 and 2 results is large for low SNR. The gap between two decreases as the SNR increases. Figure 6 shows the 2-D DOA performance in the same scenario in terms of the number of snapshots. SNR is set as 15dB. While PULA-2 azimuth and elevation RMSE is small and very close to CRB, there is a significant difference between PULA-1 and CRB especially as the number of snapshots increases. This shows that PULA-2 is an consistent method for 2-D DOA estimation. In Fig. 7, the 2-D DOA estimates of the proposed method are marked for each trial to see the angular distribution for PULA-1. SNR is set to SNR = 15 dB. As it is seen from the figure, this algorithm localizes the multiple sources effectively.

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Root Mean Square Error (Deg.)

10

CRB PULA−1, φ PULA−2, φ 0

10

−1

10

−2

10

−3

10

0

5

10

15

20

25

30

35

40

SNR (dB) Fig. 4 2-D paired azimuth DOA performance for different SNR levels when there are three sources 1

Root Mean Square Error (Deg.)

10

CRB PULA PULA−1, θ PULA−2, θ 0

10

−1

10

−2

10

−3

10

0

5

10

15

20

25

30

35

40

SNR (dB) Fig. 5 2-D paired elevation DOA performance for different SNR levels when there are three sources

Figure 8 shows the azimuth DOA performance when there are two coherent sources and SNR = 15 dB. One source is at (φ1 = 100◦ , θ1 = 40◦ ). The elevation angle for the second source is fixed at θ2 = 60◦ . The azimuth angle is swept between 130◦ and 260◦ in half degree resolution. This figure shows the algorithm performance of PULA-1, PULA-2 and the classical 2-D MUSIC algorithm for the closely spaced sources. It is seen that PULA-1 algorithm is not efficient but correctly solves the angles. The elevation performance (PULA-1, θ ) is better than the azimuth performance (PULA-1, φ) which is based on the selected baselines in the used array structure. It is also seen that PULA-2 algorithm effectively solves the coupling and

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Root Mean Square Error (Deg.)

101 PULA−1, φ PULA−1, θ PULA−2, φ PULA−2, θ CRB, φ CRB, θ

0

10

−1

10

−2

10

0

200

400

600

800

1000

1200

1400

1600

Number of samples Fig. 6 2-D paired azimuth and elevation DOA estimates for different number of snapshots when there are three sources and SNR = 15dB

Elevation Angle

70 60 50 40 30 20 10

105

120

140

160

190

210 220

Azimuth Angle Fig. 7 2-D DOA distribution of PULA-1 when there are three coherent sources and SNR = 15 dB

multipath problem and its accuracy is good. In addition, no outlier is observed for 500 trials and the algorithm accurately solves the pairing problem. On the other hand, the performance of the PULA-2 is also compared with the search based classical 2-D MUSIC algorithm in the same figure. It is also possible to apply 2-D MUSIC with a computationally efficient way as in Xiaofei et al. (2014) which only requires the one-dimensional local searches. As it is known, it is not possible to directly apply MUSIC algorithm in case of unknown mutual coupling (C) between antennas. Hence the C matrix is assumed to be known while applying the MUSIC algorithm and it is seen that both the 2-D MUSIC and PULA-2 algorithms have almost the same performances. Figure 9 shows the elevation DOA performance when there are two coherent sources and SNR = 15 dB. One source is at (φ1 = 100◦ , θ1 = 40◦ ). The azimuth angle for the second source is fixed at φ2 = 180◦ . The elevation angle is swept between 10◦ and 80◦ in half degree resolution. It is seen that the elevation performance of the PULA-1 is better than

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Multidim Syst Sign Process 1

Root Mean Square Error (Deg.)

10

CRB, φ PULA−1, φ PULA−2, φ PULA−1, θ 2−D MUSIC, φ 0

10

−1

10

−2

10

140

160

180

200

220

240

260

Azimuth DOA (degree) Fig. 8 2-D paired azimuth DOA performance for two coherent sources for PULA-1 and PULA-2. SNR is 15 dB. There is unknown mutual coupling between antennas

1

Root Mean Square Error (Deg.)

10

CRB, θ PULA−1, θ PULA−2, θ PULA−1, φ 2−D MUSIC, θ 0

10

−1

10

−2

10

10

20

30

40

50

60

70

80

Elevation DOA (degree) Fig. 9 2-D paired elevation DOA performance for two coherent sources for PULA-1 and PULA-2. SNR is 15 dB. There is unknown mutual coupling between antennas

the azimuth performance. This figure also shows that PULA-2 algorithm performs well for elevation angle estimation as well.

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Multidim Syst Sign Process

5 Conclusion In this paper, 2-D search-free fast DOA estimation in case of mutual coupling and multipath signals is considered. A special array geometry which consists of two parallel ULA is employed. Different sub-array configurations are used in order to obtain the same MCM leading to paired 2-D DOA estimates. The overlapping triplet structure are further used to employ the spatial smoothing algorithm for multipath signals. Two new algorithms PULA-1 and PULA-2 are proposed. It is shown that PULA-1 is an effective algorithm which solves the 2-D DOA estimation problem for multiple sources. PULA-2 algorithm is proposed in order to improve the performance of PULA-1 by using the same array outputs with different sub-arrays which have larger baselines. It is also shown that the PULA-2 attains the CRB.

Appendix 1: The MCM structure of PULA In this part, the structure of the mutual coupling matrix (MCM) of the PULA is given. The sub-matrices C1 and C2 in (4), are composed as, ⎤ ⎡ 1 c 1 c2 0 ··· 0 c 2 c1 ⎢ 0 c 2 c1 1 c1 c2 · · · 0 ⎥ ⎥ ⎢ C1 = ⎢ (73) ⎥ . . . .. .. .. ... ... ⎦ ⎣ 1 c1 c2 P × P 0 ··· 0 c2 c1 2 2 ⎡ ⎤ 0 c 3 c4 c3 0 0 ··· 0 ⎢ 0 0 c 3 c4 c3 0 ··· 0⎥ ⎢ ⎥ (74) C2 = ⎢ ⎥ . . . .. .. .. ⎣ ⎦ 0 ··· 0 0 c3 c4 c3 0 P × P 2

2

c1 and c2 are the coupling coefficients in the nearest neighborhood of the sensors and c3 and c4 are the cross terms for the parallel ULA. As it is seen, the MCM structure for the PULA is in the block banded Toeplitz structure. The sub-matrices, C(k),m in (13) for m = 1, 2, 3 are in the form of % $ R 0 M×(P−2M−k+1) M×P C(k),1 = 0 M×(k−1) C LM×M C M×M (75) % $ R 0 M×(P−2M−k) M×P C(k),2 = 0 M×k C LM×M C M×M (76) $ % R C LM×M 0 M×(P−2M−k+1) M×P C(k),3 = 0 M×(k−1) C M×M (77) where C L and C R are the left and right coupling matrices respectively and defined as, ⎡ ⎤ c 2 c1 1 c 1 c2 0 ··· 0 ⎢ 0 c2 c 1 1 c 1 c2 · · · 0 ⎥ ⎢ ⎥ CL = ⎢ ⎥ . . . .. .. .. ... ... ⎣ ⎦ 1 c1 c2 M×M 0 ··· 0 c2 c1 ⎤ ⎡ 0 c 3 c4 c3 0 0 ··· 0 ⎢ 0 0 c 3 c4 c3 0 ··· 0⎥ ⎥ ⎢ CR = ⎢ . (78) ⎥ . . . .. .. .. ... ⎦ ⎣ 0 ··· 0 0 c3 c4 c3 0 M×M

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As we can see, C(k),2 is the shifted version of the C(k),1 and in C(k),3 , C L and C R matrices are swapped according to the C(k),1 . These relations are formulated in (14) and (15). References Aksoy, T., & Tuncer, T. E. (2012). Measurement reduction for mutual coupling calibration in DOA estimation. Radio Science, 47(RS3004), 2012. doi:10.1029/2011RS004904. FEKO User’s Manuel, EM Software and Systems, S.A. (Pty) Ltd., July 2008 [Online]. Filik, T., & Tuncer, T. E. (2009). Closed-form automatically paired 2-D direction-of-arrival estimation with arbitrary arrays, European Signal Processing Conference (EUSIPCO-2009), Glasgow, Scotland, 24 August 2009. Filik, T., & Tuncer, T. E. (2010). A fast and automatically paired 2-D direction-of-arrival estimation with and without estimating the mutual coupling coefficients. Radio Science, 45, RS3009. doi:10.1029/ 2009RS004260. Filik, T., & Tuncer, T. E. (2011). 2-D paired direction-of-arrival angle estimation with two parallel uniform linear arrays. International Journal of Innovative Computing, Information and Control, 7(6), 3269–3279. Gershman, A. B., Rubsamen, M., & Pesavento, M. (2010). One- and two-dimensional direction-of-arrival estimation: An overview of search-free techniques. Signal Processing, 90, 1338–1349. Goossens, R., & Rogier, H. (2007). A hybrid UCA-RARE/Root-MUSIC approach for 2-D direction of arrival estimation in uniform circular arrays in the presence of mutual coupling. IEEE Transactions on Antennas and Propagation, 55(3), 841–849. Kailath, T., & Roy, R. H. (1990). ESPRIT-estimation of signal parameters via rotational invariance techniques. Optical Engineering, 29(4), 296–313. Kikuchi, S., Tsuji, H., & Sano, A. (2006). Pair-matching method for estimating 2-D angle of arrival with a cross-correlation matrix. IEEE Antennas and Wireless Propagation Letters, 5, 35–40. Liu, Z., Ruan, X., & He, J. (2013). Efficient 2-D DOA estimation for coherent sources with a sparse acoustic vector-sensor array. Multidimensional Systems and Signal Processing, 24(1), 105–120. Nielsen, R. O. (1994). Azimuth and elevation angle estimation with a three-dimensional array. IEEE Journal of Oceanic Engineering, 19(1), 84–86. Shan, T.-J., Wax, M., & Kailath, T. (1985). On spatial smoothing for direction-of-arrival estimation of coherent signals. IEEE Transactions on Acoustics, Speech, and Signal Processing, 33(4), 806–811. Tuncer, T. E., & Friedlander, B. (2009). Classical and modern direction-of-arrival estimation. New York: Academic Press. Wang, B. H., & Hui, H. T. (2010). Decoupled 2D direction of arrival estimation using compact uniform circular arrays in the presence of elevation-dependent mutual coupling. IEEE Transactions on Antennas and Propagation, 58(3), 747–755. Weiss, A. J., & Friedlander, B. (1993). On the Cramer-Rao bound for direction finding of correlated signals. IEEE Transactions on Signal Processing, 41(1), 495–499. Wu, Y., Liao, G., & So, H. C. (2003). A fast algorithm for 2-D direction-of-arrival estimation. Signal Processing, 83, 1827–1831. Xia, T., Zheng, Y., Wan, Q., & Wang, X. (2007). Decoupled estimation of 2-d angles of arrival using two parallel uniform linear arrays. IEEE Transactions on Antennas and Propagation, 55, 2627–2632. Xiaofei, Z., Ming, Z., Han, C., & Jianfeng, L. (2014). Two-dimensional DOA estimation for acoustic vectorsensor array using a successive MUSIC. Multidimensional Systems and Signal Processing, 25(3), 583–600. Ye, Z., & Liu, C. (2008). On the resiliency of MUSIC direction finding against antenna sensor coupling. IEEE Transactions on Antennas & Propagation, 56, 371–380.

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Multidim Syst Sign Process Tansu Filik received the B.S. degree from the Anadolu University in 2002, the integrated M.S. and Ph.D. degree from the Middle East Technical University (METU) in 2010, all in electrical engineering. From 2002 to 2010, he was with Sensor Array and Multichannel Signal Processing (SAM) Group of METU. From 2010 to 2012, he was with Aselsan Inc., at Radar, Electronic Warfare and Intelligence Systems Division. Currently, he is with Anadolu University at Department of Electrical Engineering. His research interest includes array signal processing, statistical signal processing and signal processing for communications.

T. Engin Tuncer received the B.S. and M.S. degrees in electrical and electronics engineering from the Middle East Technical University (METU), Ankara, Turkey, in 1987 and 1989, respectively, and the Ph.D. degree in electrical engineering from Boston University, Boston, MA, in 1993. From 1991 to 1993, he was an instructor and Adjunct Professor at Boston University. He joined the Electronics Engineering Department, Ankara University, in 1993. Since 1998, he has been with METU, where he is currently a Professor in the Electrical and Electronics Engineering Department. His research interests include array signal processing, antenna and array design for direction finding and target localization, multichannel systems and blind identification, statistical signal processing, and signal processing for communications

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