sensors Communication

Fast Noncircular 2D-DOA Estimation for Rectangular Planar Array Lingyun Xu 1,2, * and Fangqing Wen 3 1 2 3

*

College of Science, Nanjing University of Aeronautics & Astronautics, Nanjing 210016, China Department of Electrical and Computer Engineering, University of Delaware, Newark, DE 19716, USA Electronic and Information School of Yangtze University, Jingzhou 434023, China; [email protected] Correspondence: [email protected] or [email protected]; Tel.: +86-139-5203-5864

Academic Editors: Zhiguo Shi, Yujie Gu and Rongxing Lu Received: 20 February 2017; Accepted: 31 March 2017; Published: 12 April 2017

Abstract: A novel scheme is proposed for direction finding with uniform rectangular planar array. First, the characteristics of noncircular signals and Euler’s formula are exploited to construct a new real-valued rectangular array data. Then, the rotational invariance relations for real-valued signal space are depicted in a new way. Finally the real-valued propagator method is utilized to estimate the pairing two-dimensional direction of arrival (2D-DOA). The proposed algorithm provides better angle estimation performance and can discern more sources than the 2D propagator method. At the same time, it has very close angle estimation performance to the noncircular propagator method (NC-PM) with reduced computational complexity. Keywords: two-dimensional direction of arrival (2D-DOA); real-valued propagator method; noncircular; rectangular planar array

1. Introduction Two-dimensional direction of arrival (2D-DOA) estimation has been widely used in mobile communication systems, sonar, navigation, radar, etc. [1–5], which is an important research branch in array signal processing. Many 2D-DOA estimation algorithms have sprung up in recent years in order to improve the performance of angle estimation, which include the two dimensional multiple signal classification(2D MUSIC) algorithm [6], the 2D Unitary estimation of signal parameters via rotational invariance techniques (ESPRIT) algorithm [7], the modified 2D-ESPRIT algorithm [8], the matrix pencil method [9], the maximum likelihood method [10,11], the parallel factor (PARAFAC) algorithm [12], and so on [13–20]. However, those 2D-DOA estimation algorithms are confronted with the problem of the high computational complexity generally and they are very difficult to apply in engineering practice. As is known to us, the propagator method (PM) algorithm uses linear operations to replace the eigenvalue decomposition of the covariance matrix [21], and it has a great advantage in resolving the amount of calculation. Therefore, the 2D-DOA estimation based on PM is becoming a hot spot of research. For example, Wu et al. have developed the 2D-DOA estimation algorithm via the rotational invariance property of propagator matrix [22]. In [23], an improved PM algorithm is proposed for 2D-DOA estimation, which not only reduces the computational complexity, but also avoids the aperture loss. Unfortunately, all the algorithms mentioned above did not consider the characteristics of the impinging signals. In fact, many noncircular signals such as the amplitude modulated (AM), binary phase shift keying (BPSK), minimum shift keying (MSK), and Gaussian MSK (GMSK) signals are used in wireless communication or satellite systems. In recent years, some scholars use non-circular signal characteristics to improve the performance of direction estimation, which contain the noncircular MUSIC (NC-MUSIC) algorithm [24], the NC-ESPRIT algorithm [25], and the noncircular parallel Sensors 2017, 17, 840; doi:10.3390/s17040840

www.mdpi.com/journal/sensors

Sensors 2017, 17, 840

2 of 15

factor (NC-PARAFAC) algorithm [26]. On the one hand, the angle estimation performance can be achieved by the algorithms [24–26]. On the other hand, the computation loads are increased greatly due to the doubled array aperture. The noncircular rational invariance propagator method has also been proposed for angle estimation in [27], which aimed at the linear array. If it is extended to the rectangular planar array for 2D-DOA estimation, the complexity would be increased greatly. In this paper, we take advantage of the characteristics of noncircular signals and derive a novel noncircular propagator method algorithm based on the uniform rectangular planar array. The main works of this paper are listed in a straightforward manner as follows: (1) the property of the noncircular signal and Euler’s transformation are used to construct a new real-valued rectangular array data; (2) the rotational invariance relations for real-valued signal space are depicted in a new way; (3) the PM algorithm is applied to two-dimensional angle estimation for the rectangular planar array which is paired automatically; and (4) theory analysis and simulation results confirm that our algorithm has better direction finding performance and can discern more sources than 2D-PM [23]. Due to real-valued processing, it can save about 75% computational load compared with the NC-PM algorithm [27]. However, its estimation performance is close to NC-PM algorithm, which has higher computational load. 2. Data Model In order to get the two-dimensional direction finding, we consider a uniform rectangular planar array (URA) consisting of N uniform linear subarrays as shown in Figure 1, and there are M sensors in each subarray. The inter-element spacing between the two sensors is d in both the x-axis and y-axis. Suppose there are K narrowband far-field uncorrelated sources with wavelength λ impinging on the array from different directions. We also assume the noise is independent of the sources and d = λ/2. The output signal of the ith subarray xi (t) can be denoted as [26]: xi (t) = Ax Φi−1 S(t) + ni (t), i = 1, 2, · · · , N,

(1) T

where Ax = [ax (v1 ), ax (v2 ), · · · , ax (vK )] and ax (vk ) = [1, e− jπvk , · · · , e− j( M−1)πvk ] , vk = cos φk sin θk , θk is the elevation angle and φk is the azimuth angle. Φ = diag(e− jπu1 , e− jπu2 , · · · , e− jπuK ) and uk = sin φk sin θk . S(t) = [s1 (t), s2 (t), · · · , sK (t)] T is the noncircular signal vector. In addition, the vector of strictly second-order noncircular signals can be expressed as [28]: S(t) = ΛSo (t),  So (t) ∈ CK ×1 , So (t) = So ∗ (t), and Λ = diag e jϕ1 , e jϕ2 , · · · e jϕK , (e jϕ p 6= e jϕq , f or p 6= q). ni (t) is the additive white Gaussian noise vector of the ith subarray. Therefore, the whole array output is    x( t ) =   

x1 ( t ) x2 ( t ) .. . x N (t)





    =    

Ax Ax Φ .. . A x Φ N −1





    S( t ) +     

n1 (t) n2 (t) .. . n N (t)

    = AS(t) + n(t),  

(2)

where A = Ay ◦ Ax is the MN × K steering vector matrix, ◦ represents the Khatri–Rao product, and Ay

T

= [ay (u1 ), ay (u2 ), · · · , ay (uK )], ay (uk ) = [1, e− jπuk , · · · , e− j( N −1)πuk ] , and T

n(t) = [n1 (t) T , n2 (t) T , · · · , n N (t) T ] .

Sensors 2017, 2017, 17, 17, 840 840 Sensors

of 16 15 33 of

Z s k(t) k=1,2,...,K

k 0 1 ...

...

1

N-2

N-1

Y

k d

M-1 X subarray 1

subarray N

Figure 1. 1. The Thestructure structure of of planar planar array. array. Figure

3. Real-Valued Real-Valued PM PM Algorithm Algorithm for for 2D-DOA 2D-DOA Estimation Estimation 3. 3.1. Euler Euler Transformation Transformation 3.1. The real real part part and and imaginary ) can byby utilizing the real-valued property of The imaginary part part of of xx(t(t) canbebeobtained obtained utilizing the real-valued property noncircular signals and Euler’s formula as follows: of noncircular signals and Euler’s formula as follows: ∗ xRx(Rt)(t= t) + t)]/2 ARRSSoo((t t))+nn ) [x[ x((t)  x *((t)] / 2= A (t()t,), RR

(3)

x I (t) = [x(t) − x∗ (t)]/(−2j) = A I So (t) + n I (t), x I (t )  [ x (t)  x * (t)] / ( 2 j )  A I S o (t )  n I (t ) , where nR (t) = [n(t) + n∗ (t)]/2, n I (t) = [n(t) − n∗ (t)]/(−2j), * * where   n R (t )  [n(t )  n (t )] / 2 , n I (t )  [n(t )  n (t )] / (2 j ) ,

(4)

 cos ϕ1 cos ϕ2 ··· cos ϕK      cos · ·cos ·  K cos(πv  K + ϕK ) cos(πv 1 1 + ϕ1 ) coscos  2 (πv2 + ϕ 2 )     M  . . . .    . . . .  cos( v1   cos( v2  2 ) .  cos(. vK   K )  .  . 1)       M   cos[π ( M    − 1)v1 + ϕ1 ] cos[π ( M − 1)v2 + ϕ2 ] · · ·  cos[π ( M − 1)vK + ϕK ]      coscos[ (πu1 (+ ϕ11))v   ] cos[ ( M cos (πu  cos(πuK + ϕK )  2 ]+ ϕ 2 ) cos[ ( M ·1·)·v   ]   )  v M 1    1 1 2 2 K K      cos[π (v1 + u1 ) + ϕ1 )] cos[π (v2 + u2 ) + ϕ2 )] ··· cos[π (vK + uK ) + ϕK )]    cos( u1. 1 ) cos( u2  2 ) .  cos(  u  )    K K M .. ..    . .   AR =   cos[ (v1  u. 1 )  1 )] cos[ (v2  u2 )  2. )]  cos[ .(vK  u K )   K )]    .     cos[π (( M − 1)v1 + u1 ) + ϕ1 )] cos[π (( M − 1)v2 + u2 ) + ϕ2 )] · · · cos[π (( M − 1)vM K + uK ) + ϕ  AR       KR)]MN  K     .. .. .. .. .. .. ..  . . . . . . .  cos[ (( M  1) v1  u1 )  1 )] cos[ (( M  1) v2  u2 )   2 )]  cos[ (( M  1)vK  uK )   K )]      cos[π ( N − 1)u1 + ϕ1 ] cos ··· cos[π ( N − 1)uK + ϕK ]   [π( N −1)u2 + ϕ2 ]   N − 1)u1 + πv1 + ϕ1 ] cos[π ( N − 1)u2 + πv2 + ϕ2 ] ··· cos[π ( N − 1)uK +πvK + ϕK ]  cos[π (cos[   ( N . 1)u1  1 ] cos[ ( N  1)u2  .2 ] cos[  ( N  1)u K   K ]  .  .  .. . . ..    cos[ ( N  1)u2   v2 . 2 ] cos[ (.N  1)u K   vK   K ]      cos[ ( N  1)u1   v1  1 ] cos[π ( N − 1)u1 + π ( M − 1)v1 + ϕ1 ] cos[π ( N − 1)u2 + π ( M − 1)v2 + ϕ2 ] cos[π ( N − 1)uK +πM (M  − 1) v K + ϕ K ]      

 cos[ ( N  1)u1   ( M  1)v1  1 ] cos[ ( N  1)u2   ( M

(4)

                ∈ R MN ×K                 M        



   1)v2   2 ] cos[ ( N  1)u K   ( M  1)vK   K ]  sin ϕ1 sin ϕ2 ··· sin ϕK     sin(πv1 + ϕ1 ) sin(πv2 + ϕ2 ) ··· sin(πvK + ϕK ) M .. sin  K .. sin 1.. sin  2 ..     . . . .     v  ) v  ) v  ) sin(  sin(   sin(      2 1) v + ϕ ] K [ π (KM − 1) v + ϕ ] sin[π ( M1− 1)1v1 + ϕ1 ] sin[π2( M − · · · sin   2 2 K  MK    (πu2 + ϕ2 ) sin(πu1 + ϕ1 ) sin · · · sin   (πuK + ϕK )    (vK + uK ) + ϕK ]   sin[π (sin[ v1 + π1()vv22 + sin [π v11] 1 ] sin[sin  (uM1 )+1)ϕ ( M [  u22]) +ϕ2 ]sin[ ( M· ·· 1)vK   K ]  M .. .. .. ..    sin( u1.  1 ) sin( u2  2 ).  .sin( u K   K ) .     sin[π sin[ (( M − sin[π(v(( M −)1) v2 ]+ u2 ) + ϕ2 ] sin[ · ·· (vsin[uπ (() M − ]1)v + u ) + ϕ ] K K K (v11)vu11 )+u11)] + ϕ1 ] sin[  u     2 2 2 K K K .. .. .. .. .. .. .. M   R MN K    . . . . . . .

                     AI =         A   I     −1()( Mu1+ sin[π ( N − 1)u2 + ϕ2 ] ··· sin[π( N − 1)uK + ϕK ]  1] sin[ 1) vϕ  sin[π ( N 1  u1 )  1 ] sin[ (( M  1) v2  u2 )   2 ]  sin[ (( M  1) vK  uK )   K ]  sin[π ( N − 1)u1 + πv1 + ϕ1 ] sin[π ( N − 1)u2 + πv2 + ϕ2 ] ··· sin[π ( N − 1)uK + πvK + ϕK ]            .. .. .. ..   . . .  sin[ ( N  1)u1  1 ] sin[ ( N  1)u2  2 ] sin[ ( N  1)uK   K ] .     ( N − 1)u1 + π ( M − 1)v1 + ϕ1 ] sin[π ( N − 1)u2 + π ( M − 1)v2 + ϕ2 ] sin[π sin[π ( N − 1)uK + π (M − 1)vK + ϕK ] sin[ ( N  1)u2   v2  2 ] sin[ ( N  1)u K   vK   K ]     sin[ ( N  1)u1   v1  1 ]

    sin[ ( N  1)u1   ( M  1)v1  1 ] sin[ ( N  1)u2   ( M  1)v2  2 ]

(3)



M      sin[ ( N  1)uK   ( M  1)vK   K ]

Then, we define a new virtual array data as follows:



                ∈ R MN ×K                 M        

Sensors 2017, 17, 840

4 of 15

Then, we define a new virtual array data as follows: " y( t ) = "

AR AI

where B =

#

xR (t) x I (t)

#

nR (t) n I (t)

#

"

∈ R2MN ×K , nr (t) =

= BSo (t) + nr (t),

(5)

∈ R2MN ×1 .

= [I M( N −1) 0 M( N −1)× M ] ∈ R M( N −1)× MN , T2 = [0 M( N −1)× M I M( N −1) ] ∈ R M( N −1)× MN ; then, we construct two matrices J1 = [T1 + T2 , 0 M( N −1)× MN ] ∈ R M( N −1)×2MN and J2 = [0 M( N −1)× MN , T2 − T1 ] ∈ R M( N −1)×2MN , and we can get the following relationship: J2 B = J1 BG, (6)  πuK πu2 1 where G = diag tan( πu 2 ), tan( 2 ), · · · , tan( 2 ) is a real-valued matrix whose diagonal elements contain the needed angle information: Define two matrices as follows:

πu +2φ



πu +2φ

1 2 cos 1 2 1 cos πu 2 2πv1 +πu1 +2φ1 1 cos πu 2 cos 2 2 ...

               J1 B =               

2 2 cos 2 2 2 cos πu 2 2πv2 +πu2 +2φ2 2 2 cos cos πu 2 2 ...

2π ( M −1)v +πu +2φ

1 1 1 1 cos πu 2 2 3πu1 +2φ1 1 cos πu 2 cos 2 2 3πu1 +2πv1 +2φ1 1 2cos cos πu 2 2 .. .

2 cos

2cos

3πu1 +2π ( M −1)v1 +2φ1 2

1 cos πu 2

π (2N −3)u1 +2φ1 1 cos πu 2 2 π (2N −3)u1 +2πv1 +2φ1 1 cos πu 2 2

2cos

2 cos

π (2N −3)u1 +2π ( M−1)v1 +2φ1 2

2 cos

2π ( M −1)v1 +πu1 +2φ1 1 sin πu 2 2 3πu1 +2φ1 πu1 2 cos sin 2 2 3πu1 +2πv1 +2φ1 1 sin πu 2cos 2 2

2 cos

π (2N −3)u2 +2π ( M −1)v2 +2φ2 2

2 cos πu 2

2 cos

.. .

2 cos

π (2N −3)uK +2π ( M−1)vK +2φK 2 πu +2φ

2 cos K 2 K sin πu2 K 2πvK +πuK +2φK 2 cos sin πu2 K 2 .. .

3πu2 +2π ( M −1)v2 +2φ2 2 2cos sin πu 2 2

3πuK +2π ( M −1)vK +2φK 2cos sin πu2 K 2

.. .. . .

.. .. .. .. . . . .

···

.. .

π (2N −3)u2 +2φ2 2 sin πu 2 2 π (2N −3)u2 +2πv2 +2φ2 2 sin πu 2 2

2 cos 2 cos

.. .

2 cos

π (2N −3)u2 +2π ( M −1)v2 +2φ2 2 sin πu 2 2

··· ··· .. .

          

M

M

    

.. .

2 cos

π (2N −3)uK +2π ( M −1)vK +2φK sin πu2 K 2

1 0 .. . 0

                ∈ R M( N −1)×K                  M         

π (2N −3)uK +2φK sin πu2 K 2 π (2N −3)uK +2πvK +2φK sin πu2 K 2

  Similarly, define two ( M − 1) × M Toeplitz matrices as follows: T3 =   

1 1 .. . ···

                ∈ R M( N −1)×K .                  M        

0 0 1 0 .. .. . . 0 0

 ··· 0 ··· 0   ..  .. , . .  1 1

 −1 0 0 · · · 0  1 −1 0 · · · 0  .. .. .. . . ..  . . . . . .  · · · 0 0 1 −1 2( M −1) N ×2MN and we construct two # matrices J3 and J4 as follows: J3 = I2N ⊗ T3 ∈ R 0 −I N ⊗ T4 ∈ R2( M−1) N ×2MN . I N ⊗ T4 0

1   0 T4 =   ..  . 0 Then, " J4 =

     

cos πu2 K

2 cos 2 cos





M

          

π (2N −3)uK +2φK cos πu2 K 2 π (2N −3)uK +2πvK +2φK cos πu2 K 2

· · · 2 cos 2π ( M−1)vK2+πuK +2φK sin πu2 K 3πuK +2φK ··· 2 cos sin πu2 K 2 3πuK +2πvK +2φK ··· 2cos sin πu2 K 2 .. .. . .

.. .



2 cos

2π ( M −1)v2 +πu2 +2φ2 2 sin πu 2 2 3πu2 +2φ2 πu2 2 cos sin 2 2 3πu2 +2πv2 +2φ2 2 2cos sin πu 2 2

2 cos

.. .

π (2N −3)u1 +2π ( M −1)v1 +2φ1 1 sin πu 2 2

··· ··· .. .

··· ··· .. .

2 cos

2 cos

π (2N −3)u2 +2φ2 2 cos πu 2 2 π (2N −3)u2 +2πv2 +2φ2 2 cos πu 2 2

2 2 cos 2 2 2 cos πu 2 2πv2 +πu2 +2φ2 πu2 2 cos sin 2 2 .. .

.. .

3πu1 +2π ( M −1)v1 +2φ1 1 2cos sin πu 2 2 π (2N −3)u1 +2φ1 1 sin πu 2 2 π (2N −3)u1 +2πv1 +2φ1 1 sin πu 2 2

2 cos πu 2 .. .. .. .. . . . .

πu +2φ

1 2 cos 1 2 1 sin πu 2 2πv1 +πu1 +2φ1 πu1 2 cos sin 2 2 .. .

2 cos

3πu2 +2π ( M −1)v2 +2φ2 2

.. .

πu +2φ

                J2 B =               

1 cos πu 2

     

· · · 2 cos 2π ( M−1)vK2+πuK +2φK cos πu2 K 3πuK +2φK ··· 2 cos cos πu2 K 2 3πuK +2πvK +2φK ··· 2cos cos πu2 K 2 M .. ..   . .   3πuK +2π ( M −1)vK +2φK πuK  · · · 2cos cos 2 2

2 cos

.. .

2 cos

2π ( M−1)v +πu +2φ

.. .. .. . . .

πu +2φ

2 cos K 2 K cos πu2 K 2πvK +πuK +2φK 2 cos cos πu2 K 2 ...

··· ··· .. .

2 2 2 2 cos πu 2 2 3πu2 +2φ2 2 2 cos cos πu 2 2 3πu2 +2πv2 +2φ2 2 2cos cos πu 2 2 .. .

2 cos

2 cos 2 cos

T1

Sensors 2017, 17, 840

5 of 15

We also get the following relationship: J4 B = J3 BD,

(7)

 where D = diag tan( πv2 1 ), tan( πv2 2 ), · · · , tan( πv2 K ) is a real-valued matrix whose diagonal elements also contain the desired angle information. 3.2. 2D-DOA Estimation According to Equation (5), the estimation of covariance matrix R of y(t) is denoted by collecting L snapshots: 1 L R = ∑ y( t i )y H ( t i ). (8) L i =1 " # Rx1 From Equation (8), R can be denoted by R = , where Rx1 ∈ RK ×2MN , Rx2 Rx2 ∈ R(2MN −K )×2MN . In the noiseless case, Rx1 pr = Rx2 , an estimation matrix pr can be obtained by [21]: −1

H H pˆ r = (Rx1 Rx1 ) Rx1 Rx2 . (9) " # IK We construct a new matrix ps = , where IK is the identity matrix. In the noiseless case, Pˆ rH the relationship between ps and B can be obtained by a unique non-singular matrix T as

ps = BT.

(10)

Substituting Equation (10) into Equation (6), we can get J2 ps T−1 = J1 ps T−1 G.

(11)

If we define P1 = (J1 ps )† J2 ps , we then have P1 = T−1 GT.

(12)

Equation (12) shows that the diagonal elements of the matrix G can be obtained by performing the eigenvalue decomposition of P1 , and T is the corresponding eigenvector. Then, we can get the estimation of uˆ k : uˆ k = 2arctan( pˆ k )/π,

(13)

where pˆ k is the kth diagonal element of the matrix G. Similarly, Substituting Equation (10) into Equation (7), we can also get J4 ps T−1 = J3 ps T−1 D.

(14)

If we define P2 = (J4 ps )† J3 ps , we then have P2 = T−1 DT.

(15)

vˆ k = 2arctan(rˆk )/π,

(16)

Then, we get the estimation of vˆ k :

where rˆk is the kth diagonal element of the matrix D.

Sensors 2017, 17, 840

6 of 15

We note that uˆ k and vˆ k share the same eigenvector T, so the pairing is automatically formed. Thus, 2D-DOA can be obtained by q  2 2 ˆθk = sin−1 uˆ k + vˆ k , (17) φˆ k = tan−1 (uˆ k /vˆ k ).

(18)

We have now achieved the essence of the proposed algorithm. The major algorithmic steps are as follows: (1) (2) (3)

Construct the matrix y(t) from Equation (5), and compute the covariance matrix R of y(t) through Equation (8). Estimation of the propagator pr from Equation (9), and then construct the matrix ps . Construct the matrix J1 ps and J2 ps and perform the eigenvalue decomposition of

(4)

P1 = (J1 ps )† J2 ps . Similarly, construct the matrix J3 ps and J4 ps and perform the eigenvalue decomposition of P2 = (J3 ps )†eJ4 ps .

(5)

Finally, estimate the 2D-DOA through Equations (17) and (18).

Remark 1. In [23], the conventional PM algorithm divides the steering matrix A into two matrices A1 ∈ CK ×K and A2 ∈ C( MN −K )×K , and A2 is the linear transformation of A1 , i.e., A2 = Pm H A1 , Pm ∈ CK ×( MN −K ) is the propagator operator. According to Equation (1), x(t) = AS(t) + n(t), and the covariance " # matrix L R p1 of received data x(t) ∈ C MN ×1 is R p = L1 ∑ x(ti )x H (ti ). We partition it as R p = , where R p2 i =1 −1

H R p1 ∈ RK × MN , R p2 ∈ R( MN −K )× MN , and we can get the propagator estimator pˆ m = (R p1 R H p1 ) R p1 R p2 . " # xR (t) In our paper, according to Equation (5), y(t) = = BSo (t) + nr (t), and we compute the covariance x I (t)

of y(t) ∈ R2MN ×1 to estimate the propagator. Apparently, the available array aperture of the proposed algorithm can be thought of as twice that of the conventional 2D-PM [23], so it has better angle performance than 2D-PM. " Remark 2. In [23], define pc =

IK H Pˆ m

# , and then pc A1 = A, which means that the columns in pc span the

same signal subspace as the column vectors in A. Divide pc into pc1 ∈ C M( N −1)×K and pc2 ∈ C M( N −1)×K , pc1 , pc2 are the first M( N − 1) rows and the last M( N − 1) rows of pc . Then, get the relationship, P+ c1 Pc2 = A1 ΦA1 , + − jπu − jπu − jπu 2 K 1 where Φ = diag(e ,e ,··· ,e ). Perform the eigenvalue decomposition of Pc1 Pc2 to obtain the diagonal elements of the matrix Φ. Similarly, reconstruct pc to p0c , p0c1 , p0c2 being the first N ( M − 1) rows and the last N ( M − 1) rows of p0c , and perform the eigenvalue decomposition of P0c1 + P0c2 to obtain the diagonal elements of the matrix Φ x , where Φ x = diag(e− jπv1 , e− jπv2 , · · · , e− jπvK ). Finally, the 2D-DOA can be obtained from the diagonal elements of Φ and Φ x . From the above mentioned, the row dimensions of pc1 , pc2 and p0c1 , p0c2 are equal to M( N − 1), ( M − 1) N, respectively. The maximum number of the identified sources is min[ M ( N − 1), ( M − 1) N ]. In our proposed algorithm, from Equation (11) and Equation (14), the row dimensions of J1 ps and J2 ps , J3 ps and J4 ps are equal to M ( N − 1), 2( M − 1) N, respectively. Therefore, the maximum number of the identified sources is min [M( N − 1), 2( M − 1) N]. If M < N, the proposed algorithm can discern more sources than that of the conventional 2D-PM [23]. Remark "3. In the NC-PM algorithm [27], according to Equation (2), the extended array output data is denoted # X as Y = = Anc So + Nnc , where Anc ∈ C2MN ×K , J MN is the MN × MN exchange matrix J MN X∗ with ones on its anti-diagnoal and zeros elsewhere, and X∗ ∈ C MN × L stands for the complex conjugation of X, Y ∈ C2MN × L . Compute the covariance of Y to estimate the propagator pnc . Similarly, the invariance

Sensors 2017, 17, 840

7 of 15

equations for pnc are constructed to estimate the 2D-DOA. As is known to us, each computation amount of the complex multiplication is four times that of the real-valued one. In our algorithm, we use Euler transformation to convert complex arithmetic of noncircular to real arithmetic. For example, according to Equation (5), y(t) = BSo (t) + nr (t), y(t) ∈ R2MN ×1 , and the computation amounts of covariance of y(t) with snapshots L are much lower than that of Y [27]. Due to real-valued processing, our algorithm can save about 75% computational load compared with the NC-PM algorithm [27]. 4. Cramer-Rao Bounds and Analysis 4.1. CRB In this section, we give the Cramer-Rao Bounds (CRB) of noncircular signal for rectangular planar array. According to Equation (5), the received data is " y( t ) =

xR (t) x I (t)

#

= BSo (t) + nr (t),

(19)

where B = [b1 , b2 , · · · , bK ] ∈ C2MN ×K , and nr (t) is the noise vector. The Fisher information matrix (FIM) in relation to φ = [φ1 , φ2 , · · · , φK ] and θ = [θ1 , θ2 , · · · , θK ] can be calculated as follows [29]: " F=

F11 F21

#

F12 F22

.

(20)

According to [29], we know that the (i, j) ith element of F11 is given by F( θ i , θ j )

.

.

H

= 2Re[trace(Bθi So ) Γ−1 (Bθ j So )] .

.

H

= 2Re[trace(Bθ ei eiT So ) Γ−1 (Bθ e j eTj So )] .

.H

= Re[trace(SoH ei eiT Bθ Γ−1 Bθ e j eTj So )] = =

.

(21)

. .H Re[trace(eiT Bθ Γ−1 Bθ e j )(eTj So SoH ei )] . .H 2LRe[(Bθ Γ−1 Bθ )ij (RsTo )ij ]

Likely, we can give the (i, j) ith element of F12 , F21 , F22 : .H

.

.H

.

.H

.

F(θi , φj ) = 2LRe[(Bθ Γ−1 Bφ )ij (RsTo )ij ],

(22)

F(φi , θ j ) = 2LRe[(Bφ Γ−1 Bθ )ij (RsTo )ij ],

(23)

F(φi , φj ) = 2LRe[(Bφ Γ−1 Bφ )ij (RsTo )ij ], .

(24) .

∂B ∂B ∂B ∂B where ei denotes the ith column of the unit matrix, Rso = L1 So SoH , Bς i = ∂ς , Bξ = [ ∂ξ , , · · · , ∂ξ ], K 1 ∂ξ 2 i   2 σ 2MN 0 ··· 0   2 0 σ · · · 0   2MN  , σ2 is the covariance of the noise. Γ =  According to .. .. .. ..  . . . .   0 0 · · · σ2 2MN Equations (21)–(24), we can obtain: .H

.

.H

.

F11 = 2LRe[(Bθ Γ−1 Bθ ) ⊕ (RsTo )], F12 = 2LRe[(Bθ Γ−1 Bφ ) ⊕ (RsTo )],

(25) (26)

Sensors 2017, 17, 840

8 of 15

.H

.

.H

.

F21 = 2LRe[(Bφ Γ−1 Bθ ) ⊕ (RsTo )],

(27)

F22 = 2LRe[(Bφ Γ−1 Bφ ) ⊕ (RsTo )],

(28)

where ⊕ represents Hadamard product. Then, the CRB can be denoted as: CRB = F−1 .

(29)

We present the curves of CRB versus different signal to noise ratios (SNRs) and snapshots L in Figures 2 and 3. The source number K is fixed at 3 M and N represents the numbers of sensors on the x-axis and the y-axis. In Figure 2, the snapshot L is fixed at 200. It is obvious that, with the improvement of SNR, the value of CRB decreases accordingly. In Figure 3, we set SNR at 20 dB, and the curve shows that the value of CRB decreases with increase of L, and simulation results and theory Sensors 2017, 17, 840 9 of 16 analysis are17, consistent. Sensors 2017, 840 9 of 16 1

101 10

M=8,N=6 M=8,N=6 M=8,N=10 M=8,N=10 M=10,N=12 M=10,N=12 M=12,N=14 M=12,N=14

0

CRB CRB

100 10

-1

10-1 10

-2

10-2 10

-3

10-3 10 0 0

5 5

10 10 SNR SNR

15 15

20 20

Figure 2. CRB comparison versus SNR. Figure 2. CRB comparison versus SNR. Figure 2. CRB comparison versus SNR. 0

CRB CRB

100 10

M=8,N=6 M=8,N=6 M=8,N=10 M=8,N=10 M=10,N=12 M=10,N=12 M=12,N=14 M=12,N=14

-1

10-1 10

-2

10-2 10

50 50

100 100

150 150 L L

200 200

250 250

300 300

Figure 3. CRB comparison versus snapshots L. Figure 3. CRB comparison versus snapshots Figure 3. CRB comparison versus snapshots L. L.

4.2. Complexity Analysis 4.2.Complexity ComplexityAnalysis Analysis 4.2. In this section, we analyse the computational complexity of the algorithm specifically. First, In this this section, we the computational complexity of the specifically. First, 2 2 In section, weanalyse analyse the of algorithm themultiplications algorithm specifically. N2 L ) real-valued estimation of the covariance matrix Rcomputational requires O(4 M2complexity (RMS). In 2 2 O(4 M N OL()4M estimation of the of covariance matrix R requires real-valued multiplications (RMS). In First, estimation the covariance matrix R requires multiplications 2 N L) 2 real-valued 2 3 addition, the estimation of the matrix p s takes O(2 MNK2  4 M2 N2 K  K3 ) RMS. Then, the addition, the estimation of the matrix p s takes O(2 MNK  4 M N K  K ) RMS. Then, the estimation and eigenvalue decomposition of the matrix P1 and P2 totally require estimation and eigenvalue2 decomposition of3 the matrix P1 and P2 totally require 2 2 O(4 M2 NK ( N  1)  3M ( N  1) K2  8M ( M  1) N2 K  2 K3 ) RMS. Therefore, the overall computational O(4 M NK ( N  1)  3M ( N  1) K  8M ( M  1) N 2K  2 K ) RMS. Therefore, the overall computational 2 2 2 2 3 2 2 2 complexity of our algorithm is O(4 M2 N2 L  5MNK2  16 M2 N2 K  3K3  4 M2 NK  3MK2  8MN2 K ) complexity of our algorithm is O(4 M N L  5MNK  16 M N K  3K  4 M NK  3MK  8MN K )

Sensors 2017, 17, 840

9 of 15

(RMS). In addition, the estimation of the matrix ps takes O(2MNK2 + 4M2 N2 K + K3 ) RMS. Then, the estimation and eigenvalue decomposition of the matrix P1 and P2 totally require O(4M2 NK( N − 1) + 3M( N − 1)K2 + 8M( M − 1) N2 K + 2K3 ) RMS. Therefore, the overall computational complexity of our algorithm is O(4M2 N2 L + 5MNK2 + 16M2 N2 K + 3K3 − 4M2 NK − 3MK2 − 8MN2 K) RMS. As we know that each computation amount of the complex multiplication is four times that of the real-valued one, we can show the Chen’s noncircular propagator algorithm [27] needs O(16M2 N2 L + 8MNK2 + 16M2 N2 K + 3K3 + 16M( N − 1)K2 ) RMS, J’s noncircular ESPRIT [25] needs O(16M2 N2 L + M3 N3 + 8M( N − 1)K2 + 3K3 + 8N ( M − 1)K2 ) RMS, Zhang’s 2D-ESPRIT algorithm [8] needs O(4M2 N2 L + M3 N3 + 4M( N − 1)K2 + 4N ( M − 1)K2 + 6K3 ) RMS, while Li’s 2D-PM [23] requires O(4M2 N2 L + 4MNK2 + 4M2 N2 K + 6K3 + 4M( N − 1)K2 + 4N ( M − 1)K2 ) RMS. The complexity comparisons with different parameters are shown in Figures 4 and 5. In Figure 4, Sensors 2017, 2017, 17, 840 840 10 of of 16 16 Sensors 10 the numbers17, of sensor M and N on the x-axis and the y-axis are set at 8 and 6, respectively. The source number K isL fixed In From FigureFigures 5, the parameters andobserve K are the same as Figure 4,algorithm and the snapshot snapshot is set set at to 3. 100. and 5, 5, we weNcan can that the proposed proposed has much much snapshot L is to 100. From Figures 44 and observe that the algorithm has L lower is set to 100. From Figures 4 and 5, we can observe that and the proposed algorithm has much lower computational load than J’s NC-ESPRIT algorithm Chen’s NC-PM algorithm. lower computational load than J’s NC-ESPRIT algorithm and Chen’s NC-PM algorithm. computational load than J’s NC-ESPRIT algorithm and Chen’s NC-PM algorithm. complexity/the complexity/the number number of of multiplication multiplication

7

107 10

6

106 10

5

105 10

J-algorithm J-algorithm Chen-algorithm Chen-algorithm Zhang-algorithm Zhang-algorithm Li-algorithm Li-algorithm the proposed proposed algorithm algorithm the

4

104 10

00

20 20

40 40

60 60

80 80

100 100

L/snapshots L/snapshots

120 120

140 140

160 160

180 180

Figure 4. 4. Complexity Complexity comparison comparison versus versus L. L. Figure Figure 4. Complexity comparison versus L.

complexity/the complexity/the number numberof of multiplication multiplication

8

108 10

J-algorithm J-algorithm Chen-algorithm Chen-algorithm Zhang-algorithm Zhang-algorithm Li-algorithm Li-algorithm the proposed proposed algorithm algorithm the

7

107 10

6

106 10

5

105 10

44

55

66

77

88

M M

99

10 10

11 11

12 12

Figure 5.Complexity Complexitycomparison comparison versus M. Figure 5.5. versus M. Figure Complexity comparison versus M.

We can can summarize summarize the the merits merits of of the the proposed proposed algorithm algorithm as as follows: follows: We (1) The The proposed proposed algorithm algorithm has has much much lower lower computational computational load load than than the the NC-PM NC-PM and and NC-ESPRIT NC-ESPRIT (1) algorithms because the proposed algorithm uses Euler transformation to convert complex algorithms because the proposed algorithm uses Euler transformation to convert complex arithmetic of noncircular PM to real arithmetic. arithmetic of noncircular PM to real arithmetic.

Sensors 2017, 17, 840

10 of 15

We can summarize the merits of the proposed algorithm as follows: (1)

The proposed algorithm has much lower computational load than the NC-PM and NC-ESPRIT algorithms because the proposed algorithm uses Euler transformation to convert complex arithmetic of noncircular PM to real arithmetic. (2) The proposed Sensors 2017, 17, 840 algorithm has better estimation performance than the 2D-PM algorithm because 11 of 16 the array aperture is doubled according to Equation (5). 5. Simulation Results (3) The maximum number of discerned sources of our algorithm is dependent on Equation (5) and the real-valued PM method. Obviously, the maximum number of the identified sources of our In this section, we use Monte Carlo simulations to verify the performance of the algorithm. In proposed algorithm is min[ M( N − 1), 2( M − 1) N ], while 2D-PM is min[ M( N − 1), ( M − 1) N ]. the simulation, the rectangular planar array is configured with N subarrays, each subarray contains (4) The proposed no extraand matching estimated 2D-DOA M sensors, L is thealgorithm snapshotsrequires of the sources, K is thecalculation. number of The the sources. We assumecan that automatically be matched. there are K = 3 non-coherent sources, which are BPSK modulated in Figures 4–8, where

) , (2 , 2 )  (25 , 20 ) and 5.( Simulation Results 1 , 1 )  (15 ,10 o

o

o

o

(3 ,3 )  (35o ,30o ) , respectively.

mean error (RMSE) is used to forverify performance assessment, which is defined InThe thisroot section, wesquared use Monte Carlo simulations the performance of the algorithm. In theas    contains and k M for  sensors, L is the snapshots of the sources, and K is the number of the sources. We assume that there theKnth are = 3trial. non-coherent sources, which are BPSK modulated in Figures 4–8, where (θ1 , φ1 ) = (15◦ , 10◦ ), Figure gives the(angle of the proposed algorithm with 50 Monte Carlo trials, ◦ ) and (θ2 , φ2 ) = (25◦6, 20 θ3 , φ3 )pairing = (35◦ , results 30◦ ), respectively. M  8 , N  6 , L  200 and SNR is 10 dB. From Figure 4, we can observe that the where The qroot mean squared error (RMSE) is used for performance assessment, which is defined as 2D-DOAs of 1000 all three sources are localized clearly and paired automatically, which proves the 2 2 1 K 1 ˆ ˆ ˆ ˆ K ∑k=1 1000 ∑n=1 (θk,n − θk ) + (φk,n − φk ) , where θk,n , φk,n are the estimated value of θk and φk for effectiveness of our algorithm. the nth trial.



K 1000 1 1 simulation, the rectangular array is configured N subarrays, each subarray 2 planar 2 ˆ , ˆ with ˆ ˆ n1 (k ,n k )  (k ,n k ) , where k ,n k ,n are the estimated value of k K k 1 1000

40

Elevation/degree

35

30

25

20

15

10

5

10

15

20

25

30

35

40

Azimuth/degree Figure Figure6.6.Angle Angleestimation estimationresults. results.

Figure7a,b 7a,b presents presents RMSE RMSE comparison at J’s Figure at different different SNRs SNRsamong amongthe theproposed proposedalgorithm, algorithm, NC-ESPRIT algorithm [25], Chen’s NC-PM algorithm [27], Zhang’s 2D-ESPRIT algorithm [8], Li’s J’s NC-ESPRIT algorithm [25], Chen’s NC-PM algorithm [27], Zhang’s 2D-ESPRIT algorithm [8], 2D-PM algorithm [23] and In Figure 5a, we 5a, set M 6, NM = 8, 100. 5b, In weFigure change5b, the Li’s 2D-PM algorithm [23] CRB. and CRB. In Figure we= set = L6,= N = In 8, Figure L = 100. numbers sensors and snapshots M = 8, N =and 8, and = 50. From 5a,b,the we we changeofthe numbers of sensorsand andsetsnapshots setLM = 8, N =the 8, curves and L of = Figure 50. From know ofthat the5a,b, proposed algorithm has betteralgorithm RMSE performance thanperformance Li’s algorithm [23]. curves Figure we know that the proposed has better RMSE than Li’s Furthermore, has close RMSE performance to performance Chen’s algorithm [27]. However, should know algorithm [23].itFurthermore, it has close RMSE to Chen’s algorithmwe [27]. However, that our algorithm lowerhas computational amount than J’s NC-ESPRIT and we should know thathas ourmuch algorithm much lower computational amount than J’salgorithm NC-ESPRIT Chen’s NC-PM algorithm owing to the real-valued meanswhich that it means is morethat suitable algorithm and Chen’s NC-PM algorithm owing to theprocessing, real-valuedwhich processing, it is for asuitable practicalfor application more a practicalsystem. application system. Figure 8 presents RMSE performance comparisons at different snapshots L. Where M = 8, N = 6, SNR is varied from 0 dB to 20 dB. We can observe that the RMSE performance is improved with the increase of snapshot L. When L increases, we get more samples to estimate the propagator matrix more accurately, and so the angle estimation performance is enhanced.

Sensors 17, 840 Sensors 2017,2017, 17, 840 Sensors 2017, 17, 840

12 11of of16 15 12 of 16

2

10 2

the proposed algorithm

10

Chen-NC-PM[27] the proposed algorithm J-NC-ESPRIT[25] Chen-NC-PM[27] Zhang-2D-ESPRIT[8] J-NC-ESPRIT[25] Li-2D-PM[23] Zhang-2D-ESPRIT[8] CRB Li-2D-PM[23]

1

10 1

RMSE/degree RMSE/degree

10

CRB 0

10

0

10

-1

10

-1

10

-2

10 -2

10

-10

-5

-10

0

-5

0

5

(a)

2

10

5 SNR/dB SNR/dB

15

10

15

(a)

10

the proposed algorithm

2

10

Chen-NC-PM[27] the proposed algorithm J-NC-ESPRIT[25] Chen-NC-PM[27] Zhang-2D-ESPRIT[8] J-NC-ESPRIT[25] Li-2D-PM[23] Zhang-2D-ESPRIT[8] CRB Li-2D-PM[23]

1

10 1

RMSE/degree RMSE/degree

10

CRB 0

10

0

10

-1

10

-1

10

-2

10 -2

10

-10

-5

-10

0

-5

0

5

10

5 SNR/dB SNR/dB

15

10

15

(b) (b)

Figure 7. The root mean squared error (RMSE) comparison of different algorithms versus SNR. (a) M Figure 7. The root mean squared error (RMSE) comparison of different algorithms versus SNR. Figure 7. The mean = 6, N = 8, Lroot = 100; (b) squared M = 8, Nerror = 8, L(RMSE) = 50. comparison of different algorithms versus SNR. (a) M (a)= 6, MN= =6,8,NL== 8, L = 100; (b) M = 8, N = 8, L = 50. 100; (b) M = 8, N = 8, L = 50. 2

10 2

L=50 L=50L=100 L=150 L=100 L=200 L=150

10

1

10

L=200

1

RMSE/degree RMSE/degree

10

0

10

0

10

-1

10

-1

10

-2

10 -2

10

0

0

2 2

4 4

6 6

8 8

10

10 12 SNR/dB SNR/dB

12

14 14

16 16

18 18

Figure 8. RMSE comparison at different values of L. Figure 8. RMSE comparison at different values of L. Figure 8. RMSE comparison at different values of L.

20 20

Sensors 2017, 17, 840 Sensors2017, 2017,17, 17,840 840 Sensors

13 of 16 12of of16 15 13

Figures 9 and 10 present RMSE versus different values of M or N, respectively. The snapshot L is fixed at 9200. it is versus indicated that RMSE is improved when M orL N Figures and In 10 addition, present RMSE different valuesperformance of M or N, respectively. The snapshot Figure 8 presents RMSE performance comparisons at different snapshots L. Where M = 8, N = 6, Multiple sensors enhance the aperture of the performance array as well is as improved diversity gain. Therefore, is increases. fixed at 200. In addition, it is indicated that RMSE when M or N it SNR is varied from 0 dB to 20 dB. We can observe that the RMSE performance is improved with the can improve the angle estimation increases. Multiple sensors enhanceperformance. the aperture of the array as well as diversity gain. Therefore, it increase of snapshot L. When L increases, we get more samples to estimate the propagator matrix more The estimation performance for two closely spaced sources is also investigated. Figure 11 can improve the angle estimation performance. accurately, and so the angle estimation performance is enhanced. depicts the scatter plot of 2D-DOA results for two closely sources. Where M = 11 8, N The estimation performance forestimation two closely spaced sources is spaced also investigated. Figure Figures 9 and 10 present RMSE versus different values of M or N, respectively. The snapshot L is = 10, SNR = 10 dB, L is 200. It results is shown algorithm well for the depicts the scatter plotthe of snapshot 2D-DOA estimation forthat two our closely spaced works sources. Where M =closely 8, N fixed at 200. In addition, it is indicated that RMSE performance is improved when M or N increases. spaced = 10, SNRsources. = 10 dB, the snapshot L is 200. It is shown that our algorithm works well for the closely Multiple sensors enhance the aperture of the array as well as diversity gain. Therefore, it can improve spaced sources. the angle estimation performance. 2

10

M=8,N=4 M=8,N=6 M=8,N=4 M=8,N=8 M=8,N=6 M=8,N=10 M=8,N=8 M=8,N=10

2

10

1

10 1

RMSE/degree RMSE/degree

10

0

10 0

10

-1

10 -1

10

-2

10 -2

10

0

2

0

4

2

6

4

6

8 8

10

12

SNR/dB 10 12

14 14

16 16

18

20

18

20

SNR/dB Figure 9. RMSE comparison at different N with M = 8. Figure 9. RMSE comparison at different N with M = 8. Figure 9. RMSE comparison at different N with M = 8. 2

10

M=4,N=8 M=6,N=8 M=4,N=8 M=10,N=8 M=6,N=8 M=12,N=8 M=10,N=8 M=12,N=8

2

10

1

10 1

RMSE/degree RMSE/degree

10

0

10 0

10

-1

10 -1

10

-2

10 -2

10

0

0

2 2

4 4

6 6

8 8

10

SNR/dB 10 12

12

14 14

16 16

18 18

20 20

SNR/dB Figure 10. RMSE comparison at different M with N = 8. Figure 10. RMSE comparison at different M with N = 8.

Figure 10. RMSE comparison at different M with N = 8.

The estimation performance for two closely spaced sources is also investigated. Figure 11 depicts the scatter plot of 2D-DOA estimation results for two closely spaced sources. Where M = 8, N = 10, SNR = 10 dB, the snapshot L is 200. It is shown that our algorithm works well for the closely spaced sources.

Sensors2017, 2017,17, 17,840 840 Sensors

1314ofof1516

28 26

Elevation/degree

24 22 20 18 16 14 12 10

8

9

10

11

12

13

14

15

Azimuth/degree Figure Figure11. 11.Scatter Scatterplot plotwith withclosely closelyspaced spacedsources. sources.

6.6.Conclusions Conclusions We Wehave havepresented presentedaanovel noveldirection directionfinding findingalgorithm algorithmfor foruniform uniformrectangular rectangular planar planar array. array. The characteristics of noncircular signal and Euler’s transformation are exploited to get the real-valued The characteristics of noncircular signal and Euler’s transformation are exploited to get the rectangular data in a array new way. algorithm reduce the computational amount real-valuedarray rectangular dataThe in proposed a new way. The can proposed algorithm can reduce the since it does not refer to plural operation and the eigenvalues’ decomposition of the covariance matrix. computational amount since it does not refer to plural operation and the eigenvalues’ decomposition The theory analysismatrix. and simulation results verify that our algorithm is verify more suitable real-timeis of the covariance The theory analysis and simulation results that ourfor algorithm processing system in engineering. more suitable for real-time processing system in engineering. Acknowledgments: This work is supported by China Scholarship Council Grants (201606835025), National Acknowledgments: This work is supported China Scholarship Council Grants (201606835025), National Natural Science Foundation of China (61371169)by and the Electronic & Information School of Yangtze University Natural Science Foundation of China (61371169) and the Electronic & Information School of Yangtze University Innovation Foundation (2016-DXCX-05). Xiaofei Zhang participated the design of the algorithm. He gave the Innovation Foundation (2016-DXCX-05). Xiaofei Zhang participated the design of the algorithm. He gave the paper many suggestions. paper many suggestions. Author Contributions: Lingyun Xu wrote the paper and revised the manuscript. Fangqing Wen carried out the simulation. All authors read and approved manuscript. Author Contributions: Lingyun Xu wrotethe thefinal paper and revised the manuscript. Fangqing Wen carried out the simulation. Xiaofei Zhang participated designofofinterest. this algoirthm. All authors read and approved the final Conflicts of Interest: The authors declareinnothe conflict manuscript.

Abbreviations

Conflicts of Interest: The authors declare no conflict of interest. 2D-DOA Two-dimensional direction of arrival URA Uniform rectangular planar array Abbreviations PM Propagator method 2D-DOA Two-dimensional direction of arrival CRB Crame–Rao bound URA Uniform rectangular planar array RMS Real-valued multiplications PM Propagator method RMSE Root mean square error CRB Crame–Rao bound SNR Signal-to-noise ratio RMS Real-valued multiplications RMSE

References SNR 1.

Root mean square error Signal-to-noise ratio

Min, S.; Seo, D.; Lee, K.B.; Kwon, H.M.; Lee, Y.H. Direction-of-arrival tracking scheme for DS/CDMA

References systems: Direction lock loop. IEEE Trans. Wirel. Commun. 2004, 3, 191–202. [CrossRef] 2.1. 3. 2. 4. 3.

Chiang, A.C. DOA estimation theY.H. asynchronous DS-CDMAtracking system. scheme IEEE Antennas Propag. Min, S.;C.T.; Seo,Chang, D.; Lee, K.B.; Kwon, H.M.; in Lee, Direction-of-arrival for DS/CDMA 2003, 51, 40–47. [CrossRef] systems: Direction lock loop. IEEE Trans. Wirel. Commun. 2004, 3, 191–202. Yang, S.; Li, Y.; Zhang, K.; Tang, W. Multiple-parameter estimation method based on spatio-temporal 2-D Chiang, C.T.; Chang, A.C. DOA estimation in the asynchronous DS-CDMA system. IEEE Antennas Propag. processing for bistatic MIMO radar. Sensors 2015, 15, 31442–31452. [CrossRef] [PubMed] 2003, 51, 40–47. Zhang, X.F.; Zhou, M.; Li, J.F. A PARALIND decomposition-based coherent two-dimensional direction Yang, S.; Li, Y.; Zhang, K.; Tang, W. Multiple-parameter estimation method based on spatio-temporal 2-D of arrival estimation algorithm for acoustic vector-sensor arrays. Sensors 2013, 13, 5302–5316. [CrossRef] processing for bistatic MIMO radar. Sensors 2015, 15, 31442–31452. [PubMed]

Sensors 2017, 17, 840

5. 6. 7. 8. 9. 10. 11. 12. 13. 14.

15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25.

26. 27.

14 of 15

Liang, H.; Cui, C.; Dai, L.; Yu, J. Reduced-dimensional DOA estimation based on ESPRIT algorithm in MIMO radar with L-shaped array. J. Electron. Inf. Technol. 2015, 37, 1828–1835. Wax, M.; Shan, T.J.; Kailath, T. Spatio-temporal spectral analysis by eigenstructure methods. IEEE Trans. Acoust. Speech Signal Process. 1984, 32, 817–827. [CrossRef] Zoltowski, M.D.; Haardt, M.; Mathews, C.P. Closed-form 2-D angle estimation with rectangular arrays in element space or beamspace via unitary ESPRIT. IEEE Trans. Signal Process. 1996, 44, 316–328. [CrossRef] Zhang, X.; Gao, X.; Chen, W. Improved blind 2D-direction of arrival estimation with L-shaped array using shift invariance property. J. Electromagn. Waves Appl. 2009, 23, 593–606. [CrossRef] Del Rio, J.E.F.; Catedra-Perez, M.F. The matrix pencil method for two-dimensional direction of arrival estimation employing an L-shaped array. IEEE Trans. Antennas Propag. 1997, 45, 1693–1694. [CrossRef] Clark, M.P.; Scharf, L.L. Two-dimensional modal analysis based on maximum likelihood. IEEE Trans. Signal Process. 1994, 42, 1443–1452. [CrossRef] Fang, W.H.; Lee, Y.C.; Chen, Y.T. Maximum likelihood 2-D DOA estimation via signal separation and importance sampling. IEEE Antennas Wirel. Propag. Lett. 2016, 16, 746–749. [CrossRef] Zhang, X.F.; Li, J.F.; Xu, L.Y. Novel two-dimensional DOA estimation with L-shaped array. EURASIP J. Adv. Signal Process. 2011, 50, 1–7. [CrossRef] Nie, X.; Li, L.P. A computationally efficient subspace algorithm for 2-D DOA estimation with L-shaped array. IEEE Signal Process. Lett. 2014, 21, 971–974. Wang, G.M.; Xin, J.M.; Zheng, N.N.; Sano, A. Computationally efficient subspace-based method for two-dimensional direction estimation with L-shaped array. IEEE Trans. Signal Process. 2011, 59, 3197–3212. [CrossRef] Wei, Y.S.; Guo, X.J. Pair-matching method by signal covariance matrices for 2D-DOA estimation. IEEE Antennas Wirel. Propag. Lett. 2014, 13, 1199–1202. Zhang, W.; Liu, W.; Wang, J.; Wu, S.L. Computationally efficient 2D DOA estimation for unform rectangular arrays. Multidimens. Syst. Signal Process. 2014, 25, 847–857. [CrossRef] Yu, H.X.; Qiu, X.F.; Zhang, X.F. Two-dimensional direction of arrival estimation for rectangular array via compressive sensing trilinear model. Int. J. Antenna Propag. 2015, 2015, 297572. [CrossRef] Xu, X.; Ye, Z. Two-dimensional direction of arrival estimation by exploiting the symmetric configuration of uniform rectangular array. IET Radar Sonarav. Navig. 2012, 6, 307–313. [CrossRef] Wu, H.; Hou, C.P.; Chen, H.; Liu, W.; Wang, Q. Direction finding and mutual coupling estimation for uniform rectangular arrays. Signal Process. 2015, 117, 61–68. [CrossRef] Gu, J.F.; Zhu, W.P.; Swamy, M.N.S. Joint 2D DOA estimation via sparse L-shaped array. IEEE Trans. Signal Process. 2015, 63, 1171–1182. [CrossRef] Marcos, S.; Marsal, A.; Benider, M. The propagator method for sources bearing estimation. Signal Process. 1995, 42, 121–138. [CrossRef] Wu, Y.; Liao, G.S.; So, H.C. A fast algorithm for 2-D direction of-arrival estimation. Signal Process. 2003, 83, 1827–1831. [CrossRef] Li, J.F.; Zhang, X.F.; Chen, H. Improved two-dimensional DOA estimation algorithm for two-parallel uniform linear arrays using propagator method. Signal Process. 2012, 92, 3032–3038. [CrossRef] Abeida, H.; Delmas, J.P. MUSIC-like estimation of direction of arrival for noncircular sources. IEEE Trans. Signal Process. 2006, 54, 2678–2690. [CrossRef] Steinwandt, J.; Roemer, F.; Haardt, M. Performance analysis of ESPRIT-type algorithms for non-circular sources. In Proceedings of the 38th IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP 13), Vancouver, BC, Canada, 26–31 May 2013; pp. 3986–3990. Zhang, L.; Lv, W.H.; Zhang, X.F. 2D-DOA estimation of noncircular signals for uniform rectangular array via NC-PARAFAC method. Int. J. Electron. 2016, 103, 1839–1856. [CrossRef] Chen, C.Q.; Wang, C.H.; Zhang, X.F. DOA and Noncircular phase estimation of noncircular signal via an improved noncircular rotational invariance propagator method. Math. Probl. Eng. 2015, 2015, 235173. [CrossRef]

Sensors 2017, 17, 840

28. 29.

15 of 15

Picinbono, B. On circularity. IEEE Trans. Signal Process. 1994, 42, 3473–3482. [CrossRef] Stoica, P.; Nehorai, A. Performance study of conditional and unconditional direction-of-arrival estimation. IEEE Trans. Acoust. Speech Signal Process. 1990, 38, 1783–1795. [CrossRef] © 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

Fast Noncircular 2D-DOA Estimation for Rectangular Planar Array.

A novel scheme is proposed for direction finding with uniform rectangular planar array. First, the characteristics of noncircular signals and Euler's ...
2MB Sizes 2 Downloads 8 Views