IEEE SIGNAL PROCESSING LETTERS, VOL. 24, NO. 1, JANUARY 2017
7
Direction-of-Arrival Estimation and Sensor Array Error Calibration Based on Blind Signal Separation Jianfei Liu, Xiongbin Wu, William J. Emery, Fellow, IEEE, Lan Zhang, Chuan Li, and Ketao Ma
Abstract—We consider estimating the direction-of-arrival (DOA) in the presence of sensor array error. In the proposed method, a blind signal separation method, the joint approximation and diagonalization of eigenmatrices algorithm, is implemented to separate the signal vector and the mixing matrix consisting of the array manifold matrix and the sensor array error matrix. Based on a new mixing matrix and the reconstruction of the array output vector of each individual signal, we propose a novel DOA estimation and sensor array error calibration procedure. This method is independent of array phase errors and performs well against difference of SNR of signals. Numerical simulations verify the effectiveness of the proposed method. Index Terms—Array calibration, directions-of-arrival (DOA) estimation, joint approximation and diagonalization of eigenmatrices (JADE).
I. INTRODUCTION IRECTION-OF-ARRIVAL (DOA) estimation has gained increasing attention due to its significance in practical applications, such as sonar, radar, and mobile communications [1]. In particular, various super-resolution algorithms, such as MUSIC [2], ESPRIT [3], and BCS [4] have demonstrated excellent DOA estimation performance. Most of these array signal processing techniques presume that the array manifold vector can be known precisely. Consequently, the presence of sensor array error will seriously degrade the performance of the DOA estimation, resulting in spurious directions and poor angular resolution. In recent decades, some algorithms have been developed for estimating DOAs of signals and sensor array error [5]–[10]. The method in [5] estimates array gain and phase errors and DOAs simultaneously, but it is limited to small array perturbations and suffers from suboptimal convergence. Although [6]–[8] do not require an iteration process; these methods are confined to linear arrays. In [9], an eigenstructure approach is presented to estimate DOAs and array error for nonlinear arrays. This study is based on the eigen decomposition of the covariance matrix,
D
Manuscript received September 20, 2016; revised October 30, 2016; accepted November 14, 2016. Date of publication November 24, 2016; date of current version December 15, 2016. This work was supported in part by 863 High Technology Project of China under Grant 2012AA091701, in part by the National Natural Science Foundation of China under Grant 61401316, and in part by China Scholarship Council under Grant 201506270074. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Monica F. Bugallo. (Corresponding author: Jianfei Liu.) J. Liu, X. Wu, L. Zhang, C. Li, and K. Ma are with the School of Electronic Information, Wuhan University, Wuhan 430072, China (e-mail:
[email protected];
[email protected];
[email protected]; whu_friday@ whu.edu.cn;
[email protected]). W. J. Emery is with the Department of Aerospace Engineering Sciences, University of Colorado, Boulder, CO 80309 USA (e-mail:
[email protected]). Color versions of one or more of the figures in this letter are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/LSP.2016.2632750
which is derived from the dot product of a received vector with its conjugate. But it requires at least two signals with spatial separation far from each other. Cao et al. estimate the DOAs using the eigen decomposition of the Hadamard product derived from a covariance matrix with its conjugate [10]. The approach proposed here subtracts the one component, which can cause a ridge close to the diagonal line of the spatial spectrum. Hence, it does not require that the two signals are spatially far apart. This strategy, however, needs to decompose the covariance matrix many times to calculate the power of the signals. Moreover, both [9] and [10] require at least K (K − 1) + 1 receiving sensors for the DOA estimation of K signals. In this letter, we utilize a novel strategy to solve the problem of estimating signal DOAs and the associated sensor array error. A blind signal separation method, the joint approximation and diagonalization of eigenmatrices (JADE) [11] algorithm, is implemented to separate the signal vector and mixing matrix. Gain errors are estimated by a conventional method in [12]. The array output vector of each individual signal is reconstructed by the estimated mixing matrix and the signal vector. Based on a new mixing matrix derived from the estimated sensor array error matrix and the array manifold matrix, a novel two-dimensional (2-D) spatial spectrum is formed. By locating the peak from the spatial spectrum, the DOAs are estimated. Then, the phase errors are obtained from the mixing matrix and the estimated DOAs. The presented method is independent of array phase errors and performs well against difference of SNR of signals. Additionally, this method only requires K + 1 sensors for the DOA estimation of K signals. II. SIGNAL MODEL Given that K narrow-band far-field signals impinge on an M -sensor array from directions θ = [θ1 , θ2 , . . . , θK ]T , where (·)T represents the transpose operation. The array output vector X (t) is described by X (t) = ΓAS (t) + N (t) , t = 1, 2, . . . , L
(1)
and S (t) = [s1 (t) , s2 (t) , . . . , sK (t)]T denotes the signal vector with zero mean. N (t) represents the additive noise vector with zero mean and σn2 variance, which is supposed to be spatially white. Furthermore, we suppose that the first sensor of the array is the reference sensor and its location is fixed to the origin of co-ordinates. A is the ideal array manifold matrix, which is given by
where
A = [a (θ1 ) , a (θ2 ) , . . . , a (θK )]
(2)
T 2π 2π a (θk ) = 1, e−j λ d 2 k , . . . , e−j λ d mk
(3)
dm k = [xm ym ] [sin θk cos θk ]T
1070-9908 © 2016 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications standards/publications/rights/index.html for more information.
(4)
8
IEEE SIGNAL PROCESSING LETTERS, VOL. 24, NO. 1, JANUARY 2017
and λ denotes the center wavelength of the signals. (xm , ym ) are co-ordinates of the mth sensor and dm k is the spatial distance between the mth sensor and the first sensor in the direction of the kth signal. Γ = diag [γ1 ,γ2 , . . . ,γM ] is an M × M diagonal array error matrix. In addition, the array error of the mth sensor can be expressed as γm = gm ej β m
(5)
where gm and βm are gain error and phase error of the mth sensor, respectively. Moreover, assume g1 = 1 and β1 = 0. Finally, define the mixing matrix as B = [b (θ1 ) , b (θ2 ) , . . . , b (θK )] = ΓA.
(6)
In this letter, we suppose that the signals are non-Gaussian and uncorrelated with each other. All the signals are independent of the noise and have different DOAs. III. NEW METHOD A. Mixing Matrix and Signal Vector Estimation The JADE algorithm has proven to be a successful algorithm to calculate the mixing matrix and the signal vector in terms of the computational complexity and estimation accuracy [13]. We apply the JADE algorithm for the signal vector and the mixing matrix estimation based on the array output vector X (t) and the number of signals K. The detail implementation of this algorithm is discussed in [11]. The JADE algorithm statistically ˆ and a unitary matrix U. ˆ Thus, calculates a whitening matrix W the mixing matrix can be estimated as ˆ J (θ1 ) , b ˆ J (θ2 ) , . . . , b ˆ J (θK ) = W ˆ JAD E = b ˆ (7) ˆ †U B †
where (·) is the pseudoinverse. As the first sensor of the array is the reference sensor, the mixing matrix estimated by (7) should be normalized by the reference element ˆbJ,1 (θ) ˆ J (θ2 ) ˆ J (θK ) ˆ J (θ1 ) b b b ˆ= , ,..., . (8) B ˆbJ,1 (θ1 ) ˆbJ,1 (θ2 ) ˆbJ,1 (θK )
t = 1, 2, . . . , L
(9)
The sample array covariance matrix is computed from the L data samples as (10)
where [μ1 , μ2 , . . . , μM ] is the eigenvalues in descending order ˆ and [h1 , h2 . . . , hM ] is the eigenvectors of R. 2 Then, the noise variance σn can be calculated from the M − ˆ K smallest eigenvalues of R 1 = M −K
M m =K +1
μm .
ˆ (θl ) = Γa (θl ) b
(13)
ˆ (θk ) = Γa (θk ) . b
(14)
ˆ k denote the estimation of Γ with (13) and (14), ˆ l and Γ Let Γ respectively. The array error matrix can be estimated as ˆ (θl ) a∗ (θl ) ˆ l = diag b (15) Γ ˆ (θk ) a∗ (θk ) ˆ k = diag b Γ (16) where and (·)∗ represent the Hadamard product and the conjugate operation, respectively. ˆ l a (θk ), and Define a new mixing matrix as ck (θl , θk ) = Γ insert (15) into it ˆ (θl ) a∗ (θl ) · a (θk ) ck (θl , θk ) = diag b ˆ (θl ) a∗ (θl ) a (θk ) . =b
(17)
Consider the signal model in (1) with no noise and (15). The array output vector of the kth signal is defined as ˆ l a (θk ) sk (t) = ck (θl , θk ) sk (t) . xk (t) = Γ
(18)
Based on the mixing matrix and signal vector of the kth signal estimated by (8) and (9), respectively, xk (t) can be reconstructed as (19)
The covariance matrix of x ˆk (t) is estimated as L
B. Gain Error Calibration
σ ˆn2
Gain errors are not considered in the following discussion, because they can be calibrated by (12). Consider two of the K signals from the directions θk and θl , where l, k = 1, 2, . . . , K and l = k. According to (6) and (8), the mixing matrix of the kth and lth signal can be written as
ˆH ˆ ˆk = 1 R x ˆk (t) x ˆH k (t) = Es (k) Λs Es (k) L t=1
where (·)H denotes the conjugate transpose.
L M ˆ = 1 R X (t) XH (t) = μm hm hH m L t=1 m =1
C. DOA Estimation
ˆ (θk ) ˆ x ˆk (t) = b sk (t) , t = 1, 2, . . . , L.
In addition, the signal vector can be obtained as ˆ (t) = U ˆ H WX ˆ (t) , S
ˆ (m, m) denote the mth diagonal element of R. ˆ Gain Let R errors can be estimated as [12] ˆ (m, m) − σ R ˆn2 (12) . gˆm = ˆ R (1, 1) − σ ˆn2
(11)
ˆ H (k) ˆn (k) Λn E +E n
(20)
ˆs (k) and E ˆn (k) represent the signal subspace and the where E noise subspace, respectively. As a consequence, we can present a novel DOA estimation method based on subspace principle. Since the noise subspace is orthogonal to the new mixing matrix ck (θl , θk ), a new 2-D spatial spectrum can be described by
−2
ˆH
(21) Pk (θ , θ) = E n (k) ck (θ , θ)
2
where ·2 represents the 2-norm of a vector. Let θˆlk denote the DOA of the lth signal estimated by the reconstruction of the kth signal. By locating the peak from the
LIU et al.: DIRECTION-OF-ARRIVAL ESTIMATION AND SENSOR ARRAY ERROR CALIBRATION BASED ON BLIND SIGNAL SEPARATION
9
TABLE I MINIMUM SENSORS REQUIRED FOR DOA ESTIMATION AGAINST NUMBER OF SIGNALS K
2
3
4
5
6
7
8
9
10
M (Our method) M ([9] and [10])
3 3
4 7
5 13
6 21
7 31
8 43
9 57
10 73
11 91
spatial spectrum Pk (θ , θ), we can estimate the DOAs as Pk (θ , θ) . θˆkk , θˆlk = arg max (22) θ ,θ
And the lth signal can also be reconstructed to calculate the spatial spectrum Pl (θ , θ), then the DOAs are estimated as Pl (θ , θ) . θˆkl , θˆll = arg max (23)
Fig. 1. Spatial spectrum of the three methods. (a) Spatial spectrum of [9]. (b) Spatial spectrum of [10]. (c) Spatial spectrum of the new method.
θ ,θ
IV. NUMERICAL SIMULATIONS Based on the above analysis, the DOAs of the kth and lth signals can be estimated as 1 1 l k l ˆk ˆ ˆ ˆ ˆ ˆ θk , θl = (24) θ +θ , θ +θ 2 k k 2 l l D. Phase Error Calibration As the DOAs have been estimated by (24), we can estimate phase errors by (15) and (16). Hence, the phase errors β = [β1 , β2 , . . . , βM ]T are obtained as ˆ (θl ) a∗ θˆl ˆ (θk ) a∗ θˆk + ∠ b ˆ= 1 ∠ b β 2 (25) where ∠ [·] denotes the phase of a complex number. In addition, phase errors can also be derived by the conventional algorithm in [5], which is used by [9] and [10]. E. Discussion First, we note that the JADE algorithm is applicable for mixing matrix estimation with non-Gaussian signals, but there are many non-Gaussian signal models in radar systems. Four of the five famous radar detection models described in [14] are non-Gaussian signal models [8]. The method presented also has value as a practical application. Second, for a linear array, the spatial distance is dm k = xm sin θk . Consider two pairs of DOAs (θ1 , θ2 ) = (θ3 , θ4 ), a∗ (θ1 ) a (θ2 ) may be equal to a∗ (θ3 ) a (θ4 ) due to sin θ1 − sin θ2 = sin θ3 − sin θ4 . Hence, this method is applicable to nonlinear arrays. In addition, if four sensors of an array are located on each vertex of a square with its side no more than λ 4 , respectively, the DOA estimation using the presented method is unambiguous [9]. Third, the proposed method only requires K + 1 sensors to estimate K signals while [9] and [10] requires at least K (K − 1) + 1 sensors. From Table I, we know that as K increases, M , as required by [9] and [10] increases more rapidly than that of the proposed new method. Thus, this will increase the complexity of the receiving system, which means more sensors, more receiving channels, and larger array.
We carry out some representative simulations to verify the validity of the approach presented. The DOAs of the signals are within the region −90◦ to 90◦ and we assume that two signals do exist. The array √ gain-phase errors of√the sensors are generated as gm = 1 + 12σg αm and βm = 12σβ ηm , respectively. In addition, αm and ηm are independent and random variables with uniform distributions over [−0.5, 0.5]. The standard deviations of gm and βm are σg and σβ , respectively. In the simulations below, we have, σg = 0.1, the samples L = 500 and the number of the Monte Carlo trials is 500. We use the following sensor configuration to get the unambiguous DOA estimation There sensors with lo λ [9]: are seven λ λ λ λ cations at (0, 0), , 0, , , 0 , , cos 80◦ , 4λ sin 80◦ , 4 4 4 4 4 λ ◦ λ ◦ ◦ λ ◦ λ 4 cos 160 , 4 sin 160 , and 4 cos 240 , 4 sin 240 . The methods discussed in [9] and [10] are selected for a performance comparison. As the three methods apply the same gain error calibration method in [12], we do not discuss it in the experiments below. Both [9] and [10] implement the same phase error calibration method presented in [5], and we compare it with the new calibration method presented by (25). Fig. 1 shows the 2-D spatial spectrum of the three methods, where DOAs are (−25◦ , 0◦ ), SNR is 20 dB, and σβ = 40◦ . There is only one distinct peak in the spectrum of the new method compared with the spectrum in [9] and [10]. By locating the peak of the spectrum, the DOAs are estimated. Fig. 2 shows root mean square error (RMSE) of the DOA estimation and the phase errors calibration against σβ , where DOAs are (−25◦ , 0◦ ) and SNR is 20 dB. This demonstrates that the three methods are independent of array phase errors and reveals that the new approach performs better. It can be clearly seen that this new method is efficient and achieves higher accuracy. Fig. 3 displays RMSE against DOA separation, where DOAs are (θ1 , θ2 ), SNR is 20 dB, and σβ = 40◦ . θ1 = −25◦ and DOA separation is described by Δθ = θ2 − θ1 . Δθ starts from 15◦ because [9] fails when Δθ is less than 15◦ . Simulation curves in Fig. 3 indicate that [9] performs poorly while the DOA separation is small. Both the new approach and [10] perform better as the DOA separation increases, but the new estimation method has a higher accuracy. Furthermore, the new calibration method outperforms [5].
10
IEEE SIGNAL PROCESSING LETTERS, VOL. 24, NO. 1, JANUARY 2017
Fig. 2.
Fig. 3.
RSME against σ β .
Fig. 5. 95% CI of the bias and variance of the DOA (from 0◦ ) estimation of the new method (in red), [9] (in green), and [10] (in blue) against the Monte Carlo trials. The horizontal lines are the lower or upper confidence limits. The dot and plus represent the data within and outside the CI, respectively.
RSME against DOA separation.
Fig. 6. ◦
RMSE against difference of SNR.
DOAs are (−25 , 0 ) and σβ = 40◦ , where the SNR of the signal from direction −25◦ is fixed to 20 dB and SNR of the other signal varies from 20 to 0 dB. Simulation curves depicted in Fig. 6 verify that [9] tends to deteriorate with the difference of SNR increasing. The performance of [10] degrades substantially when the power difference is above 15 dB, whereas the presented strategy still maintains considerable accuracy. Additionally, the two calibration methods have almost the same performance. The new calibration method performs better when difference of SNR is less than 12 dB, whereas [5] only shows a better performance when difference of SNR is above 17 dB. Fig. 4.
RMSE against SNR.
Fig. 4 displays RMSE against SNR, where DOAs are (−25◦ , 0◦ ) and σβ = 40◦ . As seen in Fig. 4, the performance of the three methods improves as the SNR increases. In addition, the new estimation method outperforms [9] and [10], and the new calibration method performs better than [5]. Here, we also show the 95% confidence interval (CI) of the bias and variance of the DOA (from 0◦ ) estimation against the Monte Carlo trials in Fig. 5, when SNR is 15 dB. The CI of the new method is narrower than that of [9] and [10]. At last, we implement simulation to verify the performance of the new method against difference of SNR of signals when
◦
V. CONCLUSION We propose a DOA estimation and sensor array error calibration method based on blind signal separation. The JADE algorithm is applied to separate the signal vector and mixing matrix. Based on a new mixing matrix and the reconstruction of array output vector, we present a novel DOA estimation and sensor array error calibration method. The new method presented is independent of array phase errors and performs well against difference of SNR of signals. Furthermore, this method requires fewer sensors than the methods in [9] and [10] when the number of signals is more than two. Numerical simulations demonstrate that the new method is effective.
LIU et al.: DIRECTION-OF-ARRIVAL ESTIMATION AND SENSOR ARRAY ERROR CALIBRATION BASED ON BLIND SIGNAL SEPARATION
REFERENCES [1] H. Krim and M. Viberg, “Two decades of array signal processing research: The parametric approach,” Signal Process. Mag., vol. 13, no. 4, pp. 67–94, 1996. [2] R. O. Schmidt, “Multiple emitter location and signal parameter estimation,” IEEE Trans. Antennas Propag., vol. AP-34, no. 3, pp. 276–280, Mar. 1986. [3] R. Roy and T. Kailath, “ESPRIT-estimation of signal parameters via rotational invariance techniques,” IEEE Trans. Acoust., Speech, Signal Process., vol. 37, no. 7, pp. 984–995, Jul. 1989. [4] M. Carlin, P. Rocca, G. Oliveri, F. Viani, and A. Massa, “Directions-ofarrival estimation through Bayesian compressive sensing strategies,” IEEE Trans. Antennas Propag., vol. 61, no. 7, pp. 3828–3838, Jul. 2013. [5] A. J. Weiss and B. Friedlander, “Eigenstructure methods for direction finding with sensor gain and phase uncertainties,” Circuits, Syst. Signal Process., vol. 9, no. 3, pp. 271–300, 1990. [6] A. Paulraj and T. Kailath, “direction-of-arrival estimation by eigenstructure methods with unknown sensor gain and phase,” in Proc. IEEE Int. Conf. Acoust., Speech, Signal Process., vol. 10, 1985, pp. 640–643. [7] Y. Li and M. Er, “Theoretical analyses of gain and phase error calibration with optimal implementation for linear equispaced array,” IEEE Trans. Signal Process., vol. 54, no. 2, pp. 712–723, Feb. 2006.
11
[8] J. Kim, H. J. Yang, B. W. Jung, and J. Chun, “Blind calibration for a linear array with gain and phase error using independent component analysis,” IEEE Antennas Wireless Propag. Lett., vol. 9, pp. 1259–1262, 2010. [9] A. Liu, G. Liao, C. Zeng, Z. Yang, and Q. Xu, “An eigenstructure method for estimating DOA and sensor gain-phase errors,” IEEE Trans. Signal Process., vol. 59, no. 12, pp. 5944–5956, Dec. 2011. [10] S. Cao, Z. Ye, D. Xu, and X. Xu, “A Hadamard product based method for DOA estimation and gain-phase error calibration,” IEEE Trans. Aerosp. Electron. Syst., vol. 49, no. 2, pp. 1224–1233, Apr. 2013. [11] J. Cardoso and A. Souloumiac, “Blind beamforming for non-Gaussian signals,” IEE Proc. F—Radar Signal Process., vol. 140, no. 6, pp. 362– 370, Dec. 1993. [12] M. Wylie, S. Roy, and R. Schmitt, “Self-calibration of linear equi-spaced (LES) arrays,” in Proc. IEEE Int. Conf. Acoust., Speech, Signal Process., vol. 1, 1993, pp. 281–284. [13] H.-C. Wu, X. Huang, Y. Wu, and X. Wang, “Theoretical studies and efficient algorithm of semi-blind ICI equalization for OFDM,” IEEE Trans. Wireless Commun., vol. 7, no. 10, pp. 3791–3798, Oct. 2008. [14] N. Levanon, Radar Principles. New York, NY, USA: Wiley, 1988, vol. 1.
