A Novel DOA Estimation Method for Closely Spaced Multiple Sources with Large Power Differences Yang Gao and Xin Jia
Jia Xu and Teng Long
Xiang-Gen Xia
Academy of Equipment Beijing 101416, China Email:
[email protected],
[email protected]
School of Information and Electronic Beijing Institute of Technology Beijing 100081, China Email:
[email protected],
[email protected]
Department of Electrical and Computer Engineering University of Delaware Newark, DE 19716, USA Email:
[email protected]
Abstract—The direction of arrival (DOA) estimation is a key issue in array signal processing, and there are many well-known methods, e.g., Capon, MUSIC, Root-MUSIC and some other DOA estimation algorithms in the literature. Nevertheless, the effective DOA estimation for closely spaced multiple sources with large power differences is still a difficult problem due to the deteriorated resolution for the low power sources. In this paper, a novel DOA estimation method called Capon-MUSIC algorithm is proposed, which has a much higher resolution for closely spaced multiple sources with obvious power differences. Finally, some numerical experiments are provided to demonstrate the effectiveness of the proposed method.
To solve this problem, a novel DOA estimation method, called Capon-MUSIC algorithm, is proposed in this paper based on the Capon algorithm and the MUSIC algorithm. Through introducing the synthetic spatial spectrum from the ratio of MUSIC spatial spectrum to Capon spatial spectrum, the Capon-MUSIC method can normalize the peak values in the spatial spectrum, which makes the weak sources’ peak easy to identify and then improve the DOA estimation performance. Finally, some numerical experiments are provided to demonstrate the effectiveness of the proposed method.
Keywords—direction of arrival (DOA), closely spaced sources, MUSIC algorithm, Capon algorithm, Root-MUSIC algorithm, Capon-MUSIC algorithm
Assume P uncorrelated plane waves are incident on a linear equispaced array of M sensors. The observation data of the signals plus noise are given as X = AS + N (1) where A = [a1 ," , aP ] denotes the array manifold matrix,
I.
INTRODUCTION
The direction of arrival (DOA) estimation can be extensively applied in the field of radar, sonar, communication and navigation, and it is one of the most important and active subjects of array signal processing. By now there have been many DOA estimation methods in the literature, which can be classified into two kinds, beamforming techniques and subspace-based methods [1-6]. The Capon’s beamformer [3], also known as the minimum variance distortionless response (MVDR) filter, can alleviate the limitations of the conventional beamformer on resolving two signals spaced closer than a beamwidth. Also, the multiple signal classification (MUSIC) algorithm has been regarded as one of the best bearing estimation methods since its publication by Schmidt in 1979 [4]. It is known that MUSIC can obtain an arbitrary high accuracy as well as high resolution for equal power sources if the number of snapshots and the SNR of sources are sufficiently high. Root-MUSIC [5] which is used in uniform linear array only can obtain better performance than MUSIC. Nevertheless, when the powers of different sources have obvious differences, the resolution and accuracy of multiple sources will be deteriorated accordingly [1-2,6]. Normally, the eigenvalues in eigenvalue spectrum and the peaks in the spatial spectrum will become smaller and difficult to identify for the low power sources due to the mutual effect among different sources.
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II.
ARRAY SIGNAL MODEL
S = [ s1 ;" ; sP ] denotes the signal vector; N = [n1 ;" ; nM ] denotes the noise vector of Gaussian white with zero mean. The i th steering vector is written as
ai = [1, e− jwi ,", e− j ( M −1) wi ]T , i = 1, 2," , P where wi =
2π d
λ
(2)
sin θi denotes the spatial angular frequency
corresponding to the angle of incidence θi , i = 1," , P , d denotes the element spacing, λ denotes the wavelength. The sample covariance matrix R are given as P
R = E[ XX H ] = ∑ σ i2 ai aiH + σ n2 I
(3)
i =1
where E[⋅] denotes statistical expectation, H
denotes
Hermitian transpose, σ denotes the power of the i th source, 2 i
σ n2 denotes the element noise variance. We begin by writing the eigendecomposition of the covariance matrix R as M
R = ∑ λi ui uiH
(4)
i =1
are the λ1 ≥ λ2 ≥ " ≥ λP > λP +1 = " = λM = σ n2 eigenvalues of R , and ui are its orthonormal eigenvectors. Throughout the paper it will be assumed that the number of where
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sources is known. ui (i ≤ P) are the signal-space eigenvectors and the remaining ones are the noise-space eigenvectors. Then, the signal-subspace and noise-subspace are represented as U s = [u1 , u2 ," , uP ] (5) U n = [uP +1 , uP + 2 ," , uM ] (6) ˆ In fact, let R denote the estimation of sample covariance matrix R using K data snapshots, and there is 1 K Rˆ = ∑ X k X kH (7) K k =1 III.
of the signals). If requiring Monte Carlo experiment, the number of tries is 1000. A. Comparison of the spatial spectrums In this experiment, the signal-to-noise ratio (SNR) of the strong source is SNR1 = 30dB , impinging on the array from the direction θ1 = 0° , while the SNR of the weak source is SNR2 = 0 dB , impinging on the array from the direction
θ 2 = 18° in fig. 1a and θ 2 = 4° in fig. 1b. The number of data snapshots is K = 200 . The DOA estimation results are shown in the following figures.
DOA ESTIMATION ALGORITHMS
A. Capon Algorithm The Capon beamforming algorithm is known as the MVDR (Minimum Variance Distortionless Response) beamformer, and the equation of its spatial spectrum can be given as 1 (8) PCapon (θ ) = H −1 aθ R aθ
where R −1 denotes the inversion of R , aθ denotes the steering vector corresponding to the angle θ . The direction of arrival can be achieved by finding the peak in spatial power spectrum. B. MUSIC Algorithm The MUSIC algorithm derives its spatial spectrum from the noise subspace of the sample covariance matrix. The equation of its spatial spectrum can be written as: 1 PMUSIC (θ ) = H (9) aθ U nU nH aθ
Fig. 1a. Comparison of the spatial spectrums with θ 2 = 18°
where U n denotes the noise subspace. C. Capon-MUSIC Algorithm The proposed Capon-MUSIC algorithm is derived from both Capon algorithm and MUSIC algorithm. The equation of the synthetic spatial spectrum of the new method can be written as: a H R −1aθ P (θ ) (10) = Hθ PC − MUSIC (θ ) = MUSIC PCapon (θ ) aθ U nU nH aθ
The DOAs of the sources can be achieved by finding the peaks in the spatial spectrum of the equation (10). IV.
NUMERICAL EXPERIMENTS
To illustrate the performance improvement achieved by our method, we consider a uniform linear array with M = 8 sensors and half-wavelength sensor spacing. For simplicity, two far-field independent signals impinge on the array, with one strong, the other weak. In the DOA estimation experiments, three algorithms are compared for different values of the system parameters. The system parameters are Δθ (the angle separation between the DOAs of strong and weak signals), K (the number of snapshots), SNR (the SNR
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Fig. 1b. Comparison of the spatial spectrums with θ 2 = 4° Fig. 1a indicates that the three methods are all able to estimate the DOAs of the strong and weak signals, with θ 2 = 18° , the peaks are obvious in three spatial spectrums, especially for MUSIC and Capon-MUSIC algorithms. Fig. 1b shows the results when θ 2 = 4° , the spatial spectrum peak in weak signal direction is not formed in the Capon and MUSIC spatial spectrums, but can be identified in the Capon-MUSIC spatial spectrum.
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B. Comparison of the spatial resolution probabilities versus the angle separation In this experiment, the SNR of the strong source is SNR1 = 30dB , impinging on the array from the direction
θ1 = 0° , while the SNR of the weak source is SNR2 = 0dB , impinging on the array from θ 2 , which verifies from 0.5° to 20° , and the increasing stepping is 0.5° . The number of data snapshots is K = 200 . The comparison of the resolution probabilities versus the angle separation is shown in fig. 2.
Fig. 3. Comparison of the minimum resolvable angle separation versus the number of snapshots Fig. 3 indicates that as the number of snapshots increasing for 50 to 1000, the minimum resolvable angle decreases and gradually stabilizes for all four algorithms and compared with the Capon algorithm and the MUSIC algorithm, the Capon-MUSIC algorithm has better resolution performance, which is close to the Root-MUSIC algorithm.
Fig. 2. Comparison of the resolution probabilities versus the angle separation Fig. 2 indicates the different results of the three methods. When the angle separation between strong and weak signals is small, the weak signal’s peak may not be formed in the spatial spectrum. The resolution probability denotes the probability of the weak signal’s peak being formed virtually. For the smaller angle separation, MUSIC algorithm is better than Capon algorithm, and Capon-MUSIC algorithm is better than MUSIC algorithm, which is similar as Root-MUSIC algorithm, while Root-MUSIC algorithm is always used in the uniform linear array only.
D. Comparison of the minimum angle separation versus the weak signal’s SNR In this experiment, the SNR of the strong source is SNR1 = 30dB , impinging on the array from the direction θ1 = 0° , while the SNR of the weak source is SNR2 , which verifies from 0dB to 20dB, and the increasing stepping is 1dB. Let Δθ =| θ1 − θ 2 | , and the minimum Δθ denotes the minimum resolvable angle separation. The number of snapshots is K = 200 . The comparison of the minimum resolvable angle separation versus the weak signal’s SNR is shown in fig. 4.
C. Comparison of the minimum angle separation versus the number of snapshots In this experiment, the SNR of the strong source is SNR1 = 30dB , impinging on the array from the direction
θ1 = 0° , while the SNR of the weak source is SNR2 = 0dB , impinging on the array from θ 2 . Let Δθ =| θ1 − θ 2 | , and the minimum Δθ denotes the minimum resolvable angle
separation. The minimum resolvable angle separation denotes the minimum resolvable angle for the two sources forming two peaks in the spatial spectrum of each algorithm. The number of snapshots K verifies from 50 to 1000, whose increasing stepping is 50. The comparison of the minimum resolvable angle separation versus the number of snapshots is shown in fig. 3.
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Fig. 4. Comparison of the minimum angle separation versus the weak signal’s SNR Fig. 4 indicates that as the weak signal’s SNR increasing for 0dB to 20dB, the minimum resolvable angle decreases for all four algorithms. The Root-MUSIC algorithm has the best
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resolution performance, and the Capon-MUSIC algorithm follows, which is better than the Capon algorithm and the MUSIC algorithm. V.
CONCLUSIONS
For the closely spaced multiple sources identification, this paper proposed a novel method called Capon-MUSIC method by introducing the synthetic spatial spectrum from the ratio of MUSIC spatial spectrum to Capon spatial spectrum, which makes the weak sources’ peak easy to identify. It is shown that the probability of resolution of the proposed Capon-MUSIC algorithm is better than the conventional Capon algorithm and MUSIC algorithm with the same simulation scenario and similar as Root-MUSIC algorithm. Normally, with the increase of snapshots and decrease of the power difference, the minimum resolvable angle separation will decrease for all the algorithms, and the Capon-MUSIC algorithm has the performance close to the Root-MUSIC algorithm but better than the Capon algorithm and the MUSIC
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algorithm. Finally, the numerical experiments are provided to demonstrate the effectiveness of the proposed method. REFERENCES [1]
[2]
[3] [4]
[5]
[6]
H. L. Van Trees. “Optimum Array Processing: Part IV of Detection, Estimation, and Modulation Theory,” New York: John Wiley & Sons Inc., 2002. H. Krim and M. Viberg. “Two decades of array signal processing research: the parametric approach,” IEEE Signal Proc. Mag., vol. 13(4), pp. 67-94, Jul. 1996. J. Capon. “High resolution frequency-wavenumber spectrum analysis,” Proceedings of the IEEE, vol. 57, no. 8, pp. 1408-1418, Aug. 1969. R. O. Schmidt. “Multiple emitter location and signal parameter estimation”, IEEE Trans. on Antenna and Propagation, vol. AP-34, No.3, pp. 276-280, Mar. 1986. A. Barabell. “Improving the resolution performance of eigenstructurebased direction-finding algorithms,” IEEE International Conference Acoustics, Speech, and Signal Processing, ICASSP '83., vol. 8, pp. 336339, Apr. 1983. T. Engin and F. Benjamin. “Classical and modern direction-of-arrival estimation,” Burlington: Elsevier Inc., 2009.
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