New DOA algorithm using subspace projection and synthetic spatial spectrum

Lan Xiaoyu Electronic and Information Engineering College

Shenyang Aerospace University Shenyang, China

e-mail: lanxiaoyu1015@163.com

Zou Yan NO.91404 Army

Qinhuangdao, China e-mail:zou_tian@163.com

Abstract— In this paper, we propose a new direction of arrival (DOA) estimator for sensor-array processing. The estimator we propose is a modified weighted noise subspace of MUSIC algorithm (MWNSM). The algorithm is to solve the problem that the MUSIC algorithm has a strong depends on the signal-to-noise ratio (SNR) and snapshots. The new method reconstructs the spatial spectrum function with both noise subspace and signal subspace in this paper. The key idea is to apply the full information contained in covariance matrix and change the projection weights of steering vector on the noise and signal subspace by their revised eigenvalues, respectively. Comparing with the MUSIC algorithm, it does not increase any computational complexity either, and remarkably, it has the advantages of simultaneously reducing noise and keeping the high-resolution ability under low SNR and small sample sized scenarios. Computer simulation results shows that the proposed algorithm has a better performance than the classical DOA methods.

Keywords-arrays signal processing; direction of arrival; multiple signal classification; subspace projection; resolution;

I. INTRODUCTION

Direction of arrival (DOA) estimation for multiple plane waves impinging on an arbitrary array of sensors has received a significant amount of attention over the last several decades. It has typically played an important role in array signal processing areas such as modern wireless communication systems, radar, sonar, audio/speech processing systems and radio astronomy. Multiple Signal Classification (MUSIC) method [1] is paid so much attention because of its brilliant properties in estimating DOAs of incident signals, as in [2][3][4]. However, both experience and analysis show that the resolution performance of these methods degrades drastically in severe environments like low Signal-to-Noise Ratio (SNR), small number of snapshots, or when the incident waves are coming from close angles. To overcome these shortcomings above, many DOA estimation algorithms based on MUSIC have been presented. A new approach applies the properties of non-circular symmetry of the signal to improve the performance, but the stability is not high [5]. And [6] [7] use the sound characteristics of signal subspace projection to improve the algorithm performance stability under small number of snapshots, but the direction accuracy has not improved much. A Discrete Fourier

Transform (DFT) approach is proposed in [8]. In [9][10][11], the method using nonuniform sampling (NUS) is proposed, in this method, considering the character of NUS with the random time interval, NUS in time domain is equivalent to alter the relative distance between sensors and lead to the signals impinge time-varying adaptive virtual nonuniform linear array based on virtual array transformation. This method can enhance the angle resolution and estimation accuracy.

In this paper, an efficient DOA estimation algorithm that combined the characteristic of signal subspace and noise subspace is proposed. This method is not only to maintain the stability characteristics of the signal subspace, but also improves the resolution of the algorithm. Its performance is much better than that of MUSIC in the condition of low SNR and small snapshot, so it is predominant and robust.

II. SIGNAL MODEL AND CLASSICAL

Approaches

A. Signal model In the following we consider a uniform linear array

(ULA) of M antennas separated by half a wavelength. Assume that D incident waves sources impinging on the array from angles 1, , D� �� under an Additive White Gaussian Noise (AWGN) environment. Here the input signal vector ( )kx t at the k-th antenna is written as

i 1

( ) ( ) ( ) ( )D

k k i i kx t a s t n t��

� �� (1)

where 1,2, ,k M� � , the angle i� is the DOA of the i-thincident wave which changes within(-90° 90°). Besides,

( )k ia � , ( )is t , ( )kn t denote array mode vector of k-thelement, complex waveform of k-th element, the receiver noise of k-th element. The array mode vector matrix ( )tX is defined as

( ) ( ) ( )t t t� �X AS N (2) where T

1 2( ) [ ( ), ( ), , ( )]Mt x t x t x t� �X is an 1M � vector of received signals at time t. and the superscript T denotes the vector transpose; 1 2( ) [ ( ), ( ), , ( )]Dt a a a� � �� �A is M D�

matrix of the composite array response; T

1 2( ) [ ( ), ( ), , ( )]Dt S t S t S t� �S is a 1D� vector of incident

2013 6th International Conference on Information Management, Innovation Management and Industrial Engineering

978-1-4799-0245-3/13/$31.00 ©2013 IEEE

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waves sources; T1 2( ) [ ( ), ( ), , ( )]Mt n t n t n t� �N is an 1M �

additive noise vector.Using standard notation, the covariance matrix R of ( )tX is given by

� �H H 2E � � �R XX APA I (3)

where � � � �H1 2E diag , i Dp p p p� �SS � �P is the signal

covariance matrix, and ��E denotes the statistical

expectation, ip , 2 and I denote signal power of i -th,noise power and identity matrix.

In practice, though, the matrix R must be estimated from the received data and the date is limit, let R̂ represent the sample covariance matrix drawn from L samples or snapshots as

H

1

1ˆ ( ) ( )L

n

n nL �

� �X XR (4)

B. MUSIC Method The eigenvalue decomposition of R is

H H

1

M

i ii

��

� � � �R U U v v (5)

where 1 2{ , , }Mdiag � � �� � � � , in ideal conditions 2

1 2 1D D M� � � � � �� � � � � �� � ,the kv denotes eigenvector corresponding k� . Assume that the number of incident waves is know in advance to D, so R could be divided in to

H HS S S N N N� � � �R U U U U (6)

where �S 1 2, , , D� v v v�U denotes signal subspace of M D� matrix that contains the eigenvectors corresponding to D larger eigenvalues , �N 1 2, , ,D D M� �� v v v�U denotes noise subspace of ( )M M D� � matrix that contains the

eigenvectors associated with the smaller eigenvalue 2 .Using the orthogonality between arrival angles and noise

subspace, the MUSIC spectrum is given as follows

� �� �

MUSIC 2HN

1��

�Pa U

(7)

when � is equal to the direction angle of each signal, ( )�a is orthogonal to NU , the denominator of the ( )�P is 0, ( )�P trends to be infinite. Thus the � can be attained by

scanning the D peaks of the spatial spectrum.

III. PROPOSED METHOD

Note that MUSIC algorithm can give the right DOA peaks if the noise subspace NU is orthogonal to the signal subspace SU .However, for small sample sizes, low SNR, or two closely space signals, NU may be poorly estimated compared with the true noise subspace, therefore the parallel averaging of these peaks may not derive any peak at the DOAs or miss DOAs. [14] proposed a new method that reconstructed noise subspace by

weighting the noise subspace with corresponding eigenvalues. The new method redefined the modified noise subspace NE asfollows

1

2N 1 2[ , , , ]

D

Dn n nD D M

M

� � �

�

�� �

� �� �� ��� �� �� �

��

vv

E

v

(8)

where 1,D M� �v v are original eigenvectors in noise subspace

and 1, ,D M� �� � are corresponding eigenvalues. The resulting DOA function, termed the weighted noise subspace of MUSIC algorithm (WNSM), can be calculated as

2HWNSM N( ) 1/ ( )� ��P a E (9)

Where , the HN NE E can be written as

H 2 HN N

1

Mn

i i iD

��

� �E E v v (10)

This WNSM algorithm improves the resolution under the context of low SNR and small snapshots. However, how to choose n is not analyzed in the paper. Through a lot of computer simulations, we found the most appropriate n should be chosen between 0 and 1. To improve the performance of the algorithm in the situation of two closely spaced sources, it is necessary to modify the algorithm in these cases. In this paper, the noise subspace, weighted with its correspondingly revised noise eigenvalues i� �� instead of n

i� , is reconstructed. Suppose the

revised eigenvalue i�� is

i i� � �� �� (11) The modified noise subspace is modeled as

N 1 1 2 2[ , , , ]D D D D M M� � �� � � �� � � ��F v v v (12)

where 1,D M� �v v are original eigenvectors in noise subspace

and 1, ,D M� ��

� �� are revised eigenvalues. (0,1)� � is the correcting value. Concluded from a mount of experience data, the maxima to minima of noise eigenvalue ratio could not exceed 2, if information theory criterion gave source number estimation correctly. The inequality is written as

1 / 2D M� �� � (13) From (11) and (13), we obtain

1 / 2D M� � � �� � � � (14) The development in this way concentrates on increasing the

information utilization of noise eigenvalues. Utilizing weighting , 1,i i D� �v with its revised eigenvalue , 1,i i D� � � method,

it makes the contribution of different noise eigenvector corresponding to the eigenvalue is not the same in the form of spatial spectrum function. Furthermore, under some non-ideal contexts of finite data length and low SNR, the divergence degree of noise eigenvalues can be controlled by the correcting value� while � is independent with signal eigenvalues. That all guarantees the method provides more accuracy and steady results than the WNSM algorithm. Therefore, the problem at

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hand is to define the optimum value� . Here, we will give a some steps to follow

(a) estimate covariance matrix R , make eigencomposition and sort all the eigenvalue i� in descending order.

(b) estimation the source number D and calculate the minimal integer m satisfies 1 0.1 / 0.1 2D Mm m� �� � � � � � .

(c) let 0.1 m� � � and then i�� could be obtained . So

1 1H 2 H

N N 1 11

( , , )i

D D M

D D M M i ii D

M M

�� � �

�

�� �

� �� �

�

� �� �

� � �� �� � !

�� � �� �

�

vF F v v v v

v

(15)

From (7) and (15), the modified spatial spectrum function is defined as

MWNSM 2 2H 2 HN N N

1 1( )( ) ( )

�� �

� �Pa F a U"

(16)

where � �N1

M

ii D

� �� �

� ��"

As mentioned above, the superior improvement over WNSM algorithm is that this method takes use of revised noise eigenvalues i� �� instead of n

i� as the weight onNU , which

adjusts the projection component weight of ( )�a on the orthogonal subspace (noise subspace) effectively. Moreover this method has another advantage that it controls the divergence degree of noise eigenvalues by � in severe signal environment. However, the spatial spectrum function only uses the information in noise subspace, while the information in signal subspace is not utilized. In [8], a novel method employing signal subspace was proposed. The new method appears to be more robust to the effects of bias caused by finite data length and low SNR than using noise subspace. And the approach, named the reciprocal weighted signal subspace projection (RWSP) was derived. The function is

21 H HRWSP S

1( ) ( ) ( ) ( )

D

j jj

v� � � � ��

�

� ��P a a U U a"

(17) where �S 1 2, , , D� �U v v v S 1 2diag( , , , )D� � �� �"

The idea of RWSP algorithm weighting signal subspace with reciprocal of its eigenvalues is to weaken the projection of ( )�aon the greatest vector in matrix SU . For two closely spaced sources of equal power, the original signal vectors are unit orthogonal basis, as a result the spectrum curve of DOA location which is corresponding with maximum signal vector would be concave downward. That is propitious to enhance resolution ability, whereas, the estimation accuracy has not been increased. Therefore, considering the high estimation accuracy of MUSIC algorithm based on noise subspace projection, we organize the function (17) and (16). The novel spatial spectrum function at last is defined by

21 HS S

22 HN N

( )( )

( )

��

�

�

�a U

Pa U

"

" (18)

Finally, the function (18) is the ultimate function proposed, in order to give a clear description, here is a diagram to follow.

1 2 1D MD� � � � ��� � � # #� �

�

Figure.1 the processing step of novel method It can be seen that, if NU is estimated the same with the true

noise subspace; the two techniques yield identical peaks. However, for small sample sizes, low SNR, or closely spaced signals, NU maybe poorly aligned with the true noise subspace, causing MUSIC algorithm to miss DOAs or pick spurious DOAs. The new algorithm deals with subspace mismatches by weighting the noise subspace with its revised eigenvalues and using a signal subspace function in the numerator. This additional function exploits more information of the correlation matrix and appears to be more robust to the effects of finite data length and low SNR than MUSIC.

IV. SIMULATION AND ANALYSIS

In this section, we will present some figures to illustrate the performance of the proposed method, and all quantitative simulations are made under conditions that: there is a uniform linear array of 7 antennas separated by half a wavelength. Two incident waves are coming from the close angles 1 3� � � $ and 2 2� � $ , the signal environment is with White Gaussian Noise (AWGN).

To verify the performance of the proposed method, a comparison simulation of MUSIC and proposed method spatial spectrum is carried. From the Fig. 1, we can see that, when SNR is 0 dB and the number of snapshots is 100, the proposed method can detect the two close waves correctly while MUSIC only has one peak around 2°. It says that we can accurately estimate DOAs of close angles using proposed method.

Fig.2 and Fig.3 shows the behavior of resolution probability when we change SNR from -2 to 10 dB and number of snapshots from 16 to 112 respectively. For comparison, the estimation results using the WNS algorithm[12], RWSP algorithm[13] and MUSIC algorithm are shown in the figures. All the estimated data are calculated as the average of 200 trials. Seen from the figures, the resolution of the proposed method is much higher than that of any other method mentioned in this paper over all SNR, the result for the number of snapshots property is also similar.

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In order to verify the estimation accuracy of the new method, Table 1 shows the behavior of root mean square error (RMSE ) as SNR changed from 4 dB to 10 dB and the number of snapshots is 100. Each data is also calculated as the average of 200 trials. As seen from the results, we found that the proposed method can estimate DOAs more accurately than MUSIC algorithm.

Figure.1 The spectrum of MUSIC and proposed method

Figure.2 Resolution probability versus SNR

Figure.3 Resolution probability versus snapshots

Table 1 RMSE comparison of two algorithm for 2 DOAs estimation

SNR(dB) DOA(°) MUSICRMSE(°)

Proposed RMSE(°)

41� 0.43 0.37

2� 0.45 0.39

61� 0.37 0.35

2� 0.34 0.32

SNR(dB) DOA(°) MUSICRMSE(°)

Proposed RMSE(°)

81� 0.27 0.19

2� 0.25 0.18

10 1� 0.19 0.13

2� 0.17 0.10

V. CONCLUSION

In this paper, we proposed a high resolution DOA estimation method. Through reconstructing spatial spectrum by weighting the noise subspace and signal subspace separately with the secondary eigenvalues and the principal eigenvalues, the new approach can make full use of the information contained in signal and noise subspace. Furthermore, the simulation results are included to demonstrate the performance improvement of the proposed algorithm compared to the exiting MUSIC method.

ACKNOWLEDGMENT

This paper is supported by the National Natural Science Foundation of China (Grant No 61101161).

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