DOA Estimation of Coherent Signals in Heavy Noise ... - IERI

2012 International Conference on Environmental Engineering and Technology
Advances in Biomedical Engineering, Vol.8
DOA Estimation of Coherent Signals in Heavy Noise Environment
Shaowei Chen, Yingying Wei and Xinjian Shi
Department of Electronisc and Information
Northwestern Polytechnical University
Xi’an, Shaanxi, China
Yingying-1986523@163.com
Keywords: Direction of arrival; coherent signal; singular value decomposition; SNR.
Abstract. After researching singular value decomposition (SVD) algorithm and Toeplitz algorithm,
a novel high efficient algorithm in multipath environment is proposed in the paper. First a new
Toeplitz matrix is reconstructed using the maximum eigenvectors of signal covariance matrix. Then
the DOA (Direction of Array) of incident source can be resolved by the way of matrix SVD
algorithm. Simulations results, presented in section IV, show that the new algorithm has the better
performance than the conventional SVD and Toeplitz algorithm in accuracy and stability at low
signal to noise ratio (SNR). When SNR is -7, the identification error of Toeplitz algorithm is up to
2. 50, and SVD algorithm lose effectiveness. But the identification error of new TSVD algorithm is
near to 0.
I. Introduction
DOA is one of key research issues in spatial spectrum estimation. A lot of mature algorithms have
been developed up to now, such as MEM、MVM、ESPRIT and MUSIC etc. Influenced by
multipath propagation existing in radar, received signal under actual circumstances is always
coherent. To address the issue, now there are two types of algorithms: First, Dimension-reducing
Way. It can be further classified into spatial smoothing algorithm and matrix reconstruction
algorithm, such as modified spatial smoothing [2] (MSS), singular value decomposition [3] (SVD),
etc; Second, non Dimension-reducing Way. There are Toeplitz method, dummy array
transformation method and some other new de-correlated methods [5-7]. Researching on these
algorithms, performance of Dimension -reducing Method will drop because of the disadvantage of
array aperture loss. And non Dimension-reducing Method needs to amend and judge the data in
advance, which often lead to great estimation error.
A novel DOA algorithm, based on SVD and Toeplitz, is proposed in this paper. It combines the
advantages of two algorithms, high resolution and stability. At last a large number of simulations
show the good performance of new algorithm in this paper.
2. Mathematical model
Consider a linear receiving array with M equally spaced elements and direct sources with a total of
N paths. At the array, source wave front propagation is assumed planar and source energy is
incident on the array a distinct angles {θ 1 , θ 2 , …. , θ N }. The propagation channel is
homogeneous and the signals are narrowband (i.e. source bandwidth is much smaller than the
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eciprocal of propagation time of the signal across the array). Then the received signal at the kth
time is given by
X( k) = AS( k) + N ( k), k = 1,2,..., K
(1)
Where,
X ( k) = [ x1
( k) … x ( )]
T
M
k
T
jw0
k
Sk ( ) = [ s1
( k) … sN( k) ] , si( k) = ai( ke )
A= [ a( θ ) a( θ ) … a( θ )]
1 2
a θ β π θ λ
− jβ
− j( M −1)
i
(
i) = [1 e e β ],
i
= 2 dsin i
/
Nk ( ) = [ n1
( k) n ( )]
T
M
k
Where, d is the sensor spacing, λ is the processing center wave length, ω 0 is the processing center
angular frequency
Therefore, the spatial covariance matrix is:
2
R = [ XX H ] = A [ SS H ] A H + I
H
H
= USΣSUS + UNΣ NU
N
E E σ
Where, ‘H’ denotes the complex conjugate transpose; ‘E’ stands for expectation, an ensemble
average operator; I is M×M identity matrix; U S =[e 1 e 2 … e N ] is signal subspace, consisting by
eigenvectors corresponding to largest N eigenvalue; U N =[e N+1 e N+2 … e M ] is noise subspace,
consisting by eigenvectors corresponding to smallest (M-N) eigenvalues; Σ S is diagonal matrix
consisting of large eigenvalue; Σ N is diagonal matrix consisting of smallest N eigenvalues; σ 2
denote ideal white noise power.
In direction finding problems, eigenstructure methods of estimating the directions of arrival {θ i }
are based on exploiting this structure of R. That is, R has a range basis consisting of the direction
vectors of source plane waves, and R is the sum of an N-rank matrix U S Σ S U H S and σ 2 I.
The noise subspace vectors are orthogonal to all the steering vectors. The MUSIC spectrum is a
continuous spectrum formed as
1
PMUSIC
( θ ) =
(3)
H
2
a ( θ ) U
The source directions can be estimated as the peaks of the spatial spectrum.
Conventional eigen-structure methods, however, are applicable only when the path signal
covariance matrix is non-singular. When some of the path signals are coherent (for examples, if
they originate from the same source), the path signal covariance matrix will be singular and
properties noted do not hold. Up to now, to solve this problem there are many methods for coherent
signal. SVD and Toeplitz algorithm are two kinds of typical methods.
2.1 SVD method
SVD (singular value decomposition) algorithm has been proposed as Reference [3]. The
[3]
largest eigenvector e 1 is formed as
T
e1 = [ e11, e12, e13...... e1M
]
k=
N
(4)
T
= α a( θ ) = [ x(1) x(2)... x( M)]
∑
k = 1
1
k
Therefore, the largest eigenvector e 1 contains all the source direction information. Then a new
matrix Y can be formed by utilizing e 1 .
⎡ e1,1 e1,2 e1,
p ⎤
⎢
⎥
⎢
e1,2 e1,3 e1, s+
1
Y = ⎥
(5)
⎢ ⎥
⎢
⎥
⎢⎣
e1, m
e1, r+
1
e1,
M ⎥⎦
N
N
(2)
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Where, m+p-1=M. The noise subspace can be obtained by singular value decomposition of the
matrix Y. Finally, DOA can be estimated according to formula 3.
2.2 Toeplitz approximation method
S. Y. Kung et al.[4] proposed a Toeplitz Approximation Method (TAM) which is based on a
reduced order Toeplitz approximation of an estimated spatial covariance matrix. The estimated
covariance matrix, in the case in which sources are uncorrelated and statistically stationary, is
Toeplitz. In a multipath environment, however, the source paths are fully correlated, and this
covariance matrix is not Toeplitz. The Toeplitz structure can he guaranteed by employing spatial
smoothing, which destroys cross correlation between directional components. The TAM is designed
for robustness in an arbitrary ambient noise environment.
3. Tsvd algorithm
In accordance with the foregoing analysis, a novel direction finding algorithm, based on the
advantages of SVD and TAM, is proposed in this paper. First, eigenvalue decomposition of the
spatial covariance matrix is [U, S]= eig (R), where, U denotes eigenvectors matrix; S represents
egenvaluses matrix. As previous analysis, the largest eigenvector e 1 associated with the matrix U
completely include information of received source paths. Considering correlated source paths,
decoherent e 1
'
rm
= e11 × e1
m
(6)
Obviously, the angles element still exists in the vector r m .
As we all known like given in the TAM algorithm, the Toeplitz matrix can destroy cross
correlation between directional components. Now, we form the new covariance matrix as follow:
⎡ r1 r2
rM
⎤
⎢ '
r2 r1 r ⎥
M −1
Y = ⎢
⎥
⎢ ⎥
(7)
⎢ ' '
⎥
⎣rM
rM−
1
r1
⎦
We can easily verify the new matrix Y is Toeplitz structure (conjugate symmetric matrix).
Then, perform an SVD on Y, [U, S, V]=svd(Y) and arrange the singular values in decreasing.
The SVD is
⎡S1 0 ⎤⎡V′
1⎤
Y = USV′
= [ U1 U2]
⎢
0 S
⎥⎢
2
V
⎥
(8)
⎣ ⎦⎣ 2 ′ ⎦
Where, S should ideally be a diagonal matrix of N eigenvalues; the N×N diagonal matrix S 1
contains the N largest singular values; the (M-N)×(M-N) diagonal matrix S 2 contains the (M-N)
smallest singular values. U stands for left singular matrix; V stands for right singular matrix.
Similarly, the N columns eigenvectors corresponding to S 1 of matrix U is signal subspace. The (M-
N) columns eigenvectors corresponding to S 1 of matrix U is noise subspace Un=U (:, (N+1):M).
With the equation 3, the source directions can be estimated quickly.
In summary, the five steps of the novel approach are:
Step1
Calculate the spatial covariance matrix R;
Step2
Perform eigenvalue decomposition on R. Extract the largest vector e 1 according to the largest
eigenvalue;
Step3
Reconstruct the new Toeplitz covariance matrix Y according to equation 7;
Step4
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Perform an SVD on Y, so we can easily obtain noise subspace Un;
Stpe5
Search the peak of the continuous spectrum as the DOA of source according to formula 3.
4. Simulations
The snapshot model consists of a linear array of 8 equispaced with inter-element spacing of one half
wavelengths. Two equal-powered emitters at angular locations 15° and 22°. The number of
snapshots sets to 100. In order to perform SVD 、Toeplitz and TSVD applied to the direction
finding, the temporal snapshot data and Gaussian noise data were generated artificially by a
computer. The simulation results is given as Fig.1-4.
Fig. 1 shows simulation results for SVD、Toeplitz and TSVD at SNR of 7dB. Three methods
all can resolve the emitters, but TSVD yields nearer optimal angle estimates than other two
algorithms. But SNR decrease to -7dB, Fig. 2 shows that TSVD has the accurate performance. The
SVD cannot resolve the angles, while the error of the Toeplitz is up to 2.5°.
150
Spatial spectrum (dB)
100
50
0
TSVD
SVD
Toeplitz
-50
0 10 20 30 40
Angle (°)
Figure.1 The three arrows indicate the original spectral peaks at 7dB SNR.
100
Spatial spectrum (dB)
80
60
40
20
0
-20
TSVD
SVD
Toeplitz
0 10 20 30 40
Angle (°)
Figure.2 The three arrows indicate the original spectral peaks at -7dB SNR.
In the same environment condition we carry 100 Monte Carlo trials. Fig. 3-4 respectively show
Monte Carlo simulation results of the error and success probability for SVD、Toeplitz and TSVD.
Obviously, the success probability of TSVD is near to 100% at a SNR of -10dB, in contrary SVD is
down to 0. Although Toeplitz can achieve high success probability, the error is greater than new
method. So we can summarize TSVD has optimal resolution and stability comparing to
conventional SVD and Toeplitz algorithms particularly at low SNR.
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Estimation error (°)
1.4
1.2
1
0.8
0.6
0.4
0.2
TSVD
SVD
Toeplitz
0
0 5 10 15 20
SNR (dB)
Figure.3 Estimation error for three algorithms at SNR of 0-20.1
1
0.8
TSVD
SVD
Toeplitz
Success probability
0.6
0.4
0.2
0
-10 -5 0 5 10 15 20
SNR (dB)
Figure.4 Success probability for SVD、Toeplitz and TSVD at SNR of -10 to 20.
5. Conclusion
TSVD algorithm for DOA finding problems, with application in multipath environment with linear
equispaced arrays, has been presented. The algorithm combines the advantages of two classical
algorithms, has excellent performance. At last simulations results indicate that it can be optimistic
in expecting high resolution and good accuracy when a greatly low SNR is used.
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