2020 4th International Conference on Modelling, Simulation and Applied Mathematics (MSAM 2020) ISBN: 978-1-60595-674-9
A Novel DOA Estimation Method under the Impulsive
Noise Environments
Rui-yan CAI
1,2, Wei DING
3,*and Ming-yang JU
1,2 1College of Information Engineering, Dalian University, Dalian, China
2
Communication and Networks key Laboratory, Dalian University, Dalian, China
3
Institute of Systems Engineering, Academies of Military Science, Beijing, China
*Corresponding author
Keywords: Cyclostationary, Alpha-stable distribution, DOA estimation, Cyclic correntropy, ESPRIT.
Abstract. A key task in radio monitoring is direction of arrival (DOA) estimation, and lots of algorithms were proposed in recent years. These algorithms have been designed in the assumption of Gaussian model and usually break down in the impulsive noise. Therefore, the DOA estimation of cyclostationary signal is necessary. Since the performance of the second-order cyclic statistics degenerates severely under the impulsive noise conditions, the generalized cyclic cross correntropy is defined, and a novel method is proposed to obtain the DOA estimation based on the cyclic correntropy. Simulation results illustrate that the proposed method outperforms the existing algorithms in terms of effectiveness and robustness.
Introduction
In the signal processing field, DOA estimation is an important method that has widely applications in wireless communications and radar[1]. During the development of DOA estimation technique, people have been interested in high resolution algorithms, and have made many important progress in this field. The algorithms based on subspace can estimate the direction of multiple signals at the same time and achieve higher precision. Therefore, these algorithms have come to a wide attention in recent years. Gardner [2] proposes the cyclic MUSIC algorithm and the cyclic ESPRIT algorithm by applying cyclic correlation matrix to subspace data model. The MUSIC is similar to the ESPRIT in that, it exploits the subspace information, however, as a common drawback, the spectrum peak search of the MUSIC algorithm is more complex.
The fractional lower order statistics can suppress the impulsive noise[3], however, it has no inhibitory effect on the co-channel interference. The fractional lower order cyclic statistics can suppress both the co-channel interference and the impulsive noise, but the order relies on the prior knowledge of the alpha-stable distribution. The algorithms of signal parameter estimation and wireless location based on correntropy can effectively restrain alpha-stable distributed noise even if the prior knowledge of signal and noise is unknown, but cannot restrain the co-channel interference effectively. Combining the cyclic statistics and the correntropy, the cyclic correntropy is proposed, which can suppress both the interference of the spectrum overlapping of signals and the impulsive noise and obtain good results in the application of carrier frequency estimation.
Signal and Noise Model
Signal Model
Consider a ULA with sensors and the sensor interval , where is wavelength. Suppose signals impinging on this array. The received data samples can be expressed as
( )t ( ) ( ) t ( )t
X A S U . (1)
where X( )t denotes the received data samples in the array, S( )t denotes the signal vector, A( ) denotes the steering matrix, and U( )t denotes the additive noise vector.
ESPRIT is a subspace-based DOA estimation technique that exploits the rotational invariance property of signal subspace. Consider the array as being comprised of two identical subarrays and , the signals received can be expressed as x( )n As( )n u( )n and y( )n Aψs( )n u( +1)n , where is a vector of impinging
signals as observed at the reference element of subarray , is an additive noise vector,
1 2
=diag(ej ,ej ,...,ejL)
ψ is a unitary matrix.
Alpha-stable Distribution
Alpha-stable distribution is commonly expressed by its characteristic function:
( )t exp j u u 1 j sgn( ) ( , )u u
. (2) where
tan( 2), 1 ( , )
(2 ) log , 1
u
u
. (3)
and
1, 0
sgn( ) 0, 0
1, 0
u
u u
u
. (4)
where
0, 2
is the characteristic exponent that mainly characterizes the intensity of impulsive noises.
-1,1 is the symmetry parameter. If = 0, the distribution is symmetric alpha-stable ( ) distribution.
0,
is the dispersion parameter.
,
is the location parameter.Cyclic Correntropy
The correntropy of a zero mean cyclostationary process x( )t is defined by
( , ) [ ( ( )- ( ))]
V tx E κσ x t xt . (5) where κσ is the Gaussian kernel, and is the kernel size.
Consider the correntropy function is periodic within a certain period T0,
0
( , ) ( , )
x x
V t T V t . (6) it can be represented by a Fourier series expansion as
2
( , ) ( ) j t
x x
V t V e
where is the cyclic frequency. n T/ 0, where . The cyclic correntropy function can be defined by 0 0 2 2 2 0 1
( ) T ( , ) j t
x x
T
V V t e dt
T
. (8)If two i.i.d random processes x( )t and y( )t obey , the generalized cross correntropy function can
be expressed as
;
( ( ) ( )) ( ) ( )
xy
V t E xt yt xt yt . (9)
The ESPRIT Algorithm Based on Cyclic Cross Correntropy
Generalized Cyclic Cross Correntropy Matrix
A novel ESPRIT algorithm, which is based on the cyclic cross correntropy, is proposed. The estimation matrix can be defined by
1 1 1 2 1
2 1 2 2 2
1 2 ( ) ( ) ( ) ( ) ( ) ( ) ( )= ( ) ( ) ( ) M M
M M M M
x y x y x y x y x y x y xy
x y x y x y
V V V
V V V
V V V
V . (10)
where is the cycle frequency parameter, is the lag parameter, and
1 2
( )n [ ( ),y n y n( ),...,yM( )]n T
y areM1 vecotrs. The ( , )i j entry can be written as
2
( ) ( ( ) ( )) ( ) ( )
2 2 2 2
i j
j t
x y i j i j
t
V x t y t x t y t e . (11)
where i j, 1, 2,...,M.
CCE-ESPRIT Algorithm
We first review the conception of the total least square ESPRIT (TLS-ESPRIT) and related theory. Based on them, the cyclic correntropy-based ESPRIT (CCE-ESPRIT) algorithm is proposed. The observation data can be rewritten as ( ) [ ( ),1 2( ),..., 1( )]
T M
n x n x n x n
x and ,
and the generalized cyclic cross correntropy matrix can be obtained.
Define two (M 1) M selection matrices J1 and J2 as J1[IM, 0M1] and J2[0M1,IM], it can be
easily obtained that X1J X1 [ (1),x1 x1(2),...,x N1( )] and X2J X2 [x2(1),x2(2),...,x N2( )] , where
and .
1 1 1 1
2 2 2 2
( )
( ) ( ) ( )
( )
J X A U n
JX n X n S n
J X A U n
. (12)
based on the signal model
1 2 (1,:) = ( ,:) A A A A A N
. (13)
then A2A1.
The generalized cyclic cross correntropy matrix substitutes the covariance matrix in ESPRIT, we can define matrix pencil { ,V V1 2}, the singular value decomposition (SVD) of V1 can be expressed as
1 1
1 1 2
2 2 , H H H Q
V U Q U U
Q
where 1 includes principal eigenvalues.
1 1 2 1 1 1 2 1
H H
U V V Q U V Q. (15)
It is easy to deduce that, the dimension of { ,V V1 2} can be reduced from MM topp, and { ,V V1 2} is converted into { 1, 1 2 1}
H
U V Q
. We can obtain DOA by computing SVD of { 1, 1 2 1}
H
U V Q
.
Simulation Results
To illustrate the performance of CCE-ESPRIT, we compare with fractional lower order cyclic ESPRIT (FLOC-ESPRIT), cyclic ESPRIT (Cyclic-ESPRIT) and traditional total least square ESPRIT (TLS-ESPRIT) algorithms. Since the random variable has infinite variance for 2, the generalized signal-to-noise ratio (GSNR) is utilized to evaluate the ratio of the signal power over noise dispersion. The GSNR can be expressed as
2
1 1
10 lg [ ]
N n
GSNR s n
N
. (16)In which 0 denotes dispersion parameter.
In these simulations, the performance of four algorithms for BPSK signals is evaluated. Assume a ULA with isotropic sensors and two BPSK signals impinge on this array, and the locations
1 30
and 2 40 , respectively. The underlying noise is distribution. To compare the performance, we use the root mean square error (RMSE) as the evaluation criterion, it is defined by
2
1
1
( )
N
n
RMSE n
N
. (17)where ( )n
is the estimation value of the experiment, and the curves are averaged over 500 Monte-Carlo trials.
Experiment: The effect of the Characteristic Exponent
In this experiment, the number of snapshots is fixed at 500, the GSNR is set as a constant at 15dB and the cycle frequency is 200MHz. The characteristic exponent varies from 1.2 to 2. It is easy to deduce that, with increase, the noise model transits from extremely impulsive to Gaussian environments. Fig.1 demonstrates the performance comparison of the four algorithms. We can observe that FLOC-ESPRIT, Cyclic-ESPRIT and TLS-ESPRIT algorithms gain more evident decrease in RMSE than our proposed algorithm as the characteristic exponent increase, however, in this range, the RMSE of CCE-ESPRIT is much lower than others.
Figure 1. Performance comparison as a function of .
Conclusion
This paper focuses on the DOA estimation problem in the impulsive noise. Basing on the ESPRIT method and cyclic correntropy theory, we propose CCE-ESPRIT method. This method performs better than the existing algorithms, especially in both strong impulsive noise environments, and avoids spectrum peak search. Simulation results illustrate that CCE-ESPRIT is better than the existing algorithms.
References
[1] Steinwandt, J., Roemer, F., Haardt, M. Generalized least squares for ESPRIT-type direction of arrival estimation, IEEE Signal Processing Letters 24.11 (2017) 1681-1685.
[2] Gardner, William A. Simplification of MUSIC and ESPRIT by exploitation of cyclostationarity, Proceedings of the IEEE 76.7 (1988) 845-847.
