DOA) Estimation Using RBF Neural Network in

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Robust Direction of Arrival (DOA) Estimation Using. RBF Neural Network in Impulsive Noise Environment). Hong Tang, Tianshuang Qiu, Sen Li, Ying Guo, and ...
Robust Direction of Arrival (DOA) Estimation Using RBF Neural Network in Impulsive Noise Environment* Hong Tang, Tianshuang Qiu, Sen Li, Ying Guo, and Wenrong Zhang School of Electronic and Information Engineering, Dalian University of Technology, Dalian, Liaoning 116024, China [email protected]

Abstract. The DOA problem in impulsive noise environment is approached as a mapping which can be modeled using a radial-basis function neural network (RBFNN). To improve the robustness, the input pairs are preprocessed by Fractional Low-Order Statistics (FLOS) technique. The performance of this network is compared to that of the FLOM-MUSIC for both uncorrelated and correlated source. Numerical results show the good performance of the RBFNN-based DOA estimation.

1 Introduction In sensor array processing, one of the critical problem has been the estimation of DOA of the source signals. Many high-resolution DOA estimation algorithms have been proposed. But, a common problem must be noted that the additive noise is assumed to be Gaussian distributed in these algorithms. However, this assumption is not always true in practice. There are always numerous man-made or natural electromagnetic disturbances [1] in wireless channel. The non-Gaussian impulsive noise is also be proved by experiments [2]. In open documents, FLOM-MUIC [3] is notable. But it still involve mass computational complexity. Neural networks are candidates to carry on the computation tasks required in several array processing [4]. The estimation of DOA in impulsive noise environment is treated as a nonlinear mapping from the input pairs to the output pairs.

2 The Distribution of Impulsive Noise In this paper, the alpha-stable distribution is introduced as the impulsive noise model in wireless channel. Alpha-stable distribution does not have closed form of united probability density function. It can be conveniently described by its characteristic function

*

as

ϕ( t) = e

{ jat −γ

t

α

[1+ jβ sgn (t )ω (t ,α )]}

.

Where

ω (t , α ) = tan

απ 2

,

if

α ≠1,

This work is supported by National Science Foundation of China under Grants 30170259, 60372081, and the Science Foundation of Liaoning Province under Grant 2001101057.

J. Wang, X. Liao, and Z. Yi (Eds.): ISNN 2005, LNCS 3498, pp. 332–337, 2005. © Springer-Verlag Berlin Heidelberg 2005

Robust Direction of Arrival (DOA) Estimation

ω (t , α ) =

2

π

333

log t , if α = 1 , and sgn(⋅) is a sign function. α is the characteristic expo-

nent restricted in 0 < α ≤ 2 . It controls the thickness of the tail in the distribution. γ (γ > 0) is the dispersion parameter and β (−1 ≤ β ≤ 1) is the symmetry parameter. a ( −∞ < a < ∞ ) is the location parameter. When α = 2 and β = 0 , the α -stable distribution becomes the Gaussian distribution. The theoretical fact that the alpha-stable distribution can be a general noise model is proved by the generalized central limit theorem. For jointly SαS random variables X and Y with 1 < α ≤ 2 , the fractional loworder covariance (FLOC) of X with Y is defined as c XY = E ( XY < p −1> )

(1)

1≤ p =| Y |( p−2) Y * . The FLOC is a statistics of alpha stable random variables, which in a certain conditions plays a role for SαS (symmetry α -stable) random variables analogous to the one played by covariance for Gaussian random variables.

3 RBFNN for DOA Estimation RBFNN is a method for approximating nonlinear mapping since the DOA estimation is of nonlinear nature. The nonlinear mapping from the input space to the output space can be viewed as a hypersurface Γ representing a multidimensional function of the input. During the training phase, the input-output patterns presented to the network are used to perform a fitting for Γ . The generalization phase presents an interpolation of the input data points along the surface Γ . The architecture of the neural network for DOA estimation is shown in Fig 1. The received signals from the uniform linear array can be expressed as K

x m ( n) = ∑ a k e

ω  j  k n +φk   c  − j ( m −1) d sin θ k

k =1

e

+ u m (n) m = 1,!, M n = 1,!, N

(2)

Where K is the number of source signals, M is the number of elements of a linear array. N is the number of snapshots. ak is the complex amplitude of the kth source signal, φ k the initial phase, ω k is the center frequency, and θ k is the DOA of the kth source signal. u m (n) is the additive alpha-stable noise. The RBFNN is employed to perform the inverse mapping from the space of sensor output {x(1), x(2),!, x(Q)} to the space of DOA {θ(1), θ(2),!, θ(Q)} . The input vectors are mapped through the hidden layer and each output node computes a weighted sum of the hidden layer outputs, as shown in the following. Q

θ k ( j ) = ∑ g ik h(|| x( j ) − x(i ) || 2 ) i =1

k = 1, ! , K

j = 1, ! , Q

(3)

Where Q is the number of training samples. h(⋅) can be selected as Gaussian function, g ik is the ith weight of the network.

334

Hong Tang et al. Hidden layer Input layer Signals

M

3

2

2

1

θK

z 2 M ( M −1)

Preprocessing

K

Output layer

θ3

z3

θ2

z2

θ1

z1

1

Fig. 1. The architecture of RBFNN for DOA estimation.

3.1 Data Preprocessing Generating the array output vector according to equation (3), then we transform the vector into an input vector of RBFNN and produce the estimation of DOA. Since in the DOA estimation, the initial phase of signals φ k contains no information about the DOA. It can be eliminated from the training data by forming the FLOC cij . N

(

cij = ∑ xi (n) x j (n) n =1

)

< p −1>

i, j = 1,!, M

(4)

In Gaussian noise environment, spatial correlation preprocessor can be used. But, this second-order statistics (SOS) preprocessor is not valid here. So this paper uses a more robust preprocessor FLOC (5), we call it as FLOS-based preprocessor, seen in part 4. Since for i = j , cij does not carry any information on the DOA, the rest of the elements cij can be rearranged into a new input vector b given as

[

b = c21 ,!, c M 1 , c12 ,!, c M 2 ,!, c1M ,!, c M ( M −1)

]

T M ( M −1)×1

(5)

It follows that the number of input nodes is given by M ( M − 1) . Note that we need twice as many input nodes for the neural network since RBFNN does not deal with complex numbers. Hence, the total number of input nodes should be 2M ( M − 1) . The dimension of the hidden layer is equal to the number of the Gaussian functions that can be chosen as Q if the perfect recall is desired. Obviously, the number of output node is obviously equal to the number of signals K . In the simulation performed later, the relative signal power is taken as unity though different power levels do not affect the procedure of detecting the DOA. The input vector is then normalized by its norm in the training, testing, i.e. z = b b . 3.2 Network Training and Testing Now, the steps for DOA estimation in alpha-stable noise environment can be summarized as follows: 1. Generate array output vectors {x(q), q = 1,!, Q} 2. Estimate the FLOC matrix of the array output vector {C(q), q = 1,!, Q}

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3. Form the vectors { b(q), q = 1,!, Q} using (6) and form the normalized vectors { z(q), q = 1,!, Q} using (7) 4. Form the training set {z ( q), q = 1,!, Q} and employ an appropriate training procedure to learn the training set. 5. Form the test set {z test } to evaluate the RBFNN. The main advantage with an RBFNN over other neural network approaches is that it does not require training the network with all possible combinations of input vectors. In this paper, we generate the source signals uniformly distributed from − 90 " to + 90 " to train and test a RBFNN in the simulation.

4 Flom-Music Algorithm The classical MUSIC degrades because the covariance is sensitive to the impulsive noise, i.e., the covariance is unbounded in alpha-stable noise environment. Boundedness is important for statistical signal processing in that statistical analysis cannot evaluate unbounded quantities to obtain reasonable conclusions. In open documents, the ROC-MUSIC [3] and FLOM-MUIC [4] are notable. Both algorithms are based on the fractional low-order statistics (FLOS) because the FLOS is bounded and valid to impulsiveness distribution. The vector form of (3) is given as x(n) = As(n) + u(n) , where x(n) = [x1,(n), ⋅⋅⋅, xM(n)]T is the output vector of sensor array, A is the steering matrix, s(n) = [s1,(n), ⋅⋅⋅, sK(n)]T is the source signal vector and u(n) = [u1,(n), ⋅⋅⋅, uM(n)]T is a alpha-stable noise vector. According to the definition of FLOC in (2), the FLOC matrix C can be given as C = E[ XX < p −1> ] = AΛΛ H + γ I

(6)

The detailed expression of Λ and γ can be found in [4]. From (9), it is found that the DOA information is involved in C . C is considered with full rank and sub-space techniques can be applied to extract DOA information. Perform SVD on C , and construct the M × ( M − K ) matrix E n = [e K +1 , e K + 2, !, eM ] , where [e K +1 , e K +2, !, eM ] are the left singular associated with the smallest M − K singular values of C . Compute the null spectrum S (ϕ ) = 1 a H (ϕ )E n E nH a(ϕ ) . where the M × 1 steering vector a(ϕ ) is

[1, e

− j 2π ( d 2 / λ ) sin θ

]

,! , e − j 2π ( d M / λ ) sin θ . Note that C should be computed by time average

instead of ensemble average. The peak locations of (10) are the estimation of DOAs. For DOA estimation of correlated/coherent signals, spatial smoothing is a possible way to decorrelate [5].

5 Simulation Results In the simulation performed, a Generalized Signal Noise Ratio (GSNR) is defined

( { } γ ) . In fact, the GSNR is a direct generalization from SNR.

as GSNR = 10 log E s(t )

2

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When the alpha-stable noise reduces to the Gaussian noise, the GSNR is the ratio of signal variance to noise variance. Uncorrelated signal sources: An array of M = 10 elements is used. A hidden layer of 200 nodes was chosen. A total of 1000 snapshots for training and testing are available to the simulations. The characteristic exponent of the alpha-stable noise is typically chosen to α = 1.5 . The width parameter of Gaussian function is uniformly set as 0.05. In Fig 2, the array receives two uncorrelated signals with different angular separations (∆θ = 10 " and 15" ) where the DOAs are assumed to be uniformly distributed from − 90 " to 90 " in both the training and testing phases with GSNR=0 dB. 200 input vectors were used for training. For the test phase, 50 input vectors were used for the networks simulated with ∆θ = 10 " and 15" . The results show that the network successfully produces actual outputs (+) very close to the desired DOA (solid lines). This also demonstrates that the network improved its performance through generalization and yielded satisfactory results.

Fig. 2. DOA estimation versus number of samples Q using RBFNN. Source 1 is varied from –90° to 90°, while source 2 is 10° and 15° separated from source 1.

Fig. 3. DOA estimation with two coherent signals with ∆θ = 10°.

Coherent signal sources: To study the effect of coherent signal sources on the performance of the RBFNN, the testing data was generated assuming the array receives two coherent signals with angular separation of 10°. The training data was performed with data derived from signals with GSNR=0 dB and the testing was performed after

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spatial smoothing with data contaminated by additive alpha-stable noise with GSNR= 0 dB. DOA obtained from RBFNN for coherent signals are plotted in fig 3.

6 Conclusions This paper introduces alpha-stable distribution as the impulsive noise model of wireless channel. To avoid huge computation, the problem of DOA estimation is treated as a nonlinear mapping from sensors output to the DOA. A suitable RBFNN is employed to approximate the nonlinear mapping. FLOS-based preprocessor suppresses the negative effect of impulsiveness successfully.

References 1. Nikias, C. L., Shao, M.: Signal Processing with Alpha-Stable Distribution and Application. New York, John Wiley & Sons, Inc (1995) 13-108 2. Blackard, K. L., Rappaport, T. S.: Measurements and Models of the Radio Frequency Impulsive Noise for Indoor Wireless Communications. IEEE Journal on Selected Areas in Communications (1993) 991-1001 3. Tsung-Hsien, L., Mendel, J. M.: A Subspace-Based Direction Finding Algorithm Using Fractional Lower Order Statistics. IEEE Transactions on Signal Processing (2001) 16051613 4. El Zooghby, A. H., Christodoulou, C. G., Georgiopoulos, M.: Performance of Radial-basis Function Networks for Direction of Arrival Estimation with Antenna Arrays. IEEE Transactions Antennas and Propagation (1997) 1611-1617 5. Rupi, M., Tsakalides, P., Re, E.D., Nikias, C. L.: Robust Spatial Filtering of Coherent Sources for Wireless Communications. Elsevier Signal Processing (2000) 381-396