Localization of Narrow-Band Sources in Unknown Spatially Correlated Noise
© Salah Bourennane et al. 2010
Received: 9 July 2009
Accepted: 10 February 2010
Published: 28 March 2010
In subspace-based method for direction-of-arrival (DOA) estimation of signal wavefronts, the additive noise term is often assumed to be spatially white or known to within a multiplicative scalar. When the noise is nonwhite but has a known covariance matrix, we can still handle the problem through prewhitening. However, the problem turns to be complex when the noise field is completely unknown. In this paper, we study the localization of the sources, when the noise covariance matrix is one unknown band matrix. An iterative denoising algorithm based on the noise subspace spanned by the eigenvectors associated with the smallest eigenvalues is developed. The performance of the proposed algorithm is evaluated by computer simulations. We also test the proposed algorithm with some experimental data recorded during an underwater acoustic experiment.
Array processing is used in diverse areas such as radar, sonar, communications, and seismic exploration. Usually the parameters of interest are the directions of arrival of the radiating sources. The high-resolution subspace-based methods for direction-of-arrival (DOA) estimation have been a topic of great interest. The subspace-based methods well-developed so far require a fundamental assumption, which is that the background noise is uncorrelated from sensor to sensor, or known to within a multiplicative scalar. In practice this assumption is rarely fulfilled and the noise received by the array may be a combination of multiple noise sources such as flow noise, traffic noise, or ambient noise, which is often correlated along the array [1, 2]. However, the spatial noise is estimated by measuring the spectrum of the received data when no signal is present. The data for parameter estimation is then prewhitened using the measured noise. The problem with this method is that the actual noise covariance matrix varies as a function of time in many applications. At low signal-to-noise ratio (SNR) the deviations from the assumed noise characteristics are critical and the degradation may be severe for the localization result. A maximum likelihood algorithm is presented in , where the spatial noise covariance is modeled as a function of certain unknown parameters. Also in [4, 5] a maximum likelihood estimator is analyzed. The problem of incomplete prewhitening or colored noise is circumvented by modeling the noise with a simple descriptive model. As opposed to AR and ARMA models used in  this gives the advantages of incorporating prior knowledge into the model [5, 7]. There are other approaches to the problem of spatially correlated noise: one is based on the assumption that the correlation structure of the noise field is invariant under a rotation or a translation of the array, while another is based on a certain linear transformation of the sensor output vectors [8–13].
In this paper, we present an algorithm to estimate the noise with band covariance matrix. This algorithm is based on the noise subspace spanned by the eigenvectors associated with the smallest eigenvalues of the covariance matrix of the recorded data. The goal of the present study is to investigate how the perturbations due to the noise covariance matrix affect the accuracy of the narrow-band signal DOA estimates [14–18].
The remainder of the paper is as follows. Section 2 presents the high resolution algorithm. Section 3 proposes the model of the additive noise. Section 4 introduces the covariance matrix of the band noise. Section 5 presents the proposed algorithm. Sections 6 and 7 present some comparative results concerning the proposed algorithm exemplified using simulated data and real-world signals in a noisy environment. Section 8 gives the conclusions.
2. Problem Formulation
where denotes the expectation operator, superscript represents conjugate transpose, is the noise covariance matrix, and is the signal covariance matrix. The above assumption concerning the noncorrelation of the sources means that is full rank.
3. Modeling the Noise Field
An internal noise generated by the sensors so-called thermal noise. This noise is assumed to be independent [8, 11] from sensor to sensor, but not necessarily spatially white. Then the spatial covariance matrix of this noise denoted is diagonal.
An external noise received on the sensors, whose spatial covariance matrix is assumed to have the following structure [8, 11, 12, 19], , where are unknown parameters and are complex weighting matrices, are chosen such that is positive definite and of band structure.
4. Modeling the Covariance Matrix of the Band Noise
In many applications when a uniform linear array antenna system is used, it is reasonable to assume that noise correlation is decreasing along the array (see Figure 1). This is a widely used model for colored noise. We can then obtain a specific model for noise correlation under the following assumptions:
The noise correlation model which is obtained is represented on Figure 1.
In the following section, an algorithm to estimate the band noise covariance matrix is developed for narrow-band signals.
5. Estimation of the Noise Covariance Matrix
5.1. Proposed Algorithm
Several methods have been proposed for estimating the directions of arrival of multiple sources in unknown noise fields. Initially the noise covariance matrix is measured, when signals of interest are not present. Other techniques [5, 7, 21] based on the maximum likelihood algorithm are developed, which incorporate a noise model to reduce the bias for estimating both the noise covariance matrix and the directions of arrival of the sources. In this paper, our approach is realized in two steps. Using an iterative algorithm, the noise covariance matrix is estimated, then this estimate is subtracted from the covariance matrix of the received signals.
Using (9), we obtain a constraint for estimating correctly the largest eigenvalues and the corresponding eigenvectors. Indeed by minimizing the error we can estimate the eigen-elements by an iterative algorithm. Then by using (10) we calculate the noise covariance matrix. The proposed method for estimating the noise covariance matrix can be summarized as follows.
Estimate the covariance matrix of the received signals using snapshots: . Calculate the eigendecomposition of this matrix: with and , where , and are, respectively, the eigenvalue and the corresponding eigenvector.
Eigendecomposition of the matrix . The new matrices and are calculated using the previous steps. Repeat the algorithm until a significant improvement of the estimated noise covariance matrix is obtained.
The iteration is stopped when or the Frobenius norm calculated only with the elements such that for and is a fixed threshold. Symbol stands for Frobenius norm and the superscript " " indicates the iteration.
5.2. Estimation of the Spatial Correlation Length
6. Simulation Results
In the following simulations, a uniform linear array of omnidirectional sensors with equal interelement spacing is used, where is the mid-band frequency and is the velocity of propagation. The number of independent realizations used to estimate the covariance matrix of the received signals is . The signal sources are temporally stationary zero-mean white Gaussian processes with the same frequency . Three equipower uncorrelated sources impinge on the array, with the . The noise power is taken as the average of the diagonal elements of the noise covariance matrix .
Band-Toeplitz noise covariance matrix, with each element given by a modeling function;
Band-Toeplitz noise covariance matrix with the elements arbitrary chosen;
Band noise covariance matrix used in .
6.1. Noise Covariance Matrix Estimation and Results Obtained
To localize the directions of arrival of sources and to evaluate the performance of the proposed algorithm, the high-resolution methods such as MUSIC [9, 10] are used after the preprocessing, with priori knowledge of the exact number of sources ( ).
Example 1 (Band-Toeplitz noise covariance matrix).
In each of the two studied cases ( and ), the noise covariance matrix is estimated with a fixed threshold value after a few iterations and we notice that the number of iterations for is greater than that of .
Example 2 (Band-Toeplitz noise covariance matrix with the elements arbitrary chosen).
Example 3 (Band noise covariance matrix using model in [16, Example ]).
The configuration of this experiment contains a ten-element uniform rectilinear array with three sources at , , and (angles are measured with respect to the normal of the array). One hundred snapshots of array data were taken.
where is the noise variance equal for every sensor and is the spatial correlation coefficient. Figure 3 shows the root-mean-squared error versus sensor noise correlation factor over the range from 0 to 0.9. The SNR is held constant at , the number of sensor and the noise spatial correlation length .
Figure 5 shows the RMSE performances of the tested algorithms and the CRB with respect to the number of snapshots.
Example 3 t.
In order to study the influence of the estimation of the spatial correlation length on the localisation of the sources, we have considered the same data as in the previous example. The value of is varied over the range 1 to and for each value the number of the sources (number of largest eigenvalues) is estimated after applying the proposed algorithm. Figure 6 shows that the estimation of the number of the sources is sensitive to the choice of . The number of the sources is correctly even the value of is overestimated . Figure 7 shows the variations of the bias of the estimated angles for different values of . When is equal to 5 we obtain good results— and very small value of a bias—this conclusion is observed for several scenarios.
Figure 8 shows the norm of the vector difference between the 10 elements of the principal diagonal of the simulated matrix and those of the estimated matrix for and .
Figures 9(a), 9(b), 10(a), 10(b), 11(a), and 11(b) show the localization results of the sources before and after the preprocessing. Before the preprocessing, we use directly the MUSIC method to localize the sources. Once the noise covariance matrix is estimated with the proposed algorithm, this matrix is subtracted from the initial covariance matrix of the received signals, and then we use the MUSIC method to localize the sources. The three simulated sources are , , and for Figure 9; , , and for Figure 10; , , and for Figure 11. For Figure 10, the SNR value (10 dB) is greater than those of Figures 9 and 11 ( dB).
The comparison of the results of Figures 10 and 11 comes to the conclusion that the MUSIC method cannot separate the close sources without the preprocessing when the SNR is low, so in Figure 10 we can only detect two sources before preprocessing. And for each case we can note that there is an improvement in the results obtained with the preprocessing. Comparing the results of with that of for each figure, we can also reconfirm that when increases, the estimation error increases on the estimated noise covariance matrix, so we obtain better results with the preprocessing for than for .
6.2. Performance of the Proposed Method versus Noise Spatial Correlation Length
The experimental results presented in Figure 13 show that the correlation length also influences the estimate of the DOA values.
The spatial correlation length authorized by the algorithm is a function of the number of sensors and the number of sources. Indeed, the number of parameters of the signal to be estimated is , and the number of parameters of the noise is . In order to estimate them it is necessary that and that . In the limit case: , we have , which corresponds to a bidiagonal noise covariance matrix. If the model of the noise covariance matrix is band-Toeplitz [5, 21], the convergence of the proposed algorithm is fast, and the correlation length of the noise can reach .
7. Experimental Data
Figure 16(a) shows the obtained results of the localization using MUSIC method on the covariance matrices. The DOA of the reflected signals on the two objects are not estimated. This is due to the fact that the noise is correlated.
Figure 16(b) shows the obtained results using our algorithm. The two objects are localized.
In this paper the problem of estimating the direction of arrival (DOA) of the sources in the presence of spatially correlated noise is studied. The spatial covariance matrix of the noise is modeled as a band matrix and is supposed to have a certain structure. In the numerical example, the noise covariance matrix is supposed to be the same for all sources, which covers many practical cases where the sources are enclosed.
This algorithm can be applied to the localization of the sources when the spatial-spectrum of the noise or the spatial correlation function between sensors is known. The obtained results show that the proposed algorithm improves the direction estimates compared to those given by the MUSIC algorithm without preprocessing. Several applications on synthetic data and experiment have been presented to show the limits of these estimators according to the signal-to-noise ratio, the spatial correlation length of the noise, the number of sources, and the number of sensors of the array. The motivation of this work is to reduce the effect of the additive spatially correlated noise for estimating the DOA of the sources.
The authors would like to express their thanks to the anonymous reviewers for their careful reading and their helpful remarks, which have contributed in improving the clarity of the paper.
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