Sparse spatial spectral estimation in directional noise environment

High-resolution direction of arrival (DOA) estimation methods, such as the multiple signal classification method, typically assume that additive noise is white Gaussian noise, i.e., its covariance matrix is an identity matrix multiplied by the noise power. However, the ambient noise at any two different sensors, in practice, may correlate with each other, and the spatial distribution of ambient noise intensity may be directional because of wind and shipping noise sources.¹ A linear harmonic noise model² is considered in this study to appropriately describe the ambient noise. The model is based on a truncated Fourier expansion of the ambient noise field. The noise covariance matrix is linear in the Fourier coefficients.

The compressive sensing theory³ has been developed in recent years, which can effectively solve the overdetermined linear equations. When the search grid of candidate directions is far greater than the number of signals that impinge on the array, the signal model of the array is sparse. Several efficient sparse spatial spectral estimation methods have been proposed, such as focal underdetermined system solution for multi-snapshot version,⁴ L1-singular value decomposition,⁵ and sparse spectrum fitting (SpSF) algorithm.⁶ However, these methods are all evaluated with the assumption of white Gaussian noise, thereby restricting their practical application.

In this letter, a directional background noise sparse spectrum fitting (DN-SpSF) algorithm is proposed to estimate the spatial spectral from the covariance matrix in a directional noise environment. Compared with the SpSF algorithm, the proposed algorithm is more robust to the choice of regularization parameter and shows lower sidelobe level in both simulations and experimental results.

Consider a sensor array of $N$ elements with its steering vector denoted by $a(θ)$⁠. $K$ far-field and narrow-band signals are impinged on the array from unknown directions $θ=(θ1,…,θK)$⁠. The output of the array $x(n)$ can be modeled as

where $A=[a(θ1),…,a(θK)]$ is the steering matrix of the array, $N$ is the total number of snapshots, $s(n)=[s1(n),…,sK(n)]T$ contains the source waveforms, and $e(n)$ is the noise term.

Assume that noise term $e(n)$ is uncorrelated with the sources, and the data covariance matrix $R$ is given by

where $E{}$ denotes the expectation operation, the superscript $H$ is the Hermitian transpose, $P=E{s(n)sH(n)}∈CK×K$⁠, and $Σ=E{e(n)eH(n)}$ is the covariance matrix of the additive noise term. This covariance matrix is traditionally estimated by the sample covariance matrix $R̂=1/N∑n=1Nx(n)xH(n)$⁠.

The commonly used model of the noise term is the zero-mean white Gaussian noise with covariance matrix $E{e(n)eH(n)}=σ2I$⁠. However, the ambient noise at any two different sensors, in practice, may correlate with each other, and the spatial distribution of ambient noise intensity may be directional because of wind and shipping noise sources. An appropriate noise model is established to express the ambient noise objectively. The noise intensity denoted by $v(θ,n)$ is regarded as a function of azimuth $θ$ at a given time $n$⁠. Furthermore, we assume that the random variable $v(θ,n)$ is spatially independent and has a zero-mean Gaussian distribution in the time domain, that is,

where $δnm$ is the Kronecker delta function, $δ(θ−σ)$ is the Dirac delta function, and $ε(θ)$ is the spatial power density function of noise. By using this assumption, we obtain the noise covariance as⁷ 

in which $ε(θ)$ is a periodic function; thus, it can be represented by Fourier series as follows:

where

A small number of terms, written as $L$⁠, can be used as the order of the Fourier series because the noise spectral density function is smooth and varies slowly with the azimuth $θ$ in most practical cases of interest in the underwater environment.⁸ Substituting Eqs. (5) and (4) into Eq. (2) and letting the order of Fourier series be $L$ obtain⁷ 

where $J=2L+1$⁠, and

As shown in these equations, $Σ¯j$ only depends on the steering vector and is independent of the ambient noise information. When $L=0$⁠, then $Σ=c0Σ¯0$⁠. Moreover, when the interval between any two adjacent elements is equal to the half wavelength, then $Σ=c0I$⁠. At this time, the ambient noise model is equivalent to the white Gaussian noise model.

DOA estimation methods usually involve a search over a grid of $Q$ candidate directions, which constitute a set denoted by $Θ$⁠. If the number of sources is $K$ and an $M$ element array is used, then the sources are sparse when K < M < Q,⁹ particularly when $K$ is far smaller than $Q$⁠.

The recently proposed SpSF algorithm is formulated by applying $l1$-norm penalization of fitting the source covariance model to the estimated spatial covariance.⁶ Under the condition of white Gaussian noise, covariance matrix $R$ is given by $R=APAH+σ2I$⁠. After applying the vectorization operation to $R$⁠, we obtain the following relation:

where $r=vec(R)$⁠, $vec()$ denotes the vectorization operation; $Bi$ is the $i$th column of matrix $B$⁠, in which $Bi=a∗(θi)⊗a(θi),θi∈Θ$⁠, the superscript $∗$ donates conjugate operation; and $p=diag(P)$⁠, which takes the diagonal elements of $P$ into a vector. On the basis of Eq. (9) and the sparsity-promoting $l1$-norm penalization,⁶ the DOA estimator of SpSF algorithm can be given as follows:

where $r̂=vec(R̂)$ and $λ$ is a regularization parameter. We can refer to Refs. 6 and 10 to choose an appropriate value of this parameter.

On the basis of the condition of spatial directional ambient noise, which benefits from the ambient noise model described in Eq. (7), we obtain the following expression after applying the vectorization operation:

where matrix $Γ=[vec(Σ0),vec(Σ1),…,vec(ΣJ)]$ and vector $η=[η1,…,ηJ]T$⁠. Therefore, the SpSF method in the directional ambient noise environment (written as DN-SpSF) can be given as

where $λ$ is also a regularization parameter.

The DN-SpSF method, as a parametric algorithm, uses the ambient noise structure and improves the performance of DOA estimation. White Gaussian noise is a special case of ambient noise model; thus, the DN-SpSF method can be regarded as a more generalized form of the SpSF method.

A uniform linear array with ten isotropic sensors and a half wavelength inter-element spacing is considered in this work. Two uncorrelated far-field narrow-band signals impinge on the array from the direction of 45° and 90°, respectively (the direction of 0° is defined as the endfire direction). The powers of these two signals are 0 dB. Noise spectral density function $ε(θ)$ is shown in Fig. 1(a), of which the order of Fourier series is 3, and the coefficient vector is defined as $η=[1.5,0,0.2,0,0.4,0,0.1]T$⁠. Total snapshots $N$ is 100, which is 10 times larger than the element number. The regularization parameters of both SpSF and DN-SpSF algorithms are set to 2. The observable directions range from 0° to 180° at 0.1° intervals. Therefore, $Q$ is 1801, which is far greater than the number of the signals. Signal-to-noise ratio (SNR) is defined as the rate of the power of the first signal to the noise received by the reference element, which is $SNR=10lgE{s1(n)s1∗(n)}/∫02πε(θ)dθ$⁠.

Figure 1(b) depicts the spatial spectral estimation obtained by CBF (conventional beamforming), SpSF, and DN-SpSF algorithms with an SNR of −10 dB. The order of Fourier series expansion in the DN-SpSF algorithm is exactly equal to 3, which is equal to the real order in the linear noise model. The CBF algorithm is affected by directional ambient noise and has a high sidelobe level. As the sparsity of the signal model is considered, the SpSF algorithm outperforms the CBF methods. An advantage of the linear noise model is that the DN-SpSF algorithm has the lowest sidelobe level and the highest resolution among the three DOA estimation algorithms.

Next, the SpSF and DN-SpSF methods are evaluated in 100 independent trials for each regularization parameter $λ$ between 0.5 and 3 with a step of 0.1. The SNR is equal to 0 dB, and the DOA estimate root-mean-square error (RMSE) is defined as $E{(θ1−θ̂1)2+(θ2−θ̂2)2}/2$⁠, where $θ̂1$ and $θ̂2$ are the estimates of $θ1$ and $θ2$⁠, respectively. In Fig. 1(c), we evaluate the RMSE of the SpSF and the DN-SpSF methods for each regularization parameter. The DN-SpSF method has a lower RMSE than the SpSF method in the directional noise environment. Moreover, the RMSE of the DN-SpSF algorithm varies within a small range with the change of regularization parameter. Thus, the DN-SpSF algorithm is more robust to the choice of regularization parameter.

The order of the linear noise model is also a factor to influence the performance of DN-SpSF algorithm. Here, DN-SpSF algorithm is evaluated in 100 independent trials for each order of the linear noise model ranging from 1 to 5, and the regularization parameter is set to 2. The RMSE of DN-SpSF algorithm is listed in Table 1 in two different SNR situations. When the order used in the linear noise model is greater than or close to the real order (that is 3 in this simulation), the value of RMSE is approximately invariant and significantly less than the 1-order linear noise model case. As mentioned in Eq. (5) and Eq. (6), the high order coefficients of Fourier series expansion for a smooth noise spectral density function will be close to zero. Therefore, the low order coefficients of Fourier series expansion are the main factors affecting the fitting performance. Considering that the noise spectral density function is always smooth and varies slowly with the azimuth in most practical cases of interest in the underwater environment,⁸ the order $L$ in Eq. (7) is usually a small number. It is reasonable to choose the order around 5 in the next experimental processing, since we consider that it is enough to contain the main low orders.

Some underwater source localization results based on the DN-SpSF algorithm will be presented in this section. Experimental data were collected with 32 hydrophones uniformly spaced along a horizontal line in the ocean during autumn. The inter-sensor spacing was approximately 4 m and the uniform linear array was placed at a depth of 50 m below the sea surface. Two ships, called ships A and B, moved along a straight line in the far field of the sensor array, as shown in Fig. 2. The scanning direction grid is chosen to uniformly cover the entire region of interest $Θ=[0°,180°]$ with a step size of 0.5°. The endfire direction of the array is 0°. We processed the broadband experimental data after dropping sampling at 2048 Hz considering a frequency band of 100 to 250 Hz. The total signal time to be analyzed is 1 h and the integration time is 20 s which is divided into 20 snapshots with 50% of the data overlapping in each snapshot.

The bearing time recordings (BTRs) of the CBF, SpSF, and DN-SpSF algorithms are compared in Fig. 3. A 5-order linear noise model is utilized in the DN-SpSF algorithm, and the regularization parameters of the SpSF and DN-SpSF algorithms are set to 0.5. The tracks of two ship targets, together with other interferences, are exhibited in Figs. 3(a)–3(c). According to GPS information provided by test ship, ship A drove from the direction of approximately 150° to the endfire direction of the sensor array and then drove to the direction of approximately 90°. The performance of DOA estimation degrades rapidly near the endfire of the array because of the short equivalent array aperture in the endfire direction. Meanwhile, ship B traveled from the direction of approximately 60° to 150° and from near to far distance, which is also reflected in the amplitude of the spatial spectral diagrams. As shown in Fig. 3, the BTRs of the SpSF and the DN-SpSF algorithms are superior to that of the CBF because of the application of signal sparsity. It is obvious that the estimated directions of ship B from SpSF and DN-SpSF before 30 min have much higher resolutions than that from CBF. However, the ambient noise is usually directionally distributed in the azimuth direction. After applying the linear noise model, as shown in Fig. 3(c), the DN-SpSF algorithm obtains a lower sidelobe level than the SpSF algorithm. Figure 3(d) depicts the profile of the above algorithms at $t=35min$⁠. In Fig. 3(d), the average sidelobe level of DN-SpSF algorithm in the space between Ship A and Ship B is about −36.7 dB which is significantly lower than that of SpSF algorithm with the average sidelobe level as −28 dB. The estimated power of ship A and B at $t=35min$ shown in Fig. 3(d) is −12.86 dB and −9.42 dB for SpSF algorithm and −13.02 dB and −9.32 dB for DN-SpSF algorithm, respectively. Therefore, the spatial spectral estimation of the DN-SpSF algorithm maintains a similar signal power as the SpSF algorithm but achieves lower sidelobe level, which proves that the DN-SpSF algorithm has a better adaptability.

A DN-SpSF algorithm has been proposed to estimate the spatial spectral from the covariance matrix in a directional noise environment. The simulations and experimental results demonstrate that the DN-SpSF algorithm outperforms the SpSF algorithm in terms of low sidelobe level and good DOA estimation performance in a spatially directional noise environment. Hence, our method can potentially detect a target in a complex ocean environment.

This work was supported by the National Natural Science Foundation of China under Grant No. 11527809.