Wideband direction of arrival (DOA) estimation using a sensor array plays a fundamental role in passive sonar signal processing. Although sparsity-based DOA estimation methods can attain high resolution in the condition of few snapshots and low signal-to-noise ratio, the localization accuracy is seriously affected by strong interferences. In this paper, a matrix filter with nulling (MFN) is used to pass weak targets in sector-of-interest (passband) while attenuating the out-of-sector (stopband) interferences by forming deep nulls toward the directions of interferences adaptively. Then, a method based on sparse spectrum fitting (SpSF) and MFN is proposed to localize closely spaced wideband signals in a strong interference environment. In comparison with the minimum variance distortionless response and SpSF, the proposed method achieves higher localization accuracy, which is verified by simulation and experimental results.
I. INTRODUCTION
High-resolution direction of arrival (DOA) estimation using sensor array is an important research subject in passive sonar signal processing. The minimum variance distortionless response (MVDR)1 method and multiple signal classification (MUSIC)2 algorithm are commonly used methods because of their high resolution. However, these traditional high-resolution methods suffer from some limitations to applications, (1) they require sufficient snapshots to achieve high resolution; (2) they cannot handle the problem of coherent arrivals; (3) for the MUSIC algorithm, it is difficult to decompose the covariance matrix into the signal subspace and noise subspace correctly without prior knowledge of the number of the signals, particularly in a strong interference environment. Inaccurate subspace decomposition has a negative impact on the performance of MUSIC due to its sensitivity to subspace orthogonality. Recently, sparse signal processing3–7 has attracted considerable attention in DOA estimation. Numerous high-resolution DOA estimation algorithms based on sparse signal processing have been proposed, such as the sparse spectrum fitting (SpSF) algorithm3 and -singular value decomposition.4 In contrast to the aforementioned traditional high-resolution methods, sparse signal processing algorithms can be applied in the condition of fewer snapshots, as well as lower signal-to-noise ratio (SNR) to achieve high resolution. They can also handle coherent arrivals. The properties of sparse signal processing algorithms have been proven far superior to the aforementioned traditional high-resolution methods.
Passive sonar mainly observes the acoustic radiated noise of the surroundings to achieve target localization. However, when the radiated noise levels of some targets are low, the strong interferences, such as tow-vessel noise, will affect the localization accuracy seriously or even mask the weak targets directly. Hence, source localization in a strong interference environment is a difficult problem to address and a suitable DOA algorithm is needed to estimate the directions of weak targets in the strong interference environment.
Matrix filters have been widely used in passive sonar systems to pass the signals in sector-of-interest (passband) while suppressing the out-of-sector (stopband) interferences,12,13 which improves the subsequent source localization accuracy. Vaccaro and Harrison8 proposed the design criteria for a conventional matrix filter (CMF) in the frequency domain, which can be solved by convex optimization algorithms. Over the last few years, many algorithms have been used to design matrix filters. Zhu et al.9 developed a matrix filter design method by applying a semi-infinite programming technique, which is only suitable for linear arrays. Macinnes10 designed an efficient matrix filter by formulating the design procedure as a rank-deficient linear least-squares problem. Yan and Ma11 reformulated a matrix filter design as a second-order cone (SOC) programming. Unfortunately, the abilities of the aforementioned matrix filters in interference suppression are limited by stopband attenuation (SA). If the signal-to-interference ratio (SIR) is considerably lower than SA, then the power of the residual component of the interference after filtering remains stronger than weak targets. To overcome this shortcoming, an adaptive matrix filter was proposed in Ref. 15. This type of matrix filter is driven by the received data and is capable of forming deep nulls toward the directions of the interferences adaptively to attenuate the interferences sufficiently; thus, it can be regarded as matrix filter with nulling (MFN).
A method based on SpSF and CMF (SpSF-MF) was proposed in Ref. 14 to achieve high resolution in a strong interference environment. However, when the power of the interference after filtering remains stronger than the weak targets, the residual component of the interference will have a severe effect on the localization accuracy of SpSF-MF. Therefore, MFN is used instead of CMF in this paper, and a novel method based on SpSF and MFN (SpSF-MFN) is proposed to localize multiple wideband sources in a strong interference environment. The proposed method enhances the ability of sparse signal processing in weak target localization in a strong interference environment. The properties in interference suppression and localization accuracy are also verified by simulation and experimental results.
II. SPARSE SPECTRUM FITTING ALGORITHM
In this section, a sparse wideband signal model is established, and the SpSF algorithm is introduced.
A. Signal model
Consider a uniform linear array with M elements. Once the data are received, they are partitioned into N segments. Then, an L-point fast Fourier transform (FFT) is applied to each segment. The steering vector at the lth frequency bin can be written as
where is the center frequency of the lth frequency bin, is the corresponding wavelength, d is the sensor spacing, and the superscript T is the transpose. Discretizing the space of interest [−90, 90]° into Q candidate directions, the array output at the lth frequency bin can be modeled as
where , , and represent the FFT coefficients of the received data, wideband signals, and additive noise at the nth segment, respectively; is the steering matrix; and is the vector that comprises Q candidate directions. Given that only a few sources generate the acoustic field in fact, the number of the non-zero elements of is K. Therefore, Eq. (2) is a sparse signal model.
Assuming that is uncorrelated with the signals, the covariance matrix at the lth frequency bin can be formed as
where denotes the expectation operation; superscript represents the Hermitian transpose; and are the covariance matrices of the signals and the noise at the lth frequency bin, respectively.
Further, when the white Gaussian noise is assumed, the covariance matrix in Eq. (3) can be modified as
where is the variance of the additive noise. The diagonal elements of represent the signal power and are sparse with K non-zero elements because only K non-zero elements exist in .
B. Sparse spectrum fitting algorithm
Applying the vectorization operation to Eq. (4), the following relation can be obtained:
where , in which denotes the vectorization operation; is the ith column of the matrix , in which , where the superscript * donates conjugate operation and donates Kronecker product; and , which takes the diagonal elements of into a vector. The SpSF algorithm is formulated by applying penalization to fit the source covariance model to the estimated spatial covariance. Therefore, the SpSF estimator can be given as follows:
where is a regularization parameter; and represent the and , respectively.
After solving the problem in Eq. (6) at every frequency bin, the eventual estimation can be calculated as
III. SPARSE SPECTRUM FITTING USING THE MATRIX FILTER WITH NULLING
The SpSF algorithm can realize the high resolution in DOA estimation due to the application of signal sparsity. However, the localization accuracy of this algorithm will degrade when strong interferences exist. In this section, a novel method based on SpSF and MFN (SpSF-MFN) is proposed to achieve high localization accuracy in a strong interference environment.
A. Design of conventional matrix filter
Once a data snapshot in Eq. (2) is available, the output of the matrix filter can be expressed as
where is the matrix filter operator at frequency
The purpose of the matrix filter is to pass the sources in the passband with minimal distortion while attenuating the interferences in the stopband. Hence, the spatial filtering characteristics of the desired matrix filter are
where and are the sectors of the passband and stopband, respectively. The design criteria of the CMF can be described as
where is the steering matrix of the passband bearing sector, in which and are the left and right limitations of , respectively; is selected to specify the designed SA, and represents the Frobenius norm.
Notably, the CMF design is independent of the received data, and the SA is fixed after it has been designed. When interferences in the stopband are extremely strong, the residual component of the interferences after filtering has a negative effect on DOA estimation.
Assuming that only one signal and one uncorrelated interference exist in the passband and stopband at and , respectively, the covariance matrix of in Eq. (8) can be written as
where and are the power of the signal and interference, respectively, while and are the steering vectors of the signal and interference, respectively. Equation (11) shows that will be dominated by the residual component of the interference and cannot be ignored in the following DOA estimation when the power of the residual interference is considerably higher than that of the signal.
The passband constraint of the filter requires the steering vector of the signal in the passband with slight distortion. However, the stopband constraint only guarantees that the power of the interference is decreased regardless of the steering vector. Therefore, when the residual component of the interference is considerably stronger than that of the signal, the structure of is badly destroyed due to the change in the steering vector of the interference from to , which has a negative influence on DOA estimation. If the SA is set extremely low, then the passband response error will increase. Furthermore, setting the suitable SA is difficult when weak targets are masked by strong interferences and the prior knowledge of SIR is hard to obtain in this case. Therefore, this paper utilizes MFN to preprocess the received data to avoid the shortcoming of CMF.
B. Design of matrix filter with nulling
The output power of the matrix filter can be formed as
where represents the matrix trace.
The principle of the MFN is to minimize the output power after filtering while passing the signals of interest with minimal distortion and suppressing the interferences in the stopband. Hence, the problem can be formulated as15
where is the passband response error. The problem in Eq. (13) is a convex optimization problem and can be efficiently solved by convex optimization algorithms, such as SOC programming.15 MFN can form deep nulls toward the directions of the interferences for attenuation; thus, its ability in interference suppression is not limited by the SA, which indicates that MFN provides better performance in interference suppression than CMF.
C. Sparse spectrum fitting using the matrix filter with nulling
After MFN preprocessing, the covariance matrix of the filtered output can be given by
where , and the dimension of is . In practice, is always replaced by the sample covariance matrix . After applying the vectorization operation to Eq. (14), the following equation can be obtained:
where ; is the ith column of the matrix , , in which is the ith column of the matrix ; and , which takes the diagonal elements of into a vector. Through sparsity-promoting penalization, the SpSF-MFN estimator can be represented as follows:
IV. SIMULATION AND EXPERIMENT
A. Simulation results
A uniform linear array with 32 isotropic sensors and 4 m spacing is considered in this simulation. Two uncorrelated far-field wideband signals and a wideband interference impinge on the array from −5°, −8°, and 20° (90° is defined as the endfire direction). The frequency ranges of the signals and interference are [90,180] Hz. Interference-to-noise ratio (INR) is 40 dB, while the SNRs of the two signals are both 0 dB. The received data sampled at 5120 Hz are partitioned into 30 segments, and a 1024-point FFT is applied to every segment. The CMF and MFN are designed at every frequency bin with the passband bearing sector being [−40, 0]° and the stopband bearing sector being [−90, −50]° [10, 90]°. The amplitude responses calculated as and amplitude response errors calculated by are compared among the CMF with SA = −40 dB, the CMF with SA = −15 dB and the MFN at a low frequency (90 Hz), a middle frequency (135 Hz), and a high frequency (180 Hz). The results are shown in Fig. 1. In this simulation, SIR is −40 dB. Thus, the SA of the CMF should be set lower than −40 dB to suppress the interference sufficiently; that is, the CMF with SA = −40 dB can attenuate the interference sufficiently, whereas the CMF with SA = −15 dB cannot. For MFN, considering that it mainly depends on deep null to suppress the interference, its performance will not be limited by the SA. The SA of the MFN is set to −15 dB in this simulation.
(Color online) Simulation results: (a), (c), and (e) amplitude responses of the three matrix filters at 90, 135, and 180 Hz, respectively; (b), (d), and (f) amplitude response errors of the three matrix filters at 90, 135, and 180 Hz, respectively. Two green dashed lines represent the left and right limitations of the passband bearing sector, respectively.
(Color online) Simulation results: (a), (c), and (e) amplitude responses of the three matrix filters at 90, 135, and 180 Hz, respectively; (b), (d), and (f) amplitude response errors of the three matrix filters at 90, 135, and 180 Hz, respectively. Two green dashed lines represent the left and right limitations of the passband bearing sector, respectively.
As shown in Figs. 1(a), 1(c), and 1(e), the passbands and the stopbands of the CMF with SA = −15 dB and the MFN satisfy the design requirements. Moreover, the MFN provides a deep null toward the direction of the strong interference. The amplitude response of the deep null is −50 dB, which can suppress the interference sufficiently. However, the passband bearing sector of the CMF with SA = −40 dB at the low frequency shown in Fig. 1(a) is smaller than the desired passband bearing sector. As frequency increases, the passband bearing sector becomes larger but still slightly smaller than the desired passband bearing sector. Figures 1(b), 1(d), and 1(e) show the passband response errors of three matrix filters at different frequencies. The passband response error of the CMF with SA = −15 dB is the smallest among three matrix filters. The passband response error of the MFN is slightly larger than that of the CMF with SA = −15 dB but considerably smaller than the CMF with SA = −40 dB.
Hence, this simulation indicates that the amplitude responses and amplitude response errors of the CMF and MFN are nearly the same when the SAs are set the same; however, the performance of the MFN on interference suppression is considerably better than that of the CMF because the MFN provides a deep null to attenuate the strong interference. When the SA of the CMF increases to strengthen the performance on interference suppression, the amplitude response and the passband response error of the CMF become markedly worse than that of the MFN. Overall, the performance of the MFN is far superior to that of the CMF.
Figure 1 indicates that the passband response error of the CMF becomes large rapidly with the increase of the SA. To avoid the large passband response error, the SAs of the CMF and MFN are set to −15 dB in the subsequent simulations. The amplitude responses of the CMF and MFN at all frequency bins are plotted in Fig. 2.
(Color online) Amplitude responses of the MFN and CMF in all frequency bins.
From Fig. 2, it can be seen that the sectors of the passbands and the stopbands among all frequency bins satisfy the design requirements and remain nearly unchanged. The amplitude response of the passband is approximately 0 dB in each matrix filter, and the stopband is strictly controlled under −15 dB. Moreover, the MFN at each frequency bin provides a deep null to attenuate the interference, the amplitude response of which is nearly −50 dB. Then, the MVDR, SpSF, SpSF-MF, and SpSF-MFN algorithms are applied to estimate the DOAs of two weak targets. Given that the sample covariance matrix is singular in this condition, the diagonal loading (DL) technique17 is applied to the MVDR algorithm. The DL level is set to 0.09. The DOA estimation results are shown in Fig. 3.
(Color online) Estimation results of (a) MVDR, (b) SpSF, (c) SpSF-MF, and (d) SpSF-MFN. The red dots and blue dashed lines represent the estimated DOAs in the passband and true DOAs, respectively.
(Color online) Estimation results of (a) MVDR, (b) SpSF, (c) SpSF-MF, and (d) SpSF-MFN. The red dots and blue dashed lines represent the estimated DOAs in the passband and true DOAs, respectively.
As shown in Fig. 3(a), the MVDR algorithm only forms a peak between two targets and fails to distinguish two weak targets at −8° and −5° for the lack of snapshots. In Fig. 3(b), SpSF can hardly localize two sources with only 3° spacing in the strong interference environment, and many pseudo peaks exist around the strong interference. In Fig. 3(c), the SpSF-MF algorithm becomes invalid, for the residual component of the interference after filtering is still much stronger than weak targets and destroys the structure of the sample covariance matrix, which severely impacts DOA estimation. In Fig. 3(d), the proposed method can distinguish two close sources and simultaneously suppress the strong interference.
Then, six different INR situations are considered, and 1000 independent simulations are conducted to calculate the probability of source resolution in each situation. The observable directions range from −90° to 90° at 1° intervals. For each independent simulation, the sources are considered to be distinguished successfully if the estimated DOAs , and the correct directions , are satisfied18
After finishing all simulations, the probability of source resolution can be calculated by dividing the times that the sources are distinguished successfully by the total number of the simulations.
Meanwhile, the ensemble root-mean-squared errors (RMSEs) are computed by
where is the estimated DOA of the ith signal in the sth simulation. The results are shown in Fig. 4.
(Color online) Results of (a) probability of source resolution versus INR, and (b) RMSE versus INR.
(Color online) Results of (a) probability of source resolution versus INR, and (b) RMSE versus INR.
Figure 4 clearly shows that the MVDR algorithm retains poor performance in all INR conditions and the RMSE is the largest among four types of DOA estimation algorithms due to the lack of snapshots. The performances of SpSF and SpSF-MF worsen with the increase of INR. In the condition that the INR is lower than 25 dB, the performance of the SpSF-MF algorithm is slightly better than that of the SpSF algorithm due to the application of the CMF. However, when the INR is larger than 25 dB, the residual component of the interference is considerably stronger than the weak targets, which seriously affects the performance of the SpSF-MF algorithm. Hence, in this case, SpSF-MF is worse than SpSF and its probability of source resolution is even worse than MVDR. When INR is larger than 35 dB, the SpSF-MF algorithm becomes invalid. Unlike the other three algorithms, the probabilities of source resolution of the SpSF-MFN algorithm reach 1 and the corresponding RMSEs are the lowest for all INR values in the simulations, which indicates that the SpSF-MFN algorithm achieves accurate DOA estimation for weak targets in a strong interference environment.
B. Experiment
Experimental data were collected by a tow array with 32 hydrophones uniformly spaced at 4 m. The array was placed at 50 m under the sea surface. The observed space [−90, 90]° is uniformly discretized into 181 candidate directions, where 90° is defined as the endfire direction. Two interferences moved in the far field of the array. One interference was at the direction of around −60°, and the other moved from around 20° to 15°. The direction of the tow-vessel noise was around 73°. A few weak targets were located at −29°, −25°, and −18°. The CMF and MFN are designed with the passband bearing sector as [−30, −17]° while the stopband bearing sector as [−90, −40]° [−7, 90]°. The SAs are −15 dB. The received data sampled at 2048 Hz, are transformed into the frequency domain with 2048-point FFT. The sample covariance matrix is formed by averaging over 39 snapshots with 50% overlap. The duration of every segment is 10 s, and the total analysis time is 5 min. The MVDR, SpSF, SpSF-MF, and SpSF-MFN algorithms are applied to localize the weak targets with the analyzed frequency ranges from 90 to 180 Hz. The results are shown in Fig. 5.
(Color online) BTRs of (a) MVDR, (b) SpSF, (c) SpSF-MF, (d) SpSF-MFN. (e) Profile of the preceding results at t = 190 s. Red arrows mark the estimated DOAs, and the green dashed lines in (e) represent the directions of weak targets.
(Color online) BTRs of (a) MVDR, (b) SpSF, (c) SpSF-MF, (d) SpSF-MFN. (e) Profile of the preceding results at t = 190 s. Red arrows mark the estimated DOAs, and the green dashed lines in (e) represent the directions of weak targets.
Figures 5(a)−5(e) show the DOA estimation results, where the estimated DOAs are marked by red arrows. As shown in Figs. 5(a) and 5(e), the MVDR algorithm can distinguish the weak targets at −18° and −25° but fails to distinguish the target at −29°. Meanwhile, Fig. 5(e) shows that the widths of the peaks formed by MVDR are much wider than other three algorithms, which indicates that the resolution of MVDR is the poorest among the four types of DOA algorithms. In Figs. 5(b)–5(e), SpSF is observed to hardly distinguish the weak targets at around −29° and −25° due to the existence of strong interferences, while SpSF-MF and SpSF-MFN can clearly localize three weak targets. In Fig. 5(e), it can be seen that the power of the strongest interference is around 15 dB higher than the weak targets. After the processing of the CMF, the power of the residual interference is nearly the same as that of the weak targets, thereby leading to the appearance of two pseudo peaks at around −36° and −10°, which are marked by the red arrows in Fig. 5(c). Figure 5(d) shows that SpSF-MFN sufficiently suppresses strong interferences, for the MFN can form deep nulls toward the directions of the interferences to suppress them. Therefore, the SpSF-MFN estimator achieves superior performance in a strong interference environment.
V. CONCLUSION
In this paper, a new method based on sparse spectrum fitting and matrix filter with nulling is proposed to localize wideband signals that are closely spaced in a strong interference environment, which is useful for passive sonar DOA estimation. The proposed method considers the MFN as a preprocessor to treat the received data, and strong interferences can be suppressed by the deep nulls formed by the MFN adaptively, thereby avoiding the increase of the passband response error due to the low SA design. The simulation and experimental results prove that the proposed method achieves better properties in interference suppression and localization accuracy than the SpSF and SpSF-MF algorithms. Meanwhile, in comparison with the MVDR algorithm, the proposed method can be applied in fewer snapshots and its resolution is superior to the MVDR algorithm because of the application of signal sparsity.
ACKNOWLEDGMENTS
This work was supported by the National Natural Science Foundation of China under Grant Nos. 11527809 and 51679204.