DOA Estimation of Broadband Sources Using Dimension-Reduced Matrix Filter with Deep Nulling in a Strong Interference Environment

To enhance the performance of the adaptive matrix filter with nulling, a dimension-reduced matrix filter with deep nulling (DR-MFDN) is proposed in this paper and used for direction of arrival (DOA) estimation of broadband sources in a strong interference environment. Consider the change of noise after filtering, the gaussian matrix is used for prewhitening to prevent the noise from converting into colored noise without affecting the response characteristics of the beam-space. Finally, the multiple measurement vector orthogonal matching pursuit (MOMP) algorithm is used for DOA estimation of weak targets after filtering, and the simulation results show a better DOA estimation performance compared to the OMP and MUSIC algorithms with short snapshots.


Introduction
Matrix filters [1] have been widely used in sonar and radar because of its excellent data filtering and signal separation in spatial domain.Matrix filter projects the observation space into the subspace by linear transformation, which pass the signals in sector-of-interest(passband) while suppressing the outof-sector(stopband) interference.Vaccaro and Harrison [1] proposed the design criteria for a conventional matrix filter (CMF) in the frequency domain.MacInnes [2] enhances the performance of the passband response in beam-space to apply the least mean square criterion and solve it using convex optimization, but fails to have strict control over the stopband attenuation.Yan [3][4] et al. proposed a second-order cone programming model for CMFs under the stopband constraint, passband least mean square criterion and passband minimax criterion.Han [5] et al. proposed weighted iterative least squares CMF algorithm.Hassanien et al. [6] proposed DR-CMF by constraining the stopband and minimizing the difference between the actual response and the quiescent response matrix filters.These variants of the CMF approach can strictly control the attenuation of the stopband while maintaining the passband performance, however, the data cannot be effectively filtered when strong interference is present in spatial domain.To solve the problem of weak targets DOA estimation in strong interference environments, adaptive matrix filters have been introduced.Adaptive matrix filters, represented by dimension-reduced matrix filter with nulling (DR-MFN) [6] and Matrix Filter with nulling (MFN) [7], are able to provide nulls towards directions of the interference and sufficiently attenuate it adaptively, however, the presence of noise and perturbation limits the depth of the nulls of the interference and degrades the passband performance.
To enhance the performance of the adaptive matrix filters [6][7], a dimension-reduced matrix filter with deep nulling (DR-MFDN) is proposed in this paper and applied for broadband source DOA estimation in a strong interference environment.Firstly, the MVDR algorithm is applied to pre-process the array data to confirm the bearing sector where the weak targets are located and design the DR-CMF.
The covariance matrix of the subspace steering matrix is then taken as the projection matrix within the main lobe bearing sector where the strong interference is located.Thereafter, the projection matrix is used to enhance the energy of the interference source.Finally, a DR-MFDN with deeper nulls in the interference direction is obtained based on the minimum energy criterion after the transformation of the projection matrix.Simulation results show that DR-MFDN reduces the noise influence on the passband and further improves the nulls depth compared to DR-MFN, enhancing the beam-space performance.Considering the fact that white noise at the input will be converted to colored noise, in this paper, gaussian matrix is used for prewhitening operation to prevent the noise from converting into colored noise without affecting the characteristics of beam-space.In this paper, the DR-MFDN and MOMP [8] algorithms are combined for the DOA estimation of weak targets.The simulation results show a better DOA estimation performance compared to OMP [9] and MUSIC [10] algorithms with short snapshots.

Signal model
Consider that there are K far-field narrowband signals with frequency f impinge on a uniform linear array with M elements distributed along the x-direction with spacing d at κ θ ( 1, 2, , κ = Κ  , κ θ is the angle of the th κ impinge signal with respect to the y-direction).Neglect the evanescent field, the received signal at the mth element of the array under the wavenumber integration model can be modeled as where the superscript T is the transpose.
The array output at the frequency f can be a discrete form of equation (1) written as  represent the FFT coefficients of the received data, wideband signals and noise at the frequency f , respectively; 1 ( ), , ( ) Assuming f e as white noise, the covariance matrix of the output at the frequency f can be written as where E[ ] ⋅ denotes the expectation operation; superscript H represents the Hermitian transpose;  ; 2 f σ is the variance of the white noise; M I is an identity matrix of dimension M .

Prewhitening matrix filter and MOMP algorithm
Conventional array processing methods cannot effectively estimate the DOAs of weak targets while strong interference is present, therefore, transforming the signal into the designed beam-space is necessary.The beam-space snapshot vector ' 1 ( ) as a linear transformation of the frequency domain snapshot where is the matrix filter that corresponds at the frequency f .
The beam-space response after transformation by the matrix G is not orthogonal, it will convert the white noise of the environment into colored noise, which is not convenient for following processing.Hence an orthogonal prewhitening operation is required and equation ( 5) is modified as ) However, the beam-space response changed under the transformation of equation (7).Considering that the SVD decomposition of matrix filter is done with ( ) While the orthogonal prewhitening beam-space response can be expressed as ( ) Compared with equation ( 8) and equation ( 9), the orthogonal prewhitening operation normalizes the eigenvalues of the matrix H f G , which makes the beam-space response changed.To guarantee that the transformed beam-space maintains the original characteristics and to prevent white noise from converting into colored noise, a gaussian matrix is used to perform the prewhitening operation.The gaussian prewhitening matrix filter is given by


is a gaussian matrix with each row vector obeying a standard normal distribution and uncorrelating to the others.Therefore, the characteristics of f T are identical to f G .And the frequency domain snapshot is transformed as ( ) ( ) Under the transformation of equation (10), the noise covariance matrix where (   A DOA estimation algorithm with excellent performance is required after transforming the signal into beam-space.The OMP algorithm [9], a classical algorithm that utilizes the 1 l norm of sparse signals, is widely used for DOA estimation due to its excellent signal reconstruction performance.Moreover, under the Multiple Measurement Vector (MMV) model, the estimation performance of DOA can be enhanced with joint estimation by utilizing the information between pieces of data.In this paper, the MOMP algorithm combined with matrix filter is used to solve the optimization problem shown in equation ( 12) since ( ) f n s is a vector with multiple snapshots and K-sparse.
where τ is the tolerance.After solving the sparse vector ( ) f n s at every frequency, the eventual estimation of spatial spectrum in beam-space can be calculated as max min 1 1 ( )

Dimension-reduced matrix filter with deep nulling for DOA estimation of weak targets
To enhance the performance of the adaptive matrix filter with nulling, the DR-MFDN is proposed and used for DOA estimation of weak targets.Firstly, the MVDR algorithm is applied to pre-process the array data to confirm the bearing sector where the weak targets are located and design the DR-CMF.
The DR-CMF is designed by constraining the stopband and minimizing the difference between the actual response and the quiescent response matrix filter, and the design criteria can be formulated as where F ⋅ is the frobenius norm; p Θ and s Θ are the bearing sectors of the passband and the stopband, respectively; is the quiescent response matrix filter [11]; and δ is a parameter of the stopband attenuation.The stopband attenuation level is SA 20lg( / ) M δ = , hence the value of δ can be set by SA.
The DR-MFDN proposed in this paper can be derived from DR-CMF.After enhancement of the interference sources with a projection matrix, the matrix filter with deep nulling, designed based on the minimum energy criterion, reduces the influence of perturbation and noise on the passband and obtain deeper nulls towards directions of the strong interference.Thus, the response performance of the matrix filter can be enhanced.Taking the subspace steering matrix extracted from the main lobe, where the . The design criteria for DR-MFDN can be formulated as min ( ) , , 1, , Equation ( 15) can be converted to a second-order cone programming problem, formulated as where ε is the tolerance; ; ⊗ denotes the Kronecker product.

Simulation
In the simulation, consider a uniform linear array of 32 isotropic array elements with spacing of 1.25m.Two far-field broadband signals and three far-field broadband interferences impinge on the array from 34°, 40°, -30°, -45°and -60° (90°is defined as the endfire direction).The frequency range of signals and interferences is [400, 500] Hz.The signal-to-noise ratios (SNRs) of the two signals are both 0dB, while the signal-to-interference ratios (SIRs) of all interferences are -40dB.The sampling frequency is 50 kHz and the data are partitioned into 10 segments and FFT is done for each segment.DR-CMF, DR-MFN and DR-MFDN are designed at all frequencies with passband bearing sector . The amplitude responses and amplitude response errors in beam-space are calculated as

{ }
The beam-space response characteristics of DR-CMF, DR-MFN and DR-MFDN at SA 25dB = − are compared at two frequencies, 400Hz and 500Hz, and the results are shown in figure 2. Both DR-MFN and DR-MFDN provide nulls at the interference directions and attenuate uniformly in other regions of the stopband, and the latter provide deeper nulls of the interference directions.In the passband, the amplitude response error curves of DR-MFDN and DR-CMF nearly matched, while DR-MFN has a poor performance.DR-MFDN provides a better performance than DR-MFN with deeper nulls to attenuate strong interference and lower passband distortion to allow signals to pass.
The amplitude response of DR-MFN and DR-MFDN in 400-500Hz band is shown in figure 3, and the passband amplitude response is nearly 0dB and the stopband is strictly controlled below -25dB at every frequency.Specifically, the former yields an average amplitude response of -35dB, while the latter achieves -45dB.The estimation results without filtering are shown in figure 4(a).Short snapshots resulted in a single burst at two weak targets, while the true azimuth could not be identified.Figure 4(b) shows the results of the MUSIC, OMP, MOMP algorithms using DR-MFN or DR-MFDN.With DR-MFN filtering, the spatial spectrum estimation results of the MUSIC algorithm are inaccurate for the intensity estimation of weak signal sources.After DR-MFDN filtering, the strong interference is better suppressed, which guarantees the integrity of the weak signal information.Compared to DR-MFN, the MUSIC algorithm has a more accurate estimation performance for the intensity of weak signals after using DR-MFDN.The OMP and MOMP algorithms do not provide significantly difference in the estimation results under the two filter treatments.However, without prewhitening, the estimation results of the OMP and MOMP algorithms at low SNRs have multiple peaks near the true azimuth.To compare the performance of different prewhitening operations and DOA estimation methods, only DR-MFDN is taken in the following.Estimation results after prewhitening operation by using equations ( 7) and ( 10) are shown in figure 5.In figure 5(a), the MUSIC, OMP and MOMP algorithms do not provide effective DOAs information when the orthogonal prewhitening operation is performed.This is attributed to the fact that the orthogonal prewhitening operation changes the beam-space characteristics of the matrix filter, the energy in stopband is not sufficiently suppressed, which results in no significant improvement or even degradation of estimation performance compared to the non-whitening operation.In figure 5(b), after gaussian prewhitening operation, the MUSIC algorithm performs identically to the unwhitening operation, but the OMP and MOMP algorithms have better performance than the unwhitening and orthogonal whitening operations and do not have multiple false peaks in the passband, and MOMP is superior to the other two algorithms.Overall, the MOMP algorithm has the best performance for DOA estimation under gaussian whitening treatment for short snapshots.

Conclusion
In this paper, a dimension-reduced matrix filter with deep nulling is proposed to reduce the influence of noise and perturbation in passband and to provide deeper nulls towards directions of the strong interference, then the estimation of DOA is performed under the sparse signal model.The estimation results of the MUSIC algorithm show that after DR-MFDN filtering, the strong interference is better suppressed compared to DR-MFN.Compared with the orthogonal prewhitening operation, gaussian prewhitening operation maintains the original beam-space characteristics of the matrix filter while preventing the noise from converting into colored noise.The simulation results of OMP, MOMP, and MUSIC algorithms after using matrix filters show that the MOMP algorithm has the best DOA estimation performance of the weak targets with short snapshots.However, the method in this paper is subject to certain limitations, when the bearing difference between the target and the strong interference is less than the width of the transition zone, the strong interference cannot be effectively suppressed.Therefore, a further study is needed to set the critical width of the transition zone.Furthermore, when the interference intensity is low or the number of interferences is limited, the performance of DR-MFDN is essentially comparable to that of DR-MFN.
by the array in the frequency band min max [ , ] f f are partitioned into N segments and FFT is applied for each segment.The steering vector at the frequency f can be written as

Figure 1 .
a distribution of values of the covariance matrix for f T and f G , respectively.With enough Monte Carlo trials, the covariance matrix of f T is an identity matrix.Therefore, the gaussian prewhitening operation prevents the noise from converting colored.(a) and (b) are values distributions of covariance matrices for matrices f T and f G , respectively (Monte Carlo trials 1000 times)

Figure 2 .
(a) and (b)amplitude responses of the three matrix filters at 400 and 500Hz,respectively; (c) and (d)amplitude response errors of the three matrix filters at 400 and 500Hz,respectively.Two green dashed lines represent the left and right limitations of the passband bearing sector, respectively.

Figure 3 .
Figure 3. Amplitude response of DR-MFN and DR-MFDN in 400-500Hz bandThe received data from the array are filtered using both DR-MFN and DR-MFDN, after which DOA estimation is performed.To compare the performance of DOA estimation before and after filtering with matrix fliters, the DOA estimations of CBF, MVDR and MUSIC of the received data without filtering are also given.

Figure 4 .
Estimation results (a)without filtering and (b) with DR-MFN or DR-MFDN

Figure 5 .
Estimation results using (a) orthogonal and (b)gaussian prewhitening operation