Robust adaptive monopulse algorithm based on main lobe constraints and subspace tracking
 Shuang Qiu^{1},
 Xiaofeng Ma^{1}Email author,
 Weixing Sheng^{1},
 Yubing Han^{1} and
 Renli Zhang^{1}
DOI: 10.1186/s1363401704490
© The Author(s) 2017
Received: 30 July 2016
Accepted: 26 January 2017
Published: 13 February 2017
Abstract
In continuous wave (CW) radar and high pulse repetition frequency pulseDoppler (HPRFPD) radar, the interference plus noise sample snapshots are hard to be obtained. The desired signal in the received snapshots makes the LCMVbased adaptive monopulse algorithm sensitive to pattern look direction error. A linearly constrained subarray robust adaptive monopulse algorithm based on main lobe maintenance constraint and subspace tracking is developed in this paper. The constraint of main lobe maintenance is obtained by signal subspace projection. The biiterative leastsquare (BiLS) subspace tracking is used to update the signal subspace, and a powerassociated method is developed to determine the dimension of the projection subspace automatically. The proposed robust adaptive monopulse algorithm can achieve highangle estimation accuracy and good robustness to look direction error while expending only one additional degree of freedom compared to conventional LCMVbased method.
Keywords
Robust adaptive monopulse algorithm Main lobe maintenance Subspace tracking Dimension estimation1 Background
where θ _{0} and θ _{ s } are the angle of the pattern look direction and the target direction, respectively, g(·) is the monopulse ratio, and K _{ θ } is the slope of the monopulse ratio.
In the adaptive monopulse angle tracking, the signal of interest (SOI) may be present in any direction inside the 3dB beam scope, which may cause the distortion of the sum and difference patterns formed by linearly constrained minimum variance (LCMV) algorithm [10]. Therefore, the adaptive monopulse algorithm should be robust to the look direction error. The methods in [11–21] are robust adaptive beamforming (RAB) methods. A shrinkagebased diagonal loading (DL) method is proposed in [11]. The loading level to the covariance matrix can be determined automatically. In [12], the adaptive beamforming is transformed to a fast Fourier transform based weighted pattern synthesis problem with constraints on beamwidth and peak side lobe level. Robust adaptive beamforming methods based on steering vector estimation (SVE) can be found in [13–21]. The SVEbased methods can achieve high performance by adjusting the pattern look direction automatically to the SOI direction. The robust Capon beamformer (RCB) and doubly constrained robust Capon beamformer (DCB) in [13, 14] can estimate the steering vector of SOI and the SOI power based on an uncertainty set. A modified cost function is proposed in [15] utilizing the subspaceassociated power component rather than all power components in the covariance matrix. In [16–18], the optimization cost functions for estimating the SOI steering vector are modified and then transformed into convex optimization problems. In [19], the signal steering vector is estimated based on Oracle approximating shrinkage method. In [20], the SOI steering vector is estimated with the orthonormal projection approximation subspace tracking and subspace projection. The uncertainty of the covariance matrix is considered to optimize the worstcase performance. In [21], a steering vector estimation method is developed based on maximum output power criterion and subspace rotation.
However, the previous SVEbased robust beamforming methods are unsuitable for subarray robust monopulse beamforming for the following reason. In the monopulse beamforming, the pattern look direction must be exactly known. The difference beams are formed using the look direction to guarantee the linearity of the monopulse ratios near the pattern look direction. When the previous SVEbased methods are applied to monopulse beamforming, the pattern look direction is changed. The actual look direction needs to be calculated utilizing the estimated steering vector. However, the look direction is much difficult to be calculated from the subarray steering vector. In a subarray antenna with irregular subarray geometries, the exact position of the phase center of each subarray is needed when calculating the direction. It is known that, for an irregular subarray, the phase center is changed according to the direction of the incoming signal and is difficult to locate.
Therefore, we can conclude that the robust beamforming methods which adjust the pattern look direction are not suitable for robust monopulse beamforming on subarray antenna. The main lobe constraintbased robust adaptive beamforming methods imposing multidirectional constraints [22, 23] and derivative constraints [24] will not change the look direction. However, the high cost of degrees of freedom (DOFs) makes them limited in the application to subarray monopulse beamforming.
The main propose of this paper is to design a robust monopulse algorithm which can form the antijamming sum and difference beam patterns pointing to the assumed direction while maintaining the pattern main lobes against distortion with fewer additional degrees of freedom. The constraint for main lobe maintenance is set approximating to the SOI steering vector. Thus, it can achieve good performance and cost only one DOF. Inspired by the steering vector estimation in [20], we use signal subspace projection to obtain the constraint for main lobe maintenance and subspace tracking to update the timevarying signal subspace of the data snapshots.
The method developed in this paper can be classified into signalprocessingbased beamforming methods. The eigenspacebased beamforming methods [25, 26] and reducedrank beamforming methods [27–30] which use eigenvalue decomposition to obtain the projection subspace and the reducedrank subspace, the subspace tracking based beamforming methods [20, 31] and spacetime adaptive processing methods [32–34] which utilize subspace tracking to extract the signal subspace and interference subspace, the training data processing methods [35–37] which preprocess the training data to improve the beamforming performance in nonhomogeneous clutter environments can be included in signalprocessingbased beamforming methods. The adaptive radar detectors, as the adaptive beamformer orthogonal rejection test (ABORT)like detector [38], detection schemes in mismatched signal modes [39] and twostage detection schemes [40], can utilize spatial and temporal data to adaptively discriminate the target signal from ECM signals in presence of noise. These methods can also be included in signalprocessingrelated beamforming techniques.
A crucial problem for our method is to estimate the dimension of the projection subspace especially when the number of signals is changed. For the proposed method, the optimal projection subspace is the signal subspace. The subspace tracking methods in [20] and [41–45] are not able to update the dimension of the signal subspace when subspace tracking. The dimension is fixed when subspace tracking. In this paper, we develop a powerassociated method which maximizes the output signal to noise ratio (SNR) to determine the dimension of the projection subspace. This powerassociated method is incorporated with the biiterative leastsquare (BiLS) subspace tracking [46] to adapt to the variation of the subspace dimension when subspace tracking.
2 Problem model
where X(t)=[x _{1}(t),x _{2}(t),…,x _{ M }(t)]^{ T } denotes the received signal vector.
where 0<β<1 is the forgetting factor.
where T is the subarray forming matrix with dimension M×N, and N is the number of subarrays. The nth (n=1,2,…,N) column of matrix T contains the elements summing up to the nth subarray.
For conventional LCMVbased monopulse algorithm [6], the response in the pattern look direction is constrained, and the output power of the beamformer is minimized. If there are no look direction errors, the beamformer can provide the optimal SNR. When look direction error exists, the constraint mismatches the SOI direction, and the main lobe of the beam pattern may be distorted. The distortion obviously deteriorates angle estimation accuracy. In the rest of this paper, we use coventional LCMV algorithm to represent the conventional LCMVbased monopulse algorithm presented in [6].
In monopulse angle tracking, we assume that the length of one data block is a monopulse processing period, and the angle measurements in each processing period are averaged to achieve a steady performance. Since the monopulse processing period is a short period, the target angle can be regarded unchanged. The target angle is tracked by the angle tracking loop filter from one monopulse processing period to the next. The block diagram of adaptive monopulse angle tracking loop is presented in Fig. 2 a. The pattern look direction in the current monopulse processing period is fixed and provided by the angle tracking loop filter. The loop filter utilizes the angle measurements in the current period and previous periods, and predicts the most probable target angle in the next period. Because of the dynamic target movement, the SOI may be deviated from the assumed look direction, but inside the scope of the 3 dB beam width. This condition causes look direction error. The main lobes of the sum and difference beams formed by the conventional LCMV algorithm will be distorted, and angle estimation performance will be degraded. Therefore, a robust adaptive monopulse algorithm is incorporated into the monopulse angle tracking loop, as illustrated in the dashed box in Fig. 2 a. In this paper, we only focus on the adaptive monopulse beamforming and angle estimation, and we do not discuss the angle tracking problem.
The block diagram of the proposed robust adaptive monopulse algorithm is shown in the dashed box in Fig. 2 b. In Fig. 2 b, \(\hat {\mathbf {a}}\left (t \right)\) is the constraint of main lobe maintenance; y _{ s }, y _{ u } and y _{ v } are the outputs of sum and difference beams. The signal subspace is updated through BiLS subspace tracking and the subspace dimension is determined by the powerassociated method. Thereafter, the constraint of main lobe maintenance is obtained through subspace projection. Finally, a robust LCMV beamformer with constraints on main lobe maintenance and monopulse ratio curve is developed to obtain the sum and difference beam outputs. The angle of SOI is estimated using sum and difference beam outputs.
3 The proposed robust adaptive monopulse algorithm
3.1 Robust LCMV beamformer with main lobe maintenance constraint (RMMLCMV)
where R _{ xb } can be obtained by (6) and (10).
where \(\sigma _{n}^{2}\) is the noise power; R _{ s } is the covariance matrix of the SOI and w=w _{ sr }. Therefore, the ratio \({P_{s}}/{\left \ \mathbf {w} \right \_{2}^{2}}\) can be exploited to evaluate the output SNR.
3.2 Computation of the main lobe constraint
For the subarray antenna, the constraints of main lobe maintenance need to be selected carefully to save DOFs. When expending only one additional DOF, we prove in the Appedix A that the constraint should be approximating to the steering vector of the SOI to obtain the highest output SNR. The steering vector of the SOI needs to be estimated rapidly. As is stated previously, estimating the angle of the SOI directly using the approximated steering vector is difficult in the cases that the subarray has arbitrary geometry. Therefore, the monopulse processing is still needed for angle estimation.
where U _{ s } denotes the signal subspace. (48) demonstrates the conclusion in [47, 48]. The steering vector of the SOI can be estimated by projecting a _{ b0} to the signal subspace.
where ρ is a userdefined threshold. When the scope is set as the 3 dB beamwidth, ρ can be set to be 0.707.
3.3 Subspace tracking and subspace dimension estimation
In the proposed method, the signal subspace and the beamforming weights are updated in each snapshot to handle the dynamic cases. Conventional subspace decomposition method based on eigenvector decomposition (EVD) is unable to update the signal subspace in realtime because of its high computational complexity. In this section, the BiLS subspace tracking [46] is exploited to track the dynamic signal subspace rapidly and accurately, and a powerassociated method is developed to determine the dimension of the projection subspace.
When the dimension r _{ c }=r _{ max }+1, where r _{ max } is the largest possible number of signals, U _{ B }(t) converges to the right principal singular vectors of \(\mathbf {X}_{b}^{H}\). T _{ A }(t)D _{ A }(t)T _{ B }(t) is the SVD of the r _{ c }×r _{ c } upper triangular matrix R _{ A }(t). The largest r _{ c } singular values of X _{ b } can be approximated by the r _{ c } singular values of R _{ A }(t), i.e., the diagonal elements of D _{ A }(t) [46]. The columns of U _{ B }(t) are arranged so that the corresponding singular values are in descending order. If the dimension of the signal subspace is r _{ d }(t), the signal subspace can be denoted by the first r _{ d }(t) columns of U _{ B }(t).
For our proposed robust monopulse algorithm, selecting the projection subspace from U _{ B }(t) is a key problem. It can be concluded from Appendix A and (48) that the optimal projection subspace is the signal subspace. When the dimension of the projection subspace is equal to the dimension of the signal subspace, the output SNR can be maximized.
Generally, the MDL criterion [49] can be exploited to determine the dimension of the signal subspace. But in the proposed method, the following condition may limit its application to BiLS subspace tracking. The MDL criterion needs to use all N eigenvalues of R _{ xb }(t) to determine the dimension. Assume that the eigenvalues of R _{ xb }(t) are denoted by \(\phantom {\dot {i}\!}{\lambda _{1}} \ge {\lambda _{2}} \ge \cdots \ge {\lambda _{{r_{c}}}} \ge \cdots \ge {\lambda _{N}}\). The largest r _{ c } eigenvalues of R _{ xb }(t) can be obtained from the diagonal elements of D _{ A }(t), while the smallest N−r _{ c } ones cannot be obtained by BiLS. The eigenvalues obtained from BiLS denoted by \({\hat {\lambda }_{1}} \ge {\hat {\lambda }_{2}} \ge \cdots \ge {\hat {\lambda }_{{r_{c}}}}\), satisfies \([{\hat {\lambda }_{1}}, {\hat {\lambda }_{2}}, \cdots, {\hat {\lambda }_{{r_{c}}}}] = {\mu _{\lambda } }[{\lambda _{1}}, {\lambda _{2}}, \cdots, {\lambda _{{r_{c}}}}]\), where μ _{ λ } is a scalar. Since μ _{ λ } is unknown in BiLS, we cannot use \([{\hat {\lambda }_{1}}, {\hat {\lambda }_{2}}, \cdots, {\hat {\lambda }_{{r_{c}}}}]\) to obtain the remaining N−r _{ c } eigenvalues. An alternative approach is to regard the smallest N−r _{ c } eigenvalues equal to \({\hat \lambda _{{r_{c}}}}\). However, by this approach, they are larger than the actual eigenvalues according to the eigenvalue distribution presented in [50]. This condition may fluctuate the performance of MDL criterion. The inaccurately estimated dimension of the signal subspace may affect the computation of the main lobe constraint and may further deteriorate the performance of main lobe maintenance and the output SNR.
In this section, determining the dimension of the projection subspace is transformed to the maximization of the output SNR of the proposed beamformer by a powerassociated method. When the output SNR is maximized, we can conclude from Appendix A and (48) that the projection subspace is equal to the signal subspace. Then the subspace dimension can be obtained.
In (50), w _{ k } denotes the sum beam weight when the dimension r _{ d }=k; \({{\hat {\mathbf {a}}}_{k}} = {\mathbf {U}_{Bk}}(t)\mathbf {U}_{Bk}^{H}(t){\mathbf {a}_{b0}} ={{\hat {\mathbf {a}}}_{k1}}+{\mathbf {u}_{k}}(t)\mathbf {u}_{k}^{H}(t){\mathbf {a}_{b0}}\), where U _{ Bk }(t) comprises the first k columns of U _{ B }(t) and u _{ k }(t) is the kth column. By replacing \({\hat {\mathbf {a}}}\) with \({{\hat {\mathbf {a}}}_{k}}\) in (22), (27) and (28), P _{ s }(k) and w _{ k } can be obtained. The constraint \(\left {\left ({{{\hat {\mathbf {a}}}_{k}}^{H}{\mathbf {a}_{b0}}} \right)} \left / {\left \ {{\mathbf {a}_{b0}}} \right \_{2}^{2}}\right. \right  \ge \rho \) constrains the vector \({{\hat {\mathbf {a}}}_{k}}\) to be in the main lobe, and ρ is defined in (49). The method in (50) requires 3(r _{ c }+1)N ^{2}+12(r _{ c }+1)N+O(r _{ c }) complex multiplications. Since the computation for each k is parallel, the computational complexity can be less than 3(r _{ c }+1)N ^{2}+12(r _{ c }+1)N+O(r _{ c }).
3.4 Steps of the proposed algorithm
 (1)
Track the signal subspace and determine the dimension of the projection subspace as listed in Table 2.
 (2)
The constraint for main lobe maintenance, i.e., \(\hat {\mathbf {a}}\left (t \right)\), can be obtained from step 1. \(\hat {\mathbf {a}}\left (t \right) = {{\hat {\mathbf {a}}}_{k}},\;k = {r_{d}}\left (t \right)\).
 (3)
When \(\left \ {\hat {\mathbf {a}}\left (t \right)  \hat {\mathbf {a}}\left ({t  1} \right)} \right \_{2}^{2} \le \varepsilon \) and ε is the error bound, compute the weight vector of the sum and difference beams of the proposed method at snapshot index t using (22)–(35).
 (4)
 (5)
Estimate the direction of the SOI using (39) and (40). Then average the angle measurements in one monopulse processing period.
As is stated previously, the proposed algorithm is developed for subarray adaptive monopulse in consideration of the limitations of the SVEbased and main lobe constraintbased robust beamforming methods. It can also be applied to adaptive monopulse on the full array. The low cost of DOFs makes the proposed algorithm more suitable for subarray adaptive monopulse than previous methods. The proposed algorithm can also be utilized to adaptively form the subbeams of the four quadrant monopulse approach (Section 2 in [3]) and the multiple squinted beams approach (Section 3 in [51]) to improve the robustness of these subbeams to pattern look direction error when the interferenceplusnoise covariance matrix is unavailable.
4 Numerical simulations and discussions
The performance of the proposed robust adaptive monopulse algorithm is demonstrated by simulations based on the array geometry in Fig. 1 a. The array working in Ka band consists of 960 isotropic elements and is divided into 12 subarrays shown in different symbols and different colors. The array elements are at 0.58 wavelength separation and on a triangular grid. Each element is connected to a digital controlled phase shifter. The element noise is assumed to be additive white Gaussian noise with zeromean and unit variance. For all simulations, the directions of the two interferences are (56°,0°) and (60°,40°); The assumed pattern look direction is fixed at (90°,0°). The parameter α in BiLS is set to 0.999. Three scenarios are made to simulate the performance of the proposed algorithm.
where U(t) and \(\hat {\mathbf {U}}\left (t \right)\) are the true and the tracked signal subspace respectively; ∥·∥_{ F } denotes the Frobenius norm.
Scenario 2: Firstly, we compare the output SINR of the proposed robust adaptive monopulse algorithm with those of the conventional LCMV algorithm [6], RCB [13], DCB [14], RAB with derivative constraints [24], RAB with multidirectional constraints [23] and the method in [47]. In the two simulations, the directions of SOI are (91°,1°) and (90.5°,0.5°), respectively. The input SNR at the array elements varies from 30dB to 10dB, and input INR is 35dB larger than the input SNR. T=1000 snapshots are exploited. The RAB with multidirectional constraints imposes constraints on directions (90°,0°), (90°,−1°), (90°,1°), (89°,−1°), (89°,0°), (89°,1°), (91°,−1°), (91°,0°), and (91°,1°). The RAB with derivative constraints imposes 0, 1, and 2ordered derivative constraints with respect to u and v.
The results demonstrate that the proposed method can obtain higher performance than RAB with multidirectional constraints and RAB with derivative constraints, and only costs one additional DOF compared to conventional LCMV algorithm.
5 Conclusions
In consideration of the limitations of the SVEbased and main lobe constraintbased robust beamforming methods for performing subarray monopulse beamforming, we develop a linearly constrained subarray robust adaptive monopulse algorithm, which is different from previous methods. The constraint of main lobe maintenance constructed by signal subspace projection expends only one DOF. Then, we develop a powerassociated method to determine the dimension of the projection subspace and combine it with BiLS subspace tracking to adapt to the dimensional variation of the signal subspace. The powerassociated method can achieve high performance when only the principal eigenvalues and the principal eigenvectors can be obtained in BiLS. Simulation results demonstrate that the proposed algorithm can maintain high antijamming capability and angle estimation performance when look direction error exists. The proposed algorithm can also be applied to adaptive monopulse on the full array. The low cost of DOFs makes it particularly suitable for the subarray adaptive monopulse technique.
6 Appendix A
When the constraint of main lobe maintenance is set as the steering vector of SOI, i.e., \(\hat {\mathbf {a}} = \mathbf {T}_{b}^{H}\mathbf {a}\left ({{u_{s}},{v_{s}}} \right)\), B C _{ sr }=0 and \(\mathbf {B}\hat {\mathbf {a}} = 0\). The component of SOI in \(\mathbf {W}_{q}^{H}\mathbf {R}_{xb}^{H}{\mathbf {B}^{H}}{\left ({\mathbf {B}{\mathbf {R}_{xb}}{\mathbf {B}^{H}}} \right)^{ H}}\mathbf {B}{\mathbf {X}_{b}}(t)\) is negligible. Therefore, in (61), the SOI component is fully enhanced by the quiescent weight \(\mathbf {W}_{q}^{H}{\mathbf {X}_{b}}(t)\), and the interferences and the noise are suppressed. The output SNR of RMMLCMV can be maximized.
Abbreviations
 BiLS:

Biiterative leastsquare
 BW:

Beamwidth
 INR:

Interference to noise ratio
 LCMV:

Linearly constrained minimum variance
 PAST:

Projection approximation subspace tracking
 RAB:

Robust adaptive beamformer
 SINR:

Signal to interference plus noise ratio
 SNR:

Signal to noise ratio
 SOI:

Signal of interest
 SVE:

Steering vector estimation
Declarations
Acknowledgements
This work is supported in part by the National Natural Science Foundation of China under grant 61501240, 61471196, 11273017 and 61401207; the College Graduate Scientific Research Innovation Fund in Jiangsu Province of China under Grant No. KYLX16 0447.
Authors’ contributions
SQ and XM designed the algorithm scheme. SQ and RZ performed the experiments and analyzed the experiment results. SQ, WS, and YH contributed to the manuscript drafting and critical revision. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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