A robust generalized sidelobe canceller via steering vector estimation
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Abstract
In the presence of the direction of arrival (DOA) mismatch, the performance of generalized sidelobe canceller (GSC) may suffer severe degradation due to the gain loss of the desired signal in the main array and cancellation. In this paper, one effective GSC algorithm is proposed to improve the robustness against the DOA mismatch of the desired signal. Firstly, two subspaces, which contain the desired signal’s actual steering vectors of the main and auxiliary arrays, can be obtained by using the range information of the angle which the desired signal may come from. By rotating these two subspaces, the desired signal’s actual steering vectors of the main and auxiliary arrays can be estimated based on the maximum output power criterion. Then, with the estimates of the steering vectors in the former step, the gain loss of the desired signal in the main array can be alleviated. Moreover, one adaptive weight vector with the ability to block the desired signal in the auxiliary array can be obtained simultaneously, which effectively avoids the signal of interest cancellation consequently. Cycle iterative approach is also applied to guarantee the estimation accuracy of a wide range of angle deviation. Numerical simulations demonstrate the effectiveness and applicability of the proposed method.
Keywords
The generalized sidelobe canceller (GSC) DOA mismatch Output power maximization Steering vector estimation1 Introduction
The adaptive antenna has the ability to select a set of amplitude and phase weights with which to combine the outputs from the elements to consequently produce an artificially controlled beampatttern that optimizes the reception of a desired signal. This form of array processing provides relevant improvements in antiinterference performance which has been widely applied in numerous fields, such as military radar, communication, medical imaging, and navigation in decades [1, 2, 3, 4, 5, 6]. A large scale of adaptive array is commonly utilized to obtain better resolution and interference cancellation performance, which results to the consequence that the computational load becomes the bottleneck in the implementation of an adaptive beamforming algorithm. To save computational cost, the generalized sidelobe canceller (GSC) is an effective approach generally applied in radar and communication systems where the desired signal is only presented in a fraction of time or the amplitude of the desired signal in the auxiliary array is generally very small [7, 8]. The GSC can work as an adaptive beamformer that usually improves the gain of the desired signal by forming a mainlobe toward the direction of arrival (DOA) of the desired signal and in the meanwhile suppresses the interferences by nulling at the DOAs of the interference signals. However, the low implementation complexity makes the GSC more popular than the common adaptive beamformer in the practical application [9, 10, 11]. It is well known that, in the presence of the desired signal existing in the observation data received by the auxiliary array, the GSC inclines to misread the desired signal as interference and to cause a signal of interest (SOI) cancellation consequently [12]. The blocking process is one commonly applied technique to avoid the SOI cancellation which blocks the desired components from the primary data of the auxiliary array before the adaptive cancelling. As long as the desired signal is effectively blocked from entering the interference cancelling filter, only interference cancellation occurs, thus giving a higher overall output SNR than with conventional beamforming alone. However, this approach is very sensitive to the mismatch of the desired signal’s direction of arrival that can boil down to the mismatch of the steering vector of interest (SVI). Additionally, the uncertainty of SVI will induce the gain loss of the desired signal in the main array which brings about the dramatic degradation of the output signaltointerference and noise ratio (SINR).
In these years, improving the robustness of the beamformer against the mismatch of the steering vector is becoming an essential requirement and several contributions have been proposed to work on it [13, 14, 15, 16, 17, 18, 19]. Against the DOA mismatch, the most common technique is to delimit one set of unitygain constraints for a small range of angles around the presumed look direction. Nevertheless, this technique critically sacrifices the degrees of the auxiliary array freedom and may degrade the antiinterference performance. Consequently, how to precisely estimate the actual steering vector has attracted considerable attention. In [20], the joint maximum likelihood (ML) estimators of the useful signal and interference vectors were derived and the estimation problem was casted as a semidefinite program (SDP) problem. In [21], the ML estimator was still applied which led the steering vector estimation problem into a fractional quadratically constrained quadratic problem (QCQP). Nevertheless, this method cannot be straightforwardly applied in the GSC structure. In [22], the maximum output power criterion was introduced. Together with several novel constraints which prevented the estimation converging to the interference subspace, the desired signal’s actual steering vector was obtained by solving the QCQP. However, the high computational burden prevents it from practical usage. Furthermore, analytical solution cannot be obtained. In [23], the constraints to avoid the angle ambiguity were replaced by the subspace projection. Ultimately, either the analytical solution or the low calculation complexity was realized. This work made a great motivation for the method proposed in this paper.
In our previous work [24], only SOI cancellation has been worked out. However, the gain loss of the main array may still cause the output SINR deterioration in the case of the DOA mismatch. Hence, this paper is mainly concerned with two basic issues caused by the desired signal’s DOA mismatch: the SOI cancellation and the gain loss of the desired signal in the main array. Firstly, one angular sector expressed as \( \varTheta =\left[{\theta}_0^{\prime }\varDelta \theta, {\theta}_0^{\prime }+\varDelta \theta \right] \) has to be properly selected from which the desired signal may come. \( {\theta}_0^{\prime } \) denotes the assumed DOA of the desired signal. This sector can be either obtained from the priori knowledge or the low accuracy angle measurement. Hence, two subspaces constructed from the steering vectors of DOAs within Θ over the main and auxiliary arrays can be obtained. It has been proved [23] that both SVIs (the main and auxiliary arrays) can be expressed as linear combinations of the basis vectors of these two subspaces, respectively. By utilizing the successive maximization of the array output power while limiting the SVIs within the according subspaces, we can iteratively obtain the estimations of the actual SVIs. Ultimately, the optimum weights of the auxiliary and main arrays can be worked out thereafter.
The remainder of this paper is organized as follows. In Section 2, the fundamental theory of GSC and the problem discussed in this paper are reviewed. In Section 3, one novel robust GSC is proposed. In Section 4, the computational complexity is analyzed. Simulation results are shown in Section 5, while the last section gives some concluding remarks.
2 Background
where \( {\mathbf{R}}_a=E\left[{\mathbf{x}}_a(t){\mathbf{x}}_a^H(t)\right] \) and \( {\mathbf{R}}_{am}=E\left[{\mathbf{x}}_a(t){\mathbf{x}}_m^H(t)\right] \). In practice, they are usually substituted by the sample covariance matrices \( {\widehat{\mathbf{R}}}_a=1/L{\displaystyle {\sum}_{l=1}^L{x}_a(l){x}_a^H(l)} \) and \( {\widehat{\mathbf{R}}}_{am}=1/L{\displaystyle {\sum}_{l=1}^L{x}_a(l){x}_m^H(l)} \) with L training snapshots.
However, the performance of the blocking procedure highly relies upon the accuracy of the desired signal’s DOA. Once there exists a mismatch between the presumed desired signal’s DOA and the actual one (i.e., \( {\theta}_0^{\prime}\ne {\theta}_0 \)), two major issues may be induced to the GSC system: (i) the blocking process can not sufficiently eliminate the desired signal received by the auxiliary array, which will lead to the SVI cancellation (i.e., \( \mathbf{B}\left({\theta}_0^{\prime}\right){\mathbf{a}}_{a,0}\left({\theta}_0\right)\ne \mathtt{0} \)) and (ii) the entire gain of the desired signal in the main array will decrease as a result of the main beam pointing deviation (i.e., by this time, \( \left{\mathbf{w}}_m^H{\mathbf{a}}_{m,0}\left({\theta}_0\right)\right<\left{\mathbf{w}}_m^H{\mathbf{a}}_{m,0}\left({\theta}_0^{\prime}\right)\right \)). Consequently, the ultimate output SINR of the GSC will significantly deteriorate on the account of these two issues. Therefore, it is of great importance to avoid the SVI cancellation and alleviate the gain loss of the desired signal in the main array simultaneously.
3 Proposed robust algorithm
The optimization problem of (10) can become a typical generalized eigenvalue problem. Hence, the analytical solution can be straightforwardly carried out [25]. This method is called generalize eigenvector GSC (GEGSC). However, it is highly possible to converge the estimation of the actual steering vector to the interference subspace that will lead to the antiinterference performance degradation. To avoid this phenomenon properly, constraints are imposed in [15] while the solution is finally obtained using sequential quadratic programming (QP) iteratively. Nevertheless, this method cannot lead to a closed form solution, and the high computational cost prevents it from implementation.
It is worth mentioning that the operation above needs to set the initial value of the complex weights as w _{ m } = w _{pre}, where w _{pre} is the initial quiescent weights of the main array constructed based on the presumed DOA of the desired signal. By substituting (19) into (7) and utilizing the initial quiescent weights w _{ pre }, the adaptive weights of the auxiliary array can be obtained which may achieve the goal to eliminate the SVI cancellation. However, the beampattern of the main array still points to the incorrect direction of the desired signal. As the above content mentioned, it will lead to the gain loss in the main array. Henceforth, we turn our attention to solve this problem.

Step 1: construct the subspaces U _{1} and \( {\mathbf{U}}_1^{\mathbf{\prime}} \) using (12) and (20), respectively.

Step 2: initialize the complex weights w _{pre} based on the prior information of the presumed DOA of the desired signal and let \( {\widehat{\mathbf{w}}}_m^0={\mathbf{w}}_{\mathrm{pre}} \).

Step 3: based on the complex weights \( {\widehat{\mathbf{w}}}_m^{i1} \) obtained in the former step, update the steering vector \( {\widehat{\mathbf{a}}}_{a,0}^i\left({\theta}_0^{\prime}\right) \) using (19).

Step 4: based on the steering vector \( {\widehat{\mathbf{a}}}_{a,0}^i\left({\theta}_0^{\prime}\right) \), update the complex weights \( {\widehat{\mathbf{w}}}_m^i \) using (24).

Step 5: if \( \sqrt{{\left\Vert {\widehat{\mathbf{w}}}_m^i{\widehat{\mathbf{w}}}_m^{i1}\right\Vert}^2+{\left\Vert {\widehat{\mathbf{a}}}_{a,0}^i\left(\theta \right){\widehat{\mathbf{a}}}_{a,0}^{i1}\left(\theta \right)\right\Vert}^2}\le \xi \), go to step 7; otherwise, go to step 6.

Step 6: update the cycle iteration index, i = i + 1, and then, go to step 3.

Step 7: calculate the adaptive weights of the auxiliary array using (26).
4 Complexity analysis
In this paper, we propose a robust GSC approach against the desired signal’s DOA mismatch via estimating the steering vectors of the main and auxiliary array simultaneously. Many of existing algorithms work on this issue via imposing varied novel inequality constraints, which usually cannot lead to a closet form solution and are difficult to be implemented practically. Therefore, the subspace rotating approach is imposed in our work to replace inequality constraints. Consequently, the closed form solution can be obtained. Since the adaptive weights are calculated with training snapshots in portion of the radar pulse repetition period, the implementation complexity of the method proposed above needs to be considered. In the meanwhile, comparison with four approaches mentioned in the contents is also presented.
In our work, the auxiliary weight vectors of all approaches were calculated by employing the optimization problem presented in (6) resulting in a computational cost of \( O\left({N}_a^2{N}_m\right) \). As the desired signal’s DOA mismatch exists, the antiinterference performance of the traditional GSC deteriorates due to both the signal cancellation and the gain loss of the desired signal. Thus, in Eq. (10), the auxiliary steering vector estimation can be obtained via maximizing the residue output power of the GSC with the computational cost of \( O\left({N}_a^2{N}_m\right) \). However, the unconfined estimation has a high probability of converging to the interference subspace. In [24], only the steering vector of the desired signal over the auxiliary array was estimated via maximization output criterion combined with subspace projection. Though the signal cancellation can be avoided by accurate estimation of the auxiliary steering vector of the desired signal which results in a computational cost of O(K _{1} N _{ m } N _{ a } + K _{1} N _{ m }), the gain loss of the desired signal resulted from the main beam deviation still exists. In this paper, both the weight vector of the main array and the steering vector of the auxiliary array can be obtained. Consequently, the robustness against the desired signal’s DOA mismatch turned out to be accomplished. The estimation procedure costs a computational expense of \( O\left(\left({N}_a+{K}_2\right){N}_m^2+\left(T1\right){N}_m^2\right) \), where T denotes the iteration index. Since two steering vectors need to be estimated, the implementation complexity raises a large scale. Nevertheless, in the most common scenario, the DOA mismatch stays in a small degree which is less possible to exceed the beamwidth. In this situation, the cycle iteration is no longer required (i.e., T = 1) and the computational cost reduces to \( O\left(\left({N}_a+{K}_2\right){N}_m^2\right) \) (seeing example B for more details). Moreover, the major computational burden of approach proposed in this paper is occupied by two generalized eigenvalue decompositions (GED). Therefore, by using many implementation algorithms of GED, for instance, the method proposed in [26], the computational cost can be reduced further which makes our approach more available in practical usage.
5 Numerical simulation
In the following, we present simulation results to prove the effectiveness of the proposed method. In the meanwhile, comparison with four approaches is also made to demonstrate the superiority denoted as (i) the conventional GSC given by (7); (ii) the generalize eigenvector GSC (GEGSC) given by (10); (iii) the generalize eigenvector GSC with subspace projection (GESPGSC) in [14]; and (iv) the proposed method with only one cycle iteration. In all cases, the GSC structure is composed of the main array with 16 antenna elements and the auxiliary array with 5 antenna elements. All elements are presumed to be omnidirectional and spaced half of a wave length apart. The additive noise in each antenna element is modelled as spatially and temporally independent complex Gaussian white stochastic process. Two interference sources, both with interferencetonoise ratio (INR) of 30 dB, are assumed to plane impinge on the array from the presumed direction of −30° and 25°. All signal sources are independent from each other with fixed snapshot observations of L = 1000 unless otherwise specified. The convergence factor ξ is set to be 0.5 in all simulation scenarios. To obtain each point in the simulation curves, 100 independent runs are utilized.
5.1 Simulation of beamforming
5.2 Simulation of output SINR versus DOA mismatch
5.3 Simulation of output SINR versus snapshots
In this example, the experiment has been carried out to demonstrate the output SINR versus the number of sample snapshots. The random DOA mismatch of the desired signal with uniform distribution from −8° to 8° is considered. Moreover, the random DOA mismatch changes from trial to trial but maintains fixed between snapshots. The rest of the parameters involved are the same as the first example except the number of snapshots changes from 5 to 300. From the results shown in Fig. 3, it can be seen that the proposed method provides a faster convergence rate and higher output SINR than the others. In the circumstance of random DOA mismatch, the proposed method with multiple iterations outperforms the one with single iteration about 5 dB in terms of the output SINR, meanwhile the curve is much more stable. Both of the conventional GSC and the GESPGSC have low convergence rate, and because of stochastic averaging effect, these two methods have similar performances in this scenario. The GEGSC still leads to the worst performance resulting from the inexact estimation of the actual steering vector.
5.4 Simulation of output SINR versus input SNR
5.5 Simulation of convergence curve
6 Conclusions
In this paper, we present one effective robust form of GSC against the desired signal’s steering vector caused by DOA mismatch. With the orthogonal matrices properly constructed, the estimations of steering vectors over the main and auxiliary arrays, which are prevented from converging to interferences, have been achieved. Consequently, the SOI cancellation and the gain loss of the desired signal in the main array caused by the DOA mismatch can be sufficiently solved simultaneously. Simulation results demonstrate the effectiveness of the proposed method. Comparing with the conventional GSC and some modified approaches, the performance of the proposed method is nearly optimal over a wide range of SNR and the complexity is acceptable in terms of practical implementation.
Notes
Acknowledgements
This work is supported by the National Natural Science Foundation of China (No. 61301262 and No. 61371184).
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