Application of spheroidal sequences to sidelobe control in beampattern synthesis

In radar and sonar system, with regard to achieve high interference rejection and enhance target signal detection performance, adaptive beampattern synthesis with low sidelobe levels is desired. A new approach to beampattern synthesis with sidelobe control is developed. The essence of the presented method is to consider the application of spheroidal sequences to beampattern synthesis to alleviate the computational complexity. The presented technique represents a modification of the beampattern synthesis with sidelobe control of the prescribed threshold. The spheroidal sequences approach that characterises the out-of-sector angular regions is considered to alleviate the computational cost. The resulting beampattern synthesis approach is showed to be convex, and its second-order cone formulation is given that facilitates a computationally efficient way to implement the proposed beamformer with the help of the optimal software. Simulation results are given to demonstrate the effectiveness of the presented algorithm and to show the superiority to the state-of-the-art algorithm.


Introduction
One of the fundamental problems in array signal processing is beampattern synthesis. The conventional formulation of beampattern synthesis is to design a beamforming vector to form a desired beampattern. The adaptive beamformer can maintain low sidelobe levels and automatically null out interfering sources. Unfortunately, the Capon beamformer will have unacceptably high sidelobes in the case of low sample support, which may lead to a substantial performance drop in the presence of high noise or unexpected (i.e. suddenly appearing) interferers.
Recently, several techniques to sidelobe control have been reported in the literature [1][2][3][4]. The essence of the approaches is to enforce deep nulls towards the directions of arrival (DOAs) of the strong interferences, while keep the target signal distortionless. The diagonal loading [1] is an approach by adding small omnidirectional interferences to sample covariance matrix to achieve a lower sidelobe level. Other famous approaches are the penalty function method [2] and sparse constraint on beampattern technique [4]. With regard to sidelobe control, the Capon beamformer is modified to incorporate multiple inequality constraints outside the sectors-of-interest beampattern. Among them, the diagonal loading [1] has been widely employed due to its simplicity and relatively acceptable performance. Although using the method [3], the sidelobe levels are guaranteed to be under a prescribed threshold; this approach has a high computational cost. However, the peak sidelobe level is chosen by ad hoc trials with the considerations that it should be chosen by rule and line in order to ensure the problem feasible and the constraints active.
The second-order cone (SOC) formulation of the method [3] is obtained, which can be solved using the interior point method in a computational efficient way. In antenna array design, the out-ofsector angular areas of multiple quadratic inequality constraints may be often too difficult to treat and, therefore, techniques that simplify the problem are desirable. Following the general ideal, we further reduce the complexity of the obtained SOC problem using the ideal similar to that of the spheroidal sequences approach of [5]. The first n spheroidal sequences provide a set of orthonormal sequences which concentrate most of their energy. The essence of the technique is to replace the conventional constraints outside the sector-of-interest with a much smaller number of parsimonious constraints.
The organisation of the paper is as follows. We begin in the material on signal formulation in Section 2 by describing the signal model for a uniform linear array (ULA). In Section 3, we consider the application of quadratic inequality constraints outside the sector-of-interest areas to the Capon beamforming technique that can control the peak sidelobe level. In order to reduce the computational burden, we use the spheroidal sequences approach but apply it to characterise the out-of-sector angular regions. This is followed in Section 4. Simulation results validate improved robustness of the developed algorithm.

Problem formulation
Assume that an array of N sensors receives a narrowband signal s(t). The beamformer output is modelled as where t is the time index, w is the N × 1 weight vector, and ⋅ H denotes the Hermition transpose. The array snapshot data is given by where s(t), i(t) and n(t) denote the vectors of target signal, interference signal and sensor noise, respectively. The weight vector w can be obtained by maximising the signal-to-interferenceplus-noise ratio (SINR), is the array signal vector corresponding to the target signal, and σ 2 is the array signal. The problem of maximising array output SINR is given as Since R in is not available, it is usually replaced by the sample [6]. When the number of snapshots used for covariance matrix estimation is insufficient, the Capon beamformer presents high sidelobe levels, which reduce its performance in case of unexpected interferences or increase in the noise power. Moreover, note that even in a scenario where the actual signal steering and spatial signature of the signal is known exactly, the presence of the desired signal in training data cell may substantially reduce the convergence rates of adaptive beamformer.

Adaptive beamforming with sidelobe control
Let θ i ∈ Θ s i = 1, …, J be a chosen (uniform or logarithmic) grid that approximates the out-of-sector angular areas Θ s using a finite number of angels. Adding the multiple quadratic inequality constraints outside the sector-of-interest area to the Capon beamforming technique, we obtain the following modified Capon beamforming with sidelobe control: where ε is the prescribed sidelobe level. The optimisation problem has quadratic objective function. There is a single linear equality constraint and multiple quadratic inequality constraints. Therefore, the problem (3) is convex. Then, its SOC formulation is obtained, which can be solved efficiently using the interior point method. Although using this approach, the sidelobe levels are guaranteed to be under a certain prescribed value, this algorithm has a high computational complexity. However, this approach will suppress the desired signal when the SNR is high. What's more, the desired peak sidelobe level should be chosen properly, otherwise, the problem (6) may be infeasible.

Proposed method
The aim of this work is to consider the application of spheroidal sequences approach to control the sidelobe level. In order to achieve high interference suppression, the number of quadratic constraints J should be as large as possible. At the same time, in the interest of low computational complexity and interference rejection, capability is proposed. Therefore, in practice, the parameter J should be obtained by finding satisfying trade-off between the interference suppression capability and computational complexity. In this section, an algorithm that reduces the number J of inequality constraints considerably without affect the capability of adaptively nulling the interference is developed. Following the technique in [5], we introduce a Hermitian positive definite matrix The eigen-decomposition of (4) is given as where λ j , j = 1, …, M are the eigenvalues of C in descending order, e j is the eigenvector associated with λ j , E s = e 1 , e 2 , …, e M + 1 and Λ = diag λ 1 , …, λ M + 1 denote the eigenvector matrix and eigenvalue matrix, respectively. We present to replace the J quadratic constraints by a substantially smaller number P(P ≤ M) of the spheroidal sequence-based constraints w H a(θ i ) 2 ≤ ε, i = 1, …, J Using the idea similar to that of method [7], we set P as the smallest integer such that where 0 < ξ < 1 is a certain prescribed value. Therefore, the P eigenvalues contain most of energy in the eigenvalues of C . Depended on the positive semi-definite property of sample covariance matrix R ⌢ , the output power can be given as where ∥ ⋅ ∥ denotes the Euclidean norm and L is the Cholesky factor of sample covariance matrix R ⌢ , i.e. R ⌢ = L H L. Therefore, the proposed beamforming approach can be reformulated into which can be rewritten in an equivalent SOC formulation [3] min y τ subject to 1 − w H a(θ) = 0 0 a(θ) y ∈ SOC 1 Define y = τ ε w T T , where ⋅ T denoted the transpose.
where κ is the symmetric cone corresponding to the constraints Notice that complex data and variables in the formulation (13)

Simulations
A ULA of 20 omnidirectional sensors spaced half a wavelength apart is considered. Three signals, i.e. the target signal and two uncorrected interfering sources, impinge on the array. The angel of arrival (AOA) of the TS is θ = 0°, and the AOAs of the two interfering sources are θ i1 = − 30° and θ i2 = − 50°, respectively. The number of the snapshot is 100. The interference-to-noise ratio equals to 30 dB. Additive noise is assumed to be a spatially white Gaussian noise with zero mean and unit variance. The proposed beamformer is compared with the penalty function method [2], eigenspace-based beamformer (ESB) [8], and the SOCprogramming-based method [3]. The number of constraint angles in the sidelobe areas is J = 60 and the sidelobe level is assumed to be 0.2. For the penalty function method, the symmetric T Chebychev quiescent beampattern is chosen to be −12°, 12°. For the beamformer [9] based on the semi-definite programming (SDP), Θ = −3°, 3° is used. For the proposed method, the predetermined threshold ξ = 0.95 is used. The diagonal loading method is used to remedy the antenna distortion. The ESB beamformer is known to be one of most powerful robust techniques applicable to arbitrary steering vector mismatch case. All the results are given based on 100 Monte-Carlo simulations.
In the first example, we simulate a scenario with exactly known signal steering vector and actual spatial signature of the signal is considered. In the second example corresponds to the scenario where the look direction imperfection is assumed. The assumed and true signal DOAs are 1° and 0°, respectively. It means that there is a 1° mismatch.
In the third example, we simulate the situation when the spatial signature of the target signal is distorted by local scattering effects. In the case, the steering vector of the TS is affected by a local scattering effect and modelled as where p corresponds to the direct path and b θ i , i = 1, 2, 3, 4 correspond to the coherently scattered paths. The angles θ i , i = 1, 2, 3, 4 are randomly and independently drawn in each simulation run from a uniform generator with 0° and standard deviation 2°. The angles φ i , i = 1, 2, 3, 4 are independently and uniformly taken from the interval 0, 2π in each simulation run. From Figs. 1-3, we observe that the proposed method can enjoy the low sidelobe level. In order to verify the performance comprehensively, the beampattern versus angle in the presence of arbitrary array mismatches is also investigated, and the corresponding results are shown in Fig. 4. The true ASV of the signals can be written as a = a θ p + σ 1 , …, σ N T , j = 1, …, N .
where σ j , j = 1, …, N are zero-mean independent identically distributed complex Gaussian random variables with standard deviation, σ e = 0.1. It is found that the performance of the presented method has a lower sidelobe level even when arbitrary array mismatches are considered. This performance improvement is a direct result of the application of the spheroidal sequences approach. These simulation results demonstrate that the presented beamformer shows robust operation in no-mismatch and mismatch scenarios.

Conclusion
In this letters, we present a novel approach to obtain the low computational cost and low sidelobe level simultaneously. It minimises the array output power while maintaining the distortionless response in the direction of the desired signal and searching for the predetermined sidelobe level. In order to replace the conventional constraints on the sidelobe areas with a much smaller number of constraints, we apply the spheroidal sequences approach to characterise the out-of-sector angular areas. Satisfying trade-off between the interference suppression capability and computational complexity is obtained. Simulations validate the robustness of the presented method, and enjoy the performance improvement in terms of low sidelobes and computational load.