Meng Hwa Er

Robust Adaptive Beamformers Based on Worst-Case Optimization and Constraints on Magnitude Response

In this paper, novel robust adaptive beamformers are proposed with constraints on array magnitude response. With the transformation from the array output power and the magnitude response to linear functions of the autocorrelation sequence of the array weight, the optimization of an adaptive beamformer, which is often described as a quadratic optimization problem in conventional beamforming methods, is then reformulated as a linear programming (LP) problem. Unlike conventional robust beamformers, the proposed method is able to flexibly control the robust response region with specified beamwidth and response ripple. In practice, an array has many imperfections besides steering direction error. In order to make the adaptive beamformer robust against all kinds of imperfections, worst-case optimization is exploited to reconstruct the robust beamformer. By minimizing array output power with the existence of the worst-case array imperfections, the robust beamforming can be expressed as a second-order cone programming (SOCP) problem. The resultant beamformer possesses superior robustness against arbitrary array imperfections. With the proposed methods, a large robust response region and a high signal-to-interference-plus-noise ratio (SINR) enhancement can be achieved readily. Simple implementation, flexible performance control, as well as significant SINR enhancement, support the practicability of the proposed methods.

A flexible array synthesis method using quadratic programming

A highly flexible synthesis method for an arbitrary array is proposed to best approximate a desired array pattern in a minimum-mean-square-error sense. The basic idea of the technique is to form a quadratic program with its cost function given by the mean-square error between the array response and a properly selected pattern described by a known mathematical function. This quadratic program can be a constrained or unconstrained optimization problem depending on the requirements of the desired array pattern. In formulating the quadratic program, no assumption has been made on the gain/phase response or characteristics of the individual array elements. Therefore, one can synthesize an array of arbitrary shape to any appropriate pattern with the characteristic of the array elements taken into consideration as long as one is able to model the array accurately. The proposed method is used to synthesize arrays of different shapes, linear as well as planar arrays (including rectangular and circular planar arrays), using a Chebyshev polynomial or zero function as a design template, to illustrate the feasibility of the proposed method. >

A study of the uniqueness of steering vectors in array processing

Abstract We first analyze the relationship between the severity of rank-one ambiguity of steering vectors and the inter-sensor spacing for uniform linear arrays (ULAs). We next identify a class of non-uniform linear arrays suffering from rank-one ambiguity. Subsequently, we show that by an appropriate choice of some inter-sensor spacings of a non-uniform linear array, one may remove completely rank-one ambiguity, and we propose a general approach to constructing such an array. It is interesting to note that the average inter-sensor spacing of such an array can be infinitely large. We also analyze higher-rank ambiguity associated with linear arrays and identify a class of non-uniform arrays with such ambiguity. We show analytically that if the aperture of a p -sensor linear array with arbitrary inter-sensor spacings is greater than or equal to ( p −1)λ/2, where λ is the wavelength of the signal of interest, then rank-( p −1) ambiguity exists. Although this result is well known for ULAs, its validity to general linear arrays has not been mentioned before. Finally, we propose a procedure for analyzing the closeness of steering vectors to rank-( p −1) ambiguity by computation.

Application of constrained optimization techniques to array pattern synthesis

Abstract The paper presents two design methods for synthesizing antenna array patterns with an adjustable mainlobe width and minimum average sidelobe level. One method involves matching the array response to a desired response over the mainlobe width while minimizing the mean-square value of the array response over the sidelobe regions. The other method involves minimizing the mean-square error between a desired response and the response of the antenna array over a certain mainbeam width subject to a mean-square sidelobe constraint. Numerical techniques based on matrix factorization are proposed to reduce the computational complexity. Subsequently, a set of linear constraints are used to approximate the effect of the quadratic constraint. Numerical results show that the proposed techniques are very effective in the design of antenna pattern having an adjustable mainlobe width and minimum average sidelobe level.

A Novel Adaptive Beamformer Based on Semidefinite Programming (SDP) With Magnitude Response Constraints

A novel robust adaptive beamformer, formulated as a semidefinite programming (SDP) problem, is proposed in this paper. With new constraints on the magnitude response, the beamwidth and response ripple of the robust response region can be well controlled. Moreover, only a small part of these inequality constraints on the magnitude response are active during optimization so that few degrees of freedom (DOFs) of the adaptive beamformer are consumed. Consequently, the resultant beamformer has significant improvement on signal-to-interference-plus-noise ratio (SINR). An important problem in the proposed beamformer is how to generate the array weight vector from the optimal semidefinite matrix. In this paper, a method utilizing the extended spectral factorization method is proposed to solve this problem. Simple implementation, flexible performance control as well as significant SINR enhancement support the practicability of the proposed method.

A Robust Adaptive Beamformer Based on Worst-Case Semi-Definite Programming

In this correspondence, a novel robust adaptive beamformer is proposed based on the worst-case semi-definite programming (SDP). A recent paper has reported that a beamformer robust against large steering direction error can be constructed by using linear constraints on magnitude response in SDP formulation. In practice, however, array system also suffers from many other array imperfections other than steering direction error. In order to make the adaptive beamformer robust against all kinds of array imperfections, the worst-case optimization technique is proposed to reformulate the beamformer by minimizing the array output power with respect to the worst-case array imperfections. The resultant beamformer has the mathematical form of a regularized SDP problem and possesses superior robustness against arbitrary array imperfections. Although the formulation of robust beamformer uses weighting matrix, with the help of spectral factorization approach, the weighting vector can be obtained so that the beamformer can be used for both signal power and waveform estimation. Simple implementation, flexible performance control, as well as significant signal-to-interference-plus-noise ratio (SINR) enhancement, support the practicability of the proposed method.

A robust minimum variance beamformer with new constraint on uncertainty of steering vector

A robust minimum variance beamformer, also called robust Capon beamformer (RCB), with a new constraint on the uncertainty of nominal array steering vector (ASV) is proposed in this paper. Unlike the uncertainty constraints used in literature, the proposed constraint is constructed by replacing the nominal ASV with the projected one onto the signal-plus-interference subspace. With this compact uncertainty constraint, the proposed RCB achieves higher output signal-to-noise-plus-interference ratio (SINR) compared with the conventional RCBs. Theoretical analysis and simulation results show the effectiveness of the proposed RCB.

A novel robust adaptive beamformer based on worst-case linear optimization

Recently, a novel robust adaptive beamformers with constraints on the array magnitude response based on linear optimization is proposed. The optimization is carried out on the autocorrelation of the array weight instead on the array weight as the conventional beamformers. This kind of adaptive beamformer is able to flexibly control the robust region with a specific ripple. In practice, adaptive beamformers suffer from not only steering direction error, but also many other imperfections. In order to make an adaptive beamformer robust against all kinds of array imperfections, in this paper, we propose a new beamformer based on worst-case optimization. The resultant design possesses superior robustness against arbitrary array imperfections. With the proposed method, a large robust response region and a high signal-to-interference-plus-noise ratio (SINR) enhancement can be achieved.

A robust method for broadband beamforming in the presence of pointing error

Abstract This paper presents a robust technique for broadband beamforming in the presence of pointing error. Based on the fact that the output power of an optimized beamformer achieves a maximum if the steering vector coincides with that of the desired signal, the technique iteratively searches for the correct ‘steering vector’ of a broadband source with a known bandwidth. By approximating the ‘steering vector’ by its first order Taylor series expansion in terms of the steering angle, the maximization process reduces to a one-dimensional optimization problem. Numerical results are presented to illustrate the performance achievable.

Array pattern synthesis with a controlled mean-square sidelobe level

The author presents a technique for synthesizing an antenna pattern with a controlled mean-square sidelobe level and a smallest possible beamwidth. The basic idea is to minimize the mean-square error between the array response and the desired response over a mainlobe width subject to a mean-square sidelobe constraint. This formulation results in a quadratically constrained minimization problem. An efficient numerical technique to obtain the optimum weights is presented. Numerical results showed that, under high interference-to-white-noise ratio, the new design approach performs better, on the average, than the Chebyshev technique, in terms of interference rejection. >