Lloyd J. Griffiths

A projection approach for robust adaptive beamforming

It is well known that calibration errors can seriously degrade performance in adaptive arrays, particularly when the input signal-to-noise ratio is large. The effect is caused by the perturbation of the presumed steering vector from its optimal value. Although it is not as widely known, similar degradation occurs in sampled matrix inversion processing when the covariance matrix is estimated while the desired signal is present in the snapshot data. Under these conditions, performance loss is due to errors in the estimated covariance matrix and occurs even when the steering vector is known exactly. In the paper, a new method based of modification of the steering vector is proposed to overcome both the problems of perturbation and of sample covariance errors. The method involves projection of the presumed steering vector onto the observed signal-plus-interference subspace. An analysis is also presented illustrating that the sample covariance errors can be viewed as a particular type of perturbation error and a simple approximation is derived for the expected beamformer performance as a function of the number of data snapshots. Both analytical and experimental results are presented that illustrate the advantages of the proposed method. >

A simple algorithm to achieve desired patterns for arbitrary arrays

A simple iterative algorithm which can be used to find array weights that produce array patterns with a given look direction and an arbitrary sidelobe specification is presented. The method can be applied to nonuniform array geometries in which the individual elements have arbitrary (and differing) radiation patterns. The method is iterative and uses sequential updating to ensure that peak sidelobe levels in the array meet the specification. Computation of each successive pattern is based on the solution of a linearly constrained least-squares problem. The constraints ensure that the magnitude of the sidelobes at the locations of the previous peaks takes on the prespecified values. Phase values for the sidelobes do not change during this process, and problems associated with choosing a specific phase value are therefore avoided. Experimental evidence suggests that the procedure terminates in remarkably few iterations, even for arrays with significant numbers of elements. >

Broad-band signal-subspace spatial-spectrum (BASS-ALE) estimation

An approach to spatial-spectrum estimation of broadband sources is described. The approach, termed broadband signal-subspace spatial spectrum (BASS-ALE) estimation, is based on the eigenstructure of a broadband spatial/temporal covariance matrix and is justified by identifying the low-rank character of spatial/temporal observations of broadband sources. BASS-ALE estimators are described which incorporate: source focusing, spatial/temporal noise decorrelating, a signal (or noise-only) subspace generated from the eigenstructure of the transformed spatial/temporal covariance matrix, and broadband source models. Through discussion and simulation, BASS-ALE estimators are compared to the coherent signal-subspace processor (a similar subspace method), and relative advantages of the two are identified. >

Quiescent pattern control in linearly constrained adaptive arrays

When an adaptive array operates in the presence of white noise only, the resulting beam pattern is referred to as the quiescent response. Typically, these patterns have mainlobe and sidelobe shapes differing from those designed for use in deterministic, nonadaptive arrays. This paper describes a simple method which allows nearly arbitrary specification of the quiescent response in a linearly constrained power minimization adaptation array. The only restriction on the quiescent is that it must meet the constraints defined for the adaptive array. Since many well-known deterministic designs such as Chebychev are not likely to meet the linear constraint conditions used in adaptive arrays for mainlobe and other pattern control functions, a procedure is presented which modifies the deterministic design to force it to meet the linear constraints in a least-squares manner. Once this has been accomplished, the methods outlined in this paper can be used to cause the modified deterministic design to become the quiescent response of the adaptive array. As a result, the adaptive array can be configured to closely resemble a deterministic array when the noise is white. Under conditions of correlated interference, or jamming, however, the response changes so as to effectively steer nulls in the appropriate directions. The method is based on the use of a generalized side-lobe canceller and requires one additional linear constraint for both narrow-band and broad-band arrays.

A unified approach to the design of linear constraints in minimum variance adaptive beamformers

A simple, systematic procedure for designing linear constraints in minimum-variance beamformers which allows an arbitrary specification of the quiescent response (the beamformer response when only white noise is present) is described. In this approach, the first constraint is dedicated to the imposition of a desired quiescent response, and additional constraints are included to assure proper reception of the desired signal. These additional constraints make the overall beamformer response equal to the quiescent response in the desired signal region so that the signal is not cancelled when it is present. Optionally, the response can be fixed in other regions of interest by adding more constraints. This design procedure demonstrates that the key to designing efficient constraints is finding the weighting coefficients which specify the desired quiescent response, a problem identical to the synthesis of desired beam patterns for nonadaptive arrays. The effectiveness of the procedure is illustrated by examples in both narrowband and broadband arrays. >

Steering vector estimation in uncalibrated arrays

This paper presents an iterative algorithm for estimating the signal steering vectors and associated power levels received by an array of uncalibrated isotropic sensors. The inputs are assumed to consist of narrowband, uncorrelated directional signals in the presence of additive white noise. An iterative algorithm is employed to successively search for estimates that simultaneously satisfy a signal subspace and an orthogonality condition. These conditions are shown to be both necessary and sufficient for identification of the underlying steering vectors in the case when the data covariance matrix is known exactly, i.e. for the case of infinite data observation. The iterative method employs minimum distance criterion (projections) to successively map the solution between three constraint sets until a stable point is determined. Two examples are presented which illustrate the application of the algorithm in direction finding and beamforming. >

A Systematic Procedure For Implementing The Blocking Matrix In Decomposed Form

The Generalized Sidelobe Canceller (GSC) is an important beam- forming structure for implementing adaptive linearly constrained beamformers. A significant feature of the GSC is that it effi- ciently eliminates the constraints such that the weights can be updated unconstrainedly in a reduce-dimensional space. The elimination of constraints is achieved by specifying a blocking matrix whose columns form a basis orthogonal to the constraints. For a given linearly-constrained minimization problem, the specification of the blocking matrix is not unique. However, the com- putational complexity of GSC depends entirely on the detailed composition of the blocking matrix. This paper presents a systematic procedure to construct the blocking matrix with effective implementation structure.

Estimation of signal steering vectors in uncalibrated arrays

This paper considers the problem of extracting the signal powers and steering vectors from the signal covariance matrix without the knowledge of array manifold. Under the assumption of narrowband and uncorrelated signals, it is shown that it is necessary and sufficient for the signal powers and steering vectors to satisfy two conditions, termed the signal subspace and orthogonality conditions, in order for them to match the signal covariance matrix asymptotically. Based upon these two conditions, an algorithm is derived to iteratively search for the signal powers and steering vectors which closely match the signal covariance matrix estimated from the observed data. A real-data example is presented to illustrate the robustness of the proposed algorithm.<<ETX>>

A Novel Approach for Designing Linear Constraints in Adaptive Arrays

This paper describes an approach for designing linear constraints in adaptive arrays which allows arbitrary specification of the quiescent response (the array response when only white noise is present). The first step of this approach is to use available pattern synthesis techniques for determining weights which result in a desired quiescent pattern. A set of linear constraints is then derived which, when the array operates in the adaptive mode, produces this quiescent response and also prevents signals of interest from being cancelled. This design method also suggests that the key to designing linear constraints in adaptive arrays lies in developing efficient algorithms to achieve well-behaved array responses for nonadaptive arrays.

Sensitivity of the linearly-constrained constant-modulus cost function

The authors present results from the application of the linearly constrained constant-modulus (LCCM) adaptive algorithm to a perturbed array environment. Assuming a narrowband signal model, analytic expressions are derived for the feasible gain values on the signal in the presence of general array perturbations and ambient white noise. The feasible gain values are found to be the roots of a cubic equation that is parameterized by the number of array elements, the signal-to-noise ratio, the modulus factor, the kurtosis of the signal of interest, and the angle between the true signal steering vector, and the perturbed steering vector. A principal result of the analysis is that LCCM is robust in the presence of array perturbations but is sensitive to the modulus factor value. A simulation is included that compares the sensitivity to perturbations of the techniques of linearly constrained minimum power, unconstrained constant-modulus, and LCCM. Experimental results are presented on a four-element square array with half-wavelength spacing which is used to receive a QPSK signal arriving from a presumably known direction.<<ETX>>