Array Manifold Vector Research Articles

Unified Array Manifold Decomposition Based on Spherical Harmonics and 2-D Fourier Basis

In this paper, we derive a unified framework for orthonormal decomposition of the array manifold on scalar fields. Such fields are encountered in cases where polarization is not considered, e.g., single polarized radio waves and acoustic pressure. The results generalize and unify different decompositions of the array manifold, found in recent literature, to arrays of arbitrary geometry including conformal arrays. The concept of equivalence matrix is introduced, establishing a link between the spherical harmonics and 2-D Fourier basis functions. Under some mild assumptions that typically hold in practice, a one-to-one relationship between spherical harmonic spectra and 2-D Fourier spectra may be established. Additionally, it is shown that the rows of the equivalence matrix and the 2-D Fourier spectra of the array manifold span the same subspace. With such results the spherical harmonic and 2-D Fourier decompositions of the array manifold vector, i.e., Wavefield Modeling and 2-D Effective Aperture Distribution Function (EADF) are shown to be equivalent. Results on the modeling capabilities of both orthonormal decompositions are obtained. Moreover, the equivalence matrix is shown to facilitate noise attenuation. A fast spherical harmonic transform with complexity <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O</i> ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Q</i> log <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Q</i> ) can be obtained by exploiting the equivalence matrix, where <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Q</i> represents the total number of points on the sphere. Finally, the equivalence matrix allows to gain more insight into the relation between rotating a function on the sphere and on the torus. These contributions facilitate high-resolution array processing both in elevation and azimuth irrespective of the array structure or imperfections.

Adaptive near-field beamforming techniques for sound source imaging

Phased array signal processing techniques such as beamforming have a long history in applications such as sonar for detection and localization of far-field sound sources. Two sometimes competing challenges arise in any type of spatial processing; these are to minimize contributions from directions other than the look direction and minimize the width of the main lobe. To tackle this problem a large body of work has been devoted to the development of adaptive procedures that attempt to minimize side lobe contributions to the spatial processor output. In this paper, two adaptive beamforming procedures-minimum variance distorsionless response and weight optimization to minimize maximum side lobes--are modified for use in source visualization applications to estimate beamforming pressure and intensity using near-field pressure measurements. These adaptive techniques are compared to a fixed near-field focusing technique (both techniques use near-field beamforming weightings focusing at source locations estimated based on spherical wave array manifold vectors with spatial windows). Sound source resolution accuracies of near-field imaging procedures with different weighting strategies are compared using numerical simulations both in anechoic and reverberant environments with random measurement noise. Also, experimental results are given for near-field sound pressure measurements of an enclosed loudspeaker.

Robust array pattern synthesis with uncertain manifold vector

The knowledge of the array manifold vectors of an acoustic array can be imprecise, which is often the case in practice. This may cause undesirably high sidelobes for a nominal optimal beamformer where the array manifold vectors are assumed to be known exactly. Although the norm constraint on beamformer weights can be imposed to improve the robustness of the optimal beamformer, it is not clear how to choose the optimal constrained parameter based on the known level of uncertainty of the array manifold vectors. A pattern synthesis approach to arbitrary arrays with robustness against array manifold vectors errors is developed. Our technique optimizes the worst-case performance by minimizing the worst-case sidelobe level while maintaining a distortionless response to the worst-case signal steering vector. The parameters can be optimally chosen based on the uncertainty of the array manifold vector. The robust beamformer problem is shown to be convex, which can be efficiently solved using second-order cone programming. A simple lower bound on how much worse the robust optimal beamformer will be compared to the nominal one is also derived. Computer simulations show better performance of the proposed robust beamformer.

Crossed-Dipole Arrays for Asynchronous DS-CDMA Systems

In this paper, the use of a crossed-dipole array is proposed in joint space-time channel estimation for asynchronous multipath direct sequence code division multiple access (DS-CDMA) systems. The polarization diversity offered by such an array, unlike linearly polarized arrays, is able to detect and estimate any arbitrary completely polarized signal path. By utilizing the polarization information inherent in the received signal to construct the polar-spatio-temporal array (polar-STAR) manifold vector, the accuracy and resolution of the polar-STAR parameters' estimation are significantly improved, and its signal detection capability is enhanced. To alleviate the need for a multidimensional search in the polarization space, a computationally efficient joint polarization-angle-delay channel parameter estimation algorithm is proposed for a desired user that operates in an asynchronous multiuser and multipath environment. The proposed algorithm, which can be seen as an application of MUSIC-type techniques, is based on combining a two-dimensional STAR-Subspace type technique with a set of analytical equations and is supported by representative examples and computer simulation studies.

Source localization using recursively applied and projected (RAP) MUSIC

A new method for source localization is described that is based on a modification of the well-known MUSIC algorithm. In classical MUSIC, the array manifold vector is projected onto an estimate of the signal subspace. Errors in the estimate of the signal subspace can make localization of multiple sources difficult. Recursively applied and projected (RAP) MUSIC uses each successively located source to form an intermediate array gain matrix and projects both the array manifold and the signal subspace estimate into its orthogonal complement. The MUSIC projection to find the next source is then performed in this reduced subspace. Special assumptions about the array manifold structure, such as Vandermonde or shift invariance, are not required. Using the metric of principal angles, we describe a general form of the RAP-MUSIC algorithm for the case of diversely polarized sources. Through a uniform linear array simulation with two highly correlated sources, we demonstrate the improved Monte Carlo error performance of RAP-MUSIC relative to MUSIC and two other sequential subspace methods: S and IES-MUSIC. We then demonstrate the more general utility of this algorithm for multidimensional array manifolds in a magnetoencephalography (MEG) source localization simulation.

Efficient wideband source localization using beamforming invariance technique

A novel scheme for wideband direction-of-arrival (DOA) estimation is proposed. The technique performs coherent signal subspace transformation by a set of judiciously constructed beamforming matrices. The beamformers are chosen to transform each of the narrowband array manifold vectors into the one corresponding to the reference frequency, regardless of the actual spatial distribution of the sources. The focused data correlation matrix can thus be obtained without any preliminary DOA estimation or iteration. A simplified version of the beamspace Root-MUSIC algorithm is developed and used in conjunction with the proposed method to efficiently localize multiple wideband sources with a linear, equally spaced array. Numerical simulations are conducted to demonstrate the efficacy of the new scheme.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Coherent wide-band processing for arbitrary array geometry

We present a new method for coherent wide-band direction finding of far-field sources impinging on a two-dimensional array with a known arbitrary geometry. This method, termed array manifold interpolation (AMI), is based on obtaining the array manifold at a desired frequency f <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> , by linear interpolation of the array manifold at a given frequency, f. We use a separable representation of the array manifold vector, which separates the array geometry and the frequency from the direction θ, in order to derive the required array manifold interpolation matrix. The AMI method is practical, computationally efficient, and robust. For the special case of a uniform circular array, we present a fast implementation of the AMI method, which utilizes the FFT algorithm.

The mathematical basis for element and fourier beamspace MUSIC and root-MUSIC algorithms

Signal subspace spectral estimation algorithms of the MUSIC type [l-3] are particularly effective for estimating the direction of arrival (DOA) of incoming plane wave signals arising from point emission sources in the far field of spatial phased arrays. The subspace algorithms require eigenanalysis of an estimated form of the covariance matrix. A linear system of N distinct sensors with complex signal outputs can be mathematically formulated in terms of linear operators in an N-dimensional complex Hilbert space, e (iv]. The covariance matrices used in the analyses of the system response are N-dimensional Hermitian matrices. Eigenanalysis of the covariance matrix is required to separate the signal subspace from the noise subspace. The source DOAs are estimated from the array manifold vectors that exhibit minimum projections on the noise subspace. Eigenanalysis of N-dimensional matrices, which cannot take advantage of simplifying structures, typically requires 0 ( N3) operations. This computational burden often precludes the conventional application of signal subspace algorithms to large-order array systems. The computational complexity of signal subspace methods in the case of a large-order array system may be substantially reduced by operating in the beamspace domain [ 4-101. Here the spatial spectrum (angular region of interest) is divided into subbands (angular sectors). A particular spatial frequency subband is established by forming a number of overlapping beams to encompass the angular sector. In the simplest case, the beam outputs are computed as successive values of an N-point discrete Fourier transform (DFT) of the snapshot vector composed of the N sensor outputs at a particular sampling instant. Q successive N-point DFT beams subtend an angular fraction Q/N of the total spatial spectrum. Different subbands are selected by selecting different sets of Q successive N-point DFT beams; subbands may be processed simultaneously in parallel. Processing within each subband requires eigenanalysis of a Q-dimensional beamspace covariance matrix to determine the respective signal subspace and noise subspace associated with the subband. The DOAs of sources within the subband are determined as those for which the norm of the projection of the corresponding beamspace manifold on the noise subspace is minimum. In this way, a large problem of order N is reduced to a number of smaller problems of order Q. For a uniformly spaced linear array ( ULA) , the element space array manifold vectors possess a simple Vundernonde structure. The beamspace transformation causes an important change as the transformed array manifold vectors no longer exhibit a Vandermonde structure. This beamspace property was originally demonstrated in [lo] and is discussed in detail in Section IV of the present work. As a consequence of this change in structure, the class of algorithms that employ forward and backward spatial averaging is not applicable in beamspace. This renders the beamspace approach inoperative in the case of a single snapshot (single realization of a time series). In general, beamspace signal subspace methods have developed from a largely heuristic point of view. An alternative means of reducing computational complexity is motivated by the subbanding and multirate processing techniques employed in a number of application areas including speech processing, digital communications, etc. In this mode of operation, one ultimately works in a pseudo-element space as opposed to beamspace. This