There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

The quintessential goal of sensor array signal processing is the estimation of parameters by fusing temporal and spatial information, captured via sampling a wavefield with a set of judiciously placed antenna sensors. The wavefield is assumed to be generated by a finite number of emitters, and contains information about signal parameters characterizing the emitters. A review of the area of array processing is given. The focus is on parameter estimation methods, and many relevant problems are only briefly mentioned. We emphasize the relatively more recent subspace-based methods in relation to beamforming. The article consists of background material and of the basic problem formulation. Then we introduce spectral-based algorithmic solutions to the signal parameter estimation problem. We contrast these suboptimal solutions to parametric methods. Techniques derived from maximum likelihood principles as well as geometric arguments are covered. Later, a number of more specialized research topics are briefly reviewed. Then, we look at a number of real-world problems for which sensor array processing methods have been applied. We also include an example with real experimental data involving closely spaced emitters and highly correlated signals, as well as a manufacturing application example.

Two decades of array signal processing research: the parametric approach

1. Basic Concepts. 2. Nonparametric Methods. 3. Parametric Methods for Rational Spectra. 4. Parametric Methods for Line Spectra. 5. Filter Bank Methods. 6. Spatial Methods. Appendix A: Linear Algebra and Matrix Analysis Tools. Appendix B: Cramer-Rao Bound Tools. Appendix C: Model Order Selection Tools. Appendix D: Answers to Selected Exercises. Bibliography. References Grouped by Subject. Subject Index.

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Spectral analysis of signals

We present a source localization method based on a sparse representation of sensor measurements with an overcomplete basis composed of samples from the array manifold. We enforce sparsity by imposing penalties based on the /spl lscr//sub 1/-norm. A number of recent theoretical results on sparsifying properties of /spl lscr//sub 1/ penalties justify this choice. Explicitly enforcing the sparsity of the representation is motivated by a desire to obtain a sharp estimate of the spatial spectrum that exhibits super-resolution. We propose to use the singular value decomposition (SVD) of the data matrix to summarize multiple time or frequency samples. Our formulation leads to an optimization problem, which we solve efficiently in a second-order cone (SOC) programming framework by an interior point implementation. We propose a grid refinement method to mitigate the effects of limiting estimates to a grid of spatial locations and introduce an automatic selection criterion for the regularization parameter involved in our approach. We demonstrate the effectiveness of the method on simulated data by plots of spatial spectra and by comparing the estimator variance to the Crame/spl acute/r-Rao bound (CRB). We observe that our approach has a number of advantages over other source localization techniques, including increased resolution, improved robustness to noise, limitations in data quantity, and correlation of the sources, as well as not requiring an accurate initialization.

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A sparse signal reconstruction perspective for source localization with sensor arrays

Array processing involves manipulation of signals induced on various antenna elements. Its capabilities of steering nulls to reduce cochannel interferences and pointing independent beams toward various mobiles, as well as its ability to provide estimates of directions of radiating sources, make it attractive to a mobile communications system designer. Array processing is expected to play an important role in fulfilling the increased demands of various mobile communications services. Part I of this paper showed how an array could be utilized in different configurations to improve the performance of mobile communications systems, with references to various studies where feasibility of apt array system for mobile communications is considered. This paper provides a comprehensive and detailed treatment of different beam-forming schemes, adaptive algorithms to adjust the required weighting on antennas, direction-of-arrival estimation methods-including their performance comparison-and effects of errors on the performance of an array system, as well as schemes to alleviate them. This paper brings together almost all aspects of array signal processing.

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Application of antenna arrays to mobile communications. II. Beam-forming and direction-of-arrival considerations

An eigenstructure-based method for direction finding in the presence of sensor mutual coupling, gain, and phase uncertainties is presented. The method provides estimates of the directions-of-arrival (DOA) of all the radiating sources as well as calibration of the gain and phase of each sensor and the mutual coupling in the receiving array. The proposed algorithm is able to calibrate the array parameters without prior knowledge of the array manifold, using only signals of opportunity and avoiding the need for deploying auxiliary sources at known locations. The algorithm is described in detail, and its behavior is illustrated by numerical examples. >

Direction finding in the presence of mutual coupling

Time delay estimation of a noise-like random signal observed at two or more spatially separated receivers is a problem of considerable practical interest in passive radar/sonar applications. A new method is presented to analyze the mean-square error performance of delay estimation schemes based on a modified (improved) version of the Ziv-Zakai lower bound (ZZLB). This technique is shown to yield the tightest results on the attainable system performance for a wide range of signal-to-noise ratio (SNR) conditions. For delay estimation using narrow-band (ambiguity-prone) signals, the fundamental result of this study is illustrated in Fig. 3. The entire domain of SNR is divided into several disjoint segments indicating several distinct modes of operation. If the available SNR does not exceed SNR 1 , signal observations from the receiver outputs are completely dominated by noise thus essentially useless for the delay estimation. As a result, the attainable mean-square error \bar{\epsilon}^{2} is bounded only by the a priori parameter domain. If SNR 1 2 , the modified ZZLB coincides with the Barankin bound. In this regime differential delay observations are subject to ambiguities. If SNR > SNR 3 the modified ZZLB coincides with the Cramer-Rao lower bound indicating that the ambiguity in the differential delay estimation can essentially be resolved. The transition from the ambiguity-dominated mode of operation to the ambiguity-free mode of operation starts at SNR 2 and ends at SNR 3 . This is the threshold phenomenon in time delay estimation. The various deflection points SNR i and the various segments of the bound (Fig. 3) are given as functions of such important system parameters as time-bandwidth product (WT), signal bandwidth to center frequency ratio (W/ω 0 ) and the number of half wavelengths of the signal center frequency contained in the spacing between receivers. With this information the composite bound illustrated in Fig. 3 provides the most complete characterization of the attainable system performance under any prespecified SNR conditions.

Fundamental limitations in passive time delay estimation--Part I: Narrow-band systems

The most common methods for location of communications or radar transmitters are based on measuring a specified parameter such as signal angle of arrival (AOA) or time of arrival (TOA). The measured parameters are then used to estimate the transmitter location. Since the AOA/TOA measurements are done at each base station separately, without using the constraint that all measurements must correspond to the same transmitter, they are suboptimal. We propose a technique that uses exactly the same data as the common methods, except that the estimation of location is based on exact maximum likelihood, and the location determination is direct. Although there are many stray parameters, including the attenuation coefficients and the signal waveform, the method requires only a two-dimensional search. Monte Carlo simulations indicate that the accuracy is equivalent to AOA, TOA, and their combination for high SNR, while for low SNR, the accuracy of the proposed method is superior.

Direct position determination of narrowband radio frequency transmitters

The performance of two well-known bearing-only location techniques, the maximum likelihood (ML) and the Stansfield estimators, is examined. Analytical expressions are obtained for the bias and the covariance matrix of the estimation error, which permit performance comparison for any case of interest. It is shown that the Stansfield algorithm provides biased estimates even for large numbers of measurements, in contrast with the ML method. The RMS error of the Stansfield technique is not necessarily larger than the RMS of the ML technique. However, it is shown that the ML technique is superior to the Stansfield method when the number of measurements is large enough. Simulation results verify the predicted theoretical performance. >

Performance analysis of bearing-only target location algorithms

An eigenstructure-based method for direction finding in the presence of sensor gain and phase uncertainties is presented. The method provides estimates of the Directions of Arrival (DOA) of all the radiating sources as well as calibration of the gain and phase of each sensor in the observing array. The technique is not limited to a specific array configuration and can be implemented in a'ny eigenstructure-based DOA system to improve its performance.

Anthony J. Weiss

Papers

Direction finding in the presence of mutual coupling

Fundamental limitations in passive time delay estimation--Part I: Narrow-band systems

Direct position determination of narrowband radio frequency transmitters

Performance analysis of bearing-only target location algorithms

Eigenstructure methods for direction finding with sensor gain and phase uncertainties