Acoustic vector-sensor array processing
TL;DR: The authors derive a compact expression for the Cramer-Rao bound on the estimation errors of the source direction-of-arrival (DOA) parameters in the multi-source multi-vector-sensor model.
Abstract: A method is presented for localizing acoustic sources using an array of sensors, the output of each being a vector consisting of the acoustic pressure and acoustic particle velocity. The authors derive a compact expression for the Cramer-Rao bound (CRB) on the estimation errors of the source direction-of-arrival (DOA) parameters in the multi-source multi-vector-sensor model. An explicit expression is found for the mean-square angular error (MSAE) bound for source localization with a single vector sensor. The authors present two simple algorithms for estimating the source DOA with this sensor, along with their statistical performance analyses. >
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TL;DR: The authors present a new approach for localizing electromagnetic sources using sensors where the output of each is a vector consisting of the complete six electric and magnetic field components.
Abstract: The authors present a new approach for localizing electromagnetic sources using sensors where the output of each is a vector consisting of the complete six electric and magnetic field components. Two types of source transmissions are considered: (1) single signal transmission (SST), and (2) dual signal transmission (DST). The model is given in terms of several parameters, including the wave direction of arrival (DOA) and state of polarization. A compact expression is derived for the Cramer-Rao bound (CRB) on the estimation errors of these parameters for the multi-source multi-vector-sensor model. Quality measures including mean-square angular error (MSAE) and covariance of vector angular error (CVAE) are introduced, and their lower bounds are derived. The advantage of using vector sensors is highlighted by explicit evaluation of the MSAE and CVAE bounds for source localization with a single vector sensor. A simple algorithm for estimating the source DOA with this sensor is presented along with its statistical performance analysis. >
632 citations
01 Jan 2011
TL;DR: In this article, the authors used the irrationals as a numerical tool when they determined that the frequency of a vibrating string was proportional to the square root of its cross-sectional area and further added to the quantitative tradition of acoustics with conclusions such as: "The velocity of sound is greater than the velocity of cannon balls and equals 230 six-foot intervals per second".
Abstract: The origin of computational and numericalacoustics coincides with the emergence of theoretical physics [1] as an intellectual endeavor. Pythagoras developed the theory of the (Western) musical scale in terms of a device called a monochord in which adjacent consonant notes of the musical scale were obtained by plucking two string segments whose relative lengths were ratios of the small integers 1, 2, and 3. He recognized that the lengths of these strings were inversely proportional to the frequency of sound generated when plucked. Since that time, computational methods in acoustics have expanded to use more numbers than these first three integers. Mersenne [2] in the seventeenth century added the irrationals as a numerical tool when he determined that the frequency of a vibrating string was proportional to the square root of its cross-sectional area. He further added to the quantitative tradition of acoustics with conclusions such as: “The velocity of sound is greater than the velocity of cannon balls and equals 230 six-foot intervals per second.” Although the former statement is also probably true for sound propagating in water, Mersenne’s contributions to the understanding of underwater acoustics are suspect judging from his speculation that sound travels more slowly in water than air because the density of water is greater than air.
362 citations
TL;DR: The concept of quaternionic signal is introduced, and the SVDQ allows to calculate the best rank-α approximation of a quaternion matrix and can be used in subspace method for wave separation over vector-sensor array.
353 citations
Cites background from "Acoustic vector-sensor array proces..."
...Classical approaches in vector-sensor signal processing propose to concatenate the components into a long vector before processing [1,24,25]....
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...Introduction Vector-sensor signal processing concerns many areas such as acoustic [25], seismic [1,18], communications [2,16] and electromagnetics [24,25]....
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...Quaternionic model for vector-sensor signals Signals recorded on a vector-sensor array are often arranged in a long-vector form [1,24,25] before processing, or sometimes, processed component-wise....
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TL;DR: It is shown that a vector-sensor array's smaller estimation error is a result of two distinct phenomena: an effective increase in signal-to-noise ratio due to a greater number of measurements of phase delays between sensors and direct measurement of the DOA information contained in the structure of the velocity field due to the vector sensors' directional sensitivity.
Abstract: We examine the improvement attained by using acoustic vector-sensors for direction-of-arrival (DOA) estimation, instead of traditional pressure sensors, via optimal performance bounds and particular estimators. By examining the Cramer-Rao bound in the case of a single source, we show that a vector-sensor array's smaller estimation error is a result of two distinct phenomena: (1) an effective increase in signal-to-noise ratio due to a greater number of measurements of phase delays between sensors and (2) direct measurement of the DOA information contained in the structure of the velocity field due to the vector sensors' directional sensitivity. Separate analysis of these two phenomena allows us to determine the array size, array shape, and SNR conditions under which the use of a vector-sensor array is most advantageous and to quantify that advantage. By extending the beamforming and Capon (1969) direction estimators to vector-sensors, we find that the vector-sensors' directional sensitivity removes all bearing ambiguities. In particular, even simple structures such as linear arrays can determine both azimuth and elevation, and spatially undersampled regularly spaced arrays may be employed to increase the aperture and, hence, the performance. Large sample approximations to the mean-square error matrices of the estimators are derived and their validity is assessed by Monte Carlo simulation.
313 citations
TL;DR: It is shown quantitatively how assumptions about the parameters can fundamentally affect the maximum number of identifiable sources in various acoustic and electromagnetic vector-sensor models.
Abstract: We present a bound on the number of sources identifiable in a class of array processing models with multiple parameters and signals per source. The bound is applied to determine the maximum number of uniquely resolvable plane-wave sources in various acoustic and electromagnetic vector-sensor models. We examine the use of a priori information about the sources, the effects of known and unknown noise characteristics, and the presence of nuisance parameters. Connections between identifiability and existence of the Cramer-Rao bound (CRB) are investigated. We show quantitatively how assumptions about the parameters can fundamentally affect the maximum number of identifiable sources.
182 citations
References
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Book•
01 Jan 1990TL;DR: In this paper, a comprehensive introduction to probability theory covering laws of large numbers, central limit theorem, random walks, martingales, Markov chains, ergodic theorems, and Brownian motion is presented.
Abstract: This book is an introduction to probability theory covering laws of large numbers, central limit theorems, random walks, martingales, Markov chains, ergodic theorems, and Brownian motion. It is a comprehensive treatment concentrating on the results that are the most useful for applications. Its philosophy is that the best way to learn probability is to see it in action, so there are 200 examples and 450 problems.
5,168 citations
Book•
01 Jan 2001
TL;DR: This edition of A Course in Probability Theory includes an introduction to measure theory that expands the market, as this treatment is more consistent with current courses.
Abstract: Since the publication of the first edition of this classic textbook over thirty years ago, tens of thousands of students have used A Course in Probability Theory. New in this edition is an introduction to measure theory that expands the market, as this treatment is more consistent with current courses. While there are several books on probability, Chung's book is considered a classic, original work in probability theory due to its elite level of sophistication.
2,647 citations
"Acoustic vector-sensor array proces..." refers background or methods in this paper
...of i.i.d. random vectors that have finite fir:jt absolute moment, we have by Kolmogorov's strong law of large numbers (Theorem 5.4.2 of [ 29 ]) e3E[lT'(t)l2]u = (T: . U. Thus,...
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...theorem (Theorem 4.4.6 of [ 29 ]), N1l2U converges in distribution to the limit distribution of N1&....
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...Since N112(G-u) -a;'[I- UUT]ZN converges in probability to zero, we have by Slutsky's theorem (Theorem 4.4.6 of [ 29 ]) that the scaled estimation error of U (i.e., N1/'(G - U)) converges in distiibution to the limit distribution of a;'[I - UUT]ZN, which is N(0, a~~(1- uu')cov(As(t))(I - uuT))....
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...The model (3.1) generalizes the one commonly used in scalar-sensor arrays [26], [ 27 ]....
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TL;DR: The Cramer-Rao bound (CRB) for the estimation problems is derived, and some useful properties of the CRB covariance matrix are established.
Abstract: The performance of the MUSIC and ML methods is studied, and their statistical efficiency is analyzed. The Cramer-Rao bound (CRB) for the estimation problems is derived, and some useful properties of the CRB covariance matrix are established. The relationship between the MUSIC and ML estimators is investigated as well. A numerical study is reported of the statistical efficiency of the MUSIC estimator for the problem of finding the directions of two plane waves using a uniform linear array. An exact description of the results is included. >
2,552 citations
"Acoustic vector-sensor array proces..." refers methods in this paper
...The model (3.1) generalizes the one commonly used in scalar-sensor arrays [ 26 ], [27]....
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TL;DR: In this article, the generalized inverse of matrices and its applications are discussed and discussed in terms of generalized inverse of matrix and its application in the context of generalization of matrix matrices.
Abstract: (1973). Generalized Inverse of Matrices and Its Applications. Technometrics: Vol. 15, No. 1, pp. 197-197.
2,402 citations
"Acoustic vector-sensor array proces..." refers background in this paper
...Observe that the array manifold matrix in (2.7) can be written as the Khatri-Rao product [24], [ 25 ] (if the two matrices whose kth columns are, respectively, ek ard [l, uflT....
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Book•
01 Jun 1989
TL;DR: Pierces as mentioned in this paper is a classic text on acoustics with a rich history and development of the field of sound and acoustical engineering. But he organizes it superbly and writes intelligently with a wonderful way of integrating the history and evolution of the science and the graphics are exceptionally clear and communicative.
Abstract: My Personal Review: Texts on acoustics approach the subject from many different angles and at many different levels. Pierce's text is classic, rigorous and complete. It should serve the needs of serious students of acoustics for a variety of purposes musical acoustics and sound are my particular perspective.Some writers cater their approach to electrical engineers or to mechanical engineers, assuming that by tieing everything to those disciplines they will make the effort easier for their readers. This may serve well those who come from those disciplines, but may not serve others well and may not serve all applications of acoustics equally well either. Pierce does not do so. His approach is rigorously mathematical and pure, going to the heart of the matter, rather than one of attempting to cut corners by making analogies to other fields that you may or may not know.The book is not for the faint of heart or the mildly curious, it is deep and demanding. But he organizes it superbly and writes intelligently with a wonderful way of integrating the history and development of the science, and the graphics are exceptionally clear and communicative.Highly recommended for the very serious about this subject. My favorite among the books I have consulted.
2,235 citations
"Acoustic vector-sensor array proces..." refers background in this paper
...Under assumption Al, it can be shown [3], [ 4 ] that...
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...where I(t) is the sound intensity vector (the proportionality factor above is l/poc; see, e.g., [3] or [ 4 ])....
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