Array Response Matrix Research Articles

Waveform inversion via reduced order modeling

We introduce a novel approach to waveform inversion based on a data-driven reduced order model (ROM) of the wave operator. The presentation is for the acoustic wave equation, but the approach can be extended to elastic or electromagnetic waves. The data are time resolved measurements of the pressure wave gathered by an acquisition system that probes the unknown medium with pulses and measures the generated waves. We propose to solve the inverse problem of velocity estimation by minimizing the square misfit between the ROM computed from the recorded data and the ROM computed from the modeled data, at the current guess of the velocity. We give a step by step computation of the ROM, which depends nonlinearly on the data and yet can be obtained from them in a noniterative fashion, using efficient methods from linear algebra. We also explain how to make the ROM robust to data inaccuracy. The ROM computation requires the full array response matrix gathered with colocated sources and receivers. However, we find that the computation can deal with an approximation of this matrix, obtained from towed-streamer data using interpolation and reciprocity on-the-fly. Although the full-waveform inversion approach of nonlinear least-squares data fitting is challenging without low-frequency information, due to multiple minima of the data fit objective function, we find that the ROM misfit objective function has better behavior, even for a poor initial guess. We also find by explicit computation of the objective functions in a simple setting that the ROM misfit objective function has convexity properties, whereas the least-squares data fit objective function displays multiple local minima.

Direction of Arrival Estimation of Wideband Sources Using Sparse Linear Arrays

In this paper, we study the problem of wideband direction of arrival (DoA) estimation with sparse linear arrays (SLAs), where a number of uncorrelated wideband signals impinge on an SLA and the data is collected from multiple frequency bins. To boost the performance and perform underdetermined DoA estimation, the difference co-array response matrices for all frequency bins are constructed first. Then, to merge the data from different frequency bins, we resort to the Jacobi-Anger approximation to transform the co-array response matrices of all frequency bins into a single virtual uniform linear array (ULA) response matrix. The major advantage of this approach is that the transformation matrices are all signal independent. For the special case where all sources share an identical distribution of the power spectrum, we develop two super-resolution off-the-grid DoA estimation approaches based on atomic norm minimization (ANM), one with and one without prior knowledge of the power spectrum. Our solution is able to resolve more sources than the number of antennas but also more than the number of degrees of freedom (DoF) of the difference co-array of the SLA. For the general case where each source has an arbitrary power spectrum, we propose a multi-task ANM method to exploit the joint sparsity from all frequency bins. Simulation results show that our proposed methods present a clear performance advantage over existing methods, and achieve an estimation accuracy close to the associated Cramér-Rao bounds (CRBs).

Imaging in Three-Dimensional Waveguides with Partial Aperture Data

We consider the problem of imaging extended reflectors in three-dimensional acoustic waveguides using a planar array that is parallel to the waveguide’s cross-section. Our data consist of the multiple-frequency array response matrix. To form an image, we back-propagate a projection of the data on the propagating modes in the waveguide. The projection operator is adequately defined for any array-aperture size and shape that cover fully or partially the waveguide cross-section. The properties of the imaging method are analyzed theoretically and its effectiveness is assessed by numerical experiments for scatterers and arrays of varying shape and size.

Gram–Schmidt orthogonalisation aided hybrid precoding in millimetre‐wave massive MIMO systems

The authors introduce a novel hybrid precoding algorithm based on Gram–Schmidt orthogonalisation (GSO) in millimetre-wave massive MIMO systems. Specifically, the columns of array response matrix orthogonalised by the GSO process are considered as a set of candidate analogue precoders, then traditional orthogonal matching pursuit (OMP) is utilised to find the optimal analogue and digital precoders. Since GSO is a recursive process that depends on the order in which the matrix columns are selected. A heuristic solution to the order of columns selection is suggested according to the array response vector along which the fully-digital precoder has the maximum projection. The proposed algorithm, not only constrained to uniform linear arrays, can avoid the matrix inversion in designing the digital precoder compared to OMP. Simulation results show that the spectral efficiency and bit error rate of the proposed hybrid precoding solutions are close to that obtained with fully digital architectures. Furthermore, the results indicate that the proposed hybrid precoding solutions outperform the orthogonality-based matching pursuit, which uses the columns of the DFT matrix as a set of candidate analogue precoders.

Hybrid Precoding for Massive mmWave MIMO Systems

Due to high costs and power consumptions, fully digital baseband precoding schemes are usually prohibitive in millimeter-wave massive MIMO systems. Therefore, hybrid precoding strategies become promising solutions. In this paper, we present a novel real-time yet high-performance precoding strategy. Specifically, the eigenvectors corresponding to the larger eigenvalues of the right unitary matrix after singular value decomposition on an array response matrix are used to abstract the angle information of an analog precoding matrix. As the obtained eigenvectors correspond to the larger singular values, the major phase information of channels is captured. In this way, the iterative search process for obtaining the analog precoding vectors is avoided, and thus the hybrid precoding can be realized in parallel. To further improve its spectral-efficiency, we enlarge the resultant vector set by involving more relevant vectors in terms of their correlation values with the unconstrained optimal precoder, and a hybrid precoder is thus produced by using the vector set. The simulation results show that our proposed scheme achieves near the same performance as the orthogonal matching pursuit does, whereas it costs much fewer complexities than the OMP, and thus can be realized in parallel.

Imaging Extended Reflectors in a Terminating Waveguide

We consider the problem of imaging extended reflectors in terminating waveguides. We form the image by back-propagating the array response matrix projected on the waveguide's nonevanescent modes. The projection is adequately defined for any array aperture size covering fully or partially the waveguide's vertical cross-section. We perform a resolution analysis of the imaging method and show that the resolution is determined by the central frequency, while the image's signal-to-noise ratio improves as the bandwidth increases. The robustness of the imaging method is assessed with fully nonlinear scattering data in terminating waveguides with complex geometries.

Effect of microstructural elongation on backscattered field: Intensity measurement and multiple scattering estimation with a linear transducer array

The effect of microstructural elongation on ultrasonic backscattered fields was studied. Two methods for determining the elongation direction of macrozones in titanium alloys, using the anisotropic spatial coherence of the backscattered field, are presented. Both methods use a phased array attached on a rotative holder that records the array response matrix at several angles. Two titanium alloys were investigated: TA6V and Ti17. TA6V exhibited a strong macrozone elongation, whereas Ti17 macrozones were found equiaxial. The first method is based on the measurement of backscattered intensity in function of the probe angle relative to the macrozones elongation direction. An angular dependence of backscattered intensity is observed in presence of elongated scatterers, and their elongation direction is collinear with the probe direction corresponding to a minimal intensity. This variability is linked to both piezoelectric shape and the backscattered field spatial properties. The second method is based on the measurement of the relative proportion of single to multiple scattering in a diffusive media, using a simplified version of the single scattering filter developed in Aubry and Derode (2009). It allows the measurement of the level of multiple scattering: both titanium alloys exhibited strong multiple scattering. The elongation direction was determined as the direction of minimal multiple scattering. Furthermore, these results were confirmed by the measurement of the coherent backscattering cone on both samples.

Partial-aperture array imaging in acoustic waveguides

We consider the problem of imaging extended reflectors in waveguides using partial-aperture array, i.e. an array that does not span the whole depth of the waveguide. For this imaging, we employ a method that back-propagates a weighted modal projection of the usual array response matrix. The challenge in this setup is to correctly define this projection matrix in order to maintain good energy concentration properties for the imaging method, which were obtained previously by Tsogka et al (2013 SIAM J. Imaging Sci. 6 2714–39) for the full-aperture case. In this paper we propose a way of achieving this and study the properties of the resulting imaging method.

A comparative study of data filtering methods for imaging in strongly scattering media

We consider the problem of imaging small defects embedded in strongly scattering media, often called clutter, using an active array of transducers that can play the dual role of emitters and receivers. Our data is the array response matrix collected by sending short pulses from each source and recording the response at all array elements. Imaging in strong clutter is quite challenging as the array data are dominated by noise due to the multiple scattering of the waves with the medium heterogeneities. To successfully image in this regime using simple coherent imaging functionals, we follow the methodology of coherent signal enhancement through data filtering. We study the performance of the different approaches with extensive numerical simulations, carried out in a non-destructive testing setup.

Two-Dimensional DOA and Polarization Estimation for a Mixture of Uncorrelated and Coherent Sources with Sparsely-Distributed Vector Sensor Array.

This paper presents an L-shaped sparsely-distributed vector sensor (SD-VS) array with four different antenna compositions. With the proposed SD-VS array, a novel two-dimensional (2-D) direction of arrival (DOA) and polarization estimation method is proposed to handle the scenario where uncorrelated and coherent sources coexist. The uncorrelated and coherent sources are separated based on the moduli of the eigenvalues. For the uncorrelated sources, coarse estimates are acquired by extracting the DOA information embedded in the steering vectors from estimated array response matrix of the uncorrelated sources, and they serve as coarse references to disambiguate fine estimates with cyclical ambiguity obtained from the spatial phase factors. For the coherent sources, four Hankel matrices are constructed, with which the coherent sources are resolved in a similar way as for the uncorrelated sources. The proposed SD-VS array requires only two collocated antennas for each vector sensor, thus the mutual coupling effects across the collocated antennas are reduced greatly. Moreover, the inter-sensor spacings are allowed beyond a half-wavelength, which results in an extended array aperture. Simulation results demonstrate the effectiveness and favorable performance of the proposed method.

Reduced-rank estimation for ill-conditioned stochastic linear model with high signal-to-noise ratio

Reduced-rank approach has been used for decades in robust linear estimation of both deterministic and random vector of parameters in linear model y=Hx+ϵn. In practical settings, estimation is frequently performed under incomplete or inexact model knowledge, which in the stochastic case significantly increases mean-square-error (MSE) of an estimate obtained by the linear minimum mean-square-error (MMSE) estimator, which is MSE-optimal among linear estimators in the theoretical case of perfect model knowledge. However, the improved performance of reduced-rank estimators over MMSE estimator in estimation under incomplete or inexact model knowledge has been established to date only by means of numerical simulations and arguments indicating that the reduced-rank approach may provide improved performance over MMSE estimator in certain settings. In this paper we focus on the high signal-to-noise ratio (SNR) case, which has not been previously considered as a natural area of application of reduced-rank estimators. We first show explicit sufficient conditions under which familiar reduced-rank MMSE and truncated SVD estimators achieve lower MSE than MMSE estimator if singular values of array response matrix H are perturbed. We then extend these results to the case of a generic perturbation of array response matrix H, and demonstrate why MMSE estimator frequently attains higher MSE than reduced-rank MMSE and truncated SVD estimators if H is ill-conditioned. The main results of this paper are verified in numerical simulations.

Computationally Efficient Angle and Polarization Estimation in the Presence of Multipath Propagation Using Dual-Polarization Vector Sensor Array

This paper presents a computationally efficient angle and polarization estimation method for a mixture of uncorrelated and coherent sources using a dual-polarization vector sensor array. The uncorrelated sources are separated from the coherent sources on the basis of the modulus property of eigenvalues. The angles of the uncorrelated sources are estimated by employing rotational invariance and the associated polarization is obtained from the estimate of the uncorrelated array response matrix through elementwise division. For the distinguished coherent sources, two Hankel matrices are constructed from the elements of the estimated coherent array response matrix of each coherent group, from which two rotational-invariant submatrix pairs are extracted for estimating the coherent angles with a high precision. The least-square solution to the coherent polarization equation is derived for estimating the coherent polarization parameters. For each uncorrelated source and coherent group, the proposed method estimates the associated angle and polarization parameters separately, which avoids the need of 3D spectral search. In comparison with the existing methods, the simulation results show that the proposed method yields favorable performance in terms of computational efficiency and estimation accuracy.

A quick responding quartz crystal microbalance sensor array based on molecular imprinted polyacrylic acids coating for selective identification of aldehydes in body odor

In present work, a novel quartz crystal microbalance (QCM) sensor array has been developed for prompt identification of primary aldehydes in human body odor. Molecularly imprinted polymers (MIP) are prepared using the polyacrylic acid (PAA) polymer matrix and three organic acids (propenoic acid, hexanoic acid and octanoic acid) as template molecules, and utilized as QCM surface coating layer. The performance of MIP films is characterized by 4-element QCM sensor array (three coated with MIP layers and one with pure PAA for reference) dynamic and static responses to target aldehydes: hexanal, heptanal, and nonanal in single, binary, and tertiary mixtures at distinct concentrations. The target aldehydes were selected subsequent to characterization of body odor samples with solid phase-micro extraction gas chromatography mass spectrometer (SPME-GC–MS). The hexanoic acid and octanoic acid imprinted PAA exhibit fast response, and better sensitivity, selectivity and reproducibility than the propenoic acid, and non-imprinted PAA in array. The response time and recovery time for hexanoic acid imprinted PAA are obtained as 5s and 12s respectively to typical concentrations of binary and tertiary mixtures of aldehydes using the static response. Dynamic sensor array response matrix has been processed with principal component analysis (PCA) for visual, and support vector machine (SVM) classifier for quantitative identification of target odors. Aldehyde odors were identified successfully in principal component (PC) space. SVM classifier results maximum recognition rate 79% for three classes of binary odors and 83% including single, binary, and tertiary odor classes in 3-fold cross validation.

Quaternized magnetic nanoparticles–fluorescent polymer system for detection and identification of bacteria

Nanomaterial-based ‘chemical nose’ sensor with sufficient sensing specificity is a useful analytical tool for the detection of toxicologically important substances in complicated biological systems. A sensor array containing three quaternized magnetic nanoparticles (q-MNPs)–fluorescent polymer systems has been designed to identify and quantify bacteria. The bacterial cell membranes disrupt the q-MNP–fluorescent polymer, generating unique fluorescence response array. The response intensity of the array is dependent on the level of displacement determined by the relative q-MNP–fluorescent polymer binding strength and bacteria cells–MNP interaction. These characteristic responses show a highly repeatable bacteria cells and can be differentiated by linear discriminant analysis (LDA). Based on the array response matrix from LDA, our approach has been used to measure bacteria with an accuracy of 87.5% for 107cfumL−1 within 20min. Combined with UV–vis measurement, the method can be successfully performed to identify and detect eight different pathogen samples with an accuracy of 96.8%. The measurement system has a potential for further applications and provides a facile and simple method for the rapid analysis of protein, DNA, and pathogens.

Invariants of the time reversal operator and characterization of solid media: An overview

The invariants of the time reversal operators for solid media have been the object of several studies in the past 15 years in the context of multi-element array imaging. These invariants are obtained from the decomposition of the array response matrix. Their analysis was applied to the detection of flaws using bulk waves, or Rayleigh and Lamb guided more. It was also applied to the characterization of shells using the radiation of circumferential guided modes. A review of the different solutions that have been proposed to improve this analysis will be given. Then, selected experimental examples of time reversal invariants will be discussed.

Comparison between experimental and 2-D numerical studies of multiple scattering in Inconel600® by means of array probes

Ultrasonic non-destructive testing of polycrystalline structures can be disturbed by scattering at grain boundaries. Understanding and modeling this so-called “structural noise” is crucial for characterization as well as detection purposes. Structural noise can be considered as a fingerprint of the material under investigation, since it contains information about its microstructure. The interpretation of experimental data necessitates an accurate comprehension of complex phenomena that occur in multiple scattering media and thus robust scattering models. In particular, numerical models can offer the opportunity to realize parametrical studies on controlled microstructures. However, the ability of the model to simulate wave propagation in complex media must be validated. In that perspective, the main objective of the present work is to evaluate the ability of the finite-element code ATHENA 2D to reproduce typical features of multiple wave scattering in the context of ultrasonic non-destructive evaluation, with an array of sources and receivers. Experiments were carried out with a 64-element array, around 2MHz. The sample was a mock-up of Inconel600® exhibiting a coarse grain structure with a known grain size distribution. The numerical model of this microstructure is based on Voronoi diagrams. Two physical parameters were used to compare numerical and experimental data: the coherent backscattering peak, and the singular value distribution of the array response matrix. Though the simulations are 2-D, a good agreement was found between simulated and experimental data.

Active detection of a moving target in a waveguide with strong masking echoes

In shallow water, active detection of a small moving target can be difficult because of strong echoes from large fixed obstacles. To cancel strong unwanted echoes, differences between successive acquisitions can be achieved, however they are very sensitive to fluctuations. A projection method combined with a fast acquisition technique is proposed as a robust alternative. An ultrasonic experiment is presented: a 64 transducers linear vertical array is used to detect a small target moving above a large obstacle in a waveguide. To reduce acquisition time, eight groups of adjacent elements transmit linear frequency modulations with increasing delays in a single emission. The 8x64 array response matrix is then obtained by correlations and time windowing. The projection is achieved between two acquisitions obtained while the target is moving, in order to remove the obstacle’s contribution. Namely, the second acquired matrix is projected on the space orthogonal to the eight singular vectors of the first acquired matrix. Then, it is shown that the first singular vector of the projected matrix focuses on the second target’s position. Comparisons are made with the decomposition of the time reversal operator in differential mode and conventional beamforming.

Robust imaging of localized scatterers using the singular value decomposition and ℓ1 minimization

We consider narrow band, active array imaging of localized scatterers in a homogeneous medium with and without additive noise. We consider both single and multiple illuminations and study ℓ1 minimization-based imaging methods. We show that for large arrays, with array diameter comparable to range, and when scatterers are sparse and well separated, ℓ1 minimization using a single illumination and without additive noise can recover the location and reflectivity of the scatterers exactly. For multiple illuminations, we introduce a hybrid method which combines the singular value decomposition and ℓ1 minimization. This method can be used when the essential singular vectors of the array response matrix are available. We show that with this hybrid method we can recover the location and reflectivity of the scatterers exactly when there is no noise in the data. Numerical simulations indicate that the hybrid method is, in addition, robust to noise in the data. We also compare the ℓ1 minimization-based methods with others including Kirchhoff migration, ℓ2 minimization and multiple signal classification.

Selective Imaging of Extended Reflectors in Two-Dimensional Waveguides

We consider the problem of selective imaging extended reflectors in waveguides using the response matrix of the scattered field obtained with an active array. Selective imaging amounts to being able to focus at the edges of a reflector, which typically give rise to weaker echoes than those coming from its main body. To this end, we propose a selective imaging method that uses projections on low-rank subspaces of a weighted modal projection of the array response matrix, $\widehat{\mathbb{P}}(\omega)$. We analyze theoretically our imaging method for a simplified model problem where the scatterer is a vertical one-dimensional perfect reflector. In this case, we show that the rank of $\widehat{\mathbb{P}}(\omega)$ equals the size of the reflector divided by the cross-range array resolution. We also derive analytic expressions for the singular vectors of $\widehat{\mathbb{P}}(\omega)$ and carry out a detailed theoretical analysis of our selective imaging functional. Our numerical simulations are in very good agreement with the theory and illustrate the robustness of our imaging functional for reflectors of various shapes.

A statistical approach to target detection and localization in the presence of noise

The problem addressed in this paper is the combined detection and localization of a point reflector embedded in a medium by sensor array imaging when the array response matrix is obtained in a noisy environment. We consider additive measurement noise or a clutter noise in the multiple scattering regime. We study a detection test based on reverse-time migration of the array response matrix that is the most powerful for a given false alarm rate and compare it with a test based on the singular values of the response matrix. Moreover, we show that reflector localization should be performed with reverse-time migration rather than any other form of weighted-subspace migration and we give the standard deviation of the localization error.