Efficient Aperture Fill Time Correction for Wideband Sparse Array Using Improved Variable Fractional Delay Filters.

To solve the problem of aperture fill time (AFT) for wideband sparse arrays, variable fractional delay (VFD) FIR filters are applied to eliminate linear coupling between spatial and time domains. However, the large dimensions of the filter coefficient matrix result in high system complexity. To alleviate the computational burden of solving VFD filter coefficients, a novel multi-regultion minimax (MRMM) model utilizing the sparse representation technique has been presented. The error function is constrained by the introduction of L2-norm and L1-norm regularizations within the minimax criterion. The L2-norm effectively resolves the problems of overfitting and non-unique solutions that arise in the sparse optimization of traditional minimax (MM) models. Meanwhile, the use of multiple L1-norms enables the optimal design of the smallest sub-filter number and order of the VFD filter. To solve the established nonconvex model, an improved sequential-alternating direction method of multipliers (S-ADMM) algorithm for filter coefficients is proposed, which utilizes sequential alternation to iteratively update multiple soft-thresholding problems. The experimental results show that the optimized VFD filter reduces system complexity significantly and corrects AFT effectively in a wideband sparse array.

Optimal Design of Sparse Matrix Phased Array Using Simulated Annealing for Volumetric Ultrasonic Imaging with Total Focusing Method.

The total focusing method (TFM) is often considered to be the 'gold standard' for ultrasonic imaging in the field of nondestructive testing. The use of matrix phased arrays as probes allows for high-resolution volumetric TFM imaging. Conventional TFM imaging involves the use of full matrix capture (FMC) for ultrasonic signals acquisition, but in the case of a matrix phased array, this approach is associated with a huge volume of data to be acquired and processed. This severely limits the frame rate of volumetric imaging with 2D probes and necessitates the use of high-end equipment. Thus, the aim of this research was to develop a novel design method for determining the optimal sparse 2D probe configuration for specific conditions of ultrasonic imaging. The developed approach is based on simulated annealing and involves implementing the solution of the sparse matrix phased array layout optimization problem. In order to implement simulated annealing for the aforementioned task, its parameters were set, the acceptance function was introduced, and the approaches were proposed to compute beam directivity diagrams of sparse matrix phased arrays in TFM imaging. Experimental studies have shown that the proposed approach provides high-quality volumetric imaging with a decrease in data volume of up to 84% compared to that obtained using the FMC data acquisition method.

Direction of Arrival Estimation of Coherent Wideband Sources Using Nested Array.

Due to their ability to achieve higher DOA estimation accuracy and larger degrees of freedom (DOF) using a fixed number of antennas, sparse arrays, etc., nested and coprime arrays have attracted a lot of attention in relation to research into direction of arrival (DOA) estimation. However, the usage of the sparse array is based on the assumption that the signals are independent of each other, which is hard to guarantee in practice due to the complex propagation environment. To address the challenge of sparse arrays struggling to handle coherent wideband signals, we propose the following method. Firstly, we exploit the coherent signal subspace method (CSSM) to focus the wideband signals on the reference frequency and assist in the decorrelation process, which can be implemented without any pre-estimations. Then, we virtualize the covariance matrix of sparse array due to the decorrelation operation. Next, an enhanced spatial smoothing algorithm is applied to make full use of the information available in the data covariance matrix, as well as to improve the decorrelation effect, after which stage the multiple signal classification (MUSIC) algorithm is used to obtain DOA estimations. In the simulation, with reference to the root mean square error (RMSE) that varies in tandem with the signal-to-noise ratio (SNR), the algorithm achieves satisfactory results compared to other state-of-the-art algorithms, including sparse arrays using the traditional incoherent signal subspace method (ISSM), the coherent signal subspace method (CSSM), spatial smoothing algorithms, etc. Furthermore, the proposed method is also validated via real data tests, and the error value is only 0.2 degrees in real data tests, which is lower than those of the other methods in real data tests.

Colocated MIMO Radar Waveform-Array Joint Optimization for Sparse Array.

Colocated multiple-input multiple-output (MIMO) radar can transmit a group of distinct waveforms via its colocated transmit antennas and the waveform diversity leads to several advantages in contrast to conventional phased-array radar. The performance depends highly on the degrees available, and element spacing can be deemed as another source of degrees of freedom. In this paper, we study the joint waveform and element spacing optimization problem. A joint waveform and array optimization criterion is proposed to match the transmit beampattern, the suppression range, and the angular sidelobes, under the constraints of minimal element spacing and total array aperture. Meanwhile, the effect of receive beamforming on suppressing mutual correlation between returns from different spatial directions is also incorporated into the optimization criterion. The optimization problem is solved by the sequential quadratic programming algorithm. Numerical results indicate that with more degrees of freedom from array spacings, colocated MIMO radar achieves a better transmit beampattern matching performance and a lower sidelobe level, compared with a fixed half-wavelength spaced array, but the benefits from additional degrees of freedom from array spacing optimization have a limit.

A Sparse-Array Design Method Using Q Uniform Linear Arrays for Direction-of-Arrival Estimation.

Nowadays, sparse arrays have been a hotspot for research in the direction of arrival (DOA). In order to achieve a big value for degrees of freedom (DOFs) using spatial smoothing methods, researchers try to use multiple uniform linear arrays (ULAs) to construct sparse arrays. But, with the number of subarrays increasing, the complexity also increases. Hence, in this paper, a design method, named as the cross-coarray consecutive-connected (4C) criterion, and the sparse array using Q ULAs (SA-UQ) are proposed. We first analyze the virtual sensor distribution of SA-U2 and extend the conclusions to SA-UQ, which is the 4C criterion. Then, we give an algorithm to solve the displacement between subarrays under the given Q ULAs. At last, we consider a special case, SA-U3. Through the analysis of DOFs, SA-UQ can find underdetermined signals. Moreover, SA-U3 can obtain DOFs close to other sparse arrays using three ULAs. The simulation experiments prove the performance of SA-UQ.

Research on Underdetermined DOA Estimation Method with Unknown Number of Sources Based on Improved CNN.

This paper proposes a joint estimation method for source number and DOA based on an improved convolutional neural network for unknown source number and undetermined DOA estimation. By analyzing the signal model, the paper designs a convolutional neural network model based on the existence of a mapping relationship between the covariance matrix and both the source number and DOA estimation. The model, which discards the pooling layer to avoid data loss and introduces the dropout method to improve generalization, takes the signal covariance matrix as input and the two branches of source number estimation and DOA estimation as outputs, and achieves the unfixed number of DOA estimation by filling in invalid values. Simulation experiments and analysis of the results show that the algorithm can effectively achieve the joint estimation of source number and DOA. Under the conditions of high SNR and a large snapshot number, both the proposed algorithm and the traditional algorithm have high estimation accuracy, while under the conditions of low SNR and a small snapshot, the algorithm is better than the traditional algorithm, and under the underdetermined conditions, where the traditional algorithm often fails, the algorithm can still achieve the joint estimation.

Direction of Arrival Estimation of Generalized Nested Array via Difference-Sum Co-Array.

To address the weakness that the difference co-array (DCA) only enhances the degrees of freedom (DOFs) to a limited extent, a new configuration called the generalized nested array via difference-sum co-array (GNA-DSCA) is proposed for direction of arrival (DOA) estimation. We consider both the temporal and spatial information of the array output to construct the DSCA model, based on which the DCA and sum co-array (SCA) of the GNA are systematically analyzed. The closed-form expression of the DOFs for the GNA-DSCA is derived under the determined dilation factors. The optimal results show that the GNA-DSCA has a more flexible configuration and more DOFs than the GNA-DCA. Moreover, the larger dilation factors yield significantly wider virtual aperture, which indicates that it is more attractive than the reported DSCA-based sparse arrays. Finally, a hole-filling strategy based on atomic norm minimization (ANM) is utilized to overcome the degradation of the estimation performance due to the non-uniform virtual array, thus achieving accurate DOA estimation. The simulation results verify the superiority of the proposed configuration in terms of virtual array properties and estimation performance.

Hole-Free Nested Array with Three Sub-ULAs for Direction of Arrival Estimation.

Sparse arrays are of deep concern due to their ability to identify more sources than the number of sensors, among which the hole-free difference co-array (DCA) with large degrees of freedom (DOFs) is a topic worth discussing. In this paper, we propose a novel hole-free nested array with three sub-uniform line arrays (NA-TS). The one-dimensional (1D) and two-dimensional (2D) representations demonstrate the detailed configuration of NA-TS, which indicates that both nested array (NA) and improved nested array (INA) are special cases of NA-TS. We subsequently derive the closed-form expressions for the optimal configuration and the available number of DOFs, concluding that the DOFs of NA-TS is a function of the number of sensors and the number of the third sub-ULA. The NA-TS possesses more DOFs than several previously proposed hole-free nested arrays. Finally, the superior direction of arrival (DOA) estimation performance based on the NA-TS is supported by numerical examples.

Transmit-Receive Sparse Synthesis of Linear Frequency Diverse Array in Range-Angle Space Using Genetic Algorithm.

Unlike conventional phased array (PA), frequency diversity array (FDA) can perform the beampattern synthesis not only in an angle dimension but also in a range dimension by introducing an additional frequency offset (FO) across the array aperture, thus greatly enhancing the beamforming flexibility of an array antenna. Nevertheless, an FDA with uniform inter-element spacing that consists of a huge number of elements is required when a high resolution is needed, which results in a high cost. To substantially reduce the cost while almost maintaining the antenna resolution, it is important to conduct a sparse synthesis of FDA. Under these circumstances, this paper investigated the transmit-receive beamforming of a sparse-fda in range and angle dimensions. In particular, the joint transmit-receive signal formula was first derived and analyzed to resolve the inherent time-varying characteristics of FDA based on a cost-effective signal processing diagram. In the sequel, the GA-based low sidelobe level (SLL) transmit-receive beamforming of the sparse-fda was proposed to generate a focused main lobe in a range-angle space, where the array element positions were incorporated into the optimization problem. Numerical results showed that 50% of the elements can be saved for the two linear FDAs with sinusoidally and logarithmically varying frequency offsets, respectively termed as sin-FO linear-FDA and log-FO linear-FDA, with only a less than 1 dB increment in SLL. The resultant SLLs are below -9.6 dB, and -12.9 dB for these two linear FDAs, respectively.

An L-Shaped Three-Level and Single Common Element Sparse Sensor Array for 2-D DOA Estimation.

The degree of freedom (DOF) is an important performance metric for evaluating the design of a sparse array structure. Designing novel sparse arrays with higher degrees of freedom, while ensuring that the array structure can be mathematically represented, is a crucial research direction in the field of direction of arrival (DOA) estimation. In this paper, we propose a novel L-shaped sparse sensor array by adjusting the physical placement of the sensors in the sparse array. The proposed L-shaped sparse array consists of two sets of three-level and single-element sparse arrays (TSESAs), which estimate the azimuth and elevation angles, respectively, through one-dimensional (1-D) spatial spectrum search. Each TSESA is composed of a uniform linear subarray and two sparse subarrays, with one single common element in the two sparse subarrays. Compared to existing L-shaped sparse arrays, the proposed array achieves higher degrees of freedom, up to 4Q1Q2+8Q1-5, when estimating DOA using the received signal covariance. To facilitate the correct matching of azimuth and elevation angles, the cross-covariance between the two TSESA arrays is utilized for estimation. By comparing and analyzing performance parameters with commonly used L-shaped and other sparse arrays, it is found that the proposed L-shaped TSESA has higher degrees of freedom and array aperture, leading to improved two-dimensional (2-D) DOA estimation results. Finally, simulation experiments validate the excellent performance of the L-shaped TSESA in 2-D DOA estimation.

Efficient Beampattern Synthesis for Sparse Frequency Diverse Array via Matrix Pencil Method.

Due to the introduction of frequency offsets, the pattern synthesis problem of sparse Frequency diverse array (FDA) becomes more complicated than that of the phased array. A typical way to solve this problem is to use a global optimization algorithm, but this is usually time-consuming. In this paper, we propose an efficient non-iterative beampattern synthesis approach for sparse FDA. For a given reference pattern, which can be generated by other synthesis methods, we first sample it uniformly and construct the Hankel matrix with the sampled data. By low-rank processing, a low-rank approximation version of the Hankel matrix can then be obtained. Finally, the matrix enhancement and matrix pencil (MEMP) and matrix pencil (MP) methods are applied to estimate the antenna positions, frequency offsets, and excitations of the obtained array from the approximated matrix. Besides this, two typical FDA frameworks including multi-carrier FDA (MCFDA) and standard FDA (SFDA) are considered. Numerical simulation results prove that the proposed method outperforms the existing methods in terms of synthesis error, average runtime, and percentage of saving elements.

"Conical" Frustum Multi-Beam Phased Arrays for Air Traffic Control Radars.

The design of conical frustum phased array antennas for air traffic control (ATC) radar systems is addressed. The array architecture, which is controlled by a fully digital beam-forming (DBF) network, is composed by a set of equal vertical modules. Each module consists of a linear sparse array that generates on receive multiple instantaneous beams pointing along different directions in elevation. To reach the best trade-off between the antenna complexity (i.e., minimum number of array elements and/or radio frequency components) and radiation performance (i.e., matching a set of reference patterns), the synthesis problem is formulated in the Compressive Sampling (CS) framework. Then, the positions of the array elements and the complex excitations for generating each single beam are jointly determined through a customized version of the Bayesian CS (BCS) tool. Representative numerical results, concerned with ideal as well as real antenna models, are reported both to validate the proposed design strategy and to assess the effectiveness of the synthesized modular sparse array architecture also in comparison with conventional arrays with uniformly-spaced elements.

Design of Ultrasonic Synthetic Aperture Imaging Systems Based on a Non-Grid 2D Sparse Array.

This work provides a guide to design ultrasonic synthetic aperture systems for non-grid two-dimensional sparse arrays such as spirals or annular segmented arrays. It presents an algorithm that identifies which elements have a more significant impact on the beampattern characteristics and uses this information to reduce the number of signals, the number of emitters and the number of parallel receiver channels involved in the beamforming process. Consequently, we can optimise the 3D synthetic aperture ultrasonic imaging system for a specific sparse array, reducing the computational cost, the hardware requirements and the system complexity. Simulations using a Fermat spiral array and experimental data based on an annular segmented array with 64 elements are used to assess this algorithm.

Low Discrepancy Sparse Phased Array Antennas.

Sparse arrays have grating lobes in the far field pattern due to the large spacing of elements residing in a rectangular or triangular grid. Random element spacing removes the grating lobes but produces large variations in element density across the aperture. In fact, some areas are so dense that the elements overlap. This paper introduces a low discrepancy sequence (LDS) for generating the element locations in sparse planar arrays without grating lobes. This nonrandom alternative finds an element layout that reduces the grating lobes while keeping the elements far enough apart for practical construction. Our studies consider uniform sparse LDS arrays with 86% less elements than a fully populated array, and numerical results are presented that show these sampling techniques are capable of completely removing the grating lobes of sparse arrays. We present the mathematical formulation for implementing an LDS generated element lattice for sparse planar arrays, and present numerical results on their performance. Multiple array configurations are studied, and we show that these LDS techniques are not impacted by the type/shape of the planar array. Moreover, in comparison between the LDS techniques, we show that the Poisson disk sampling technique outperforms all other approaches and is the recommended LDS technique for sparse arrays.

Subsampling Approaches for Compressed Sensing with Ultrasound Arrays in Non-Destructive Testing.

Full Matrix Capture is a multi-channel data acquisition method which enables flexible, high resolution imaging using ultrasound arrays. However, the measurement time and data volume are increased considerably. Both of these costs can be circumvented via compressed sensing, which exploits prior knowledge of the underlying model and its sparsity to reduce the amount of data needed to produce a high resolution image. In order to design compression matrices that are physically realizable without sophisticated hardware constraints, structured subsampling patterns are designed and evaluated in this work. The design is based on the analysis of the Cramér-Rao Bound of a single scatterer in a homogeneous, isotropic medium. A numerical comparison of the point spread functions obtained with different compression matrices and the Fast Iterative Shrinkage/Thresholding Algorithm shows that the best performance is achieved when each transmit event can use a different subset of receiving elements and each receiving element uses a different section of the echo signal spectrum. Such a design has the advantage of outperforming other structured patterns to the extent that suboptimal selection matrices provide a good performance and can be efficiently computed with greedy approaches.

Design of 2D Sparse Array Transducers for Anomaly Detection in Medical Phantoms.

Aperiodic sparse 2D ultrasonic array configurations, including random array, log spiral array, and sunflower array, have been considered for their potential as conformable transducers able to image within a focal range of 30-80 mm, at an operating frequency of 2 MHz. Optimisation of the imaging performance of potential array patterns has been undertaken based on their simulated far field directivity functions. Two evaluation criteria, peak sidelobe level (PSL) and integrated sidelobe ratio (ISLR), are used to access the performance of each array configuration. Subsequently, a log spiral array pattern with -19.33 dB PSL and 2.71 dB ISLR has been selected as the overall optimal design. Two prototype transducers with the selected log spiral array pattern have been fabricated and characterised, one using a fibre composite element composite array transducer (CECAT) structure, the other using a conventional 1-3 composite (C1-3) structure. The CECAT device demonstrates improved coupling coefficient (0.64 to 0.59), reduced mechanical cross-talk between neighbouring array elements (by 10 dB) and improved operational bandwidth (by 16.5%), while the C1-3 device performs better in terms of sensitivity (~50%). Image processing algorithms, such as Hough transform and morphological opening, have been implemented to automatically detect and dimension particles located within a fluid-filled tube structure, in a variety of experimental scenarios, including bespoke phantoms using tissue mimicking material. Experiments using the fabricated CECAT log spiral 2D array transducer demonstrated that this algorithmic approach was able to detect the walls of the tube structure and stationary anomalies within the tube with a precision of ~0.1 mm.

A Priori-Based Subarray Selection Algorithm for DOA Estimation.

A finer direction-of-arrival (DOA) estimation result needs a large and dense array; it may, however, encounter the mutual coupling effect, which degrades the performance of DOA estimation. There is a new approach to mitigating this effect by using a nonuniform array to achieve DOA estimation. In this paper, we consider a priori DOA estimation, which is easily obtained from tracking results. The a priori DOA requires us to pay close attention to the high possibility of where the DOA will appear; then, a weight according to the prior probability distribution of DOA is added to each direction, which leads the sensing matrix of DOA estimation to be near low-rank. Thus, according to the low-rank matrix approximation theory, an optimal low-rank approximate matrix is obtained and an algorithm is proposed to select the elements of the original array according to right singular vectors of the approximate matrix. After that, the impacts of different weights are analyzed, and a mixed weight is presented which has flexibility for common use. Finally, a number of numerical simulations are carried out, and the results verify the effectiveness of the proposed methods.

An MIMO Radar System Based on the Sparse-Array and Its Frequency Migration Calibration Method.

In this paper, a Multiple Input Multiple Output (MIMO) radar system based on a sparse-array is proposed. In order to reduce the side-lobe level, a genetic algorithm (GA) is used to optimize the array arrangement. To reduce the complexity of the system, time-division multiplexing (TDM) technology is adopted. Since the signals are received in different periods, a frequency migration will emerge if the target is in motion, which will lead to the lower direction-of-arrival (DOA) performance of the system. To solve this problem, a stretching transformation method in the fast-frequency slow-time domain is proposed, in order to eliminate frequency migration. Only minor adjustments need to be implemented for the signal processing, and the root-mean-square error (RMSE) of the DOA estimation will be reduced by about 90%, compared with the one of an uncalibrated system. For example, a uniform linear array (ULA) MIMO system with 2 transmitters and 20 receivers can be replaced by the proposed system with 2 transmitters and 12 receivers, achieving the same DOA performance. The calibration formulations are given, and the simulation results of the automotive radar system are also provided, which validate the theory.

Generalized L-Shaped Nested Array Concept Based on the Fourth-Order Difference Co-Array.

In this paper, a generalized L-shaped nested array based on the fourth-order difference co-array is proposed for two-dimensional (2D) directions' estimation. The new structure framework makes full use of the physical sensor locations to form a virtual uniform rectangular array (URA) as large as possible. As it utilizes the fourth-order difference instead of the traditional second-order difference result, this structure framework can acquire a much higher degree-of-freedom (DOF) than the existing 2D sparse arrays. The proposed structures have two advantages. One is that the subarrays can be chosen as any nested-class arrays, which makes the sparse array design more flexible. We can choose arbitrary subarray structures for DOF enhancement purposes. Another advantage is that the relative position of two subarrays can be set as any integral multiple of half wavelength. This means that two subarrays can be located as far as possible so that the relative influence between two physical subarrays can be ignored. The DOFs of several typical generalized L-shaped nested arrays (GLNAs) are compared in this paper. By setting the subarrays as different types and the relative position as a special value, a special GLNA is presented. Simulations show that GLNAs have obvious superiority in 2D direction-of-arrival estimation.

Ultrasonic Phased Array Sparse-TFM Imaging Based on Sparse Array Optimization and New Edge-Directed Interpolation.

The ultrasonic phased array total focusing method (TFM) has the advantages of full-range dynamic focusing and high imaging resolution, but the problem of long imaging time limits its practically industrial applications. To reduce the imaging calculation demand of TFM, the locations of active array elements in the sparse array are optimized by combining almost different sets with the genetic algorithm (ADSGA), and corrected based on the consistency of the effective aperture with the equivalent point diffusion function. At the same time, to further increase the imaging efficiency, a sparse-TFM image with lower resolution is obtained by reducing the number of focus points and then interpolated by the new edge-directed interpolation algorithm (NEDI) to obtain a high quality sparse-TFM image. Compared with TFM, the experimental results show that the quantitative accuracy of the proposed method is only decreased by 1.09% when the number of sparse transmitting elements reaches 8 for a 32-element transducer, and the imaging speed is improved by about 16 times with the same final pixel resolution.