Direction Finding Using Coprime Array with Sensor Gain and Phase Errors

Acoustic signals received by platform mounted sonar arrays can be spatially processed to enhance the detection of targets in the presence of both ambient and platform generated (self) noise. Ambient noise in the ocean, such as that due to distant shipping or biological choruses, are known to be spatially correlated. The platform generated noise will be of near-field origin and may not be received by all elements in the array. In this paper we investigate the performance of the minimum variance distortionless response (MVDR) beamformer and the recently introduced Fourier integral method (FIM) and compare their performances with the conventional beamformer. Real passive sonar data, obtained from a platform mounted sparse linear array of hydrophones, is used to study the performance of the beamformers in a typical sonar environment. It is shown that in the absence of self noise, when the array is accurately calibrated the MVDR beamformer will perform very well, but when sensor gain or phase errors are present the performance of the MVDR beamformer is degraded. Further, the MVDR beamformer is unable to reject the self noise which is not "seen" by the entire array. FIM however seems to perform well and a modified version of FIM, which we call weighted FIM (WFIM), is shown to perform better and is at worst comparable to a well calibrated MVDR beamformer.

Distorted sensors could occur randomly and may lead to the breakdown of a sensor array system. We consider an array model within which a small number of sensors are distorted by unknown sensor gain and phase errors. With such an array model, the problem of joint direction-of-arrival (DOA) estimation and distorted sensor detection is formulated under the framework of low-rank and row-sparse decomposition. We derive an iteratively reweighted least squares (IRLS) algorithm to solve the resulting problem. The convergence property of the IRLS algorithm is analyzed by means of the monotonicity and boundedness of the objective function. Extensive simulations are conducted regarding parameter selection, convergence speed, computational complexity, and performances of DOA estimation as well as distorted sensor detection. Even though the IRLS algorithm is slightly worse than the alternating direction method of multipliers in detecting the distorted sensors, the results show that our approach outperforms several state-of-the-art techniques in terms of convergence speed, computational cost, and DOA estimation performance.

In this paper, we use transmit/receive co-prime arrays for direction finding of mixed coherent and uncorrelated targets based on the concept of sum coarrays. The sum coarray is defined as the set of pair-wise sums of the position vectors of the elements in the transmit and receive apertures. The data measurements from the transmit/receive co-prime array are used to emulate observations at a virtual array, whose element positions are given by the sum coarray corresponding to the co-prime array. A group sparse reconstruction problem is formulated employing multiple snapshots, where each group corresponding to a specific angle of arrival extends across all the time samples. The number of resolvable targets for the proposed sparsity-based approach is limited by the number of elements in the sum coarray. Supporting simulation results are provided, which validate the effectiveness of the proposed direction finding method.

Sparse Bayesian learning (SBL) is applied to the coprime array for underdetermined wideband direction of arrival (DOA) estimation. Using the augmented covariance matrix, the coprime array can achieve a higher number of degrees of freedom (DOFs) to resolve more sources than the number of physical sensors. The sparse-based DOA estimation can deteriorate the detection and estimation performance because the sources may be off the search grid no matter how fine the grid is. This dictionary mismatch problem can be well resolved by the SBL using fixed point updates. The SBL can automatically choose sparsity and approximately resolve the non-convex optimizaton problem. Numerical simulations are conducted to validate the effectiveness of the underdetermined wideband DOA estimation via SBL based on coprime array. It is clear that SBL can obtain good performance in detection and estimation compared to least absolute shrinkage and selection operator (LASSO), simultaneous orthogonal matching pursuit least squares (SOMP-LS) , simultaneous orthogonal matching pursuit total least squares (SOMP-TLS) and off-grid sparse Bayesian inference (OGSBI).

A coprime array is capable of achieving more degrees-of-freedom for direction-of-arrival (DOA) estimation than a uniform linear array when utilizing the same number of sensors. However, existing algorithms exploiting coprime array usually adopt predefined spatial sampling grids for optimization problem design or include spectrum peak search process for DOA estimation, resulting in the contradiction between estimation performance and computational complexity. To address this problem, we introduce the Estimation of Signal Parameters via Rotational Invariance Techniques (ESPRIT) to the coprime coarray domain, and propose a novel coarray ESPRIT-based DOA estimation algorithm to efficiently retrieve the off-grid DOAs. Specifically, the coprime coarray statistics are derived according to the received signals from a coprime array to ensure the degrees-of-freedom (DOF) superiority, where a pair of shift invariant uniform linear subarrays is extracted. The rotational invariance of the signal subspaces corresponding to the underlying subarrays is then investigated based on the coprime coarray covariance matrix, and the incorporation of ESPRIT in the coarray domain makes it feasible to formulate the closed-form solution for DOA estimation. Theoretical analyses and simulation results verify the efficiency and the effectiveness of the proposed DOA estimation algorithm.

In this letter, we address the problem of direction finding using coprime array, which is one of the most preferred sparse array configurations. Motivated by the fact that non-uniform element spacing hinders full utilization of the underlying information in the receive signals, we propose a direction-of-arrival (DoA) estimation algorithm based on low-rank reconstruction of the Toeplitz covariance matrix. The atomic-norm representation of the measurements from the interpolated virtual array is considered, and the equivalent dual-variable rank minimization problem is formulated and solved using a cyclic optimization approach. The recovered covariance matrix enables the application of conventional subspace-based spectral estimation algorithms, such as MUSIC, to achieve enhanced DoA estimation performance. The estimation performance of the proposed approach, in terms of the degrees-of-freedom and spatial resolution, is examined. We also show the superiority of the proposed method over the competitive approaches in the root-mean-square error sense.

The problem of determining the direction-of-arrival (DOA) of multiple sources impinging upon a LES array in the presence of sensor gain and phase errors is addressed. We propose a simple and fast calibration algorithm that does not require any a priori knowledge of the DOA of a signal. Numerical examples are provided to demonstrate the effectiveness of the proposed algorithm.

In this paper, we propose a unified coprime array by extending two subarrays in it from one-period to multi-period. One subarray can have arbitrary number of periods and the other has more periods than it. Therefore, more than one element of the unified structure coincide. The resulting configuration treats the nested array as well as the existing coprime structure, such as the extended one and CACIS as its special case. The comparisons of different coprime arrays are provided. The direction-of-arrival estimation performance and the coarray structures of these coprime arrays are examined through numerical simulations.

To address the challenges of limited degrees of freedom (DOF) and mutual coupling effects in sparse array-based Direction-of-arrival (DOA) estimation, this paper proposes a novel array configuration termed the Golden ratio Inspired Coprime Array (GICA). This design integrates the golden ratio (ϕ≈1.618) into the geometric arrangement of three hierarchically structured subarrays to achieve enhanced difference coarray properties. Theoretical analysis demonstrates that the proposed configuration, through strategic sensor placement and virtual domain processing, significantly increases the achievable DOF. Comprehensive simulations show that this design exhibits competitive estimation performance, achieving reduced root mean square error (RMSE) across most signal-to-noise ratio (SNR) regimes and snapshot conditions when compared with contemporary coprime array configurations. Additionally, quantitative mutual coupling analysis reveals that the proposed structure achieves superior electromagnetic compatibility, demonstrating the lowest coupling leakage coefficient among tested configurations. Experimental validation under varying coupling strengths shows that this array design maintains stable estimation performance with minimal degradation. These results confirm the proposed configuration as an effective sparse array solution that simultaneously enhances DOA estimation accuracy and mutual coupling robustness.

Gain and phase errors of the transmitter and receiver arrays will impact the amplitude and phase of the received echo signal, so that the multiple-input multiple-output (MIMO) radar system cannot obtain high-quality target images with theoretical super-resolution in practical applications. To solve this problem, a generalized Estimation of Signal Parameters using Rotational Invariance Technique (GESPRIT) imaging algorithm based on the spatial spectrum estimation theory is proposed for twodimensional (2D) super-resolution target imaging in this paper, and the azimuth statistical resolution limit (SRL) of the MIMO radar is derived. Firstly, a 2D echo signal model of MIMO radar with gain and phase errors is constructed by using geometric approximation and phase centre approximation (PCA). Then, nonlinear least square (NLS) optimization is used to reconstruct the rotation invariant properties of the transmitter and receiver arrays destroyed by gain and phase errors, and the 2D position of each target scattering center is obtained via the ESPRIT algorithm. Finally, a hypothesis test approach based on the approximate generalized likelihood ratio test (AGLRT) is proposed to resolve two closely-separated scattering centers using the first order approximation model. Based on this, the azimuth resolution performance of MIMO radar is derived and analyzed, the closed solution of resolution probability and azimuth SRL is obtained. Simulation results demonstrate the effectiveness of the proposed algorithm and the correctness of the derived SRL.

Low sidelobe levels may be a crucial factor in the success of achieving high gain using linear receiving arrays, since the level of the sidelobes determines the degree to which noisy nearby sources can be prohibited from interfering with the detection of a weak source on a different azimuth. In real world experiments at sea, one frequently operates with an array which has not only errors in amplitude and phase between channels but also dead elements. The effect of random errors in both amplitude and phase is to yield a sidelobe level of the average beam pattern which is the sum of the error-free pattern and an additional level due to the random errors. The sidelobe level due to the random errors depends upon the details of the amplitude shading function, the distribution function from which the random errors are drawn, and increases with the mean square value of the errors. Thus, given rms values of amplitude and phase errors, there is a practical limit to which the sidelobes can be reduced by amplitude shading the array. The effect of element failure on the sidelobe level can be drastic and depends on the sum of the amplitude weights of the dead elements. Thus, a failure of a highly-weighted element will have a greater effect on the sidelobe level than the failure of two or three whose combined weights are less. [Work supported by NAVMAT.]

In this paper, the problem of passive direction finding is addressed using an acoustic vector sensor array (AVS), which may be deployed either in free space or near a reflecting boundary. Building upon the \(4 \times 1\) vector field measured by an AVS, the particle-velocity coarray augmentation (PVCA) is proposed to admit the underdetermined direction finding using the spatial difference coarray derived from the vectorization of the array covariance matrix. Unlike the widely used spatial coarray Toeplitz recovery technique, the PVCA is applicable to arbitrary array geometries and imposes no reduction of the spatial difference coarray aperture. For the array located at or near a reflecting boundary, the PVCA allows resolving up to 13 sources, while for the array located in free space, the PVCA can identify 9 sources at most. By applying to the systematically designed nonuniform arrays, such as coprime arrays and nested arrays, the PVCA can be coupled with the spatial smoothing technique to get the number of resolvable sources multiplied. Finally, the effectiveness of the PVCA is verified by numerical simulations.

Self-calibrating dividers at industrial frequencies

Comprehensive error analysis of ultra-wideband direct conversion receivers

Underwater Inflatable Co-prime Sonar Array (UICSA) is a compact array assembly, which upon deployment in the ocean expands to its predetermined length and acquires sensing data. The UICSA employs an inflatable array support structure, which can be compressed into a compact package. The employed the sparse array configuration – co-prime array also reduces the number of sensors typically required for a uniform linear configuration. With the "two-way compression", storing, handling, and transporting the compactly designed UICSA system is convenient, particularly for the autonomous vehicles with limited payload space. In this paper, we propose four designs of UICSA based on different support structure constructions. We fabricated the prototypes and deployed them in an acoustic test tank. The experimental data were analyzed to evaluate the system performance. The results validate the capability of the UICSA for direction finding and also reveal that the underwater inflatable structure as a platform can have broad prospects in the ocean-related applications.