Coprime Sensor Arrays Research Articles

CAWE-ACNN Algorithm for Coprime Sensor Array Adaptive Beamforming.

This paper presents a robust adaptive beamforming algorithm based on an attention convolutional neural network (ACNN) for coprime sensor arrays, named the CAWE-ACNN algorithm. In the proposed algorithm, via a spatial and channel attention unit, an ACNN model is constructed to enhance the features contributing to beamforming weight vector estimation and to improve the signal-to-interference-plus-noise ratio (SINR) performance, respectively. Then, an interference-plus-noise covariance matrix reconstruction algorithm is used to obtain an appropriate label for the proposed ACNN model. By the calculated label and the sample signals received from the coprime sensor arrays, the ACNN is well-trained and capable of accurately and efficiently outputting the beamforming weight vector. The simulation results verify that the proposed algorithm achieves excellent SINR performance and high computation efficiency.

Far-Field DOA Estimation of Uncorrelated RADAR Signals through Coprime Arrays in Low SNR Regime by Implementing Cuckoo Search Algorithm

For the purpose of attaining a high degree of freedom (DOF) for the direction of arrival (DOA) estimations in radar technology, coprime sensor arrays (CSAs) are evaluated in this paper. In addition, the global and local minima of extremely non-linear functions are investigated, aiming to improve DOF. The optimization features of the cuckoo search (CS) algorithm are utilized for DOA estimation of far-field sources in a low signal-to-noise ratio (SNR) environment. The analytical approach of the proposed CSAs, CS and global and local minima in terms of cumulative distribution function (CDF), fitness function and SNR for DOA accuracy are presented. The parameters like root mean square error (RMSE) for frequency distribution, RMSE variability analysis, estimation accuracy, RMSE for CDF, robustness against snapshots and noise and RMSE for Monte Carlo simulation runs are explored for proposed model performance estimation. In conclusion, the proposed DOA estimation in radar technology through CS and CSA achievements are contrasted with existing tools such as particle swarm optimization (PSO).

DOA Estimation in Low SNR Environment through Coprime Antenna Arrays: An Innovative Approach by Applying Flower Pollination Algorithm

The design of the modern computing paradigm of heuristics is an innovative development for parameter estimation of direction of arrival (DOA) using sparse antenna arrays. In this study, the optimization strength of the flower pollination algorithm (FPA) is exploited for the DOA estimation in a low signal to noise ratio (SNR) regime by applying coprime sensor arrays (CSA). The enhanced degree of freedom (DOF) is achieved with FPA by investigating the global minima of highly nonlinear cost function with multiple local minimas. The sparse structure of CSA demonstrates that the DOF up to O(MN) is achieved by employing M+N CSA elements, where M and N are the numbers of antenna elements used to construct the CSA. Performance analysis is conducted for estimation accuracy, robustness against noise, robustness against snapshots, frequency distribution of root mean square error (RMSE), variability analysis of RMSE, cumulative distribution function (CDF) of RMSE over Monte Carlo runs and the comparative studies of particle swarm optimization (PSO). These reveal the worth of the proposed methodology for estimating DOA.

Passive beamforming in co-prime sensor arrays using NSCB method under original pattern perturbations minimization

The co-prime sensor array (CSA), which is an effective sparse configuration, has the drawback of generating grating lobes in passive beamforming. To mitigate this problem, we propose a novel method based on null synthesis with cancellation beams (NSCBs) under original pattern perturbations minimization. In this paper we regard the grating lobes as interference signals with known directions. Firstly, the angular location of the grating lobes in the CSA subarrays are analytically extracted. Then, using the NSCBs method, the grating lobes of the subarrays nulled by adaptive weighting of the CSA elements. We have also shown that the original pattern perturbations minimization problem in mean square sense reduces to minimizing the perturbation of the array excitation. The proposed method will be formulated as a quadratic optimization problem, which can be efficiently solved using the method of Lagrange multipliers. The analytical and simulated results demonstrate that the power pattern obtained by the proposed method in the CSA, in terms of side lobe levels, peak side lobe ratio and integrated side lobe ratio, is better than the previous methods, so called product and min processors.

Detecting Gaussian Signals Using Coprime Sensor Arrays in Spatially Correlated Gaussian Noise

Coprime sensor arrays (CSAs) can estimate the directions of arrival of O(MN) narrowband planewave sources using only O(M + N) sensors with the CSA product processor. All previous investigations on the product processed CSA's performance for detecting Gaussian signals assumed spatially white Gaussian noise. Considering the product and conventional delay-and-sum beamforming processors applied to the CSA geometry, this paper derives the detection gain for each processor under the deflection metric when the background noise is spatially correlated. The conditional probability distribution functions are also derived in closed form to evaluate and compare the receiver operating characteristic performances. Despite the nonlinear processing, the product processed CSA's detection performance closely rivals its conventional beamforming (CBF) counterpart, while also demonstrating greater robustness than the CBF to spatially correlated Gaussian noise.

Nested sensor array extension factors required to match the peak sidelobe height of a uniform linear array

Nested sensor arrays (NSAs) are a type of non-uniform linear array (NULA) that reduce the total number of sensors used for a given aperture and retain the same resolution of a uniform linear array (ULA) [Pal and Vaidyanathan, 2010]. The peak sidelobe (PSL) height fails to fall below the acceptable standard of -13 dB associated with ULAs for basic NSA configurations. The basic NSA can be extended by a factor to improve the PSL height. Coprime sensor arrays, a related NULA design, have established sensor extension factors that have been found to reduce the PSL height to this -13 dB standard [Adhikari et al., 2014]. For NSAs, such an extension factor has not been established. This research finds the parameters required to extend an NSA to adequately reduce the PSLs. NSAs can significantly reduce the number of sensors required to yield a satisfactory beampattern, as low as twenty-five percent of the total aperture for some cases. Considering various array apertures, certain configurations successfully use a distinctly small aperture but require a higher percentage of those to be active. To verify our results, we construct a sixty-four sensor microphone array with which we test the optimal configurations. [Work supported by Louisiana Tech University.]

Processor dependent bias of spatial spectral estimates from coprime sensor arrays.

Coprime sensor arrays (CSAs) can estimate the directions of arrival of O(MN) narrowband plane wave sources using only O(M+N) sensors with the CSA product processor. Processing data from a finite aperture array effectively smears the true spatial power spectral density (PSD) with a kernel function determined by both the array geometry and the processing of the signals observed by the array. This paper examines the asymptotic behaviors of the kernel functions resulting from two different processors applied to a CSA sampling geometry in the limit of large aperture. The kernel functions of the product processed CSA and conventionally beamformed coprime sensor arrays (CBF CSA) are compared to the baseline of the kernel of a densely populated uniform line array (ULA) of similar aperture. At the limit of large aperture, the product processed CSA estimate is asymptotically unbiased like the ULA, while the CBF CSA estimate is not. The PSD estimates computed from the CSA processors are compared when spatially correlated Gaussian noise is an input to the array to highlight the bias issues.

N-tuple coprime sensor arrays.

Until now, coprime sensor arrays have used two sparsely spaced subarrays to emulate the performance of a single uniform array with many more sensors (generally on the order of the product of each subarrays' number of sensors). This allows for similar results with fewer sensors, or the observation of higher frequencies (above the Nyquist limit) with a similar number of sensors. The theory rests on the cross-referencing (using directional filter banks) or cancellation (using product processing) of the M grating lobes in one subarray's beampattern and N grating lobes in the other, where M and N are coprime integers. Sets of coprime integers can consist of more than two integers, however, and introducing another coprime factor theoretically multiplies observable frequency (or further decreases the number of array elements needed for the same frequency). Any amount, n, of coprime integers and corresponding subarrays may be used. In this work, "n-tuple coprime sensor array" theory is expounded and implemented. Experimentally measured beampattern results of a triple coprime sensor array (with three subarrays) are shown, using an extension of the authors' previously established product processing. Results also confirm that the usable range of an n-tuple coprime array extends below its design frequency.

Comparing the effect of aperture extension on the peak sidelobe level of sparse arrays.

The reduced number of sensors in sparse arrays create high peak sidelobe levels (PSLs). This letter compares the PSLs of minimum redundancy arrays (MRAs), minimum hole arrays (MHAs), and co-prime sensor arrays (CSAs) (conventionally beamformed and product processed) with fully populated uniform linear arrays (ULAs) as a function of aperture using both numerical simulations and experimental data. This letter finds that PSLs of MRAs, MHAs, and conventionally processed CSAs are much higher than the ULA PSL and are largely insensitive to aperture extension. In contrast, CSA product processing decreases the PSL with increasing aperture, eventually matching the ULA PSL.

Statistical characterization of coprime sensor arrays: Array gain vs. spatially correlated noise

Sensor arrays suppress spatially-uncorrelated noise through conventional beamforming (CBF). Array Gain (AG) quantifies the SNR improvement at the CBF output compared to the sensor-level SNR. The AG for a CBF Uniform Line Array (ULA) in uncorrelated noise is the number of sensors. A Coprime Sensor Array (CSA) is a sparse array geometry interleaving two undersampled ULAs, multiplying the subarray CBF outputs to achieve the same resolution as a fully populated ULA of the same aperture using fewer sensors [Vaidyanathan & Pal, 2010]. The CSA product process AG in uncorrelated noise is asymptotically equal to the number of sensors for large input SNR [Adhikari & Buck, 2015]. This research derives the AGs for CSAs and ULAs using the traditional SNR definition and deflection statistics [Cox, 1973] for a spatial first-order autoregressive process. This process introduces spatial correlation and is a simple model for turbulent flow noise over a towed array. Although the CSA AG is lower than the ULA for uncorrelated noise, the CSA's AG degrades more slowly than the ULA's AG with increasing noise correlation, due to larger spacing in the CSA subarrays. The CSA is more robust to correlated noise than the ULA AG. [Funded by NUWC & ONR.]

Estimating the spatial spectra of Gaussian processes with co-prime sensor arrays

A co-prime sensor array (CSA) is a nonuniform line array formed by interleaving two undersampled uniform line arrays. The CSA requires fewer sensors to span the same aperture as a densely sampled uniform line array (ULA), allowing the CSA to match the resolution of the ULA for direction of arrival estimation of narrowband planewaves. However, each CSA subarray suffers from aliasing, or grating lobes, due to the spatial undersampling. Vaidyanathan and Pal (2011) proved that if the subarray undersampling factors are co-prime, the aliasing can be unambiguously resolved by multiplying the spatial spectra of the subarrays. This product spatial spectra is the spatial cross-spectral density between the arrays, and is an estimate of the spatial power spectral density (PSD). In this talk, we extend the classic results of Jenkins and Watts (1968) on the periodogram PSD estimator for Gaussian processes to obtain the product processor's bias for spatially wide-sense stationary processes, and the processor's covariance for spatially white Gaussians. Additionally, we demonstrate that the CSA's product PSD estimate is not necessarily positive definite. Consequently, the CSA product spectrum may fail to detect weak signals in the presence of strong interferers. [Work supported by ONR BRC Program.]

Design of a coprime array for the North Elba sea trial

Vaidyanathan and Pal [IEEE Trans. Signal Process. 2011] proposed the use of Coprime Sensor Arrays (CSAs) to sample spatial fields using fewer elements than a Uniform Line Array (ULA) spanning the same aperture. A CSA consists of two interleaved uniform subarrays that are undersampled by coprime factors M and N. The subarrays are processed independently and then their scanned responses are multiplied to obtain a unaliased output. Although the CSA achieves resolution comparable to that of a fully populated ULA, the CSA beampattern has higher sidelobes. Adhikari et al. [Proc. ICASSP, 2013] showed that extending the subarrays and applying spatial tapers could reduce CSA sidelobes. This paper considers the problem of designing a CSA for the North Elba Sea Trial described by Gingras [SACLANT Tech. Report, 1994]. The experimental dataset consists of receptions recorded by a 48-element vertical ULA in a shallow water environment for two different source frequencies: 170 Hz and 335 Hz. This paper considers all possible coprime subsamplings for this array and selects the configuration that provides the best tradeoff between number of sensors and performance. Results are shown for both simulated and experimental data. [Work supported by ONR Basic Research Challenge Program.]

Extending coprime sensor arrays to achieve the peak side lobe height of a full uniform linear array

A coprime sensor array (CSA) is a non-uniform linear array obtained by interleaving two uniform linear arrays (ULAs) that are undersampled by coprime factors. A CSA provides the resolution of a fully populated ULA of the same aperture using fewer sensors. However, the peak side lobe level in a CSA is higher than the peak side lobe of the equivalent full ULA with the same resolution. Adding more sensors to a CSA can reduce its peak side lobe level. This paper derives analytical expressions for the number of extra sensors to be added to a CSA to guarantee that the CSA peak side lobe height is less than that of the full ULA with the same aperture. The analytical expressions are derived and compared for the uniform, Hann, Hamming, and Dolph-Chebyshev shadings.

Detection performance of coprime sensor arrays

Coprime sensor arrays (CSAs) achieve the resolution of a fully populated uniform linear array (ULA) with the same aperture using fewer sensors. The CSA's reduced number of sensors diminishes its ability to attenuate white noise, and consequently also reduces its array gain. Assuming narrowband independent signal and white noise, when the signal and the noise are modeled by circular complex zero mean Gaussian distributions, the detection statistic distribution for the ULA simplifies to an exponential distribution with its parameter dependent on the input SNR. The detection statistic for the CSA is the product of the output of one subarray with the complex conjugate of the output of the second subarray. Being the product of two independent complex Gaussian random variables, the CSA detection statistic has a distribution proportional to a modified zeroth order Bessel function of the second kind [O'Donoughue and Moura, IEEE Signal Process. (2012)]. Manipulating the ULA and CSA detection statistic distributions provides analytical expressions for probabilities of detection and false alarm and also mean discriminating information. Evaluating these expressions confirms that a CSA pays a detection gain penalty of about 5 dB over a wide range of SNRs and array sizes. [Work supported by ONR.]

Theory of Sparse Coprime Sensing in Multiple Dimensions

Coprime sampling and coprime sensor arrays have been introduced recently for the one-dimensional (1-D) case, and applications in beamforming and direction finding discussed. A pair of coprime arrays can be used to sample a wide-sense stationary signal sparsely, and then reconstruct the autocorrelation at a significantly denser set of points. All applications based on autocorrelation (e.g., spectrum and DOA estimation) benefit from this property. It was also shown in the past that coprimality can be exploited in the frequency domain by using a pair of coprime DFT filter banks, to produce the effect of a much denser tiling in the frequency domain, compared to what the two filter banks can individually achieve. This paper extends these ideas to multiple dimensions. In the 1-D case the samples or sensors lie on a pair of uniform grids, whereas in the multidimensional case, they lie on a pair of multidimensional lattices, not necessarily rectangular. This makes the developments mathematically more intricate. First several properties of coarrays of lattices are derived. It is shown how one can get dense coarrays from sparse arrays on non rectangular lattices. This requires that the lattice generating matrices M and N be commuting and coprime (to be defined). Multidimensional DFT filter banks for applications such as beamforming, with commuting coprime lattice arrays, are then described, and it is shown that a very dense tiling of the frequency plane can be obtained from the two sparse lattice arrays. A particular family of commuting coprime matrices called adjugate pairs are considered in some detail, and shown to have attractive properties. A brief review of the 1-D case is included at the beginning for convenience.