Coprime Sampling Research Articles

A Novel Fast Iterative STAP Method with a Coprime Sampling Structure.

In space-time adaptive processing (STAP), the coprime sampling structure can obtain better clutter suppression capabilities at a lower hardware cost than the uniform linear sampling structure. However, in practical applications, the performance of the algorithm is often limited by the number of training samples. To solve this problem, this paper proposes a fast iterative coprime STAP algorithm based on truncated kernel norm minimization (TKNM). This method establishes a virtual clutter covariance matrix (CCM), introduces truncated kernel norm regularization technology to ensure the low rank of the CCM, and transforms the non-convex problem into a convex optimization problem. Finally, a fast iterative solution method based on the alternating direction method is presented. The effectiveness and accuracy of the proposed algorithm are verified through simulation experiments.

A Fast Power Spectrum Sensing Solution for Generalized Coprime Sampling

With the growing scarcity of spectrum resources, wideband spectrum sensing is necessary to process a large volume of data at a high sampling rate. For some applications, only second-order statistics are required for spectrum estimation. In this case, a fast power spectrum sensing solution is proposed based on the generalized coprime sampling. The solution involves the inherent structure of the sensing vector to reconstruct the autocorrelation sequence of inputs from sub-Nyquist samples, which requires only parallel Fourier transform and simple multiplication operations. Thus, it takes less time than the state-of-the-art methods while maintaining the same performance, and it achieves higher performance than the existing methods within the same execution time without the need to pre-estimate the number of inputs. Furthermore, the influence of the model mismatch has only a minor impact on the estimation performance, allowing for more efficient use of the spectrum resource in a distributed swarm scenario. Simulation results demonstrate the low complexity in sampling and computation, thus making it a more practical solution for real-time and distributed wideband spectrum sensing applications.

Time-Delay Estimation of Ground-Penetrating Radar Using Co-Prime Sampling Strategy via Atomic Norm Minimization

Time-Delay Estimation of Ground-Penetrating Radar Using Co-Prime Sampling Strategy via Atomic Norm Minimization

Fast DOA Estimation Algorithms via Positive Incremental Modified Cholesky Decomposition for Augmented Coprime Array Sensors.

This paper proposes a fast direction of arrival (DOA) estimation method based on positive incremental modified Cholesky decomposition atomic norm minimization (PI-CANM) for augmented coprime array sensors. The approach incorporates coprime sampling on the augmented array to generate a non-uniform, discontinuous virtual array. It then utilizes interpolation to convert this into a uniform, continuous virtual array. Based on this, the problem of DOA estimation is equivalently formulated as a gridless optimization problem, which is solved via atomic norm minimization to reconstruct a Hermitian Toeplitz covariance matrix. Furthermore, by positive incremental modified Cholesky decomposition, the covariance matrix is transformed from positive semi-definite to positive definite, which simplifies the constraint of optimization problem and reduces the complexity of the solution. Finally, the Multiple Signal Classification method is utilized to carry out statistical signal processing on the reconstructed covariance matrix, yielding initial DOA angle estimates. Experimental outcomes highlight that the PI-CANM algorithm surpasses other algorithms in estimation accuracy, demonstrating stability in difficult circumstances such as low signal-to-noise ratios and limited snapshots. Additionally, it boasts an impressive computational speed. This method enhances both the accuracy and computational efficiency of DOA estimation, showing potential for broad applicability.

Generalized Spatial-Temporal Coprime Sampling for Joint DOA and Doppler Estimation

Joint direction of arrival (DOA) and Doppler frequency estimation plays an important role in radar and wireless communication systems. However, the number of antennas and snapshots could be limited in real applications, so that it is necessary to design the array structure and sampling method to improve the estimation performance. We utilize coprime array and coprime sampler at the spatial and temporal domains under a constrained number of total snapshots. We propose to allow different temporal sampling parameters for the coprime samplers at different antennas, called STCS (Spatial-Temporal Coprime Sampling with possibly different temporal sampling parameters). The spatial coprime array and temporal coprime sampler parameters are optimized to maximize the degrees of freedom (DOFs). Based on the optimized coprime array and samplers, compressed sensing theory is used to jointly estimate the DOA and Doppler frequency. Numerical examples are presented to illustrate that the proposed STCS can provide the improved DOF and better estimation performance.

On the Foundation of Sparsity Constrained Sensing— Part I: Sampling Theory and Robust Remainder Problem

In the first part of the series papers, we set out to answer the following fundamental question: for constrained sampling, what kind of signal can be uniquely represented or recovered by the (distributed) discrete sample sequence(s) obtained? We term this study as <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">sparsity constrained sensing</i> or <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">sparse sensing</i> . It is different from <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">compressed sensing</i> , which exploits the sparse representation of a signal to reduce sample complexity (compressed sampling or acquisition). We use <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">sparsity constrained sensing</i> to denote a class of methods which are devoted to improving the efficiency and reducing the cost of sampling implementation itself. The “sparsity” here is referred to as sampling at a low temporal or spatial rate, which captures applications of cheaper hardware such as of lower power, less memory and throughput to implement sampling. We take frequency and direction of arrival (DoA) estimation as concrete examples and give the necessary and sufficient conditions of the sampling strategy. Interestingly, we prove that these problems can be reduced to some (multiple) remainder model, where it is equivalent to studying the residue representation of a signal. As a straightforward corollary, we supplement and complete the theory of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">co-prime sampling</i> , which receives considerable attention over last decade. Our results also connect the two classic parameter estimation frameworks, the Chinese Remainder Theorem (CRT) method and the co-prime sensing, with a unified interpretation. On the other aspect, we advance the understanding of the robust remainder problem, which models the case when sampling with noise. A sharpened tradeoff between the parameter dynamic range and the error bound is derived. We prove that, for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$N$</tex-math></inline-formula> -frequency estimation in either complex or real waveforms, once the least common multiple (lcm) of the sampling rates selected is sufficiently large, a constant error tolerance bound independent of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$N$</tex-math></inline-formula> is approached.

A Novel Sparse Framework for Angle and Frequency Estimation.

The topic of joint angle and frequency estimation (JAFE) has aroused extensive interests in the past decades. Current estimation algorithms mainly rely on the Nyquist sampling criterion. In order not to cause ambiguity for parameter estimation, the space-time intervals must be smaller than given thresholds, which results in complicated hardware costs and a huge computational burden. This paper aims to reduce the complexity for JAFE, and a novel sparsity-aware framework is proposed. Unlike the current uniform sampling architectures, the incoming narrow-band singles are sampled by a series of space-time coprime samplers. An improved rotational invariance estimator is introduced, which offers closed-form solutions for both angle and frequency estimation. The mathematical treatments indicate that our methodology is inherent in larger spatial/temporal aperture than the uniform sampling architectures; hence, it provides more accurate JAFE compared to alternative approaches relying on uniform sampling. Additionally, it attains nearly the same complexity as the current rotational invariance approach. Numerical results agree with the theoretical advantages of our methodology.

Spectrum sensing based on delayed coprime sampling

The interference environment faced by Ultra-wideband (UWB) cognitive radar has the characteristics of wide bandwidth and extensive dynamic range. For that, a higher sampling rate is required for spectrum sensing. The current coprime sampling scheme can significantly reduce the sampling rate and show a high degree of freedom. However, the virtual co-array generated by coprime sampling has certain missing elements, leading to errors and the lack of spectral estimation information. This paper proposes a delayed coprime sampling scheme to supplement missing elements in virtual co-arrays to increase degrees of freedom. The regularity of the holes and the selection of suitable delay factors are also presented. The simulation results show that the proposed method can further improve the degree of freedom and theoretically achieve the maximum degree of freedom of a fixed sample. The precision of the spectrum estimation is also improved since the data generated by the proposed method can be used to estimate the existing elements in the virtual co-array.

Achieving full grating-lobe-free field of view with low-complexity co-prime photonic beamforming transceivers

Integrated photonic active beamforming can significantly reduce the size and cost of coherent imagers for LiDAR and medical imaging applications. In current architectures, the complexity of photonic and electronic circuitry linearly increases with the desired imaging resolution. We propose a novel photonic transceiver architecture based on co-prime sampling techniques that breaks this trade-off and achieves the full (radiating-element-limited) field of view (FOV) for a 2D aperture with a single-frequency laser. Using only order-of- N radiating elements, this architecture achieves beamwidth and sidelobe level (SLL) performance equivalent to a transceiver with order-of- N 2 elements with half-wavelength spacing. Furthermore, we incorporate a pulse amplitude modulation (PAM) row–column drive methodology to reduce the number of required electrical drivers for this architecture from order of N to order of N . A silicon photonics implementation of this architecture using two 64-element apertures, one for transmitting and one for receiving, requires only 34 PAM electrical drivers and achieves a transceiver SLL of − 11.3 dB with 1026 total resolvable spots, and 0.6° beamwidth within a 23 ° × 16.3 ° FOV.

Fast and Accurate Spectrum Estimation via Virtual Coarray Interpolation Based on Truncated Nuclear Norm Regularization

Compared with the Nyquist sampling scheme, the coprime sampling scheme reduces the requirement of sampling rate significantly and simultaneously shows high degrees of freedom. Actually, the information received by the sampler is not fully utilized due to the non-uniformity of derived second-order virtual coarray. In addition, the virtual signals corresponding to the non-uniform virtual coarray may be contaminated by noise, which could affect the accuracy of spectrum estimation. In this letter, we propose a novel virtual coarray interpolation method for the inexact Toeplitz covariance matrix based on truncated nuclear norm regularization. The missing elements and exact covariance matrix can be obtained simultaneously by solving the covariance matrix completion problem that contains the error term and non-Toeplitz errors propagation control term. Simulation results demonstrate the superiorities of the proposed spectrum estimation method in terms of accuracy and computational complexity.

Coprime Sensing for Airborne Array Interferometric SAR Tomography

In airborne array interferometric SAR (Array-InSAR) tomography, the measurements acquired by conventional uniform sampling array are always restricted by the number of physical baseline elements and the size of baseline aperture. It is desirable to capture new acquisitions and enlarge the aperture with virtual signal processing instead of actually adding array baselines. For this motivation, we utilize the disparity of a pair of coprime sampling sub-arrays to enlarge the baseline aperture and construct new observations virtually. The generation of virtual measurements is equal to estimating cross-correlation matrices in real SAR data. Due to the spatial target variation, we adopted an adaptive filtering method to estimate the cross-correlation matrix. We call the above-mentioned processing of generating virtual measurements as a <i>coprime sensing technique</i>. The newly generated virtual measurements have more degrees of freedom, a larger baseline aperture, and a higher signal-to-noise ratio (SNR) than the physical measurements. These advantages offer the possibility to obtain competitive three-dimensional (3-D) imaging results without increasing the hardware cost of the Array-InSAR. We demonstrate the effectiveness of the proposed method by the coprime acquisitions selected from AIRCAS Array-InSAR data.

SADOE: Sequential‐based angle‐Doppler off‐grid estimation with coprime sampling structures for space‐time adaptive processing

SADOE: Sequential‐based angle‐Doppler off‐grid estimation with coprime sampling structures for space‐time adaptive processing

STAP for Airborne Radar Based on Dual-Frequency Space-Time Coprime Sampling Structure

To overcome the space-time adaptive processing (STAP) performance loss caused by discarding the non-uniform parts of difference coarray and copulse for coprime sampling structure under the condition of single-frequency operation, this paper proposes a new STAP algorithm to improve the precision of filter weight vector estimation by the dual-frequency operation. By selecting a single proper additional operation frequency, we can obtain the different coarray and copulse, thus fill two missed virtual sensors and pulses of the difference structures in the single-frequency operation simultaneously. Compared with the single-frequency operation, the dual-frequency operation method can acquire the bigger uniform linear array (ULA) and coherent processing interval (CPI) pulse train to improve the system degrees of freedom (DOF), and result in higher angle-Doppler resolution. In addition, the coprime array has the little mutual coupling effect because of the larger inter-sensors spacing. Therefore, the resulting method can alleviate mutual coupling and enhance the system DOF.

Multi-rate coprime sampling for frequency estimation with increased degrees of freedom

For frequency estimation, the recently proposed coprime sampling scheme receives increasing interest as it reduces sampling rate and exhibits high degrees of freedom. However, the virtual coarray generated by coprime configuration is a linear virtual array structure with some missing elements. This leads to information loss if only the contiguous part of the virtual coarray is used. In this paper, we propose a new approach to fix this problem. A multi-rate coprime sampling mechanism is designed to fill all the holes in the classical coprime virtual coarray. This is achieved by constructing new virtual coarrays containing the hole elements which can be selected to fill all the holes in the original virtual coarray. Furthermore, by properly setting the multi-rate coefficients to positive integers, our approach allows to reuse some samples obtained with the classical coprime sampling configuration. The closed-form expression of the positions of the holes is also given, which can be used to choose the appropriate multi-rate coefficients. Simulation results show that the proposed approach can increase the degrees of freedom without requiring additional samples. The estimation accuracy is also improved because our proposed approach fully exploits the information from the available samples.

Segmented discrete polynomial‐phase transform with coprime sampling

Segmented discrete polynomial-phase transform (DPT) integrates input signals to enable detection and parameter estimation of weak linear frequency modulated (LFM) signals. However, conventional DPT approaches suffer from a low unambiguously detectable range of the chirp rates because the segmentation effectively reduces the sampling rate between adjacent segments. To enable unique detection of LFM signals with a high chirp rate estimation, the authors propose the use of multiple segmentation sets where the respective segment lengths are governed by a coprime relationship. As such, the ambiguity of the estimated chirp rates that arise from a single segmentation set is eliminated through the fusion of the multiple segmentation sets using the Chinese Remainder Theorem. The effectiveness of the proposed method for the estimation of LFM signals with a high chirp rate is validated by simulation results.

GHz sampling hardware implementation with sub-Nyquist coprime sampling rates.

A sub-Nyquist coprime sampling system for sparse signals is implemented in this article. The proposed system is composed of coprime sampling hardware and a multicoset signal reconstruction algorithm. A pair of uniform samplers is utilized in the hardware to sample a wideband spare analog signal with an uncertain difference in start times. A time difference acquisition module embedded into a field-programmable gate array and a pulse-expanding circuit are then used to measure the difference in start times. Owing to the different frequencies of the two samplers, the coprime sample sets obtained are nonuniform. Before they are used as input to the multicoset signal reconstruction algorithm, these coprime sample sets need to be regrouped into multicoset sample sets according to the sample pattern. The results of experiments indicate that the signals can be reconstructed at an equivalent rate of the order of gigahertz from sub-Nyquist samples acquired by the designed coprime acquisition system.

L-shaped coprime array structures for DOA estimation

This paper proposes a new sparse array geometry for 2-D (azimuth and elevation) direction-of-arrival (DOA) estimation based on coprime sampling. The proposed array structure is L-shaped coprime array (LCA) whose each portion is one dimensional coprime linear arrays in y- and z-dimensions. Each portion of the array is used separately for 1-D azimuth and elevation angle estimation. In order to obtain the paired DOA estimates the cross-covariance matrix of two portion of the array is utilized and the paired DOA angles are estimated. LCA provides to estimate $$K \le MN$$ source directions with $$2M+N-1$$ sensors in each portion and totally $$4M+2N-3$$ sensor elements. The proposed method is evaluated through numerical simulations and its performance is compared with other coprime planar array structures. It is shown that LCA has less computational complexity together with less real sensor elements and it provides superior performance as compared to the conventional 2-D coprime planar arrays.

Coprime sampling with embedded random delays

Spectrum sensing over a broad frequency band plays an important role in cognitive radio, but it also leads to high sampling rate when the bandwidth is large. The recently proposed coprime sampling scheme has been recognized as an attractive mechanism because it allows to significantly reduce the sampling rate. However in some cases, classical high resolution spectrum estimation methods fail when coprime sampling data are used. This situation has never been considered in the open literature under the framework of practical coprime sampling. In this paper, firstly, the conditions of this phenomenon are specified. Then, a new coprime sampling scheme based on embedded random delays is proposed to prevent this phenomenon. Simulation results show the effectiveness of the proposed coprime sampling scheme.

Wideband Spectrum Sensing via Derived Correlation Matrix Completion Based on Generalized Coprime Sampling

Wideband spectrum sensing is a popular topic in signal processing, especially for many radar and communication applications. What we face is a high sampling rate and a large volume of samples, in which demand of reducing the sampling rate without sacrificing the sensing resolution and quality. The generalized coprime sampling can break the limitation of the Nyquist sampling theorem with both characteristics of sparse sensing and coprime numbers. To fully utilize all the information received of the derived correlation matrix constructed by the different time delays, the matrix completion method is exploited. The theory of matrix completion is an extension of compressive sensing, though, which is not restrained by the sparsity and the restricted isometry property. The interpolation-based method presented via the convex framework of the nuclear norm minimization has no extra fine-tuned parameters, which different from techniques like compressive covariance sampling, positive definite Toeplitz matrix completion, and so on. Moreover, compared to the selection-based method under a continuous set, the proposed method improves the spectral resolution and estimation accuracy to avoid the information losing. The Simulation results indicate the performance of the algorithm.

Exploiting platform motion for passive source localization with a co-prime sampled large aperture array.

Co-prime array geometries have received a great deal of attention due to their ability to discriminate O(MN) sources with only O(M + N) sensors. This has been demonstrated both theoretically and in simulation. However, there are many practical limitations that make it difficult to realize the enhanced degrees of freedom when applying co-prime geometries to real acoustic data taken on a horizontal line array. For instance, co-prime sampling leads to grating lobes that can obscure lower signal-to-noise-ratio acoustic signals making them difficult to detect. In this work, a synthetic aperture (SA) method is presented for filling in holes and increasing redundancy in the difference co-array by exploiting array motion. The SA method is applied to acoustic data collected off the Southeastern shore of Florida on a fixed large aperture horizontal array. Array motion is simulated by taking a co-prime sampled subarray and virtually moving it along the horizontal aperture of the fixed array. It is demonstrated that SA processing on real acoustic data results in reduced side-lobe and grating lobe levels compared to that of the physical co-prime aperture.