Steering Matrices Research Articles

Semi-blind receivers based on a coupled nested Tucker-PARAFAC model for dual-polarized MIMO systems using combined TST and MSMKron codings

Over the past two decades, tensor-based approaches have generated ever-increasing interest for designing multiple-input multiple-output (MIMO) communication systems. In this paper, we propose a new double directional dual-polarized (DD-DP) MIMO system, equipped with uniform rectangular arrays (URAs) at both ends of the link, which combines a multiple Kronecker product of symbol matrices with a tensor space-time (TST) coding, called TST-MSMKron coding. It is first shown that the channel tensor is represented by a fifth-order PARAFAC decomposition, separated into two parts associated with vertically and horizontally polarized receive antennas, respectively. Then, it is established that the tensor of received signals satisfies a new coupled nested Tucker-PARAFAC model whose core tensor is the coding tensor. Capitalizing on this tensor modeling and assuming the tensor coding is known at the reception, two semi-blind receivers composed of two stages are derived for estimating the transmitted symbols and channel parameters (directions of departure (DoD) and arrival (DoA) angles, path loss coefficients). A new rectification method is proposed to ensure the Vandermonde structure of the estimated steering matrices. Parameter identifiability and computational complexity are analyzed for each receiver. Monte Carlo simulation results are provided to illustrate the effectiveness of the proposed TST-MSMKron coding and semi-blind receivers, in terms of symbol error rate (SER) and channel parameters estimation.

Ergodic Achievable Rate Analysis and Optimization of RIS-Assisted Millimeter-Wave MIMO Communication Systems

Reconfigurable intelligent surfaces (RISs) have emerged as a prospective technology for next-generation wireless networks due to their potential in coverage and capacity enhancement. Previous works on achievable rate analysis of RIS-assisted communication systems have mainly focused on the rich-scattering environment where Rayleigh and Rician channel models can be applied. This work studies the ergodic achievable rate of RIS-assisted multiple-input multiple-output communication systems in millimeter-wave band with limited scattering under the Saleh-Valenzuela channel model. Firstly, we derive an upper bound of the ergodic achievable rate by means of majorization theory and Jensen's inequality. The upper bound shows that the ergodic achievable rate increases logarithmically with the number of antennas at the base station (BS) and user, the number of the reflection units at the RIS, and the eigenvalues of the steering matrices associated with the BS, user and RIS. Then, we aim to maximize the ergodic achievable rate by jointly optimizing the transmit covariance matrix at the BS and the reflection coefficients at the RIS. Specifically, the transmit covariance matrix is optimized by the water-filling algorithm and the reflection coefficients are optimized using the Riemannian conjugate gradient algorithm. Simulation results validate the effectiveness of the proposed optimization algorithms.

Non-Euclidean estimation of acoustic cross-spectral density matrices: Evaluation by matched-field localization in an ocean waveguide.

A set of non-Euclidean cross-spectral density matrix (CSDM) estimators based on the Frechet mean and median is applied to spatially localize an acoustic source in a stochastic shallow water environment driven by internal gravity waves. The resulting geometric-based matrix estimates are incorporated into matched-field processors defined through steering matrices using intrinsic distance measures between pairs of CSDMs on a Riemannian manifold for each replica source location. Their performance is evaluated using the probability of correct localization to determine any improvement in source localization over the conventional (maximum likelihood) statistical approach for CSDM estimation.

Semi-blind joint symbols and multipath parameters estimation of MIMO systems using KRST/MKRSM coding

In this paper, we propose a new MIMO communication system in a time-varying multipath environment, using a Khatri-Rao space-time (KRST) coding combined with a multiple Khatri-Rao product of symbol matrices (MKRSM). It is shown the signals received at the receiver form a tensor which satisfies a (M+2)-order nested PARAFAC model, where (M−1) denotes the number of symbol matrices considered for MKRSM coding. Such a generalization of the nested PARAFAC model to (M+N)-order tensors is first studied from a general point of view, with the discussion of some parameter estimation methods depending on the a priori knowledge on the model. Then, a semi-blind receiver composed of three stages, is developed for jointly estimating the transmitted symbols and the multipath parameters.In the first stage, the transmitted symbols and a matrix unfolding of the effective channel including the fading coefficients and the steering matrices, are estimated using closed-form algorithms based on Khatri-Rao factorizations. In the second one, the channel estimation is refined by means of a simplified least-squares algorithm which takes the column orthonormality assumption on the coding matrix into account. In the third one, an alternating least-squares algorithm, combined with a rectification for the Vandermonde factors containing the spatial steering vectors at the transmitter and receiver sides, is applied to estimate the multipath parameters from the estimated channel. A complexity analysis is made for the receivers, and an expected Cramer-Rao bound related to channel estimation is established. Extensive Monte Carlo simulation results show that the semi-blind receiver which combines the channel estimation refinement with the rectification technique exhibit very interesting performance.

Novel DOAs estimation method based on Doppler aided Chinese remainder theorem with all phase DFT for multiple targets in sparse array

In this study, the authors take further insight into the sparse geometry which offers a larger array aperture than uniform linear array with the same number of physical sensors. A novel direction of arrivals (DOAs) estimation model with flexible sparse geometry, which possesses the potential to significantly improve the estimation performance especially when the placed space and the weight of carrier such as airborne radar are restricted, is proposed to offer a larger aperture compared with co-prime array. The proposed algorithm can estimate DOAs by solving phase ambiguity. To improve the capability of spectrum analysis in frequency domain, all phase discrete Fourier transform (DFT), which can effectively alleviate spectrum leakage compared with traditional DFT, is proposed to apply into DOAs estimation. Additionally, the performance on degrees of freedom can be considerably improved compared with the state of the art where all the targets can be distinguished by Doppler information of received echo signal. More importantly, the proposed algorithm can effectively deal with DOA-closely-spaced targets because the proposed algorithm does not require to estimate signal subspace with ill-conditioned steering matrices. Both the theoretical analysis and simulation results demonstrate that the proposed algorithm significantly improves DOAs estimation precision with less computation cost.

Tensor-Based Angle and Range Estimation Method in Monostatic FDA-MIMO Radar

In the paper, joint angle and range estimation issue for monostatic frequency diverse array multiple-input multiple-output (FDA-MIMO) is proposed, and a tensor-based framework is addressed to solve it. The proposed method exploits the multidimensional structure of matched filters in FDA-MIMO radar. Firstly, stack the received data to form a third-order tensor so that the multidimensional structure information of the received data can be acquired. Then, the steering matrices contain the angle and rang information are estimated by using the parallel factor (PARAFAC) decomposition. Finally, the angle and range are achieved by utilizing the phase characteristic of the steering matrices. Due to exploiting the multidimensional structure of the received data to further suppress the effect of noise, the proposed method performs better in angle and range estimation than the existing algorithms based on ESPRIT, simulation results can prove the proposed method’s effectiveness.

Mixed-Field Source Localization Based on the Non-Hermitian Matrix

In this paper, an efficient high-order multiple signal classification (MUSIC)-like method is proposed for mixed-field source localization. Firstly, a non-Hermitian matrix is designed based on a high-order cumulant. One of the steering matrices, that is related only with the directions of arrival (DOA), is proved to be orthogonal with the eigenvectors corresponding to the zero eigenvalues. The other steering matrix that contains the information of both the DOA and range is proved to span the same column subspace with the eigenvectors corresponding to the non-zero eigenvalues. By applying the Gram–Schmidt orthogonalization, the range estimation can be achieved one by one after substituting each estimated DOA. The analysis shows that the computational complexity of the proposed method is lower than other methods, and the effectiveness of the proposed method is shown with some simulation results.

On the Ergodic Capacity of mmWave Systems Under Finite-Dimensional Channels

Due to the underlying sparse structure of the mmWave channels, which indeed makes the exact closed-form capacity expressions inherently hard to derive, there has been less research on the ergodic capacity of mmWave systems. To overcome this problem, by means of the majorization theory, this paper analyzes the ergodic capacity of point-to-point mmWave communication systems under finite-dimensional channel model. In particular, we derive several closed-form ergodic capacity approximations, which exhibit excellent tightness in spite of whether the steering matrices are singular or not. Then, several Jensen’s approximations and bounds of the ergodic capacity are also derived. The results indicate that the ergodic capacity seems to increase logarithmically with the number of antennas, the transmit SNR per antenna, and the eigenvalues of the steering matrix products. Besides, the DFT matrices can effectively characterize the spatial directions of mmWave channels when the number of antennas grows large. After that, high-SNR ergodic capacity, high-SNR slope, and power offset are also analyzed. It indicates that for a finite-dimensional channel, the maximum multiplexing gain increases with the number of paths instead of the number of antennas in Rayleigh channels. Numerical simulations are performed to validate the results.

Joint calibration of array shape and sensor gain/phase for highly deformed arrays using wideband signals

A joint sensor position and gain/phase calibration method for highly deformed arrays is proposed. It applies to the case where the sensor gain is omnidirectional and frequency-invariant, and the sensor phase is directional and frequency-invariant. The method is initialized with the raw estimates of the sensor gains, which are simply obtained from the main diagonal elements of the sample covariance matrix. Then, it alternately updates either the sensor gain estimates or the sensor position and phase estimates while fixing the other. When the sensor gain estimates are available, the steering matrices for all the frequency bins are estimated using the constant modulus method. By combining all these steering matrices, the estimates of the sensor positions and the frequency-invariant sensor phases are straightforwardly derived. Then, the sensor gain estimates are refined for the next iteration by using the orthogonality of the subspaces. To resolve the phase ambiguities due to the large position mismatches of highly deformed arrays, the nominal intersensor distance and the array bending angle are used. The Cramer–Rao lower bounds (CRLBs) for the unknown sensor gain/phase and position are also derived. The performance of the proposed method is evaluated using simulation data and compared with the CRLBs.

Constrained Waveform Design for Colocated MIMO Radar With Uncertain Steering Matrices

This paper deals with the robust waveform design of multiple-input multiple-output radar to improve target detectability embedded in signal-dependent interferences. Two iterative algorithms with ensuring convergence properties are introduced to maximize the worst case signal-to-interference-plus-noise ratio (SINR) over steering matrix mismatches under the constant modulus and similarity constraints. Each iteration of the proposed algorithms splits the high-dimensional problem into multiple one dimensional problems, to which the optimal solutions can be found in polynomial times. Numerical examples are provided to assess the capabilities of the proposed techniques in comparison with the existing methods in terms of the SINR and the computational times for both the continuous and discrete phase cases of the probing signal.

Joint 2-D DOA and Doppler Estimation for L-Shaped Array via Dual PARAFAC With Triple Matching Implementation

A dual parallel factor (PARAFAC)-based approach to jointly estimating the two-dimensional direction of arrival (2-D DOA) and Doppler is proposed in this paper, where an L-shaped array consisting of acoustic vector-sensor is used. First, we apply the PARAFAC decomposition to the data model formed by concatenating the outputs of multi-level delays of the observations, and we get the parameter matrix H, which accomplish the 2-D DOA estimation and pairing automatically, then the dual PARAFAC decomposition is applied to the achieved composite steering matrix from the first PARAFAC decomposition, and thus, the same permutation matrices link the estimates of steering matrices and delay matrices from X-subarray and Y-subarray, respectively. Following this, the Doppler and 2-D DOA matching information are obtained via triple matching implementation, e.g. 2-D DOA and frequency matching. Finally, Doppler is estimated by delay matrices. The proposed algorithm is computationally effective for both uniform and non-uniform L-shaped array as SNR exceeds 15dB, and its performance outperforms the joint angle and Doppler shift ESPRIT (JAD-ESPRIT) algorithm and the joint angle and Doppler shift PM (JAD-PM) algorithm. The simulation results justified the effectiveness of the proposed algorithm.

Low-Complexity High-Order Propagator Method for Near-Field Source Localization.

In this paper, an efficient high-order propagator method is proposed to localize near-field sources. We construct a specific non-Hermitian matrix based on the high-order cumulant of the received signals. With its columns and rows, we can obtain two subspaces orthogonal to all the columns of two steering matrices, respectively, with which the estimation of the directions of arrival (DOA) and ranges of near-field sources can be achieved. Different from other methods, the proposed method needs only one matrix for estimating two parameters separately, therefore leading to a smaller computational burden. Simulation results show that the proposed method achieves the same performance as the other high order statistics-based methods with a lower complexity.

Two-Dimensional DOA Estimation for Incoherently Distributed Sources with Uniform Rectangular Arrays.

Aiming at the two-dimensional (2D) incoherently distributed (ID) sources, we explore a direction-of-arrival (DOA) estimation algorithm based on uniform rectangular arrays (URA). By means of Taylor series expansion of steering vector, rotational invariance relations with regard to nominal azimuth and nominal elevation between subarrays are deduced under the assumption of small angular spreads and small sensors distance firstly; then received signal vectors can be described by generalized steering matrices and generalized signal vectors; thus, an estimation of signal parameters via rotational invariance techniques (ESPRIT) like algorithm is proposed to estimate nominal elevation and nominal azimuth respectively using covariance matrices of constructed subarrays. Angle matching method is proposed by virtue of Capon principle lastly. The proposed method can estimate multiple 2D ID sources without spectral searching and without information of angular power distribution function of sources. Investigating different SNR, sources with different angular power density functions, sources in boundary region, distance between sensors and number of sources, simulations are conducted to investigate the effectiveness of the proposed method.

Stochastic matched-field processing using polynomial chaos expansions to model environmental uncertainty

A stochastic generalization of matched-field processing (MFP) is presented that incorporates stochastic steering matrices rather than steering vectors. This provides a natural framework for including environmental uncertainty associated with incompletely known environmental knowledge. Parametric probabilities allow use of polynomial chaos (PC) expansions to describe environmental uncertainty in a rigorous manner, yielding efficient representations of the stochastic steering matrices through the application of sparse sampling methods. In particular, PC methods apply to the uncertainty of sediment variability that can be modeled via horizontal spectral methods. With moderate variability over scales justified by data in the New Jersey shelf region, the sediment uncertainty can be modeled with relatively few PC expansion coefficients, which depend only on the mean amplitude of spectral components. The coefficients express the nonlinear dependence of the acoustic field on the sediment variability, essentially orthonormalizing higher moments. The coefficients lead immediately to stochastic steering matrices, which are then compared via various MFP processors against acoustic data for MFP source localization. [Work supported by the U.S. Office of Naval Research.]

Stochastic matched-field localization of an acoustic source based on principles of Riemannian geometry.

Passive localization of acoustic sources is treated within a geometric framework where non-Euclidean distance measures are computed between a cross-spectral density estimate of received data on a vertical array and a set of stochastic replica steering matrices, rather than traditional replica steering vectors. A processing scheme involving matrix-matrix comparisons where steering matrices, as functions of the replica source coordinates, naturally incorporate environmental variability or uncertainty provides a general framework for considering the acoustic inverse source problem in an ocean waveguide. Within this context a subset of matched-field processors is examined, based on recent advances in the application of non-Euclidean geometry to statistical classification of data feature clusters. The matrices are interpreted abstractly as points in a Riemannian manifold, and an appropriately defined distance measure between pairs of matrices on this manifold defines a matched-field processor for estimating source location. Acoustic simulations are performed for a waveguide comprising both a depth-dependent sound-speed profile perturbed by linear internal gravity waves and a depth-correlated surface noise field, providing an example of the viability of this approach to passive source localization in the presence of sound-speed variability.

DOA Estimation for Generalized Three Dimensional Coprime Arrays: A Fast-Convergence Quadrilinear Decomposition Algorithm

In this paper, we construct the generalized 3-D coprime array (G-TDCA) configuration, which consists of two 2-D uniform subarrays, by selecting three pairs of coprime integers to obtain the extension of inter-element spacing. Meanwhile, we derive the analytical expression of Cramer–Rao bound for G-TDCA and verify that G-TDCA outperforms the conventional 3-D uniform array (TDUA) in the direction of arrival (DOA) (i.e., azimuth and elevation angles) estimation performance and resolution with numerical simulations. In addition, we propose a fast convergence quadrilinear decomposition (FC-QD) algorithm to extract the DOA estimates for G-TDCA. In the FC-QD algorithm, we first employ propagator method (PM) to achieve rough DOA estimates that are utilized to initialize the steering matrices. Subsequently, the received signal is constructed as two quadrilinear models and quadrilinear alternating least square algorithm is operated to attain fine DOA estimates. Moreover, phase ambiguity problem, caused by large adjacent distance between array sensors, is eliminated based on coprime property, where the uniqueness of DOA estimates can be achieved. Specifically, the proposed FC-QD algorithm has a fast convergence due to initialization via PM, and hence, FC-QD can significantly lower the computational cost without degrading the DOA estimation performance, which is demonstrated by extensive simulation results.

Joint DOD and DOA Estimation for Bistatic MIMO Radar in Unknown Correlated Noise

In this paper, joint direction-of-departure (DOD) and direction-of-arrival (DOA) estimation for bistatic multiple-input–multi-output (MIMO) radar in an unknown spatially correlated noise environment is investigated. The signal model is based on the assumption that the waveforms are transmitted by two separated subarrays having $M_1$ and $M_2$ sensors and received by two separated subarrays having $N_1$ and $N_2$ sensors, respectively. The received data are pulse-compressed using a matching matrix consisting of $M=M_1+M_2$ orthogonally transmitted waveforms. The joint covariance matrix of unknown correlated noise is analyzed. A novel algorithm is proposed by jointly estimating the DOD and DOA with transmitter and receiver subarrays in unknown noise. The joint estimation algorithm is based on the canonical correlation decomposition (CCD) and exploits the shift-invariance properties in the Kronecker product structure of each column of the various steering matrices. The estimated DOA and DOD can be automatically paired correspondingly. In addition, the formulas of stochastic Cramer–Rao bounds (CRB) for DOD and DOA estimation are derived. Simulations show that our method can effectively improve the performance of estimation in unknown correlated noise environments and is insensitive to having different noise environments in the two subarrays.

Stochastic-basis matched-field processing

Matched-field processing (MFP) suffers serious degradation due to environmental mismatch between received acoustic-field vectors and modeled replica vectors. Physical reasons for degradation include uncertainty caused by incomplete descriptions of the parameters and fields necessary for correct specification of the acoustic waveguide (i.e., environmental uncertainty), and system uncertainty associated with incomplete knowledge of the array configuration, source depth, etc. Recent research in the theory of stochastic-basis expansions (polynomial chaos) provides a mathematically consistent way of incorporating both types of uncertainty into MFP. Such expansions are used efficiently to construct replica matrices that steer high-rank subspaces capable of capturing signals with uncertain wavefronts. When combined with cross-spectral density matrices, stochastic-basis steering matrices enable the design of new processors with properties not previously evaluated in a MFP context. A maximum likelihood processor is developed which incorporates environmental uncertainty through polynomial chaos expansions. The processor can be written as a Frobenius product between an estimated cross-spectral density matrix and the inverse of a stochastic-basis replica matrix. This talk will outline the theoretical foundation of stochastic-basis MFP, and illustrate the method with some simulations. [Work supported by the Office of Naval Research.]

Wideband DOA Estimation with Interpolated Focusing KR Product Matrix

A new direction-of-arrival estimation method for uncorrelated wideband sources is proposed. Firstly, the covariance matrices of different frequency bins are transformed into higher dimensional vectors through Khatri-Rao product to produce the new models of the array outputs. Secondly, the higher dimensional noiseless signal subspace at each frequency is focused to a certain reference frequency and a single correlation matrix is constructed, where the focusing matrices are designed by interpolating the virtual steering matrices of the new model. Finally, the DOA estimates are acquired through using narrowband MUSIC method. The proposed method is able to increase the degrees of freedom of a uniform linear array and enable us to handle more sources than sensors. Numerical results demonstrate the performance of the proposed method.

A Cumulant-Based Method for Direction Finding in Uniform Linear Arrays With Mutual Coupling

The problem of direction finding in uniform linear arrays (ULAs) with unknown mutual coupling is considered. A new fourth-order cumulant (FOC)-based method for estimating directions of arrivals (DOAs) of far-field signals is developed. In our proposed method, an FOC matrix is first derived. Then, it is shown that the null space of the Khatri-Rao (KR) product of the steering matrices can be obtained via the eigen decomposition of the FOC matrix. Finally, by taking advantage of the banded complex symmetric Toeplitz structure of the mutual coupling matrix (MCM), a determinant-based spatial spectrum is introduced to determine the DOAs. Computer simulations for different signal-to-noise ratios (SNRs) and numbers of snapshots are included to demonstrate the superior performance of the proposed method over its conventional counterpart.