Coprime array exhibits many advantages over the uniform linear array (ULA) with the same number of physical sensors in resolution performance and interference suppression capability. In this study, ...the authors take the advantages of coprime array to improve the robustness of adaptive beamformer. In the coprime virtual ULA (CV-ULA), they prove that a constructed Toeplitz matrix can be taken as the sample covariance matrix from the perspective of virtual signal characteristics. The CV-ULA Capon spectrum estimator is modified to obtain the directions and powers of all impinging signals. Since the real directions of all impinging signals are located at different angular sectors, they form independent signal subspace for each impinging signal. They also assign independent steering vector mismatches for different impinging signals to obtain their real steering vectors. The steering vector mismatch of each impinging signal is independently obtained by solving its own convex optimisation problem. They reconstruct the interference-plus-noise covariance matrix (INCM) with precise steering vectors and powers of interference signals. The proposed weight vector is computed by combining the desired signal steering vector and the reconstructed INCM. Extensive simulations show that the proposed algorithm provides robustness against many types of model mismatches.
In this work, a Sinc-Galerkin method is considered and analyzed for solving the fourth-order partial integro-differential equation with a weakly singular kernel. The time derivative and ...Riemann-Liouville fractional integral term are approximated via the Crank-Nicolson method and the trapezoidal convolution quadrature rule, respectively. Then a fully discrete scheme is formulated via using the Sinc-Galerkin approximation. The exponential convergence rate in space of proposed method are derived. In addition, some properties of the Toeplitz matrix generated by the composite Sinc function at the Sinc node are extended to the cases of arbitrary order in the preliminary knowledge. Finally, some numerical examples are calculated to verify the accuracy and effectiveness of our method.
In this paper, we answer an open conjecture concerning complex symmetric matrices and truncated Toeplitz operators. We study matrix representations of truncated Toeplitz operators with respect to ...orthonormal bases which are invariant under a canonical conjugation map. In particular, we determine necessary and sufficient conditions for when a symmetric matrix is the matrix representation of a truncated Toeplitz operator with respect to a given conjugation invariant orthonormal basis. We specialise our result to the case when the conjugation invariant orthonormal basis is a modified Clark basis. With this specialisation, we answer an open conjecture in the negative, and show not every unitary equivalence between a complex symmetric matrix and a truncated Toeplitz operator arises from modified Clark basis representations. We pose a new refined conjecture for how to realise a model theory for symmetric matrices through the use of truncated Toeplitz operators, and we show this conjecture is equivalent to a specified system of polynomial equations being satisfied.
•To fill a low-rank Toeplitz matrix from a small subset of its entries.•Definition of “Toeplitz structure smoothing operator”.•The augmented Lagrange multiplier algorithm with smoothing for TMC ...problem.•Approximation matrices may hold on Toeplitz structure throughout the iteration.
Toplitz matrix completion (TMC) is to fill a low-rank Toeplitz matrix from a small subset of its entries. Based on the augmented Lagrange multiplier (ALM) algorithm for matrix completion, in this paper, we propose a new algorithm for the TMC problem using the smoothing technique of the approximation matrices. The completion matrices generated by the new algorithm are of Toeplitz structure throughout iteration, which save computational cost of the singular value decomposition (SVD) and approximate well the solution. Convergence results of the new algorithm are proved. Finally, the numerical experiments show that the augmented Lagrange multiplier algorithm with smoothing is more effective than the original ALM and the accelerated proximal gradient (APG) algorithms.
This paper investigates the symbol error probability (SEP) of point-to-point massive multiple-input multiple-output (MIMO) systems using equally likely PAM, PSK, and square QAM signallings in the ...presence of transmitter correlation. The receiver has perfect knowledge of the channel coefficients, while the transmitter only knows first- and second-order channel statistics. With a zero-forcing (ZF) detector implemented at the receiver side, we design and derive closed-form expressions of the optimal precoders at the transmitter that minimizes the average SEP over channel statistics for various modulation schemes. We then unveil some nice structures on the resulting minimum average SEP expressions, which naturally motivate us to explore the use of two useful mathematical tools to systematically study their asymptotic behaviors. The first tool is the Szegö's theorem on large Hermitian Toeplitz matrices and the second tool is the well-known limit: <inline-formula> <tex-math notation="LaTeX">\lim _{x\to \infty }(1+1/x)^{x}=e </tex-math></inline-formula>. The application of these two tools enables us to attain very simple expressions of the SEP limits as the number of the transmitter antennas goes to infinity. A major advantage of our asymptotic analysis is that the asymptotic SEP converges to the true SEP when the number of antennas is moderately large. As such, the obtained expressions can serve as an effective SEP approximations for massive MIMO systems even when the number of antennas is not very large. For the widely used exponential correlation model, we derive closed-form expressions for the SEP limits of both optimally precoded and uniformly precoded systems. Extensive simulations are provided to demonstrate the effectiveness of our asymptotic analysis and compare the performance limit of optimally precoded and uniformly precoded systems.
In this work, we present a high-order numerical method for the two-dimensional nonlinear space-fractional complex Ginzburg-Landau equation (FCGLE). Firstly, a fourth-order approximation is adopted to ...discretize the spatial Riesz fractional derivatives that leads to a semi-linear system of ordinary differential equations (ODEs), whose coefficient matrix has the complex block Toeplitz structure. Then a fourth-order exponential integrator method is used to solve the corresponding semi-linear ODEs system. In light of the results in theory, the proposed algorithm is fourth-order accuracy in both time and space. In the specific implementation of the proposed algorithm, due to the special structure of the coefficient matrix, the products of some φ-functions of matrices (related to the matrix exponential) and vectors are computed by the shift-invert Lanczos technique in the exponential integrator. In order to calculate the linear system of equations arising from the shift-invert Lanczos procedure, two classes of efficient preconditioners including Strang's circulant preconditioner/τ matrix preconditioner are constructed and implemented by fast Fourier transform and fast sine transform, respectively. Numerical examples with and without exact solutions are implemented to confirm the effectiveness of the current algorithm.
In this paper, we develop a fast numerical method for solving the time-dependent Riesz space fractional diffusion equations with a nonlinear source term in the convex domain. An implicit finite ...difference method is employed to discretize the Riesz space fractional diffusion equations with a penalty term in a rectangular region by the volume-penalization approach. The stability and the convergence of the proposed method are studied. As the coefficient matrix is with the Toeplitz-like structure, the generalized minimum residual method with a preconditioner based on the sine transform is exploited to solve the discretized linear system, where the preconditioner is constructed in view of the combination of two approximate inverse τ matrices, which can be diagonalized by the sine transform. The spectrum of the preconditioned matrix is also investigated. Numerical experiments are carried out to demonstrate the efficiency of the proposed method.
In this paper, we concentrate on design a bilateral preconditioning for all-at-once system from multidimensional time-space non-local evolution equations with a weakly singular kernel. Firstly, we ...propose an implicit difference scheme for this equation by employing an L2-type formula. Then, we prove the stability and the convergence of the proposed scheme. Based on such a numerical scheme, an L2-type all-at-once system is derived and solved in a parallel-in-space pattern. Based on the special structure of its coefficient matrix, we propose a bilateral preconditioning strategy to accelerate the convergence of Krylov subspace solvers. We theoretically show that the condition number of the preconditioned matrix is uniformly bounded by a constant. Several numerical examples are provided to test the proposed scheme and preconditioning technique.
In this paper, we consider three-way split formulas for binary polynomial multiplication and Toeplitz matrix vector product (TMVP). We first recall the best known three-way split formulas for ...polynomial multiplication: the formulas with six recursive multiplications given by Sunar in a 2006 IEEE Transactions on Computers paper and the formula with five recursive multiplications proposed by Bernstein at CRYPTO 2009. Second, we propose a new set of three-way split formulas for polynomial multiplication that are an optimization of Sunar's formulas. Then, we present formulas with five recursive multiplications based on field extension. In addition, we extend the latter formulas to TMVP. We evaluate the space and delay complexities when computations are performed in parallel and provide a comparison with best known methods.
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