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In compressive sensing theory, the measurement matrix plays a crucial role in compressive observation of sparse signals. The bipolar Toeplitz measurement matrix constructed based on chaotic map has advantages such as generating fewer free elements and supporting fast algorithms, making it widely used. While optimizing the measurement matrix can effectively improve its compressive sensing reconstruction performance, existing optimization algorithms are not suitable for the bipolar Toeplitz measurement matrix due to its structural and bipolar properties. To address this issue, this paper proposes an optimization method for the bipolar Toeplitz measurement matrix based on cosine-exponential (CE) chaotic map sequences and an improved Abolghasemi algorithm. Using an enhanced CE chaotic map to generate chaotic sequences with greater chaos and randomness, we construct the measurement matrix and optimize it using the structure matrix and the improved Abolghasemi algorithm, which preserves the matrix's bipolarity without altering its structure. We also introduce constraints on the generated sequence values during the optimization process. Through simulation experiments, the effectiveness of our optimization algorithm is verified, as the optimized bipolar Toeplitz measurement matrix significantly reduces reconstruction error and improves reconstruction probability. [ABSTRACT FROM AUTHOR]

The Toeplitz matrix reconstruction algorithms exploit the row vector of an array output covariance matrix to reconstruct Toeplitz matrix, which provide the directionof-arrival (DOA) estimation of coherent signals. However, the Toeplitz matrix reconstruction method based on any row vector of the array output covariance matrix suffers from signal correlation, it results in poor robustness. The methods based on multi-row vectors suffer serious performance degradation when in the low signal-to-noise ratio (SNR) owing to the noise energy is the square of the input noise energy. To solve the above problems, we propose an improved method that exploits all rows of the time-space correlation matrix to reconstruct the Toeplitz matrix, namely TS-MTOEP. This method firstly uses the coherence of the narrowband signal and the uncorrelated noise at different snapshots to construct the time-space correlation matrix, it effectively eliminates the influence of noise. Then, the Toeplitz matrix is reconstructed via all rows of the time-space correlation matrix, which effectively improves the energy of the signal, and further results in the improvement of the SNR. Finally, the DOAs can be obtained by combining it with the subspace-based methods. The theoretical analysis and simulation results indicate that compared with the existing Toeplitz and spatial smoothing methods, the proposed method in this paper provides good performance on estimation and resolution in cases with low input signal-to-noise due to time-space correlation matrix processing. Furthermore, in cases where the DOAs between the coherent sources are closely spaced and the snapshot number is low, our proposed method significantly improves the performance of the DOA estimation. We also provide the code to realize the reproducibility of the proposed method. [ABSTRACT FROM AUTHOR]

Direction-Of-Arrival (DOA) estimation in environments with coherent sources is a challenging problem, when the number of narrowband sources is more than elements and current coherent algorithms cannot accurately overcome such underdetermined problems. In this paper, we propose the Sparse-based Toeplitz Covariance Reconstruction (STCR) algorithm, which obtains a hole-free extended-aperture array with increased Degrees of Freedom (DOF) by exploiting a minimum redundancy array. This method provides superiority in estimating coherent sources by Toeplitz covariance matrix reconstruction. Also, using a Sparse Signal Representation (SSR) framework makes the DOA estimator stronger in source detection and robust to the noise power. Numerical results show the effectiveness of the proposed method even at low SNRs for over- and underdetermined DOA estimators compared to well-known methods. Also, it has high accuracy in DOA estimation of closely-located coherent sources using a small number of snapshots. [ABSTRACT FROM AUTHOR]

Abstract Sparse array is an irregular array with a high degree of freedom in its difference coarray domain. Nested array and coprime array are two important one dimensional sparse arrays. For fixed number of sensors, nested array has a high degree of freedom, resolution and accuracy than coprime array. 1D nested arrays do not provide 2D scanning. 2D nested array which is an extension of 1D nested array provides both 2D scanning as well as high degree of freedom, however direction of arrival (DoA) estimation techniques used for 2D nested array such as 2D unitary ESPRIT and 2D spatial smoothing MUSIC has high computational complexity. This paper shows that spatial smoothing is not necessary to apply 2D MUSIC in the difference coarray domain of 2D nested array. A new Hermitian Toeplitz matrix is constructed instead of spatial smoothing. 2D MUSIC on this Hermitian Toeplitz matrix can resolve the DoA of sources with less computations. [ABSTRACT FROM AUTHOR]

We developed a robust, quasi high-pass filter for edge detection in medical images. The kernel-based algorithm of our detector was similar to that of conventional edge detectors. The proposed edge detector has a mathematical form of local variance and is adaptive in nature. The mathematical formulation of the detector was exploited and re-expressed as a quadratic form of the Toeplitz matrix. The detector has a highly structured internal architecture with abundant spatial isotropic symmetricity. The proposed operator can greatly reduce problems frequently encountered in edge detection including fragmentation, position dislocation, and thinness loss. The detector is robust to noise and can efficiently extract crucial edge features. We named this new operator as the WL operator (Wang and Lin). The performance of the WL operator was compared to that of other edge detectors by using Pratt’s figure of merits. In addition, the performance was confirmed with experts by using visual analog scale scores. The results obtained using the WL operator for different medical imaging modalities including X-ray, CT, and MRI are promising. Therefore, the WL operator warrants further investigation. [ABSTRACT FROM AUTHOR]

For a hyponormal operator, C.R. Putnam's inequality gives an upper bound on the norm of its self-commutator. In the special case of a Toeplitz operator with analytic symbol in the Smirnov space of a domain, there is also a geometric lower bound shown by D. Khavinson (1985) that when combined with Putnam's inequality implies the classical isoperimetric inequality. For a nontrivial domain, we com-pare these estimates to exact results. Then we consider such operators acting on the Bergman space of a domain, and we obtain lower bounds that also reflect the geometry of the domain. When combined with Putnam's inequality they give rise to the Faber-Krahn inequality for the fundamental frequency of a domain and the Saint-Venant inequality for the torsional rigidity (but with non-sharp constants). We conjecture an improved version of Putnam's inequality within this restricted setting. [ABSTRACT FROM AUTHOR]

We will show the necessary and sufficient condition of separability for a subclass of circulant states in the ℂ³ ⊗ ℂ³ system. This sub-class includes the models proposed by several authors as special cases. Moreover, it will be proved that our separability condition equals to the failure condition of quasi-distillation of entanglement. [ABSTRACT FROM AUTHOR]

Abstract: Multilevel Toeplitz linear systems appear in a wide range of scientific and engineering applications. While several fast direct solvers exist for the basic 1-level Toeplitz matrices, in the multilevel case an iterative solver provides the most general and practical solution. This paper proposes several parallelization techniques that enable an efficient implementation of the conjugate gradient algorithm for solving multilevel Toeplitz systems on distributed-memory machines. The two major differences between this implementation and that for a general sparse linear solver are (1) a communication-efficient approach to handle data expansion and truncation and data transpose simultaneously; (2) the interleaving of matrix-vector multiplications and vector inner product calculations to reduce synchronization cost and latency. Similar ideas can be applied to the implementation of other iterative methods for Toeplitz systems that are not necessarily symmetric positive definite. Scaling results are shown to demonstrate the usefulness of the proposed techniques. [Copyright &y& Elsevier]

Abstract: A finite sequence of complex numbers is the set of trigonometric moments of a measure with a polynomial of degree , zero-free in , providing only that the Hermitian Toeplitz matrix having as its top row is positive-definite. The analogous representation in two dimensions for a rectangular array of numbers requires additional conditions, discovered by J.S. Geronimo and H.J. Woerdeman. We use orthogonal decomposition to give an alternate proof of their important result. [Copyright &y& Elsevier]

In this note we present a modified Newton's method for computing the smallest eigenvalue λ1 of a symmetric positive definite Toeplitz matrix (SPDTM) . This method based of the Taylor series characteristic polynomial of a SPDTM and respectively the even and odd characteristic polynomials of a SPDTM. The characteristic polynomial (and respectively the even and odd characteristic polynomials) has been approximated with the polynomial of the second order from the Taylor series because it is easy to develop pn(λ)" (pen(λ)" & pon(λ)") from specific structure Toeplitz matrix from Gohberg-Simencul formulae. [ABSTRACT FROM AUTHOR]

In this paper the necessary and sufficient conditions are given for the solution of a system of parameter varying linear inequalities of the form A(t) x ≥ b (t) for all t ∈ T, where T is an arbitrary set, x is the unknown vector, A(t) is a known triangular Toeplitz matrix and b (t) is a known vector. For every t ∈ T the corresponding inequality defines a polyhedron, in which the solution should exist. The solution of the linear system is the intersection of the corresponding polyhedrons for every t ∈ T . A general modular decomposition method has been developed, which is based on the successive reduction of the initial system of inequalities by reducing iteratively the number of variables and by considering an equivalent system of inequalities. [ABSTRACT FROM AUTHOR]

In this paper the necessary and sufficient conditions are given for the solution of a system of parameter varying linear inequalities of the form A(t) x ≥ b (t) for all t ∈ T, where T is an arbitrary set, x is the unknown vector, A(t) is a known triangular Toeplitz matrix and b (t) is a known vector. For every t ∈ T the corresponding inequality defines a polyhedron, in which the solution should exist. The solution of the linear system is the intersection of the corresponding polyhedrons for every t ∈ T . A general modular decomposition method has been developed, which is based on the successive reduction of the initial system of inequalities by reducing iteratively the number of variables and by considering an equivalent system of inequalities. [ABSTRACT FROM AUTHOR]

The study of the common intracluster correlation in simple linear regression is well developed ([1] and [2]). For the situation involving various intracluster correlation coefficients, the issues become more complicated. The prime objective of this study is to compare the loss of efficiency in using the intracluster correlation structure of common form to that in Toeplitz form for simple linear regression using generalised least squares. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a technique for improving power distribution characteristics of space time block codes (STBCs) which include peak to average power ratio (PAPR), average to minimum power ratio (Ave/min), and probability of transmitting "zero" by antenna. We consider STBCs in multiple input multiple output (MIMO) and multiple input single output (MISO) systems that achieve full diversity for linear or maximum likelihood (ML) receivers. It is proved that by multiplying a specific non-singular square matrix by the code matrix, a new code is obtained that achieves full diversity and the power distribution characteristics are improved. The proposed technique is general can be applied to any type of full diversity STBCs such as Toeplitz codes, orthogonal STBCs (OSTBCs), overlapped Alamouti codes (OACs), group orthogonal Toeplitz codes (GOTCs), and embedded Alamouti codes (EACs), and can ease their practical implementation. We systematically present closed-form expressions for the matrices multiplied by Toeplitz codes, OACs, and EACs. Because of the existence of these codes for any number of antennas, the closed-form can be used in 5G which needs huge number of antennas. The performance of the proposed technique is evaluated in terms of bit error rate (BER) and power distribution characteristics. The introduced method provides a trade-off between PAPR, Ave/min, probability of transmitting zero, and BER, to cope with the practical requirements. We also investigate the effect of the proposed method on the PAPR of output signal of shaping filter which is referred to as analog domain PAPR. The results indicate the efficiency of the introduced technique. The proposed method can balance the tradeoff between PAPR, spectral efficiency, and BER. [ABSTRACT FROM AUTHOR]