In this paper, we consider secure downlink transmission in a multicell massive multiple-input multiple-output (MIMO) system where the numbers of base station (BS) antennas, mobile terminals, and ...eavesdropper antennas are asymptotically large. The channel state information of the eavesdropper is assumed to be unavailable at the BS and hence, linear precoding of data and artificial noise (AN) are employed for secrecy enhancement. Four different data precoders (i.e., selfish zero-forcing (ZF)/regularized channel inversion (RCI) and collaborative ZF/RCI precoders) and three different AN precoders (i.e., random, selfish/collaborative null-space-based precoders) are investigated and the corresponding achievable ergodic secrecy rates are analyzed. Our analysis includes the effects of uplink channel estimation, pilot contamination, multicell interference, and path-loss. Furthermore, to strike a balance between complexity and performance, linear precoders that are based on matrix polynomials are proposed for both data and AN precoding. The polynomial coefficients of the data and AN precoders are optimized, respectively, for minimization of the sum-mean-squared-error of and the AN leakage to the mobile terminals in the cell of interest using tools from free probability and random matrix theory. Our analytical and simulation results provide interesting insights for the design of secure multicell massive MIMO systems and reveal that the proposed polynomial data and AN precoders closely approach the performance of selfish RCI data and null-space-based AN precoders, respectively.
The distance from a given n×n regular matrix polynomial to a nearest matrix polynomial of normal rank at most r for a specified positive integer r⩽n−1, is considered in two different norm settings. ...Particular emphasis is given on the distance to nearest singular matrix polynomials. The problem is shown to be equivalent to computing smallest structure preserving perturbations such that certain convolution matrices associated with the matrix polynomial become suitably rank deficient. In particular, the distance to singularity is seen to be equivalent to the rank deficiency of two different types of block Toeplitz matrices. This leads to new characterizations of the distance. Upper and lower bounds as well as information about the minimal indices of nearest singular matrix polynomials follow from these characterizations. The distances are also established to be reciprocals of certain generalized structured singular values, thus showing that computing the distance to singularity may be an NP hard problem. Based on the results, a strategy to compute the distance to singularity is devised and implemented via numerical algorithm based on BFGS and Matlab's globalsearch.m. Numerical experiments show that the computed distances compare favourably with values obtained in the literature and the bounds are tight.
The notion of standard triples plays a central role in the theory of matrix polynomials. We study such triples for matrix polynomials P(λ) with structure S, where S is the Hermitian, symmetric, ...★-even, ★-odd, ★-palindromic or ★-antipalindromic structure (with ★=∗,T). We introduce the notion of S-structured standard triple. With the exception of T-(anti)palindromic matrix polynomials of even degree with both -1 and 1 as eigenvalues, we show that P(λ) has structure S if and only if P(λ) admits an S-structured standard triple, and moreover that every standard triple of a matrix polynomial with structure S is S-structured. We investigate the important special case of S-structured Jordan triples.
The standard way of solving a polynomial eigenvalue problem associated with a matrix polynomial starts by embedding the matrix coefficients of the polynomial into a matrix pencil, known as a strong ...linearization. This process transforms the problem into an equivalent generalized eigenvalue problem. However, there are some situations in which is more convenient to replace linearizations by other low degree matrix polynomials. This has motivated the idea of a strong ℓ-ification of a matrix polynomial, which is a matrix polynomial of degree at most ℓ having the same finite and infinite elementary divisors, and the same numbers of left and right minimal indices as the original matrix polynomial. We present in this work a novel method for constructing strong ℓ-ifications of matrix polynomials of size m×n and grade d when ℓ<d, and ℓ divides nd or md. This method is based on a family called “strong block minimal bases matrix polynomials”, and relies heavily on properties of dual minimal bases. We show how strong block minimal bases ℓ-ifications can be constructed from the coefficients of a given matrix polynomial P(λ). We also show that these ℓ-ifications satisfy many desirable properties for numerical applications: they are strong ℓ-ifications regardless of whether P(λ) is regular or singular, the minimal indices of the ℓ-ifications are related to those of P(λ) via constant uniform shifts, and eigenvectors and minimal bases of P(λ) can be recovered from those of any of the strong block minimal bases ℓ-ifications. In the special case where ℓ divides d, we introduce a subfamily of strong block minimal bases matrix polynomials named “block Kronecker matrix polynomials”, which is shown to be a fruitful source of companion ℓ-ifications.
Many applications give rise to nonlinear eigenvalue problems with an underlying structured matrix polynomial. In this paper several useful classes of structured polynomials (e.g., palindromic, even, ...odd) are identified and the relationships between them explored. A special class of linearizations which reflect the structure of these polynomials, and therefore preserve symmetries in their spectra, is introduced and investigated. We analyze the existence and uniqueness of such linearizations and show how they may be systematically constructed.
In this paper, new conditions of stability and stabilization are proposed for periodic piecewise linear systems. A continuous Lyapunov function is constructed with a time-dependent homogeneous ...Lyapunov matrix polynomial. The exponential stability problem is studied first using square matricial representation and sum of squares form of homogeneous matrix polynomial. Constraints on the exponential order of each subsystem used in previous work are relaxed. State-feedback controllers with time-varying polynomial controller gain are designed to stabilize an unstable periodic piecewise system. The proposed stabilizing controller can be solved directly and effectively, which is applicable to more general situations than those previously covered. Numerical examples are given to illustrate the effectiveness of the proposed method.
We discuss Möbius transformations for general matrix polynomials over arbitrary fields, analyzing their influence on regularity, rank, determinant, constructs such as compound matrices, and on ...structural features including sparsity and symmetry. Results on the preservation of spectral information contained in elementary divisors, partial multiplicity sequences, invariant pairs, and minimal indices are presented. The effect on canonical forms such as Smith forms and local Smith forms, on relationships of strict equivalence and spectral equivalence, and on the property of being a linearization or quadratification are investigated. We show that many important transformations are special instances of Möbius transformations, and analyze a Möbius connection between alternating and palindromic matrix polynomials. Finally, the use of Möbius transformations in solving polynomial inverse eigenproblems is illustrated.
In this article, we propose a quasi-Newton algorithm to solve a matrix polynomial equation, which can be seen as a generalization of the algorithm of the same type to solve the matrix quadratic ...equation proposed in Macías et al. (2016). The proposed algorithm reduces the computational cost of the Newton–Schur method traditionally used to solve this type of equations. We show that this algorithm is local and even quadratically convergent. Finally, we present numerical experiments that ratify the theoretical results developed.
•Two new Taylor algorithms for the computation of the matrix exponential are proposed.•They are based on matrix polynomial evaluation methods more efficient than Paterson–Stockmeyer method.•In tests ...they showed higher efficiency and accuracy than state-of-the-art Padé based methods.•They also showed a higher efficiency than state-of-the-art Taylor methods, and higher accuracy in some tests.
This paper presents new Taylor algorithms for the computation of the matrix exponential based on recent new matrix polynomial evaluation methods. Those methods are more efficient than the well known Paterson–Stockmeyer method. The cost of the proposed algorithms is reduced with respect to previous algorithms based on Taylor approximations. Tests have been performed to compare the MATLAB implementations of the new algorithms to a state-of-the-art Padé algorithm for the computation of the matrix exponential, providing higher accuracy and cost performances.
Let \(\mathbb{C}^{m\times m}\) be the set of all \(m\times m\) matrices whose entries are in \(\mathbb{C},\) the set of complex numbers. Then \(P(z):=\sum\limits_{j=0}^nA_jz^j,\) \(A_j\in ...\mathbb{C}^{m\times m},\) \(0\leq j\leq n\) is called a matrix polynomial. If \(A_{n}\neq 0\), then \(P(z)\) is said to be a matrix polynomial of degree \(n\). In this paper we prove some results for the bound estimates of the eigenvalues of some lacunary type of matrix polynomials.
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