[53–4] On the Eigen-values of Certain Hermitian Forms

Kac, Mark; Murdock, W. L.; Szegö, Gábor

doi:10.1007/978-1-4612-5785-1_21

Cited by 12 publications

(28 citation statements)

References 29 publications

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“…The pseudospectra of non-Hermitian Toeplitz matrices with variable coefficients were considered in Section 5 of [34] but without any analysis and without noting that the pseudomodes have the form of wave packets. The classic paper on the spectra of variable-coefficient Toeplitz matrices in the Hermitian case is by Kac, Murdock, and Szegö [27].…”

Section: Introductionmentioning

confidence: 99%

Wave packet pseudomodes of twisted Toeplitz matrices

Trefethen

Chapman²

2004

Comm Pure Appl Math

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The pseudospectra of banded, nonsymmetric Toeplitz or circulant matrices with varying coefficients are considered. Such matrices are characterized by a symbol that depends on both position (x) and wave number (k). It is shown that when a certain winding number or twist condition is satisfied, related to Hörmander's commutator condition for partial differential equations, ε-pseudoeigenvectors of such matrices for exponentially small values of ε exist in the form of localized wave packets. The symbol need not be smooth with respect to x, just differentiable at a point (or less).

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Section: Introductionmentioning

confidence: 99%

Wave packet pseudomodes of twisted Toeplitz matrices

Trefethen

Chapman²

2004

Comm Pure Appl Math

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“…The reason relies again in formula (16). In fact for larger g since the point of minimum is unique we recall that Kac et al [31] gave the expression of c 2 as the second derivative of s in the minimum point which has to be positive by local convexity. Therefore, the explanation of the latter phenomenon could be given in terms of c 2 = c 2 (g): if c 2 (g) is positive but rapidly converging to zero as a function of g, then the quantity maxs(x)/mins(x) really captures the conditioning of T n .…”

Section: Analysis Of the Conditioning And Of The Extremal Spectrummentioning

confidence: 97%

Spectral analysis and preconditioning techniques for radial basis function collocation matrices

Cavoretto

Rossi

Donatelli

et al. 2011

Numerical Linear Algebra App

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SUMMARYMeshless collocation methods based on radial basis functions lead to structured linear systems which, for equispaced grid points, have almost a multilevel Toeplitz structure. In particular, if we consider partial differential equations (PDEs) in two dimensions then we find almost (up to a "low-rank" correction given by the boundary conditions) two-level Toeplitz matrices, i.e., block Toeplitz with Toeplitz blocks structures, where both the number of blocks and the block-size grow with the number of collocation points. In [D. Bini, A. De Rossi, B. Gabutti, Linear Algebra Appl., 428 (2008), 508-519] upper bounds for the condition numbers of the Toeplitz matrices approximating a one-dimensional model problem were proved. Here we refine the one-dimensional results, by explaining some numerics reported in the previous paper, and we show a preliminary analysis concerning conditioning, extremal spectral behavior, and global spectral results in the two-dimensional case for the structured part. By exploiting recent tools in the literature, a global distribution theorem in the sense of Weyl is proved also for the complete matrix-sequence, where the low-rank correction due to the boundary conditions is taken into consideration. The provided spectral analysis is then applied to design effective preconditioning techniques in order to overcome the ill-conditioning of the matrices. A wide numerical experimentation, both in the one and two-dimensional case, confirms our analysis and the robustness of the proposed preconditioners.

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“…We now show that Theorem 2 can be used for the SEP evaluation of precoded massive MIMO with correlated antennas. In particular, when the channel covariance matrix Σ is the nonsymmetric Kac-Murdock-Szegö (KMS) matrix that has been used widely in the literature [27]- [34], [51], the (m, n)-th entry of which is denoted by σ(m − n), i.e.,…”

Section: Asymptotic Sep Analysis For Massive Mimo With Toeplitz Comentioning

confidence: 99%

“…The first term in (27) can be replaced by (20). Similarly, Q 2 (·) function also has a very nice formula [54],…”

Section: + 3ηmentioning

confidence: 99%

“…For example, when the user device is unobstructed and the BS is surrounded by local scatterers. To show the effect of fading correlation more explicitly, we specifically consider an exponential correlation model [27]- [34], which is a simplified one-parameter model in practical environment that can capture the main phenomenon of spatial correlation between antennas, especially for a uniform linear array (ULA). To better understand the channel correlation in such massive MIMO fading channels, we consider both the precoder design and the corresponding performance analysis.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Average SEP-Optimal Precoding for Correlated Massive MIMO With ZF Detection: An Asymptotic Analysis

Dong

Zhang

Chen

2019

IEEE Trans. Commun.

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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: limx→∞(1 + 1/x) x = e. 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 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.

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[53–4] On the Eigen-values of Certain Hermitian Forms

Cited by 12 publications

References 29 publications

Wave packet pseudomodes of twisted Toeplitz matrices

Wave packet pseudomodes of twisted Toeplitz matrices

Spectral analysis and preconditioning techniques for radial basis function collocation matrices

Average SEP-Optimal Precoding for Correlated Massive MIMO With ZF Detection: An Asymptotic Analysis

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