scholarly journals A consistent estimator for spectral density matrix of a discrete time periodically correlated process

2018 ◽  
Vol 38 (1) ◽  
pp. 225-242
Author(s):  
Majid Azimmohseni ◽  
Ahmad Reza Soltani ◽  
Mahnaz Khalafi ◽  
Naeemeh Akbari Ghalesary
Keyword(s):  
Density Matrix ◽  
Spectral Density ◽  
Discrete Time ◽  
Simulated Data ◽  
Wishart Distribution ◽  
Consistent Estimator ◽  
Finite Linear Combination ◽  
Asymptotic Consistency ◽  
Spectral Density Matrix ◽  
Finite Linear

In this article, we introduce a weighted periodogram in the class of smoothed periodograms as a consistent estimator for the spectral density matrix of a periodically correlated process. We derive its limiting distribution that appears to be a certain finite linear combination of Wishart distribution. We also provide numerical derivations for our smoothed periodogram and exhibit its asymptotic consistency using simulated data.

scholarly journals Modal Participation Estimated from the Response Correlation Matrix

2019 ◽  
Vol 2019 ◽  
pp. 1-10
Author(s):  
Rune Brincker ◽  
Sandro D. R. Amador ◽  
Martin Juul ◽  
Manuel Lopez-Aenelle
Keyword(s):  
Density Matrix ◽  
Spectral Density ◽  
Correlation Matrix ◽  
Analytical Form ◽  
Mode Shapes ◽  
Transformation Matrices ◽  
Spectral Density Matrix ◽  
Spectral Density Functions ◽  
Input Load ◽  
The Time Domain

In this paper, we are considering the case of estimating the modal participation vectors from the operating response of a structure. Normally, this is done using a fitting technique either in the time domain using the correlation function matrix or in the frequency domain using the spectral density matrix. In this paper, a more simple approach is proposed based on estimating the modal participation from the correlation matrix of the operating responses. For the case of general damping, it is shown how the response correlation matrix is formed by the mode shape matrix and two transformation matrices T1 and T1 that contain information about the modal parameters, the generalized modal masses, and the input load spectral density matrix Gx. For the case of real mode shapes, it is shown how the response correlation matrix can be given a simple analytical form where the corresponding real modal participation vectors can be obtained in a simple way. Finally, it is shown how the real version of the modal participation vectors can be used to synthesize empirical spectral density functions.

scholarly journals Some Elements of Operational Modal Analysis

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Rune Brincker
Keyword(s):  
Density Matrix ◽  
Spectral Density ◽  
Modal Analysis ◽  
Mode Shape ◽  
Fourier Integral ◽  
Operational Modal Analysis ◽  
Difficult Case ◽  
Domain Identification ◽  
Spectral Density Matrix ◽  
Random Responses

This paper gives an overview of the main components of operational modal analysis (OMA) and can serve as a tutorial for research oriented OMA applications. The paper gives a short introduction to the modeling of random responses and to the transforms often used in OMA such as the Fourier series, the Fourier integral, the Laplace transform, and the Z-transform. Then the paper introduces the spectral density matrix of the random responses and presents the theoretical solutions for correlation function and spectral density matrix under white noise loading. Some important guidelines for testing are mentioned and the most common techniques for signal processing of the operating signals are presented. The algorithms of some of the commonly used time domain and frequency domain identification techniques are presented and finally some issues are discussed such as mode shape scaling, and mode shape expansion. The different techniques are illustrated on the difficult case of identifying the three first closely spaced modes of the Heritage Court Tower building.

Complex Coherence Constraint for the Definition of the Spectral Density Matrix

2018 ◽  
Vol 61 (1) ◽  
pp. 7-19
Author(s):  
Zhihua Liu ◽  
Chenguang Cai ◽  
Yan Xia ◽  
Ming Yang
Keyword(s):  
Density Matrix ◽  
Spectral Density ◽  
Random Vibration ◽  
Multiple Input Multiple Output ◽  
New Approach ◽  
Spectral Density Matrix ◽  
Multiple Input ◽  
Input Multiple Output ◽  
Recursive Formulas ◽  
Definition Of

Abstract The cross spectral density (CSD) for a multiple-input/multiple-output (MIMO) random vibration is typically defined by the complex coherence consisting of the modulus and the phase. The purpose of this paper is to present a constraint for the complex coherence to allow the CSD to be defined more easily. The study of the complex coherence constraint is based on Cholesky decomposition of the spectral density matrix (SDM). The complex coherence must be bounded in the interior or on the boundary of a constraint circle to ensure a physically realizable random vibration. This paper proposes a new approach to define the complex coherences of the SDM by using recursive formulas based on the constraint circle.

Solutions of the Singular Stochastic Regulator Problem

1973 ◽  
Vol 95 (4) ◽  
pp. 414-417 ◽  
Author(s):  
M. F. Hutton
Keyword(s):  
Steady State ◽  
Density Matrix ◽  
Spectral Density ◽  
Positive Definite Matrix ◽  
Positive Definite ◽  
Linear Process ◽  
Weighting Matrix ◽  
Spectral Density Matrix ◽  
State Expectation ◽  
Observation Noise

The limiting form of the optimum stochastic regulator is determined which minimizes the steady-state expectation, Es{x′Qx + u′Ru}, for the linear process, x˙ = Ax˙ + Bu + Gv, given noisy observations y = Hx + w (with v and w being independent white noise processes) when either the control weighting matrix, R, or the spectral density matrix, W, of the observation noise, w, is singular. It is shown that as R tends from a positive-definite matrix to a non-negative definite matrix, the optimum regulator can be synthesized by a system using at most n − ks integrators, where n is the order of the system and ks equals the rank of B minus the rank of BR. Similarly, when W tends from a positive-definite matrix to a non-negative definite matrix, the optimum regulator can sometimes be synthesized by a system using at most n − rs integrators, where rs equals the rank of H minus the rank of WH. The structure of the regulator is given for each of these cases.

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