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Circularly-Symmetric Complex Normal Ratio Distribution for Scalar Transmissibility Functions. Part I: Fundamentals

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Circularly-Symmetric Complex Normal Ratio Distribution for Scalar Transmissibility Functions. Part I: Fundamentals Mechanical Systems and Signal Processing 80 (2016) 58–77 Contents lists available at ScienceDirect Mechanical Systems and Signal Processing journal homepage: www.elsevier.com/locate/ymssp Circularly-symmetric complex normal ratio distribution for scalar transmissibility functions. Part I: Fundamentals Wang-Ji Yan, Wei-Xin Ren n Department of Civil Engineering, Hefei University of Technology, Hefei, Anhui Province 23009, China article info abstract Article history: Recent advances in signal processing and structural dynamics have spurred the adoption Received 28 May 2015 of transmissibility functions in academia and industry alike. Due to the inherent ran- Received in revised form domness of measurement and variability of environmental conditions, uncertainty 3 February 2016 impacts its applications. This study is focused on statistical inference for raw scalar Accepted 27 February 2016 transmissibility functions modeled as complex ratio random variables. The goal is Available online 22 March 2016 achieved through companion papers. This paper (Part I) is dedicated to dealing with a Keywords: formal mathematical proof. New theorems on multivariate circularly-symmetric complex Transmissibility normal ratio distribution are proved on the basis of principle of probabilistic transfor- Uncertainty quantification mation of continuous random vectors. The closed-form distributional formulas for mul- Ratio distribution tivariate ratios of correlated circularly-symmetric complex normal random variables are Multivariate statistics Complex statistics analytically derived. Afterwards, several properties are deduced as corollaries and lemmas to the new theorems. Monte Carlo simulation (MCS) is utilized to verify the accuracy of some representative cases. This work lays the mathematical groundwork to find prob- abilistic models for raw scalar transmissibility functions, which are to be expounded in detail in Part II of this study. & 2016 Elsevier Ltd. All rights reserved. 1. Introduction Various signal processing techniques have been introduced into the structural dynamics and structural health mon- itoring (SHM) literature, notably fast Fourier transform (FFT) in the frequency domain for stationary response and wavelet transform in the time-frequency domain for non-stationary response. The application of FFT in the field of structural dynamics has gained substantial interest over the past few decades, and a paradigm in the open literature is transmissibility function. Known as a mathematical representation of the output-to-output relationship in the frequency domain, trans- missibility function is used to characterize the dynamics of a system [1]. There are two kinds of definitions for transmissibility functions in the frequency domain [2]. The first one is a scalar transmissibility function [3] defined as the ratio of an arbitrary response to a reference response, while the second one is a transmissibility matrix [4–6] which formulates the relationship between two sets of response vectors. The application of transmissibility function has been attracting much attention, culminating in various publications. For example, it has been widely regarded to be a good candidate for structural damage detection by different research groups including the team of Worden [7–12], Schulz [13,14], Maia [15] and Adams [16–18], etc. In addition, several new strategies were also proposed n Corresponding author. Tel.: þ86 551 62901432. E-mail address: [email protected] (W.-X. Ren). http://dx.doi.org/10.1016/j.ymssp.2016.02.052 0888-3270/& 2016 Elsevier Ltd. All rights reserved. W.-J. Yan, W.-X. Ren / Mechanical Systems and Signal Processing 80 (2016) 58–77 59 more recently [19–22]. A summary related to the subject of structural damage detection using transmissibility functions can be found in [22]. The concept of transmissibility was also employed for operational modal analysis (OMA) by Devriendt and Guillaume [3,23]. This work was believed to be very appealing due to its insensitivity to colored excitation by a recent literature review [24]. Following this original idea, new approaches [25–27] have been proposed to seek improvements in terms of real applications. The approach to identify modal parameters from scalar transmissibility functions was also generalized to transmissibility matrix [28–30]. Power spectral density transmissibility (PSDT) was proposed by authors for OMA [31,32]. PSDT was further expanded to a continuous wavelet transmissibility (CWTR) based algorithm [33] and a singular value decomposition (SVD)-driven algorithm [34]. Also, transmissibility functions were proved to be a good alternative in model updating when the excitation is not measured [35,36]. In addition to the work mentioned above, transmissibility function has extensive applications in other subjects such as evaluation of FRFs [2] and transfer path analysis [37], etc. Most of the efforts aforementioned are based on deterministic assumptions. However, there are multiple uncertainties (e.g. the inherent randomness of measurement, variability of environmental condition and estimation error, etc.) existing in transmissibility measurements, which inevitably introduce variability and lead to misinterpreted results. Therefore, there is a need to develop theoretically sound approaches to assess the uncertainty behavior of transmissibility functions. The most serious difficulty in addressing the issue is on how to explore their proper probability density function (PDF). Recently, Mao and Todd presented a new probabilistic model for quantifying uncertainty in the estimation of a scalar transmissibility function via a Chi-square bivariate approach [1,38]. Test results could be compared to baseline statistics so as to find sta- tistical damage significance under certain confidence level [1,21]. Conventionally, a scalar transmissibility function is estimated via power spectral density (PSD) which requires averaging, overlapping and windowing. PSD estimation method such as Welch’s technique has been widely applied for more than 30 years, but thorough probabilistic analysis of the technique still remains a challenging problem due to its complicated sta- tistical structure that is dependent on algorithmic parameters (e.g. number of samples by which the sections overlap, FFT length determining the frequencies at which the PSD is estimated and windowing function, etc.). In addition, the prob- abilistic model of a scalar transmissibility function is oriented towards its magnitude, which is constrained to working in the framework of real-valued statistics. A scalar transmissibility function is defined as the ratio of FFT coefficient of two responses, which is a complex-valued random variable composed of both real and imaginary parts which are correlated with each other. More importantly, for a dynamic system with multiple outputs, one can formulate multivariate scalar transmissibility functions (transmissibility vector) given a reference output. In current stage, however, rare statistical model is able to accommodate the correlations among different transmissibility functions. In contrast to the approaches estimating transmissibility functions via PSD, the primary concern of this study is raw scalar transmissibility functions without resorting to any post-processing, thus avoiding selecting algorithmic parameters of PSD estimation. On the basis of raw FFT data, it can fully utilize a one-to-one relationship between the time-domain data and its counterpart in the frequency domain [39]. It has been proved strictly that raw FFT vector at an arbitrary frequency line approximately follows multivariate circularly-symmetric complex normal distribution [40–42]. Therefore, a scalar transmissibility function at an arbitrary frequency line can be modeled as a ratio of two correlated circularly-symmetric complex normal random variables. Correspondingly, we refer to a raw transmissibility vector at an arbitrary frequency line as multivariate circularly-symmetric complex normal ratio distribution. The distributions of ratios of random variables arise in many applied problems such as Mendelian inheritance ratios in genetics, mass to energy ratios in nuclear physics, and inventory ratios in economics, etc. [43]. Two test-statistics com- monly-taught in statistical textbooks (i.e. t-distribution and F-distribution) also fall into this category. As popular statistical models, ratio random variables have been studied extensively by a number of researchers over the past decades [43–51]. Most of the efforts are devoted to real-valued or univariate case, while the distributional properties of multivariate ratios of correlated complex random variables are not well developed. This study aims at providing a rigorous formulation for deriving fundamental statistics of raw scalar transmissibility functions, which is achieved through companion papers. In this paper (Part I), new theorems on multivariate circularly- symmetric complex normal ratio distribution as well as some useful properties deduced as corollaries and lemmas are proved mathematically. Closed-form formulas are derived to compute the PDF of circularly-symmetric complex normal ratio distribution. The marginal PDFs of the real part, the imaginary part, the magnitude and the phase of a univariate ratio random variable are also derived analytically. Main results are verified using Monte Carlo simulation (MCS). Using the theoretical findings of this paper, the probabilistic models will be further developed for scalar transmissibility functions in Part II [52]. Their accuracy will also be verified using simulated and field data from full-scale structures in Part II. 2. Notations and result outlines As will be seen in Sections 4 and 5, the mathematical derivations are very lengthy. However, the results are remarkably simple. For the ease of reading, we summarize the notations of this study and outline the results before giving a formal mathematical proof. Download English Version: https://daneshyari.com/en/article/560074 Download Persian Version: https://daneshyari.com/article/560074 Daneshyari.com.
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