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Positive Time–Frequency Distributions
Published in Antonia Papandreou-Suppappola, Applications in Time-Frequency Signal Processing, 2018
Most would agree that this property, namely, weak finite support, is a reasonable requirement; and we point out that, like the Wigner distribution, the positive distributions also satisfy this finite support property. Let us take our considerations further. Suppose now that a finite-duration signal is also zero during brief intervals over its duration. For example, suppose that we record someone counting numbers, “one” … “two” … etc., and suppose further that in between each spoken digit, the speaker pauses briefly so that no sound is made. Thus when we look at the instantaneous power, we see that it is large during the utterance of each digit, and it is zero in between each sound. Further, it is zero before and after the speaker started and ended the counting. Is it not reasonable to expect that the distribution should be zero not only before and after the speaker started counting but also during the brief intervals between sounds, when nothing was being said? (Let us ignore, as is done when making the usual argument for finite support, that some noise will occur in our recording, so that we can focus on the essence of the argument, namely, that when no signal power exists, is it not reasonable to expect that the distribution reflect this fact?) This is the essence of the strong finite support property. The Wigner distribution and the spectrogram do not satisfy this property. Thus, for example, in the recorded speech example, the Wigner distribution is zero prior to the start and after the end of the entire utterance, but it is not zero during the intervals of silence.
Introduction
Published in Hebertt Sira-Ramírez, Sunil K. Agrawal, Differentially Flat Systems, 2018
Hebertt Sira-Ramírez, Sunil K. Agrawal
Flat systems, or differentially flat systems, exhibit a property which is highly reminiscent of the property exhibited by the previous elementary under-determined algebraic example. It is not surprising that such a property may exist in controlled systems as they constitute, generally speaking, under-determined systems of equations themselves. The number of control inputs being responsible for the under-determination. It is clear that the invertibility of A and the full rank of the matrix B in the linear algebraic example will find much more stringent, but still natural, conditions in the case of controlled differential equations. In essence, these conditions will be summarized by the controllability property of the given system.
Approximation Methods
Published in Edward F. Kuester, David C. Chang, Electromagnetic Boundary Problems, 2015
Edward F. Kuester, David C. Chang
are all equivalent ways of expressing the functional F if f is the solution of (8.29). In essence, this property is the analog of Lorentz reciprocity. Indeed, Lorentz reciprocity is (8.48) when L is chosen to be the Maxwell differential operator. To see this, write Maxwell’s curl equations in the matrix-operator form () Lf≡Df−ωMf=g
First-passage probability analysis of Wiener process using different methods and its applications in the evaluation of structural durability degradation
Published in European Journal of Environmental and Civil Engineering, 2021
Zhenhao Zhang, Mingliao Zhou, Ming Fang
The symmetry-based method and Markov property-based method solve the problem in a completely different way from the Poisson process method. Depending on the utilisation of the special properties of the Wiener process, these two methods can derive the analytical solution of the first-passage probability of the Wiener process without the use of any mathematical assumptions. And the two methods share one thing in common: the full probability equation and the one-dimensional probability distribution of the Wiener process are the important theoretical tools in utilising the special properties of the Wiener process. The symmetry-based analytical method yields fully exact solution. And the Markov property-based analytical method should in essence produce exact solution. However, the calculations of this method involve the integration of non-elementary functions and require the use of numerical calculations. Therefore, the solution produced by this method is not entirely exact, but it is a high approximation of the exact solution, which can be seen from the comparison of the Equations (43) and (31). Although the two methods give almost the same result, they show different ways to get the first-passage probability of the Wiener process. And additionally, the Markov property-based method may provide enlightenment of solving the first-passage problem of a Markov process in a new way. Finally, it can be seen that the results of the symmetry-based method and Markov property-based method are explicit analytical expression, so it is convenient and accurate to apply the results to practical engineering.
New quantitative risk prediction method of deepwater blowout: Case study of Macondo well accident
Published in Energy Sources, Part A: Recovery, Utilization, and Environmental Effects, 2021
Cheng Li, Yequan Jin, Zhichuan Guan, Zhigang Dang, Jingyang Jin, Bo Zhang
Bayesian network is one of the most widely used risk assessment methods (Kabir, Taleb-Berrouane, and Papadopoulos et al. 2019). This method uses the statistical probability of an accident in the past (prior probability) to infer the probability of this accident in the future (posterior probability) (Cai et al. 2013a). This method has strong objectivity, which breaks through the description of 0 and 1 in the accident occurrence of the previous methods, and has advantages in real-time risk assessment of complex systems (Bhandari et al. 2015). However, the essence of this method is still the statistics of accident frequency (Khakzad, Khan, and Amyotte 2013). Moreover, the prior probability reflects the accidents frequency under the influence of previous environment; thus, the results may ignore the influence of the current environmental characteristics.
Microstructure tuning enables synergistic improvements in strength and ductility of wire-arc additive manufactured commercial Al-Zn-Mg-Cu alloys
Published in Virtual and Physical Prototyping, 2022
Yueling Guo, Qifei Han, Wenjun Lu, Fengchao An, Jinlong Hu, Yangyu Yan, Changmeng Liu
Controlling metallurgical defects and tuning microstructures are of the essence for mechanical property improvement. In contrast with laser additive manufacturing (Martin et al. 2017), severe hot cracking is not frequently found in WAAM 7xxx alloys, which is associated with its relatively low temperature gradient of about 103∼104 K/m (Ding et al. 2011; Klein et al. 2021b). Unfortunately, the gas porosity caused by the rapidly decreased hydrogen solubility in aluminium is indeed problematic for WAAM-processed aluminium alloys, and the mechanical property is deteriorated resultantly (Hauser et al. 2021; Zhong et al. 2019). Still, it is an open question that how gas porosity affects the fracture behaviour of WAAM-processed Al-Zn-Mg-Cu alloys.