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Global, explicit representations of nonlinearities are desirable when implementing nonlinear Wave Digital (WD) structures, as they free us from the burden of managing look-up tables, performing data interpolation and/or using iterative solvers. In this paper we present a method that, starting from certain parameterized PieceWise-Linear (PWL) curves in the Kirchhoff domain, allows us to express them...
Sallen-Key filters are widespread in audio circuits. Therefore, accurate and efficient digital models of such filters are highly desirable in audio Virtual Analog applications. In this paper, we will discuss a possible strategy, based on Wave Digital Filters (WDFs), for implementing all the analog filters described in the historical 1955 manuscript by Sallen and Key. In particular, we will group the...
Accommodating multiple nonlinearities in a modular fashion and with no use of delay elements is a relevant unsolved problem in the literature on Wave Digital (WD) networks. In this work we present a method for adapting instantaneous NonLinear (NL) bipoles characterized by monotonic increasing curves. The method relies on the fact that the characteristic of a NL bipole can be described by a line, which...
The idea of random sets is introduced by showing that three different extensions to the classical probabilistic information lead to an equivalent mathematical structure. A formal definition is then given, followed by different ways to describe the same information. A random set gives upper and lower bounds on the probability of subsets in a space of events. These non-additive and monotone (with...
The basic definitions of a probability space are briefly reviewed, thus introducing the notation useful for the theoretical developments presented in the book. Particular attention is given to continuous and discrete random variables and to the concept of expectation of a random variable, defined through both Lebesque and Stieltjes integrals. The theory is extended to joint probability spaces and...
This chapter investigates the concepts of weak and strong inclusion between random sets or relations. Approximations to random sets and relations are constructed by including given random sets or relations into random sets and relations that are easier to deal with from a computational viewpoint: these approximations yield validated outer bounds on the probability of events. Finally, mappings of random...
The theory of probability provides no straightforward answer to the problem of combining two probability measures on the same space, because it is not a problem of mathematics, but of judgment. As a consequence, random set theory too does not provide any prescriptive method for combining two random sets. In Section 6.1, some possible ways of combining or updating information on the same space are...
In this chapter, the notions introduced in Chapter 3 are extended to the case in which the uncertain information is assigned by means of marginal random sets on several different spaces, Si, or by means of a random relation on the Cartesian product S = ×Si. The multifold concept of independence is firstly introduced within the general framework of imprecise probabilities,...
Before embarking on studying the following chapters, motivations are provided as to why random sets are useful to formalize uncertainty in civil engineering. Pros and cons in using the theory of random sets are contrasted to more familiar theories such as, for example, the theory of random variables.
Taking an engineering, rather than a mathematical, approach, Bounding uncertainty in Civil Engineering - Theoretical Background deals with the mathematical theories that use convex sets of probability distributions to describe the input data and/or the final response of systems. The particular point of view of the authors is centered on the applications to civil engineering problems, and the theory...
Wave Digital Structures (WDS), with their inherent stability and robustness, would be particularly suitable for nonlinear (NL) circuit modeling in Virtual Analog applications, if it were possible for them to accommodate multi-port nonlinearities. In this work we present a method for modeling a rather general class of multi-port NL elements in the WD domain, which are obtained as the interconnection...
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