Complex Uncertain Variable

In the field of electrical engineering, information science, fluid mechanics and soon, some quantities are usually modeled by complex numbers, such as the alternatingcurrents, periodic signals and two-dimensional potential flow. Because of lack of exper-imental conditions, measuring methods or something else, these quantities are usuallyuncertain, however, they are still needed to use. Thus we only can determine them bysubjective methods. Complex random variable and fuzzy complex number have beenemployed to model these quantities. However, a lot of survey show that probability the-ory and fuzzy theory are not enough for all the problems. In2007, Liu proposed anuncertainty theory, which is a new tool for studying uncertain phenomena. Under theframework of uncertainty theory, this dissertation proposes the concept of complex un-certain variable and studies some properties, such as independence, distribution, expectedvalue and variance.Firstly, we introduce some basic concepts in uncertainty theory. Then we prove atheorem on uncertain measure, verify that a product uncertain measure is an uncertainmeasure and give a theorem on null set in a product uncertainty space. Next, we proposethe concept of complex uncertain variable, its expected value and variance, study theirproperties, discuss the independence of complex uncertain variables and study the rela-tionship between complex uncertain variable and uncertain variable. At last, we give asufcient and necessary condition of complex uncertainty distribution. This dissertationcontributes to the research field of uncertainty theory in the following aspects:(a) a def-inition of complex uncertain variable is proposed to describe complex-valued quantities.(b) a distribution of complex uncertain variable is given, and a sufcient and necessarycondition of complex uncertainty distribution is given.(c) the independence of complexuncertainty variables is given and some properties are studied.(d) the expected value ofcomplex uncertain variable is given and some properties of expected value are studied.