| The concept of fuzzy set was first put forward by professor Zadeh in "Fuzzy Sets" in 1965. Then, in 1972, Chang and Zadeh proposed the concept of fuzzy numbers to study the properties of probability functions. With the development and perfect of theories and applications of fuzzy numbers, it becomes more and more important. For example, In 2009, Wang, Shi and Paul protested the representation of uncertain multichannel digital signal spaces and study of pattern recognition based on metrics and difference values on fuzzy n-cell number space. In 2013, Ramesh and Jena discussed the fuzzy clustering with Gaussian-type membership function.For the convenience of application, the concepts of the triangle fuzzy numbers and trapezoid fuzzy numbers were defined, and the theories and applications of them have developed very well. However, it is not always suitable that using linear function to repre-sent the membership function of a fuzzy quantity. Although the trapezoid fuzzy numbers and triangle fuzzy numbers can be conveniently used in applications, sometimes, there are some defects in accuracy as they are used to represent some uncertain or imprecise digital signals. For example, if we use triangle fuzzy number u with normal point 172 (unit:cm) and support set [166,178] (unit:cm) to represent fuzzy quantity "the height of man of medium height (in a certain area)", then u(171)=0.83, it means that the degree of membership of a man with height 171 cm belonging to "the height of man of medium height" is only 0.83, which seems a bit small compared with the actual situation. In this paper, we discuss Gaussian type fuzzy numbers, which not only inherit the convenience of trapezoid fuzzy numbers and triangle fuzzy numbers, but also can be used more pre-cisely to represent some uncertain or imprecise digital signals. The specific arrangements of this paper are as follows:In Section 1, we mainly introduce the research background, purpose and meaning.In Section 2, we briefly review some basic notions, definitions and results about fuzzy numbers.In Section 3, first, we give the definition of Gaussian type fuzzy numbers, study their properties and obtain algorithm about Gaussian type fuzzy numbers; and we point out that the usual scalar multiplication which introduced by Zadeh preserve the closeness of the operation, the addition do not preserve the closeness of the operation; Then, we discuss that the addition which introduced by Zadeh of Gaussian type fuzzy numbers preserve the closeness of the operation in which case. Second, for the sake of convenience in applica-tion, we work out the calculation formulas (which can be easy realized by computer pro-gram in applications) of the means, discrete degrees as they are respectively restricted to the Gaussian type fuzzy number space. Thirdly, we define several distances on the Gaus-sian type fuzzy number space and discuss their characteristics and properties, so that we can choose the right one to use quickly. Fourthly, we discuss several orders on Gaussian type fuzzy number space and discuss their properties. Fifth, we work out the calculation formulas of the difference values and approximation relationship as they are respectively restricted to the Gaussian type fuzzy number space for automotive application cases.In Section 4, we mainly give a method of constructing Gaussian type fuzzy numbers to represent uncertain or imprecise digital signal information. Then, we give some exam-ples to demonstrate how to rank and classify for Gaussian type fuzzy number. At last, we give a practical example to demonstrate the application of Gaussian type fuzzy numbers in pattern recognition.In Section 5,we main summarize the article and talk about the future research work. |
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