Signal and System as Vectors
Jin Keun Seo, Eung Je Woo · 2012
To solve forward and inverse problems, this chapter need mathematical tools to formulate them. Since it is convenient to use vectors to represent multiple system parameters, variables, inputs (or excitations) and outputs (or measurements), the chapter first reviews vector spaces. Then, the chapter introduces vector calculus to express interrelations among them based on the underlying physical principles. To solve a nonlinear inverse problem, the chapter often linearizes an associated forward problem to utilize the numerous mathematical tools of the linear system. After introducing such an approximation method, it reviews mathematical techniques to deal with a linear system of equations and linear transformations. In a nonlinear inverse problem, the chapter adopts a linearization approach in its solution-seeking process. It also reviews Taylor’s expansion as a tool to perform such an approximation. The chapter introduces the Fourier transform as a linear transform since it is widely used in the inverse problem area. Controlled Vocabulary Terms discrete Fourier transforms; linear systems; vectors