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Direction-of-Arrival Estimation in Mobile Communication Environments
Published in Lal Chand Godara, Handbook of Antennas in Wireless Communications, 2018
Mats Viberg, Thomas Svantesson
The derivation of the measured voltages considered only one incident wave. When several waves are incident on the array, the superposition principle can be applied if the antenna elements and the receiver are linear. To simplify the notation in the following analysis, the waves are assumed to arrive in the xy plane (θ = 90°) (see Fig. 20.1). The model when p uniform plane waves are incident on an array of m elements can thus be written as () x(t)=∑l=1pa(ϕl)sl(t)=A(ϕ)s(t)
Systems Theory and Optimal Control
Published in Larry W. Mays, Optimal Control of Hydrosystems, 1997
The outputs from a static system only depend on the current values of the inputs, whereas outputs from a dynamic system depend on the current and the previous values of the inputs. The inputs and outputs for a linear system satisfy the superposition theorem, whereas a system is nonlinear if it does not satisfy the superposition theorem. The superposition theorem includes two parts, additivity and homogeneity. Additivity is that the output due to the sum of inputs is equal to the sum of the outputs due to each of the inputs or ϕ(∑n=1Nun)=(∑n=1Nϕun). Homogeneity is ϕ(cun) = c(ϕun) in which c is a constant.
Two-Dimensional Steady-State Conduction
Published in Randall F. Barron, Gregory F. Nellis, Cryogenic Heat Transfer, 2017
Randall F. Barron, Gregory F. Nellis
The governing differential equations in many heat transfer problems are linear and homogeneous; therefore, we may add together (superimpose) solutions to simple problems to obtain the solution for more complicated ones. This property is used in separation of variables when the solutions associated with each of the eigenfunctions are added together in order to obtain a series solution to the problem. Superposition uses this property of linear systems to determine the solution to a complex problem by breaking it into several, simpler problems that are solved individually and then added together. Care must be taken to ensure that the boundary conditions for the individual problems properly add to satisfy the desired boundary condition.
Volterra and Wiener Model Based Temporally and Spatio-Temporally Coupled Nonlinear System Identification: A Synthesized Review
Published in IETE Technical Review, 2021
Saurav Gupta, Ajit Kumar Sahoo, Upendra Kumar Sahoo
The linearThis article has been corrected with minor changes. These changes do not impact the academic content of the article. approach to describe and analyze system dynamics has dominated systems theory for a long period of time due to its advantages of principles of superposition and proportionality (homogeneity) involved [1]. However, numerous physical systems are neither linear nor even nearly linear, and fall outside the traditional linear way of analysis. These are very common in our day-today life, hence are considered by many researchers, engineers, physicist, mathematicians and scientists. These systems exhibit nonlinearity to some extent in the input–output relation. To study the nonlinear phenomena involved in these systems, a new apparatus of theoretical and practical method of analysis has been developed. It overcomes the difficulties brought by the nonlinear approach, and the fact that superposition and homogeneity laws are not applicable to nonlinear systems [1, 2]. An important part of nonlinear system theory is system modeling which is quite arduous, but is an essential step in control, analysis and prediction of any process behavior. Modeling often provides a better insight into the system behavior and can also improve plant control performance. Hence, it is continuously getting importance in many control applications. Finding mathematical models which can describe physical phenomena was the principal objective of many research studies in the last few decades. Extensive efforts have been devoted by many researchers to determine ordinary differential equation models for different classes of nonlinear systems.