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Analytical Techniques for Ultra-Wideband Signals
Published in James D. Taylor, Introduction to Ultra-Wideband Radar Systems, 2020
Muriladhar Rangaswamy, Tapan K. Sarkar
The Laplace transform has a long history of application to problems of electrical engineering (EE) and is perhaps the mathematical signature of the electrical engineer. It changes some of the most important differential equations of physics into algebraic equations, which are generally easier to solve. The connection between the Fourier transform used in EE and Laplace transform is intimate, but they are not equivalent. The Fourier transform is useful in finding the steady-state output of a linear circuit in response to a periodic input, the Laplace transform can provide both the steady-state and transient responses for periodic and nonperiodic inputs. The two-sided Laplace transform can be used in the analysis of linear time-invariant systems specified by constant coefficient differential equations. An inadequate knowledge of fundamental mathematics may lead one to conclude that time domain solutions contain more information and are better than frequency domain techniques or vice versa.
The Random Variable
Published in X. Rong Li, Probability, Random Signals, and Statistics, 2017
where two-sided Laplace transform is defined exactly the same as the Laplace transform except that the integral is over the entire real line. Since the characteristic (or moment generating) function and the PDF are a Fourier (or two-sided Laplace) transform pair, they carry the same information about the RV: they are both complete descriptions of the RV.
Solving optimal control problems of the time-delayed systems by a neural network framework
Published in Connection Science, 2019
Alireza Nazemi, Ensieh Fayyazi, Marzieh Mortezaee
Consider the time-delayed optimal control problem (Lee, 1993) subject to Here , and are the state, control and unknown parameter vectors, respectively, represents the terminal inequality constraints, denotes the condition of the variable concerned at the end time and σ is the delay-time associated with the state vector x. For the sake of simplicity, we will confine our discussion to the case of a single time delay σ. However, all the results can be extended in a straightforward manner to the case of multiple time delays. The time-delayed optimal control problem (1)–(4) can be transformed to one without a time-delayed argument using the following approximation scheme. Let be a two-sided Laplace transform (Pol & Bremmer, 1955) of : Here is defined over the strip of convergence , where denotes the real part of s, and denotes a two-sided Laplace transformation. The two-sided Laplace transform is used because for .
Risk preference evaluation – a fourth dimension of the application of the Laplace transform
Published in International Journal of Production Research, 2018
So, our conclusion is that Equation (22) produces the value of the CME, when we interpret as the bilateral (two-sided) Laplace transform with the frequency s exchanged for the absolute risk aversion γ.