Estimation of the Mathematical Parameters of Double-Exponential Pulses Using the Nelder–Mead Algorithm

Transient pulses for electromagnetic compatibility problems, such as the high-altitude electromagnetic pulse and ultrawideband pulses, are often described by a double-exponential pulse. Such a pulse shape is specified physically by the three characteristic parameters rise time tr, pulsewidth t_fwhm (full-width at half-maximum), and maximum amplitude E_max. The mathematical description is a double-exponential function with the parameters α, β, and E₀. In practice, it is often necessary to transform the two groups of parameters into each other. This paper shows a novel relationship between the physical parameters tr and t_fwhm on the one hand and the mathematical parameters α and β on the other. It is shown that the least-squares method in combination with the Nelder-Mead simplex algorithm is appropriate to determine an approximate closed-form formula between these parameters. Therefore, the extensive analysis of double-exponential pulses is possible in a considerably shorter computation time. The overall approximation error is less than 3.8%.