Shape Properties of Pulses Described by Double Exponential Function and Its Modified Forms

Double exponential function and its modified forms are widely used in high-power electromagnetics such as high-altitude electromagnetic pulse and ultrawide-band pulse study. Physical parameters of the pulse, typically the rise time t_r, full width at half maximum t_w, and/or fall time t_f , usually need to be transformed into mathematical characteristic parameters of the functions, commonly denoted as α and β. This paper discusses the dependences of pulse shape properties, represented by ratios of t_w/ t_r and t_f/ t_r , on a dimensionless parameter A = β/α or B = α/β; and focuses on their limit correlations associated with the mathematical forms. It has been proven that pulses with t_w/ t_r <; 4.29 cannot be expressed by the commonly used difference of double exponentials function. This limit can be mitigated partially by the latest proposed p-power of double exponentials function with a well-chosen p parameter. A novel form, difference of double Gaussian functions is also proposed to describe pulses with low t_w/ t_r ratios better. Quotient of double exponentials, however, is shown to be the most flexible function for describing transient pulses with arbitrary t_w/ t_r ratios, despite of its intrinsic drawbacks. All these functions are applied for several examples and further compared in both time and frequency domains.

I. Introduction

Double exponential function and its modified forms are widely used to describe high-power electromagnetic environments and their driven sources for simulation, such as high-altitude electromagnetic pulse (HEMP) [1]–[4] and ultrawide-band (UWB) pulses. For instance, difference of double exponentials (DEXP) and quotient of double exponentials (QEXP) have been used as a mathematical description of HEMP environments [1]–[4], as well as pulse shapes of the driven gamma-ray source in numerical simulations [5], [6]. A modified description, namely -power of double exponentials (PEXP) has been proposed in recent years to improve the description of HEMP [2], lightning discharge [7], and near-filed electrostatic discharge environments [8]. Physical parameters of these EMP-like pulses, typically the rise time , full width at half maximum (FWHM) , and/or fall time , usually need to be transformed with high precisions into mathematical characteristic parameters of the functions, commonly denoted as α and β. The least squares and Nelder–Mead algorithms have been applied for DEXP pulses to estimate the physical parameters from mathematical ones, and two complicated functions, sum of four exponential functions, are chosen to get approximate and values in a small ranging area of (α, β) [9], which cannot realize the inverse transform. Through numerical calculations and statistical means, novel correlations between the two groups of parameters, i.e., several sectional linear fit functions with gradational slopes between /, , and β/α, are established for a far-ranging area of (α, β) [10], [11]. As many as fifteen linear functions are formulated to express a correlation with typical overall fitting errors < 0.5% [11]. A limit property that a pulse shape can be fitted with DEXP function only for / > 4.291 has been noted [11], but without an explanation or a solution for pulses with lower / ratios.