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 alpha and beta. 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 Lambda = beta/alpha or B = alpha/beta; 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.