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.