Question

A frequently used time-domain waveform used to represent an electromagnetic pulse (emp) from a nuclear detonation is the so-called double-exponential pulse given by

x(t) = 1/(β-α) (e^(-αt) - e^(βt))

determine the fourier transform of this pulse
[1/{αβ+j(α+β)ω-ω²}]

Answer 1

The Fourier transform of a double-exponential pulse, like that representing an EMP from a nuclear detonation, involves complex integrals using the Fourier analysis techniques and results in the frequency domain representation of the signal.

Step-by-step explanation:

The question pertains to taking the Fourier transform of a double-exponential pulse, which is a representation of an electromagnetic pulse (EMP) commonly associated with nuclear detonation. The double-exponential pulse is provided as x(t) = 1/(β-α) (e(-αt) - e(βt)). Determining the Fourier transform of this pulse involves complex integrals and the application of Fourier analysis techniques.

To begin, we use the definition of the Fourier transform, which transforms a time domain signal into its frequency domain representation. The general form of the Fourier transform of a signal x(t) is given by X(ω) = ∫ x(t)e(-jωt)dt, where ω is the angular frequency in radians per second, and j is the imaginary unit.

When we apply the Fourier transform to the given double-exponential function, we exploit properties of the Fourier transform, such as linearity and the transforms of elemental exponentials. After integrating and simplifying, we would obtain the given result X(ω) = 1/{αβ + j(α + β)ω - ω2}, confirming the validity of the expression for the Fourier transform of the EMP waveform.

This result is important in understanding the frequency characteristics of the EMP, which is critical for various applications in electromagnetic theory and signal processing.