After we have understood the Fourier Transformation of the Heaviside Function, we can now apply our knowledge to the spectrum of a pulsed electric field. This can be done by the Fourier transform of a Heaviside function with underlying oscillating electric field.

Problem Statement

We have a rectangular shaped pulse with central frequency \(\omega_0\) given by:

\[E(t)=\cos(\omega_0t)\Theta\left(1-\left|\frac{t}{a}\right|\right).\]

Here the Heaviside-function \(\Theta(x)\) is used, which is defined as:

\begin{eqnarray*}\Theta(x)=\left\{\begin{array}{ccc}1 &\mathrm{if} & x>0 \\0 &\mathrm{else} &\end{array}\right.\end{eqnarray*}

Calculate the spectrum of the pulse \(\ {E}(\omega)\) using Fourier-Transformation.

 

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