pulse

[logistic_guy]

The following expression may be viewed as an approximate representation of a pulse with finite rise time:

\(\displaystyle g(t) = \frac{1}{\tau}\int_{t - T}^{t + T} \text{exp} \left(-\frac{\pi u^2}{\tau^2}\right)\ du\)

where it is assumed that \(\displaystyle T \gg \tau\). Determine the Fourier transform of \(\displaystyle g(t)\). What happens to this transform when we allow \(\displaystyle \tau\) to become zero? Hint: Express \(\displaystyle g(t)\) as the superposition of two signals, one corresponding to integration from \(\displaystyle t - T\) to \(\displaystyle 0\), and the other from \(\displaystyle 0\) to \(\displaystyle t + T\).

 

[khansaheb]

The following expression may be viewed as an approximate representation of a pulse with finite rise time:

\(\displaystyle g(t) = \frac{1}{\tau}\int_{t - T}^{t + T} \text{exp} \left(-\frac{\pi u^2}{\tau^2}\right)\ du\)

where it is assumed that \(\displaystyle T \gg \tau\). Determine the Fourier transform of \(\displaystyle g(t)\). What happens to this transform when we allow \(\displaystyle \tau\) to become zero? Hint: Express \(\displaystyle g(t)\) as the superposition of two signals, one corresponding to integration from \(\displaystyle t - T\) to \(\displaystyle 0\), and the other from \(\displaystyle 0\) to \(\displaystyle t + T\).

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