Fourier Transform of Pulse.

Cuthbert Nyack

As a very simple example of finding the Fourier transform, consider the nonperiodic even pulse opposite. Pulse is defined by the function:-
f(t) = 1 for - t t + t
and zero elsewhere.

The Fourier transform can be found by integrating the equation for g(w) shown opposite:-

Substituting for f(t) and integrating the equation for g(w) gives:-

Evaluating the limits and simplifing the expression for g(w) gives the sinc function multiplied by 2t (the area of the pulse). Since the pulse is even, the resulting transform is real.

As a second simple example the diagram opposite shows an odd nonperiodic pulse defined by the expression
f(t) = - 1 for - t t < + 0
and
f(t) = + 1 for 0 t + t.

Substituting the expression for f(t) into the equation for g(w) gives:-

Integrating produces:-

After Simplification the final expression for g(w) is obtained. Since f(t) is odd then g(w) is imaginary.

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COPYRIGHT 1996 Cuthbert A. Nyack.