Fourier Transform of Triangle Pulse.

Cuthbert Nyack

The diagram opposite shows an even nonperiodic triangular pulse. The function f(t) can be written as:-
f(t) = t/t2 + 1/t for - t t 0
f(t) = - t/t2 + 1/t for 0 t + t
and f(t) = 0 for all other values of t.

The Fourier Transform g(w) can be obtained from f(t) by using the standard equation opposite.

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

Since t is a constant then the equation for g(w) can be re-expressed as:-->

The integrals can be evaluated using integration by parts. For t e- j w t integration by parts gives the equation opposite.

Carrying out the integration and evaluating the limits results in:-->

After simplification the sinc squared function is obtained as the Fourier transform of a triangular pulse with unit area. Since f(t) is even then g(w) is real.

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COPYRIGHT 1996 Cuthbert A. Nyack.