Fourier Transform of one and two pulses.

Cuthbert Nyack

The equations describing the Fourier transform G(w) of a function f(t) defined in the interval - ¥ to + ¥ are shown opposite. For an aperiodic signal, the Fourier transform G(w) is a continuous function of frequency w. This contrasts with a periodic signal, whose Fourier Series is a discrete function. Because of the complex exponential in the definition then the Fourier transform G(w) is generally a complex function with real and imaginary parts. The real part of G(w) corresponds to the transform of the even part of f(t) and the imaginary part to the transform of the odd part of f(t). Note that other definitions of the Fourier Transform are possible but the one shown opposite is normally used in signal analysis.

The diagram opposite shows a piecewise continuous function which can be used to study the Fourier Transform of a wide variety of pulse shapes by varying the values of a, b, c, d, e and f.

An even pulse is obtained by setting
a = b = c = d and e = f

and an odd pulse by setting
a = b = - c = - d and e = f.

The diagram opposite shows a pulse shape which can be used to study pulses consisting of 2 separate pieces (analog of the double slit) by varying a, b, c, d, e and f. f determines the separation of the pulses, e the width of each pulse Setting c = d = 0 produces an even pulse and setting a = b = 0 produces an odd pulse shape.

The Applet below shows the Fourier Transform for a single pulse, a double pulse and related pulses.
Fn = 1 shows a single pulse whose shape can be changed by 'a' to 'f'.
Fn = 2 to 7 show special cases of Fn = 1.
Fn = 8 shows a double pulse whose shape can be changed by 'a' to 'f'.
Fn = 9 to 14 show special cases of Fn = 8.
Fn = 15 shows a pulse whose shape can be changed by 'a' to 'k'.
Fn = 16 to 21 show special cases of Fn = 15.
Fn = 15 shows a pulse whose shape can be changed by 'a' to 'k'.
Fn = 16 to 21 show special cases of Fn = 15.
Fn = 22 shows a pulse whose shape can be changed by 'a' to 'k'.
Fn = 23 to 26 show special cases of Fn = 22.
Fn = 27 shows a double pulse whose shape can be changed by 'a' to 'f'.
Fn = 28 to 31 show special cases of Fn = 27.
Fn = 32 shows a multiple pulse whose shape can be changed by 'a' to 'g'.
Fn = 33 to 34 show special cases of Fn = 32.
Fn = 35 shows a multiple pulse whose shape can be changed by 'a' to 'g'.
Fn = 36 to 39 show special cases of Fn = 35.
Fn = 40 shows the contributions to the Fourier Transform Integral for a single even pulse. Fn = 41 to 55 show special cases of Fn = 40.
Fn = 56 shows the contributions to the Fourier Transform Integral for a single odd pulse. Fn = 57 to 74 show special cases of Fn = 56.

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