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 above applet shows the real and imaginary parts of the Fourier transform as the pulse shape is changed. f(t) is the purple line( applet vertical is ± 1, horizontal is ± 2). The frequency spectrum is usually complex with real and imaginary parts. The real part of the spectrum is shown by the red line (vertical is ± 1, horizontal is ± 4p when the horizontal gain is 1). The imaginary part of the spectrum is shown by the orange line (same scale as real part). More of the spectrum can be seen by changing the horizontal gain. If the signal is even, then its spectrum is real with the characteristic sinc shape, while it is imaginary if the signal is odd.
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 above applet shows the real and imaginary parts of the Fourier transform as the pulse shape is changed. f(t) is the purple line( applet vertical is ± 1, horizontal is ± 2). The frequency spectrum is usually complex with real and imaginary parts. The real part of the spectrum is shown by the red line (vertical is ± 1, horizontal is ± 4p when the horizontal gain is 1). The imaginary part of the spectrum is shown by the orange line (same scale as real part). More of the spectrum can be seen by changing the horizontal gain. If the signal is even, then its spectrum is real, while it is imaginary if the signal is odd.
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Copyright © Cuthbert A. Nyack.