Fourier Transform Damped Sinusoid

Cuthbert Nyack

A damped sine signal can be represented by the following expression.

Expressing the sine in terms of complex exponentials and finding the Fourier Transform yields the following.

The damped cosine is represented by the expression.

Representing the cosine in terms of complex exponentials and solving the Fourier Integral produces the following expression for the Fourier Transform of the damped cosine.

The applet below shows how the Fourier transform of the sinusoid depends on 'a' and w. In the applet, the damped sinusoid is shown in purple. Vertical axis for f(t) is from -1 to +1. The phase of the sinusoid can be changed by changing the phase parameter. As the phase changes from 0 to 1 the sinusoid changes from a sine to a cosine. When FGain is 1, the horizontal range for t is from 0 to 10s. Frequency parameter is in units of radians/sec. For the spectrum, the following applies:-
The origin is in the CENTER of the plot.
Magnitude of spectrum is in green.
Vertical axis for spectrum is from -2 to +2 with VGain = 1.
Phase is in red.
Vertical axis for phase is from - p to + p.
Real part(even) of spectrum is in cyan and Imaginary part(odd) in yellow.
With SGain = 1, the horizontal scale for frequency is from -2.5rad/s to +2.5rad/s.

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