Fourier Transform Time Limited Sinusoid

Cuthbert Nyack

Consider the function f(t) below, which is a cosine within the time interval -t/2 to +t/2 and zero elsewhere.

Expressing the cosine as a sum of complex exponentials, the Fourier Integral can be evaluated to give the following real and even transform:-

Similarly the the function below consisting of a time limited sine

can be shown to have the following imaginary and odd transform.

The applet below shows the Fourier transform of a time limited sinusoid. f(t) is shown in magenta with the origin at the CENTER of the plot. The T parameter controls the width of the pulse. With HGain = 1, the horizontal axis for time goes from -5s to +5s. For the time function, the vertical axis is from -1 to +1. Frequency is in rad/s. The phase of the sinusoid can be changed by the 'Ph' parameter. As 'Ph' goes from 0 to 1 the sinusoid changes from a cosine to a sine.
SGain changes the horizontal axis for the spectrum, HGain changes the horizontal axis for f(t) and VGain changes the vertical axis for the spectrum.
For the spectrum, the following applies:-
The origin is in the CENTER of the plot.
Magnitude of the spectrum is in green.
With VGain = 1 Vertical scale for the spectrum is from -1 to +1.
Phase is in gray.
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 -12.5rad/s to +12.5rad/s.
Sometimes the curve for the magnitude covers that for the real part and the curve for the imaginary part covers that for the magnitude.

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