# 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.
Fn = 1 shows a time limited sine width t and frequency wo.
The odd spectrum is in pink.
Fn = 2 to 7 show special cases of Fn = 1.
Fn = 8 shows a time limited cosine.
Fn = 9 to 13 show special cases of Fn = 8.
Fn = 14 shows a time limited sinusoid which can be changed from sine to cosine by Ph.
Fn = 15 to 17 show special cases of Fn = 14.
Fn = 18 shows a time limited sine with edges modified by a triangle.
Fn = 19 to 23 show special cases of Fn = 18.
Fn = 24 shows a time limited cosine with edges modified by a triangle.
Fn = 25 to 29 show special cases of Fn = 24.
27 and 28 are shown with several values wo.
of t.
29 is shown with several values of Fn = 30 shows a time limited sinusoid which can be changed from sine to cosine and with edges modified by a triangle.
Fn = 31 to 33 show special cases of Fn = 30.
Fn = 34 shows a time limited sine with edges modified by a raised cosine.
Fn = 35 to 41 show special cases of Fn = 34.
Fn = 42 shows a time limited cosine with edges modified by a raised cosine.
Fn = 43 shows a special case of Fn = 42.
Fn = 44 shows a time limited sinusoid which can be changed from sine to cosine and with edges modified by a raised cosine.
Fn = 45 shows a special cases of Fn = 44.
Fn = 46 shows a time limited cosine being reconstructed from its spectrum.
Fn = 47 to 51 show special cases of Fn = 46.
Fn = 52 shows a time limited cosine with triangle edge being reconstructed from its spectrum.
Fn = 53 to 56 show special cases of Fn = 52.