# Fourier Transform Damped Exponent, Sinusoid and related functions.

Cuthbert Nyack
The damped Exponent is given by the expression below. Its Fourier Transform can be readily found and is:- 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 damped exponent, sinusoid and related functions.

Fn sets the function of the applet.

Fn = 1 shows the transform of damped exponent f(t) = e-at.
Fn = 2 to 6 show special cases of Fn = 1.
Fn = 5 and 6 shows the function reconstructed from its spectrum. Orange curve shows the reconstructed function superimposed on f(t) in Green. The white x's show the range of frequencies used in the reconstruction.
Fn = 7 shows the transform of damped exponent f(t) = e-at for 0 « t « T and f(t) = 0 for t > T.
Fn = 8 to 14 show special cases of Fn = 7.
Fn = 15 shows the transform of growing exponent f(t) = eat for 0 « t « T and f(t) = 0 for t > T.
Fn = 16 to 19 show special cases of Fn = 15.
Fn = 20 shows the transform of damped exponent f(t) = e-at - e-bt for 0 « t « T and f(t) = 0 for t > T.
Fn = 21 to 25 show special cases of Fn = 20.
Fn = 26 shows the transform of damped exponent f(t) = e-at for 0 « t « t and f(t) = e-bt for t < t « T. The second function is adjusted to join with the first.
Fn = 27 to 31 show special cases of Fn = 26.
Fn = 32 shows the transform of damped exponent f(t) = te-at for 0 « t « T and f(t) = 0 for t > T.
Fn = 33 to 37 show special cases of Fn = 32.
Fn = 38 shows the transform of function f(t) = 1. for 0 « t « t and f(t) = e-at for t < t « T.
Fn = 39 to 43 show special cases of Fn = 38.
Fn = 44 shows the transform of function f(t) = t/t. for 0 « t « t and f(t) = e-at for t < t « T.
Fn = 45 to 47 show special cases of Fn = 44.
Fn = 48 shows the transform of function f(t) = 1 - e-at. for 0 « t « t and f(t) = e-bt for t < t « T.
The second function is adjusted to join with the first.
Fn = 49 to 52 show special cases of Fn = 48.
Fn = 53 shows the transform of damped cosine f(t) = e-atcos(wo t).
Fn = 54 to 57 show special cases of Fn = 53.
Fn = 56 and 57 shows the function reconstructed from its spectrum. Orange curve shows the reconstructed function superimposed on f(t) in Green. The white x's show the range of frequencies used in the reconstruction.
Fn = 58 shows the transform of damped sine f(t) = e-atsin(wo t).
Fn = 59 to 62 show special cases of Fn = 58.
Fn = 61 and 62 shows the function reconstructed from its spectrum. Orange curve shows the reconstructed function superimposed on f(t) in Green. The white x's show the range of frequencies used in the reconstruction.
Fn = 63 shows the transform of damped cosine f(t) = e-atcos(wo t).
for 0 « t « T and f(t) = 0 for t > T.
Fn = 64 to 68 show special cases of Fn = 63.
Fn = 67 and 68 shows the function reconstructed from its spectrum. Orange curve shows the reconstructed function superimposed on f(t) in Green. The white x's show the range of frequencies used in the reconstruction.
Fn = 69 shows the transform of damped cosine f(t) = (1 - e-at)cos(wo t). for 0 « t « t and
f(t) = e-btcos(wo t) for t < t « T.
The second function is adjusted to join with the first.
Fn = 70 to 76 show special cases of Fn = 69.
Fn = 73 to 76 shows the function reconstructed from its spectrum. Orange curve shows the reconstructed function superimposed on f(t) in Green. The white x's show the range of frequencies used in the reconstruction.