The damped Exponent is given by the expression below.

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

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

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

Fn = 8 to 14 show special cases of Fn = 7.

Fn = 15 shows the transform of growing exponent f(t) = e

Fn = 16 to 19 show special cases of Fn = 15.

Fn = 20 shows the transform of damped exponent f(t) = e

Fn = 21 to 25 show special cases of Fn = 20.

Fn = 26 shows the transform of damped exponent f(t) = e

Fn = 27 to 31 show special cases of Fn = 26.

Fn = 32 shows the transform of damped exponent f(t) = te

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

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

Fn = 45 to 47 show special cases of Fn = 44.

Fn = 48 shows the transform of function f(t) = 1 - e

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

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

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

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

f(t) = e

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.

COPYRIGHT © 1996, 2012 Cuthbert Nyack.