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.
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COPYRIGHT © 1996, 2012 Cuthbert Nyack.