# DTFT, Damped Sinusoid, Aliasing.

Cuthbert Nyack

The Z transform of a damped sine f(t) = e^{-at}
sinwt is shown below:-

The applet below shows the frequency spectrum of the damped
sine over the range from 0 to the sampling frequency.
The magnitude of the spectrum is shown in
green.
The phase of the spectrum is in red.
Real part is in
cyan.
and imaginary part in
yellow.
The damped sine is in peach, the samples are
in magenta and the light yellow green and blue magenta curves show the frequencies
w_{s} -
w and
w_{s} +
w.

The applet shows that the spectrum is reflected about half the sampling
frequency. Between ½ w_{s} and
w_{s}, the real part is the same as
for the range 0 to w_{s} but the
imaginary part is inverted.

Aliasing is the term used to refer to
the phenomenon whereby frequencies in the sampled signal higher
than ½ w_{s} appear in
the range 0 to ½ w_{s}.
The frequency of the sine in rad/s is given
by the "Freq" parameter and the effect of aliasing can be seen by
leaving the sampling time at 1(sampling frequency = 2
p rad/s) and varying the Freq
parameter up to its maximum which takes it past the sampling frequency.
Noticeable aliasing begins to occur when the frequency goes past ¼
of the sampling frequency. When the freguency is between
½ w_{s} and
w_{s}, (3.14 to 6.28) the
frequency in the range 0 to w_{s}
is no longer w but
w_{s} -
w as can be seen from the lower
set of curves(notice phase is inverted).
As the sine increases from 1 to 1½ times
w_{s} (6.28 to 9.42)
then the component
in the range 0 to ½ w_{s}
is now w -
w_{s}.

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Copyright 2000 © Cuthbert A. Nyack.