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 ws - w and ws + w.

The applet shows that the spectrum is reflected about half the sampling frequency. Between ½ ws and ws, the real part is the same as for the range 0 to ws 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 ws appear in the range 0 to ½ ws. 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 ws and ws, (3.14 to 6.28) the frequency in the range 0 to ws is no longer w but ws - w as can be seen from the lower set of curves(notice phase is inverted). As the sine increases from 1 to 1½ times ws (6.28 to 9.42) then the component in the range 0 to ½ ws is now w - ws.
Return to main page
Return to page index
Copyright 2000 © Cuthbert A. Nyack.