Powers of z
Choose any two complex numbers and
, and form the sequence
What are the properties of this signal? Writing the complex numbers as
![\begin{eqnarray*}
z_0 &=& A e^{j\phi} \\
z_1 &=& e^{sT} = e^{(\sigma + j\omega)T},
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/mdft/img617.png)
we see that the signal is always a discrete-time
generalized (exponentially enveloped) complex sinusoid:
![$\displaystyle x(n) = A e^{\sigma n T} e^{j(\omega n T + \phi)}
$](http://www.dsprelated.com/josimages_new/mdft/img618.png)
Figure 4.17 shows a plot of a generalized (exponentially
decaying, ) complex sinusoid versus time.
Note that the left projection (onto the plane) is a decaying spiral,
the lower projection (real-part vs. time) is an exponentially decaying
cosine, and the upper projection (imaginary-part vs. time) is an
exponentially enveloped sine wave.
Next Section:
Phasor and Carrier Components of Sinusoids
Previous Section:
Sampled Sinusoids