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
we see that the signal is always a discrete-time generalized (exponentially enveloped) complex sinusoid:
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.
Phasor and Carrier Components of Sinusoids