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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.

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Phasor and Carrier Components of Sinusoids

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Sampled Sinusoids