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## Powers of

Choose any two complex numbers and , and form the sequence

What are the properties of this signal? Expressing the two complex numbers as

we see that the signal is always a discrete-time generalized complexsinusoid, i.e., an exponentially enveloped complex sinusoid.Figure 5.9 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.