Choose any two
complex numbers 
and

, and form the sequence

 |
(4.10) |
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.
Figure 4.17:
Exponentially decaying
complex sinusoid and projections.
![\includegraphics[scale=0.8]{eps/circledecaying}](http://www.dsprelated.com/josimages_new/mdft/img620.png) |
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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