- ... Sinusoids
^{1}
- This is a new section slated for the second edition
of
**Mathematics of the DFT** [7].
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- ...
practice.
^{2}
- An important variant of FM called
*feedback FM*, in which a single oscillator phase-modulates
itself, simply does not work if true frequency modulation is
implemented.
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- ... section.
^{3}
- The
mathematical derivation of FM spectra is included here as a side note.
No further use will be made of it in this course.
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- ...Watson44,
^{4}
- Existence of the
Laurent expansion follows from the fact that the generating function is a
product of an exponential function,
, and an
exponential function inverted with respect to the unit circle,
. It is readily verified by direct differentiation
in the complex plane that the exponential is an
*entire function* of (analytic at all finite points in the complex
plane) [2], and therefore the inverted exponential is analytic
everywhere except at .
The desired Laurent expansion may be obtained, in principle,
by multiplying one-sided series for the exponential and inverted exponential
together. The exponential series has the well known form
. The series for the inverted exponential
can be obtained by inverting again (
), obtaining the
appropriate exponential series, and inverting each term.
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