... Sinusoids1
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, $ \exp(\beta z/2)$, and an exponential function inverted with respect to the unit circle, $ \exp(-0.5\beta/z)$. It is readily verified by direct differentiation in the complex plane that the exponential is an entire function of $ z$ (analytic at all finite points in the complex plane) [2], and therefore the inverted exponential is analytic everywhere except at $ z=0$. 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 $ \exp(z) =
1+z+z^2/2!+z^3/3!+\cdots\,$. The series for the inverted exponential can be obtained by inverting again ( $ z\leftarrow 1/z$), obtaining the appropriate exponential series, and inverting each term.
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