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Another interesting class of waveshaping transfer functions is the sinusoids:
\begin{displaymath} f(x) = \cos(x + \phi) \end{displaymath}
which include the cosine and sine functions (got by choosing $\phi=0$ and
$\phi=-\pi/2,ドル respectively). These functions, one being even and the
other odd, give rise to even and odd harmonic spectra, which turn out to be:
\begin{displaymath} \cos(a \cos(\omega n)) = {J_0}(a) - 2 {J_2}(a) \cos(2 \om... ...a) \cos(4 \omega n) - 2 {J_6}(a) \cos(6 \omega n) \pm \cdots \end{displaymath}
\begin{displaymath} \sin(a \cos(\omega n)) = 2 {J_1}(a) \cos(\omega n) - 2{J_3}(a) \cos(3 \omega n) + 2{J_5}(a) \cos(5 \omega n) \mp \cdots \end{displaymath}
The functions ${J_k}(a)$ are the
Bessel functions
of the first kind, which
engineers sometimes use to solve problems about vibrations or heat flow on
discs. For other values of $\phi,ドル we can expand the expression for $f$:
\begin{displaymath} f(x) = \cos(x) \cos(\phi) - \sin(x) \sin(\phi) \end{displaymath}
so the result is a mix between the even and the odd harmonics, with $\phi$
controlling the relative amplitudes of the two. This is demonstrated in Patch
E07.evenodd.pd, shown in Figure
5.14.
Figure 5.14:
Using an additive offset to a cosine transfer function to alter the
symmetry between even and odd. With no offset the symmetry is even. For odd symmetry, a quarter cycle is added
to the phase. Smaller offsets give a mixture of even and odd.
next
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Next: Phase modulation and FM
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Miller Puckette
2006年12月30日
