Sinusoidal waveshaping: evenness and oddness

Another interesting class of waveshaping transfer functions is the sinusoids:

$$f(x) = \cos(x + \phi)$$

which include the cosine and sine functions (by choosing $\phi=0$ and $\phi=-\pi/2$, respectively.) These functions, one being even and the other odd, give rise to even and odd harmonic spectra:

$$\cos(a \cos(\omega n)) = {J_0}(a) - 2 {J_2}(a) \cos(2 \om... ...a) \cos(4 \omega n) - 2 {J_6}(a) \cos(6 \omega n) \pm \cdots$$

$$\sin(a \cos(\omega n)) = 2 {J_1}(a) \cos(\omega n) - 2{J_3}(a) \cos(3 \omega n) + 2{J_5}(a) \cos(5 \omega n) \mp \cdots$$

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 $\phi$, we can expand the expression for $f$:

$$f(x) = \cos(x) \cos(\phi) - \sin(x) \sin(\phi)$$

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.