Vibrato simulation

Vibrato is a term used to describe a wavering of pitch.

Vibrato (in varying amounts) occurs very naturally in the singing voice and in many ``sustained'' instruments (where the musician has control after the note has been played), such as the violin, wind instruments, the theremin, etc.).

In Vibrato, the frequency does not change linearly but rather sinusoidally, creating a sense of a wavering pitch.

Since the instantaneous frequency of the sinusoid is the derivative of the instantaneous phase, and the derivative of a sinusoid is a sinusoid, a vibrato can be simulated by applying a sinusoid to the instantaneous phase of a carrier signal:

$$x(t) = A_c\cos(2\pi f_ct + A_m\cos(2\pi f_mt + \phi_m) + \phi_c),$$

that is, by FM synthesis.