Modulation Index cont.

Without going any further in solving this equation, it is possible to get a sense of the effect of the modulation index $I(t)$ from

$$f_i(t) = f_c - I(t)f_m\sin(2\pi f_mt + \phi_m) +$$

$$\qquad \qquad \frac{dI(t)}{dt}\cos(2\pi f_mt + \phi_m)/2\pi.$$

We may see that if $I(t)$ is a constant (and it's derivative is zero), the third term goes away and the instantaneous frequency becomes

$$f_i(t) = f_c - I(t)f_m\sin(2\pi f_mt + \phi_m).$$

Notice now that in the second term, the quantity $I(t)f_m$ multiplies a sinusoidal variation of frequency $f_m$, indicating that $I(t)$ determines the maximum amount by which the instantaneous frequency deviates from the carrier frequency $f_c$.

Since the modulating frequency $f_m$ is at audio rates, this translates to addition of harmonic content.

Since $I(t)$ is a function of time, the harmonic content, and thus the timbre, of the synthesized sound may vary with time.