### FM Synthesis of Instrument Sounds

In class you learned that with signals of the form $$x(t)=A cos(\psi(t))$$, the instantaneous frequency of the signal is the derivative of the phase $$\psi(t)$$. So, if $$\psi(t)$$ is constant, the frequency is zero. If $$\psi(t)$$ is linear, $$x(t)$$ is a sinusoid at some fixed frequency. If $$\psi(t)$$ is quadratic, $$x(t)$$ is a chirp signal whose frequency changes linearly versus time.

FM synthesis uses a more interesting $$\psi(t)$$, one that is sinusoidal. FM, meaning “frequency modulation,” refers to the fact that the frequency of $$x(t)$$ changes according to the oscillations of $$\psi(t)$$. This is useful for synthesizing instrument sounds because the proper choice of the modulating frequencies will produce a fundamental frequency and several overtones, as many instruments do.

The general equation for an FM sound synthesizer is: $$x(t)=A(t) cos[\omega_ct + I(t) cos(\omega_mt + \phi_m) + \phi_c] (1)$$ In (1), $$A(t)$$ is the signal's amplitude. It is a function of time so that the instrument sound can be made to fade out slowly or cut off quickly. Such a function is called an envelope. The frequency $$\omega_c$$ is called the “carrier” frequency. Note that when you take the derivative of $$\psi(t)$$ to find $$\omega_i(t)$$, $$\omega_c$$ will be a constant in that expression. It is the frequency that would be produced without any frequency modulation. The parameter $$\omega_m$$ is called the "modulating" frequency. It expresses the rate of oscillation of $$\omega_i(t)$$. The parameters $$\phi_m$$ and $$\phi_c$$ are arbitrary phase constants, usually both set to $$-\frac{\pi}2$$ so that $$x(0)=0$$.

The function $$I(t)$$ has a less obvious purpose than the other parameters of FM signals. It is technically called the “modulation index envelope.” To see what it does, examine the expression for the instantaneous frequency: \begin{align} \omega_i(t) & = \frac{d}{dt} \psi(t) \\ & = \frac{d}{dt} [ \omega_ct + I(t) cos(\omega_mt + \phi_m) + \phi_c] \\ & = \omega_c - I(t) \omega_m sin(\omega_mt + \phi_m) + \frac{dI}{dt} cos(\omega_mt + \phi_m) \end{align} If $$I(t)$$ is constant, then $$I(t)\omega_m$$ gives the maximum amount by which the instantaneous frequency deviates from $$\omega_c$$. Beyond that, however, it is difficult to relate $$I(t)$$ to the sound made by $$x(t)$$ without some rather complicated analysis. Nonetheless, we would like to characterize $$x(t)$$ as the sum of several sinusoids instead of a single signal whose frequency changes. In this regard, the following comments are relevant: when $$I(t)$$ is small $$I \approx 1$$),low multiples of the carrier frequency $$(\omega_c)$$ have high amplitudes. When $$I(t)$$ is large $$(I>4)$$, both low and high multiples of the carrier frequency have high amplitudes. The net result is that $$I(t)$$ can be used to vary the overtone content of the instrument sound (overtones are harmonics). When $$I(t)$$ is small, mainly low frequencies will be produced. When $$I(t)$$ is large, higher harmonic frequencies can also be produced. For more details see the paper by Chowning. *

Below are some examples of sounds that can be synthesized with the appropriate choice of $$A(t)$$, $$I(t)$$, $$\omega_c$$, and $$\omega_m$$. These sounds were originally synthesized by Robbie Griffin.

Instrument Carrier Frequency (Hz) Modulating Frequency (Hz)
Brass
900
300
Clarinet
900
600
Bell
110
210
Knocking Sound
80
55

*Ref: John M. Chowning, “The Synthesis of Complex Audio Spectra by means of Frequency Modulation,” Journal of the Audio Engineering Society, vol. 21, no. 7, Sept. 1973, pp. 526--534.

Jeff Schodorf
Tue Feb 20 17:10:04 EST 1996