6.8 Summary

• Modulation synthesis affects a property of one signal (the carrier) with another (the modulator), letting us emulate rich, time-varying spectra with only a couple of oscillators.

• Ring modulation, \(\sin(\omega_c t)\sin(\omega_m t)\), multiplies two sinusoids. A product-to-sum identity shows it produces two sidebands at \(\omega_c \pm \omega_m\), each with amplitude \(\tfrac12\). The original carrier and modulator frequencies disappear. Slow modulation is heard as tremolo, fast modulation as two separate tones.

• Every real sinusoid contains both a positive and a negative frequency. A negative frequency is audibly identical to a phase-shifted positive frequency. This makes the amplitude spectrum even and the phase spectrum odd, and it explains ring modulation’s four symmetric sidebands.

• Amplitude modulation, \(\sin(\omega_c t)[1 + \sin(\omega_m t)]\), is ring modulation plus the carrier, so it keeps the carrier at \(\omega_c\) alongside the two sidebands.

• A frequency that varies over time must be integrated (accumulated) into phase before being passed to \(\sin\). Substituting \(\omega(t)\) directly into \(\sin(\omega t)\) is incorrect. The phase-accumulation recurrence \(\theta[n] = \theta[n-1] + \omega[n]\Delta t\) implements this in \(O(N)\).

• Frequency modulation, \(\sin(2\pi f_c t + \tfrac{D}{f_m}\sin(2\pi f_m t))\), follows from integrating a sinusoidally-varying frequency. It creates an infinite series of sidebands at \(f_c + k f_m\). The ratio \(f_c/f_m\) sets harmonicity, and the index of modulation \(I = D/f_m\) sets the number of audible sidebands (about \(I + 1\) per side).